Finite Group Databasebeta

← #72323018

Subgroups — #72323018

Order 2048.

799 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72322991C212yesmaximal, complemented
2#72321038C211yesmaximal, characteristic, complemented
3#72320492C211yesmaximal, characteristic, complemented
4#72287181C212yesmaximal, complemented
5#72287042C212yesmaximal, complemented
6#72286838C211yesmaximal, characteristic, complemented
7#72286838C211yesmaximal, characteristic, complemented
8#72286323C212yesmaximal, complemented
9#72232000C211yesmaximal, characteristic
10C2×#93877C211yesmaximal, characteristic, complemented
11#72231870C211yesmaximal, characteristic, complemented
12#512761C411yescharacteristic
13#483761—21no—
14#483760—21no—
15#483499C2211yescharacteristic
16#483499C2211yescharacteristic
17#151085C411yescharacteristic
18#151085C2211yescharacteristic
19#151084C411yescharacteristic
20#151084C2211yescharacteristic
21#151083C411yescharacteristic
22#151083C411yescharacteristic
23#151083C2211yescharacteristic
24#151083C2211yescharacteristic
25#151037C411yescharacteristic
26#150062C411yescharacteristic
27#150062C411yescharacteristic
28#149502C2211yescharacteristic
29C2×#7330C2211yescharacteristic
30#122966C2212yes—
31#122965C2212yes—
32#122964—22no—
33#122963C2212yes—
34#122962C2212yes—
35#122961—22no—
36C2×#7099C2211yescharacteristic
37C2×#7099C2211yescharacteristic
38#122465—21no—
39#122465—21no—
40#122214C2211yescharacteristic
41#93877C2212yes—
42#93877C2212yes—
43#93758C2212yes—
44#93758C2212yes—
45#93757C2212yes—
46#93757C2212yes—
47#93668C2212yes—
48#93668C2212yes—
49C2×#2262C811yescharacteristic
50#33503—41no—
51#33502—41no—
52C4×#508C2×C411yescharacteristic
53C4×#508C2×C411yescharacteristic
54C8×#135—21no—
55C8×#135—21no—
56#13654C811yescharacteristic
57#13654C2×C411yescharacteristic
58#13653C811yescharacteristic
59#13653C2×C411yescharacteristic
60#13652C811yescharacteristic
61#13652C811yescharacteristic
62#13652C2×C411yescharacteristic
63#13652C2×C411yescharacteristic
64#13633C811yescharacteristic
65#13615C811yescharacteristic
66#13615C811yescharacteristic
67C4×M64—21no—
68C2×#1397C2×C411yescharacteristic
69C2×#1397C2×C411yescharacteristic
70#12064—22no—
71#12062—22no—
72#12061—22no—
73#12059—22no—
74#12051—42no—
75C16×SD16—42no—
76#12045C2×C411yescharacteristic
77#12045C2×C411yescharacteristic
78#12045C2311yescharacteristic
79#12045C2311yescharacteristic
80#12044C2×C411yescharacteristic
81#12044C2×C411yescharacteristic
82#12044C2311yescharacteristic
83#12044C2311yescharacteristic
84#12043—21no—
85#12043—21no—
86#12043—21no—
87#12043—21no—
88#12008C2×C411yescharacteristic
89#12008C2×C411yescharacteristic
90C2×#1367—21no—
91C2×#1367—21no—
92#11621—41no—
93#11621—41no—
94#11591—21no—
95#11591—21no—
96#11410D411yescharacteristic
97#11410D411yescharacteristic
98C2×#1311C2311yescharacteristic
99#7463C2×C411yescharacteristic
100#7463C2×C411yescharacteristic
101#7463C2×C411yescharacteristic
102#7463C2×C411yescharacteristic
103#7386C2×C411yescharacteristic
104#7386C2×C411yescharacteristic
105#7386C2×C411yescharacteristic
106#7386C2×C411yescharacteristic
107#7385C2×C411yescharacteristic
108#7385C2×C411yescharacteristic
109#7385C2×C411yescharacteristic
110#7385C2×C411yescharacteristic
111#7332—21no—
112#7332C2312yes—
113#7330C2×C411yescharacteristic
114#7330C2×C411yescharacteristic
115#7330C2311yescharacteristic
116#7330C2311yescharacteristic
117#7099—21no—
118#7099—21no—
119#7099C2312yes—
120#7099C2312yes—
121#7071—22no—
122#7071—22no—
123#7070—22no—
124#7070—22no—
125#7053D412yes—
126#7053C2312yes—
127C2×#508C2×C811yescharacteristic
128C2×#508C2×C811yescharacteristic
129C8×#41—41no—
130C8×#41—41no—
131C4×#135—21no—
132C4×#135—21no—
133#2262—24no—
134C2×Q64C2×C811yescharacteristic
135C2×SD64C2×C811yescharacteristic
136C2×SD64C2×C811yescharacteristic
137C2×D32C2×C811yescharacteristic
138C2×M64—41no—
139C22×C32—21no—
140C22×C32C2×C811yescharacteristic
141C22×C32C2×C811yescharacteristic
142#2183—42no—
143Q8×C16—42no—
144C4×Q32C2×C811yescharacteristic
145C4×Q32C2×C811yescharacteristic
146C4×Q32C22×C411yescharacteristic
147C4×Q32C22×C411yescharacteristic
148C4×SD32—21no—
149C4×SD32—21no—
150C4×SD32—21no—
151C4×SD32—21no—
152C4×D16C2×C811yescharacteristic
153C4×D16C2×C811yescharacteristic
154C4×D16C22×C411yescharacteristic
155C4×D16C22×C411yescharacteristic
156#2169—42no—
157#2168—82no—
158D4×C16—42no—
159D4×C16—82no—
160#2163—81no—
161#2162—21no—
162#2162—21no—
163#2162—81no—
164#2158C2×C811yescharacteristic
165#2158C2×C811yescharacteristic
166#2150—81no—
167C2×C4⋊C16—41no—
168C2×C4⋊C16—41no—
169C2×C4⋊C16—81no—
170#2138—21no—
171#2138—21no—
172C2×C4×C16C22×C411yescharacteristic
173C2×C4×C16#4111yescharacteristic
174C8×Q16—21no—
175C8×Q16—21no—
176C8×Q16—21no—
177C8×Q16—21no—
178C8×SD16—41no—
179C8×SD16—41no—
180C8×SD16—41no—
181C8×SD16—41no—
182C8×D8—21no—
183C8×D8—21no—
184C8×D8—21no—
185C8×D8—21no—
186#1564—21no—
187#1564—21no—
188#1476—41no—
189#1476—41no—
190C2×C82C2×D411yescharacteristic
191#1424C2×C811yescharacteristic
192#1424C2×C811yescharacteristic
193#1423C2×C811yescharacteristic
194#1423C2×C811yescharacteristic
195#1416C2×C811yescharacteristic
196#1416C2×C811yescharacteristic
197#1416C2×C811yescharacteristic
198#1416C2×C811yescharacteristic
199#1415C2×C811yescharacteristic
200#1415C2×C811yescharacteristic
201#1415C2×C811yescharacteristic
202#1415C2×C811yescharacteristic
203#1397—21no—
204#1397—22no—
205#1397C2×C811yescharacteristic
206#1397C2×C811yescharacteristic
207#1397C22×C411yescharacteristic
208#1397C22×C411yescharacteristic
209#1397C22×C411yescharacteristic
210#1397C22×C411yescharacteristic
211C4×C32—41no—
212#1372C22×C411yescharacteristic
213#1372C22×C411yescharacteristic
214#1372C22×C411yescharacteristic
215#1372C22×C411yescharacteristic
216#1371—21no—
217#1371—21no—
218#1367—22no—
219#1367—22no—
220#1367—41no—
221#1367—41no—
222Q8⋊C16—42no—
223Q8⋊C16—42no—
224#1332—21no—
225#1332—21no—
226#1332—21no—
227#1332—21no—
228#1331—21no—
229#1331—21no—
230#1331—21no—
231#1331—21no—
232D4⋊C16—42no—
233D4⋊C16—42no—
234#1312C2×D412yes—
235#1312C2×D412yes—
236#1311C22×C411yescharacteristic
237#1311C2×D411yescharacteristic
238#1311#4111yescharacteristic
239#1311C2411yescharacteristic
240C8×C16—21no—
241C8×C16—21no—
242C2×#135—21no—
243C2×#135—21no—
244C4×#41—41no—
245C4×#41—41no—
246#508—22no—
247#508—22no—
248#508—22no—
249#508M3214yescomplemented
250C2×Q32C22×C811yescharacteristic
251C2×Q32C22×C811yescharacteristic
252C2×SD32—21no—
253C2×SD32—21no—
254C2×D16C22×C811yescharacteristic
255C2×D16C22×C811yescharacteristic
256C2×M32—82no—
257C22×C16—21no—
258C22×C16—82no—
259C22×C16—81no—
260C22×C16—81no—
261C22×C16C22×C811yescharacteristic
262C22×C16#13111yescharacteristic
263C8×Q8—41no—
264C8×Q8—41no—
265C8×Q8—41no—
266C8×Q8—41no—
267C4×Q16—21no—
268C4×Q16—21no—
269C4×Q16—21no—
270C4×Q16—21no—
271C4×SD16—41no—
272C4×SD16—41no—
273C4×SD16—41no—
274C4×SD16—41no—
275C4×D8—21no—
276C4×D8—21no—
277C4×D8—21no—
278C4×D8—21no—
279C8×D4—41no—
280C8×D4—41no—
281C8×D4—41no—
282C8×D4—41no—
283C8×D4—81no—
284C8×D4—81no—
285C8×D4—81no—
286C8×D4—81no—
287#431—41no—
288#431—41no—
289#431—81no—
290#431—81no—
291#427—21no—
292#427—21no—
293#416—41no—
294#416—41no—
295C2×C4×C8—81no—
296C2×C4×C8—81no—
297C2×C4×C8C4×D411yescharacteristic
298C2×C4×C8AGL(1,ℤ/8)11yescharacteristic
299C2×C4×C8AGL(1,ℤ/8)11yescharacteristic
300Q64—22no—
301SD64—22no—
302SD64—22no—
303D32—22no—
304M64—82no—
305C2×C32—24no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—22no—
309C2×C32—22no—
310C2×C32—42no—
311C2×C32C22×C811yescharacteristic
312C2×C32C22×C811yescharacteristic
313C2×C32C22×C811yescharacteristic
314C2×C32C22×C811yescharacteristic
315#367—21no—
316#367—21no—
317#366C22×C811yescharacteristic
318#366C22×C811yescharacteristic
319#366C22×C811yescharacteristic
320#366C22×C811yescharacteristic
321C4⋊C16—41no—
322C4⋊C16—41no—
323C4⋊C16—42no—
324C4⋊C16—42no—
325C4⋊C16—81no—
326C4⋊C16—81no—
327C4⋊C16—82no—
328C4⋊C16—82no—
329#358—21no—
330#358—21no—
331#358—21no—
332#358—21no—
333#357—21no—
334#357—21no—
335#357—21no—
336#357—21no—
337C22⋊C16—41no—
338C22⋊C16—41no—
339C22⋊C16—82no—
340C22⋊C16—82no—
341#346—82no—
342C4×C16—22no—
343C4×C16—22no—
344C4×C16—21no—
345C4×C16—82no—
346C4×C16C4×D411yescharacteristic
347C4×C16C4×D411yescharacteristic
348C4×C16C22×C811yescharacteristic
349C4×C16#13111yescharacteristic
350C4×C16C23×C411yescharacteristic
351C4×C16C2×#4111yescharacteristic
352#335—21no—
353#335—21no—
354#335—21no—
355#335—21no—
356#334—41no—
357#334—41no—
358Q8⋊C8—41no—
359Q8⋊C8—41no—
360Q8⋊C8—41no—
361Q8⋊C8—41no—
362D4⋊C8—41no—
363D4⋊C8—41no—
364D4⋊C8—41no—
365D4⋊C8—41no—
366C82C4×D411yescharacteristic
367C82AGL(1,ℤ/8)11yescharacteristic
368C82AGL(1,ℤ/8)11yescharacteristic
369C82C22×D411yescharacteristic
370C2×#41—41no—
371C2×#41—41no—
372#135—24no—
373#135—42no—
374#135—42no—
375#135—42no—
376C2×Q16—21no—
377C2×Q16—21no—
378C2×SD16—41no—
379C2×SD16—41no—
380C2×D8—21no—
381C2×D8—21no—
382C22×C8—21no—
383C22×C8—21no—
384C22×C8—81no—
385C22×C8—81no—
386C22×C8—81no—
387C22×C8—81no—
388C22×C8C8×D411yescharacteristic
389C22×C8#44411yescharacteristic
390C4×Q8—41no—
391C4×Q8—41no—
392C4×Q8—41no—
393C4×Q8—41no—
394C4×D4—41no—
395C4×D4—41no—
396C4×D4—41no—
397C4×D4—41no—
398C4×D4—81no—
399C4×D4—81no—
400C4×D4—81no—
401C4×D4—81no—
402#117—41no—
403#117—41no—
404#117—81no—
405#117—81no—
406C2×C42—81no—
407C2×C42—81no—
408C2×C42#44211yescharacteristic
409Q32—21no—
410Q32—21no—
411Q32—21no—
412Q32C2×M3212yes—
413SD32—24no—
414SD32—42no—
415D16—21no—
416D16—21no—
417D16—21no—
418D16C2×M3212yes—
419M32—322no—
420C2×C16—22no—
421C2×C16—21no—
422C2×C16—21no—
423C2×C16—42no—
424C2×C16—42no—
425C2×C16—42no—
426C2×C16—82no—
427C2×C16—82no—
428C2×C16—322no—
429C2×C16—321no—
430C2×C16—321no—
431C2×C16C8×D411yescharacteristic
432C2×C16C8×D411yescharacteristic
433C2×C16C2×M3212yes—
434C2×C16#50412yes—
435C2×C16C23×C811yescharacteristic
436C2×C16C2×#13111yescharacteristic
437#107—21no—
438#107—21no—
439#107—21no—
440#107—21no—
441#106—41no—
442#106—41no—
443C4⋊C8—41no—
444C4⋊C8—41no—
445C4⋊C8—41no—
446C4⋊C8—41no—
447C4⋊C8—81no—
448C4⋊C8—81no—
449C4⋊C8—81no—
450C4⋊C8—81no—
451#103—41no—
452#103—41no—
453#103—41no—
454#103—41no—
455#102—41no—
456#102—41no—
457#102—41no—
458#102—41no—
459C22⋊C8—81no—
460C22⋊C8—81no—
461C22⋊C8—81no—
462C22⋊C8—81no—
463C22⋊C8—81no—
464C22⋊C8—81no—
465C4×C8—21no—
466C4×C8—21no—
467C4×C8—81no—
468C4×C8—81no—
469C4×C8—81no—
470C4×C8—81no—
471C4×C8—81no—
472C4×C8—81no—
473C4×C8C8×D411yescharacteristic
474C4×C8#44211yescharacteristic
475C4×C8#44211yescharacteristic
476C4×C8#44211yescharacteristic
477C4×C8#44411yescharacteristic
478C4×C8C2×C4×D411yescharacteristic
479C4×C8C2×AGL(1,ℤ/8)11yescharacteristic
480C4×C8C2×AGL(1,ℤ/8)11yescharacteristic
481C32—22no—
482C32—22no—
483C32—21no—
484C32—21no—
485C32—21no—
486C32—21no—
487C32—82no—
488#41—44no—
489#41—82no—
490#41—82no—
491#41—82no—
492C2×Q8—41no—
493C2×Q8—41no—
494C2×D4—41no—
495C2×D4—41no—
496C2×D4—81no—
497C2×D4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—81no—
504C22×C4#158011yescharacteristic
505C22×C4AGL(1,ℤ/16)11yescharacteristic
506C22×C4AGL(1,ℤ/16)11yescharacteristic
507Q16—22no—
508Q16—41no—
509Q16—41no—
510Q16—41no—
511SD16—44no—
512SD16—82no—
513D8—22no—
514D8—41no—
515D8—41no—
516D8—41no—
517C2×C8—22no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—81no—
523C2×C8—81no—
524C2×C8—81no—
525C2×C8—81no—
526C2×C8—81no—
527C2×C8—81no—
528C2×C8—161no—
529C2×C8—161no—
530C2×C8—161no—
531C2×C8—161no—
532C2×C8—321no—
533C2×C8—321no—
534C2×C8#158011yescharacteristic
535C2×C8#158011yescharacteristic
536C2×C8#216812yes—
537C2×C8#217012yes—
538C2×C8C2×C8×D411yescharacteristic
539C2×C8C2×#44411yescharacteristic
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—81no—
545C4⋊C4—81no—
546C4⋊C4—81no—
547C4⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C22⋊C4—81no—
551C22⋊C4—81no—
552C22⋊C4—81no—
553C22⋊C4—81no—
554C42—81no—
555C42—81no—
556C42—81no—
557C42—81no—
558C42—81no—
559C42—81no—
560C42#158011yescharacteristic
561C42AGL(1,ℤ/16)11yescharacteristic
562C42AGL(1,ℤ/16)11yescharacteristic
563C42C2×#44211yescharacteristic
564C16—21no—
565C16—21no—
566C16—41no—
567C16—41no—
568C16—82no—
569C16—82no—
570C16—162no—
571C16—162no—
572C16#216812yes—
573C16C22×M3211yescharacteristic
574C16C2×#50411yescharacteristic
575C23—81no—
576C23—81no—
577C23#1206711yescharacteristic
578Q8—42no—
579Q8—81no—
580Q8—81no—
581Q8—81no—
582D4—42no—
583D4—84no—
584D4—81no—
585D4—81no—
586D4—81no—
587D4—162no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—81no—
591C2×C4—84no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—161no—
598C2×C4—161no—
599C2×C4—161no—
600C2×C4—161no—
601C2×C4—162no—
602C2×C4—321no—
603C2×C4—321no—
604C2×C4#1206312yes—
605C2×C4#1206711yescharacteristic
606C2×C4#1206711yescharacteristic
607C2×C4#1206711yescharacteristic
608C2×C4#1206812yes—
609C2×C4C2×#158011yescharacteristic
610C2×C4C2×AGL(1,ℤ/16)11yescharacteristic
611C2×C4C2×AGL(1,ℤ/16)11yescharacteristic
612C8—21no—
613C8—21no—
614C8—41no—
615C8—41no—
616C8—41no—
617C8—41no—
618C8—161no—
619C8—161no—
620C8—161no—
621C8—161no—
622C8#1206312yes—
623C8C2×#216811yescharacteristic
624C8C2×#217011yescharacteristic
625C22—21no—
626C22—21no—
627C22—82no—
628C22—161no—
629C22—161no—
630C22—161no—
631C22—161no—
632C22—161no—
633C22#12296812yes—
634C22C2×#1206711yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—82no—
640C4—161no—
641C4—161no—
642C4—161no—
643C4—161no—
644C4—161no—
645C4#12296812yes—
646C4C2×#1206311yescharacteristic
647C4C2×#1206811yescharacteristic
648C2—22no—
649C2—162no—
650C2—321no—
651C2#7222505511yescharacteristic
652C2#7222505511yescharacteristic
653C2C2×#12296811yescharacteristic
subgroupG/H
center, socleC22C2×#12067
commutatorC2×C16C23×C8
Frattini, cosocle#1311C24
prime radicalC2×#2262C8
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ #122214 ◁ G
LECSG ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72322991C2semidirect
2. #72321038C2semidirect
3. #72320492C2semidirect
4. #72287181C2semidirect
5. #72287042C2semidirect
6. #72286838C2semidirect
7. #72286838C2semidirect
8. #72286323C2semidirect
10. C2×#93877C2semidirect
11. #72231870C2semidirect
249. #508M32semidirect

Maximal subgroups

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

subgroupindex
#723229912
#723210382
#723204922
#722871812
#722870422
#722868382
#722863232
#722320002
C2×#938772
#722318702