Finite Group Databasebeta

← #72323130

Subgroups — #72323130

Order 2048.

758 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72322395C211yesmaximal, characteristic, complemented
2#72320554C211yesmaximal, characteristic, complemented
3#72287181C211yesmaximal, characteristic, complemented
4#72287042C211yesmaximal, characteristic, complemented
5#72286966C211yesmaximal, characteristic, complemented
6#72286779C211yesmaximal, characteristic, complemented
7#72286459C211yesmaximal, characteristic, complemented
8#72286442C211yesmaximal, characteristic, complemented
9#72286323C211yesmaximal, characteristic, complemented
10#72286323C211yesmaximal, characteristic, complemented
11#72232028C211yesmaximal, characteristic
12#72231942C211yesmaximal, characteristic, complemented
13#72231887C211yesmaximal, characteristic, complemented
14#72225240C211yesmaximal, characteristic, complemented
15#72224639C211yesmaximal, characteristic, complemented
16#513008C411yescharacteristic
17#483944C2211yescharacteristic
18#483713—21no—
19#483611—21no—
20C2×#12045C2211yescharacteristic
21#151116C411yescharacteristic
22#151116C411yescharacteristic
23#151115C411yescharacteristic
24#151115C411yescharacteristic
25#151085C2211yescharacteristic
26#151085C2211yescharacteristic
27#151083C2211yescharacteristic
28#151083C2211yescharacteristic
29#151066C411yescharacteristic
30#150142C411yescharacteristic
31#150123C411yescharacteristic
32#149513C2211yescharacteristic
33#149502C2211yescharacteristic
34#122970—21no—
35#122968—21no—
36#122968—21no—
37#122967—21no—
38#122966C2211yescharacteristic
39#122966C2211yescharacteristic
40#122966C2211yescharacteristic, complemented
41#122965C2211yescharacteristic, complemented
42#122963C2211yescharacteristic
43#122963C2211yescharacteristic
44#122963C2211yescharacteristic, complemented
45#122962C2211yescharacteristic, complemented
46#122922C2211yescharacteristic
47#122919C2211yescharacteristic
48#122478—21no—
49#122470—21no—
50#122219C2211yescharacteristic
51#93877C2211yescharacteristic, complemented
52#93877C2211yescharacteristic, complemented
53#93877C2211yescharacteristic, complemented
54#93877C2211yescharacteristic, complemented
55#93758C2211yescharacteristic, complemented
56#93758C2211yescharacteristic, complemented
57#93757C2211yescharacteristic
58#93757C2211yescharacteristic
59#93757C2211yescharacteristic
60#93757C2211yescharacteristic
61#93757C2211yescharacteristic, complemented
62#93757C2211yescharacteristic, complemented
63#93668C2211yescharacteristic, complemented
64#93668C2211yescharacteristic, complemented
65#93668C2211yescharacteristic, complemented
66#93668C2211yescharacteristic, complemented
67#33735C2×C411yescharacteristic
68#33488—41no—
69#33483—41no—
70C2×C4×D16C2×C411yescharacteristic
71#17959—21no—
72C2×C8×D8—21no—
73#13714C811yescharacteristic
74#13711C811yescharacteristic
75#13707C811yescharacteristic
76#13707C811yescharacteristic
77#13689C811yescharacteristic
78#13689C811yescharacteristic
79#13664C811yescharacteristic
80#13661C811yescharacteristic
81#13654C2×C411yescharacteristic
82#13654C2×C411yescharacteristic
83#13652C2×C411yescharacteristic
84#13652C2×C411yescharacteristic
85#13640C2×C411yescharacteristic
86#13640C2×C411yescharacteristic
87C4×M64—21no—
88#12086C2×C411yescharacteristic
89#12085C2×C411yescharacteristic
90#12084—21no—
91#12064—21no—
92#12064—21no—
93#12064—21no—
94#12062—21no—
95#12062—21no—
96#12062—21no—
97#12061—21no—
98#12059—21no—
99#12057—41no—
100#12056—41no—
101#12045—21no—
102#12045—21no—
103#12045C2×C411yescharacteristic
104#12045C2×C411yescharacteristic
105#12045C2311yescharacteristic
106#12045C2311yescharacteristic
107#12045C2311yescharacteristic
108#12044C2311yescharacteristic
109#12043—21no—
110#12038—41no—
111#12038—41no—
112#12024C2×C411yescharacteristic
113#12013—21no—
114#12012—21no—
115C2×#1372C2×C411yescharacteristic
116#11673—21no—
117#11631—41no—
118#11614—41no—
119C2×#1331—21no—
120#11411D411yescharacteristic
121#11402D411yescharacteristic
122C2×#1311C2311yescharacteristic
123#7463C2×C411yescharacteristic
124#7463C2×C411yescharacteristic
125#7463C2×C411yescharacteristic
126#7463C2×C411yescharacteristic
127#7386C2×C411yescharacteristic
128#7386C2×C411yescharacteristic
129#7385C2×C411yescharacteristic
130#7385C2×C411yescharacteristic
131#7385C2×C411yescharacteristic
132#7385C2×C411yescharacteristic
133#7385C2×C411yescharacteristic
134#7385C2×C411yescharacteristic
135#7332—21no—
136#7332C2×C411yescharacteristic
137#7332C2×C411yescharacteristic
138#7332C2311yescharacteristic
139#7332C2311yescharacteristic
140#7330C2311yescharacteristic
141#7330C2311yescharacteristic
142#7099—21no—
143#7099—21no—
144#7099C2311yescharacteristic
145#7099C2311yescharacteristic
146#7099C2311yescharacteristic
147#7099C2311yescharacteristic
148#7071—21no—
149#7071—21no—
150#7070—21no—
151#7070—21no—
152#7070—21no—
153#7070—21no—
154#7070—21no—
155#7070—21no—
156#7053D411yescharacteristic
157#7053D411yescharacteristic
158#7053C2311yescharacteristic
159#7053C2311yescharacteristic
160C22×D16C2×C811yescharacteristic
161#3168—21no—
162#2966—41no—
163C2×C4×D8—21no—
164C2×C8×D4—41no—
165C2×SD64C2×C811yescharacteristic
166C2×SD64C2×C811yescharacteristic
167C2×D32C2×C811yescharacteristic
168C2×D32C2×C811yescharacteristic
169C2×M64—41no—
170#2253C2×C811yescharacteristic
171#2249C2×C811yescharacteristic
172#2242—21no—
173#2217—21no—
174#2216C2×C811yescharacteristic
175#2215C2×C811yescharacteristic
176#2183—41no—
177Q8×C16—41no—
178C4×Q32C22×C411yescharacteristic
179C4×SD32—21no—
180C4×D16—21no—
181C4×D16—21no—
182C4×D16C2×C811yescharacteristic
183C4×D16C2×C811yescharacteristic
184C4×D16C22×C411yescharacteristic
185C4×D16C22×C411yescharacteristic
186C4×D16C22×C411yescharacteristic
187#2169—41no—
188#2169—41no—
189#2169—41no—
190#2169—81no—
191#2168—81no—
192#2168—81no—
193#2168—81no—
194D4×C16—41no—
195D4×C16—41no—
196D4×C16—41no—
197#2164—81no—
198#2163—41no—
199#2162—21no—
200#2162—41no—
201C2×#366C2×C811yescharacteristic
202C2×#357—21no—
203#2112—81no—
204C2×C22⋊C16—81no—
205#2108—81no—
206C4×M32—21no—
207C2×C4×C16C22×C411yescharacteristic
208C2×C4×C16#4111yescharacteristic
209#1591—21no—
210#1590—41no—
211#1589—21no—
212C8×Q16—21no—
213C8×SD16—41no—
214C8×D8—21no—
215C8×D8—21no—
216C8×D8—21no—
217C8×D8—21no—
218C8×D8—21no—
219C8×D8—41no—
220C8×D8—41no—
221#1568—21no—
222C2×#335—21no—
223#1494—41no—
224C2×D4⋊C8—41no—
225C2×C82C2×D411yescharacteristic
226#1424C2×C811yescharacteristic
227#1424C2×C811yescharacteristic
228#1423C2×C811yescharacteristic
229#1423C2×C811yescharacteristic
230C4⋊C32—21no—
231C4⋊C32C2×C811yescharacteristic
232C4⋊C32C2×C811yescharacteristic
233#1416C2×C811yescharacteristic
234#1416C2×C811yescharacteristic
235#1415C2×C811yescharacteristic
236#1415C2×C811yescharacteristic
237#1415C2×C811yescharacteristic
238#1415C2×C811yescharacteristic
239#1415C2×C811yescharacteristic
240#1415C2×C811yescharacteristic
241C22⋊C32—21no—
242C22⋊C32C2×C811yescharacteristic
243C22⋊C32C2×C811yescharacteristic
244#1397—21no—
245#1397—21no—
246#1397C22×C411yescharacteristic
247#1397C22×C411yescharacteristic
248#1397C22×C411yescharacteristic
249#1397C22×C411yescharacteristic
250C4×C32—41no—
251#1372—21no—
252#1372C22×C411yescharacteristic
253#1372C22×C411yescharacteristic
254#1372C22×C411yescharacteristic
255#1372C22×C411yescharacteristic
256#1371—21no—
257#1370—41no—
258#1370—41no—
259#1367—21no—
260#1367—21no—
261#1367—21no—
262#1367—21no—
263Q8⋊C16—41no—
264Q8⋊C16—41no—
265#1332—21no—
266#1332—21no—
267#1331—21no—
268#1331—21no—
269#1331—21no—
270#1331—21no—
271#1331—21no—
272#1331—21no—
273D4⋊C16—41no—
274D4⋊C16—41no—
275D4⋊C16—41no—
276D4⋊C16—41no—
277D4⋊C16—41no—
278D4⋊C16—41no—
279#1312—21no—
280#1312C2×D411yescharacteristic
281#1312C2×D411yescharacteristic
282#1312C2×D411yescharacteristic
283#1312C2×D411yescharacteristic
284#1311—21no—
285#1311C22×C411yescharacteristic
286#1311C2×D411yescharacteristic
287#1311#4111yescharacteristic
288#1311C2411yescharacteristic
289C22×D8—21no—
290C23×C8—81no—
291#529—41no—
292C2×C4×D4—41no—
293C2×Q32C22×C811yescharacteristic
294C2×SD32—21no—
295C2×D16—22no—
296C2×D16—21no—
297C2×D16—41no—
298C2×D16C22×C811yescharacteristic
299C2×D16C22×C811yescharacteristic
300C2×D16C22×C811yescharacteristic
301C2×M32—41no—
302C2×M32—81no—
303C2×M32—81no—
304C2×M32—81no—
305C22×C16—21no—
306C22×C16—81no—
307C22×C16—81no—
308C22×C16—81no—
309C22×C16C22×C811yescharacteristic
310C22×C16#13111yescharacteristic
311#499—21no—
312#495—21no—
313#488—41no—
314#467—21no—
315#466—21no—
316#465—41no—
317#446—41no—
318C8×Q8—41no—
319C4×Q16—21no—
320C4×SD16—41no—
321C4×D8—21no—
322C4×D8—21no—
323C4×D8—21no—
324C4×D8—21no—
325C4×D8—21no—
326C4×D8—41no—
327C4×D8—41no—
328#436—41no—
329#435—81no—
330C8×D4—41no—
331C8×D4—41no—
332C8×D4—41no—
333C8×D4—41no—
334C8×D4—41no—
335C8×D4—81no—
336C8×D4—81no—
337C8×D4—81no—
338#433—81no—
339#432—41no—
340C2×#107—21no—
341C2×C4⋊C8—41no—
342C2×#102—41no—
343C2×C22⋊C8—81no—
344#405—81no—
345C2×C4×C8C4×D411yescharacteristic
346C2×C4×C8AGL(1,ℤ/8)11yescharacteristic
347C2×C4×C8AGL(1,ℤ/8)11yescharacteristic
348SD64—22no—
349SD64—41no—
350D32—22no—
351D32—41no—
352M64—82no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—41no—
356C2×C32—41no—
357C2×C32C22×C811yescharacteristic
358C2×C32C22×C811yescharacteristic
359C2×C32C22×C811yescharacteristic
360C2×C32C22×C811yescharacteristic
361#367—21no—
362#366—21no—
363#366C22×C811yescharacteristic
364#366C22×C811yescharacteristic
365#366C22×C811yescharacteristic
366#366C22×C811yescharacteristic
367C4⋊C16—41no—
368C4⋊C16—41no—
369C4⋊C16—41no—
370C4⋊C16—41no—
371C4⋊C16—41no—
372C4⋊C16—81no—
373C4⋊C16—81no—
374#358—21no—
375#358—21no—
376#357—21no—
377#357—21no—
378#357—21no—
379#357—21no—
380#357—21no—
381#357—21no—
382C22⋊C16—41no—
383C22⋊C16—81no—
384C22⋊C16—81no—
385C22⋊C16—81no—
386C22⋊C16—81no—
387C22⋊C16—81no—
388C22⋊C16—81no—
389C22⋊C16—81no—
390C22⋊C16—81no—
391#346—41no—
392#346—81no—
393#346—81no—
394C4×C16—21no—
395C4×C16—21no—
396C4×C16—21no—
397C4×C16—21no—
398C4×C16—21no—
399C4×C16—81no—
400C4×C16—81no—
401C4×C16C4×D411yescharacteristic
402C4×C16C4×D411yescharacteristic
403C4×C16C22×C811yescharacteristic
404C4×C16#13111yescharacteristic
405C4×C16C23×C411yescharacteristic
406C4×C16C2×#4111yescharacteristic
407#335—21no—
408#335—21no—
409#335—21no—
410#335—21no—
411#335—41no—
412#334—41no—
413Q8⋊C8—41no—
414Q8⋊C8—41no—
415D4⋊C8—41no—
416D4⋊C8—41no—
417D4⋊C8—41no—
418D4⋊C8—41no—
419D4⋊C8—41no—
420D4⋊C8—41no—
421C82C4×D411yescharacteristic
422C82AGL(1,ℤ/8)11yescharacteristic
423C82AGL(1,ℤ/8)11yescharacteristic
424C82C22×D411yescharacteristic
425C22×D4—41no—
426C23×C4—81no—
427C2×Q16—21no—
428C2×SD16—41no—
429C2×D8—21no—
430C2×D8—21no—
431C2×D8—21no—
432C2×D8—42no—
433C2×D8—41no—
434C2×D8—81no—
435C2×M16—81no—
436C22×C8—21no—
437C22×C8—21no—
438C22×C8—81no—
439C22×C8—81no—
440C22×C8—81no—
441C22×C8—81no—
442C22×C8—81no—
443C22×C8—81no—
444C22×C8—321no—
445C22×C8—321no—
446C22×C8C8×D411yescharacteristic
447C22×C8#44411yescharacteristic
448#126—81no—
449#125—41no—
450#124—41no—
451#123—81no—
452C22⋊Q8—41no—
453#121—41no—
454C4×Q8—41no—
455C4×D4—41no—
456C4×D4—41no—
457C4×D4—41no—
458C4×D4—41no—
459C4×D4—41no—
460C4×D4—81no—
461C4×D4—81no—
462C4×D4—81no—
463#117—81no—
464C2×C4⋊C4—41no—
465C2×C22⋊C4—81no—
466C2×C42#44211yescharacteristic
467Q32—21no—
468Q32—21no—
469SD32—81no—
470D16—21no—
471D16—21no—
472D16—21no—
473D16—21no—
474D16—21no—
475D16—42no—
476D16#50412yes—
477M32—161no—
478M32—162no—
479M32—321no—
480M32—321no—
481C2×C16—21no—
482C2×C16—21no—
483C2×C16—21no—
484C2×C16—21no—
485C2×C16—41no—
486C2×C16—42no—
487C2×C16—41no—
488C2×C16—41no—
489C2×C16—41no—
490C2×C16—41no—
491C2×C16—81no—
492C2×C16—81no—
493C2×C16—81no—
494C2×C16—81no—
495C2×C16—162no—
496C2×C16—321no—
497C2×C16—321no—
498C2×C16C8×D411yescharacteristic
499C2×C16C8×D411yescharacteristic
500C2×C16C23×C811yescharacteristic
501C2×C16C2×#13111yescharacteristic
502#107—21no—
503#107—21no—
504#107—21no—
505#107—21no—
506#107—41no—
507#106—41no—
508C4⋊C8—41no—
509C4⋊C8—41no—
510C4⋊C8—41no—
511C4⋊C8—41no—
512C4⋊C8—81no—
513C4⋊C8—81no—
514C4⋊C8—81no—
515#103—41no—
516#103—41no—
517#102—41no—
518#102—41no—
519#102—41no—
520#102—41no—
521#102—41no—
522#102—41no—
523C22⋊C8—81no—
524C22⋊C8—81no—
525C22⋊C8—81no—
526C22⋊C8—81no—
527C22⋊C8—81no—
528C22⋊C8—81no—
529C22⋊C8—81no—
530#97—81no—
531#97—81no—
532C4×C8—21no—
533C4×C8—21no—
534C4×C8—81no—
535C4×C8C8×D411yescharacteristic
536C4×C8#44211yescharacteristic
537C4×C8#44211yescharacteristic
538C4×C8#44211yescharacteristic
539C4×C8#44411yescharacteristic
540C4×C8C2×C4×D411yescharacteristic
541C4×C8C2×AGL(1,ℤ/8)11yescharacteristic
542C4×C8C2×AGL(1,ℤ/8)11yescharacteristic
543C32—21no—
544C32—21no—
545C32—21no—
546C32—21no—
547C32—41no—
548C32—41no—
549C32—81no—
550C32—81no—
551C24—81no—
552C2×Q8—41no—
553C2×D4—41no—
554C2×D4—41no—
555C2×D4—41no—
556C2×D4—82no—
557C2×D4—81no—
558C2×D4—81no—
559C2×D4—161no—
560C22×C4—81no—
561C22×C4—81no—
562C22×C4—81no—
563C22×C4—81no—
564C22×C4—81no—
565C22×C4—81no—
566C22×C4—321no—
567C22×C4—321no—
568C22×C4#158011yescharacteristic
569C22×C4AGL(1,ℤ/16)11yescharacteristic
570C22×C4AGL(1,ℤ/16)11yescharacteristic
571Q16—41no—
572Q16—41no—
573SD16—161no—
574D8—22no—
575D8—41no—
576D8—41no—
577D8—41no—
578D8—41no—
579D8—41no—
580D8—82no—
581M16—162no—
582C2×C8—21no—
583C2×C8—21no—
584C2×C8—21no—
585C2×C8—21no—
586C2×C8—21no—
587C2×C8—21no—
588C2×C8—41no—
589C2×C8—81no—
590C2×C8—81no—
591C2×C8—81no—
592C2×C8—81no—
593C2×C8—81no—
594C2×C8—81no—
595C2×C8—161no—
596C2×C8—161no—
597C2×C8—161no—
598C2×C8—161no—
599C2×C8—161no—
600C2×C8—161no—
601C2×C8—321no—
602C2×C8—321no—
603C2×C8—321no—
604C2×C8#158011yescharacteristic
605C2×C8#158011yescharacteristic
606C2×C8C2×C8×D411yescharacteristic
607C2×C8C2×#44411yescharacteristic
608C4⋊C4—41no—
609C4⋊C4—41no—
610C4⋊C4—41no—
611C4⋊C4—41no—
612C4⋊C4—81no—
613C4⋊C4—81no—
614C4⋊C4—81no—
615C4⋊C4—81no—
616C4⋊C4—81no—
617C22⋊C4—81no—
618C22⋊C4—81no—
619C22⋊C4—81no—
620C22⋊C4—81no—
621C22⋊C4—81no—
622C22⋊C4—81no—
623C22⋊C4—81no—
624C22⋊C4—81no—
625C22⋊C4—81no—
626C42—81no—
627C42#158011yescharacteristic
628C42AGL(1,ℤ/16)11yescharacteristic
629C42AGL(1,ℤ/16)11yescharacteristic
630C42C2×#44211yescharacteristic
631C16—22no—
632C16—21no—
633C16—21no—
634C16—21no—
635C16—41no—
636C16—81no—
637C16—81no—
638C16—81no—
639C16—81no—
640C16—161no—
641C16—161no—
642C16—161no—
643C16—161no—
644C16C2×#50411yescharacteristic
645C16#340711yescharacteristic
646C23—81no—
647C23—81no—
648C23—81no—
649C23—162no—
650C23—162no—
651C23—321no—
652C23#1206711yescharacteristic
653Q8—81no—
654Q8—81no—
655D4—42no—
656D4—81no—
657D4—81no—
658D4—81no—
659D4—81no—
660D4—81no—
661D4—162no—
662D4—321no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—81no—
668C2×C4—81no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4—81no—
672C2×C4—161no—
673C2×C4—161no—
674C2×C4—161no—
675C2×C4—161no—
676C2×C4—161no—
677C2×C4—161no—
678C2×C4—321no—
679C2×C4—321no—
680C2×C4—321no—
681C2×C4#1206711yescharacteristic
682C2×C4#1206711yescharacteristic
683C2×C4#1206711yescharacteristic
684C2×C4C2×#158011yescharacteristic
685C2×C4C2×AGL(1,ℤ/16)11yescharacteristic
686C2×C4C2×AGL(1,ℤ/16)11yescharacteristic
687C8—21no—
688C8—21no—
689C8—21no—
690C8—41no—
691C8—41no—
692C8—41no—
693C8—41no—
694C8—161no—
695C8—161no—
696C8—161no—
697C8—161no—
698C8#3343611yescharacteristic
699C8#3344211yescharacteristic
700C22—21no—
701C22—21no—
702C22—21no—
703C22—82no—
704C22—161no—
705C22—161no—
706C22—161no—
707C22—161no—
708C22—161no—
709C22—161no—
710C22—161no—
711C22—161no—
712C22—324no—
713C22C2×#1206711yescharacteristic
714C4—21no—
715C4—21no—
716C4—21no—
717C4—21no—
718C4—21no—
719C4—161no—
720C4—161no—
721C4—161no—
722C4—161no—
723C4—161no—
724C4#48360011yescharacteristic
725C4#48360211yescharacteristic
726C2—41no—
727C2—162no—
728C2—321no—
729C2—321no—
730C2#7222364611yescharacteristic
731C2#7222507511yescharacteristic
732C2#7228664311yescharacteristic
subgroupG/H
center, socleC22C2×#12067
commutatorC2×C16C23×C8
Frattini, cosocle#1311C24
prime radical#13661C8
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ #122219 ◁ G
LECSG ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #72322395C2semidirect
2. #72320554C2semidirect
3. #72287181C2semidirect
4. #72287042C2semidirect
5. #72286966C2semidirect
6. #72286779C2semidirect
7. #72286459C2semidirect
8. #72286442C2semidirect
9. #72286323C2semidirect
10. #72286323C2semidirect
12. #72231942C2semidirect
13. #72231887C2semidirect
14. #72225240C2semidirect
15. #72224639C2semidirect
40. #122966C22semidirect
41. #122965C22semidirect
44. #122963C22semidirect
45. #122962C22semidirect
51. #93877C22semidirect
52. #93877C22semidirect
53. #93877C22semidirect
54. #93877C22semidirect
55. #93758C22semidirect
56. #93758C22semidirect
61. #93757C22semidirect
62. #93757C22semidirect
63. #93668C22semidirect
64. #93668C22semidirect
65. #93668C22semidirect
66. #93668C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#723223952
#723205542
#722871812
#722870422
#722869662
#722867792
#722864592
#722864422
#722863232
#722320282
#722319422
#722318872
#722252402
#722246392