Finite Group Databasebeta

← #164530

Subgroups — #164530

Order 512.

814 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36293C211yesmaximal, characteristic
2#22590C211yesmaximal, characteristic, complemented
3#20780C211yesmaximal, characteristic, complemented
4#7721C211yesmaximal, characteristic, complemented
5#7716C211yesmaximal, characteristic, complemented
6#7696C211yesmaximal, characteristic, complemented
7#7695C211yesmaximal, characteristic, complemented
8#7692C211yesmaximal, characteristic, complemented
9#7666C211yesmaximal, characteristic, complemented
10#7618C211yesmaximal, characteristic, complemented
11#7614C211yesmaximal, characteristic, complemented
12#7606C211yesmaximal, characteristic
13#7594C211yesmaximal, characteristic, complemented
14#7592C211yesmaximal, characteristic
15#7567C211yesmaximal, characteristic, complemented
16#2861C2211yescharacteristic
17#2860C2211yescharacteristic
18#2855C2211yescharacteristic
19#2597—21no—
20#2531—21no—
21#2526—21no—
22#2513C411yescharacteristic
23#2482—21no—
24#2480—21no—
25#2479—21no—
26#2454C411yescharacteristic
27#2448C411yescharacteristic
28#2447C411yescharacteristic
29#2442C411yescharacteristic
30#2435C411yescharacteristic
31#2360C411yescharacteristic
32#2342—21no—
33#2342—21no—
34D4×C22⋊C4C411yescharacteristic
35C4×#124C2211yescharacteristic
36#2293C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1917—21no—
40#1916—21no—
41#1851—21no—
42#1840—21no—
43#1839C2211yescharacteristic, complemented
44#1831C2211yescharacteristic, complemented
45#1831C2211yescharacteristic, complemented
46#1831C2211yescharacteristic, complemented
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1798—21no—
51#1768—21no—
52#1767C2211yescharacteristic, complemented
53#1753—21no—
54#1750—21no—
55#1745C2211yescharacteristic, complemented
56#1744C2211yescharacteristic, complemented
57#1744C2211yescharacteristic, complemented
58#1728C2211yescharacteristic
59#1272C2211yescharacteristic, complemented
60#1272C2211yescharacteristic, complemented
61#1272C2211yescharacteristic, complemented
62#1272C2211yescharacteristic, complemented
63#1271C2211yescharacteristic, complemented
64#1271C2211yescharacteristic, complemented
65#1271C2211yescharacteristic, complemented
66#1271C2211yescharacteristic, complemented
67#1270C2211yescharacteristic
68#1270C2211yescharacteristic
69#1270C2211yescharacteristic
70#1270C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×D4C2×C411yescharacteristic
76C22×M16—21no—
77C2×#126—21no—
78C2×#125C2×C411yescharacteristic
79C2×#124C2×C411yescharacteristic
80C2×#124C2×C411yescharacteristic
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4C2×C411yescharacteristic
85C2×#117—21no—
86C2×#117D411yescharacteristic
87C2×#117D411yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C42C2311yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104#400—21no—
105#399—21no—
106#399—21no—
107#398—21no—
108#397—21no—
109#397—21no—
110#397—21no—
111#397—21no—
112#397—21no—
113#396—21no—
114#394—21no—
115#394—21no—
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2311yescharacteristic
119#390—21no—
120#390—21no—
121#390C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#390C2×C411yescharacteristic
124#390C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2311yescharacteristic
128#390C2311yescharacteristic
129#389—21no—
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2311yescharacteristic
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#388—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#386—21no—
154#386—21no—
155#385—21no—
156#385—21no—
157#385—21no—
158#385—21no—
159#385—21no—
160#382C2×C411yescharacteristic
161#381—21no—
162#381—21no—
163#380—21no—
164#380—21no—
165#380C2×C411yescharacteristic
166#379—21no—
167#379—21no—
168#379—21no—
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4C2×C411yescharacteristic
179C4×C22⋊C4C2×C411yescharacteristic
180C4×C22⋊C4D411yescharacteristic
181C4×C22⋊C4D411yescharacteristic
182#376D411yescharacteristic
183#376D411yescharacteristic
184C43C2311yescharacteristic
185C43C2311yescharacteristic
186#336—21no—
187#336—21no—
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C25—21no—
207C22×Q8C22×C411yescharacteristic
208C22×D4—22no—
209C22×D4—21no—
210C22×D4—41no—
211C22×D4—42no—
212C22×D4—81no—
213C22×D4C22×C411yescharacteristic
214C22×D4C22×C411yescharacteristic
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4C22×C411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C2×M16—42no—
222C2×M16—42no—
223C2×M16—81no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232#126—81no—
233#126—81no—
234#125—81no—
235#124—22no—
236#124—41no—
237#124—81no—
238C22⋊Q8—42no—
239C4×Q8—81no—
240C4×D4—22no—
241C4×D4—41no—
242C4×D4—81no—
243#117—41no—
244#117—41no—
245#117—42no—
246#117—81no—
247#117—81no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—81no—
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—81no—
296C2×C22⋊C4—81no—
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—42no—
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2411yescharacteristic
320C4⋊C8—42no—
321C4⋊C8—42no—
322C4⋊C8—42no—
323C4⋊C8—81no—
324C22⋊C8—42no—
325C22⋊C8—42no—
326C22⋊C8—42no—
327#97—42no—
328#97—81no—
329C4×C8—42no—
330C4×C8—42no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354C24—21no—
355C24—21no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—81no—
361C24—81no—
362C24—81no—
363C24C2×C22⋊C411yescharacteristic
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—81no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4#13112yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422M16—84no—
423C2×C8—22no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C22⋊C4—22no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—42no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—22no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—81no—
523C42—81no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—162no—
549C23—162no—
550C23—162no—
551C23#41012yes—
552C23C22×C22⋊C411yescharacteristic
553Q8—81no—
554Q8—81no—
555D4—42no—
556D4—42no—
557D4—81no—
558D4—81no—
559D4—84no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—84no—
623C2×C4—162no—
624C2×C4—162no—
625C2×C4#42112yes—
626C2×C4C2×#13111yescharacteristic
627C2×C4#56811yescharacteristic
628C2×C4#57511yescharacteristic
629C2×C4#57511yescharacteristic
630C2×C4#57611yescharacteristic
631C2×C4#57711yescharacteristic
632C8—42no—
633C8—81no—
634C8—81no—
635C8—81no—
636C8—81no—
637C8—81no—
638C8—81no—
639C8—81no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—168no—
662C22#288011yescharacteristic
663C22C2×#41011yescharacteristic
664C22#288811yescharacteristic
665C22#289611yescharacteristic
666C22#289711yescharacteristic
667C22C2×#42111yescharacteristic
668C22#290111yescharacteristic
669C4—21no—
670C4—21no—
671C4—22no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4—81no—
692C2—42no—
693C2—82no—
694C2—161no—
695C2#1741211yescharacteristic
696C2#1751311yescharacteristic
697C2#1752211yescharacteristic
698C2#1755411yescharacteristic
699C2#1760411yescharacteristic
700C2#1767911yescharacteristic
701C2#1776611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22590C2semidirect
3. #20780C2semidirect
4. #7721C2semidirect
5. #7716C2semidirect
6. #7696C2semidirect
7. #7695C2semidirect
8. #7692C2semidirect
9. #7666C2semidirect
10. #7618C2semidirect
11. #7614C2semidirect
13. #7594C2semidirect
15. #7567C2semidirect
43. #1839C22semidirect
44. #1831C22semidirect
45. #1831C22semidirect
46. #1831C22semidirect
52. #1767C22semidirect
55. #1745C22semidirect
56. #1744C22semidirect
57. #1744C22semidirect
59. #1272C22semidirect
60. #1272C22semidirect
61. #1272C22semidirect
62. #1272C22semidirect
63. #1271C22semidirect
64. #1271C22semidirect
65. #1271C22semidirect
66. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#362932
#225902
#207802
#77212
#77162
#76962
#76952
#76922
#76662
#76182
#76142
#76062
#75942
#75922
#75672