Finite Group Databasebeta

← #7624495

Subgroups — #7624495

Order 512.

1273 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62363C211yesmaximal, characteristic, complemented
2C2×#3393C211yesmaximal, characteristic, complemented
3#33968C214yesmaximal, complemented
4#33964C212yesmaximal, complemented
5#33961C212yesmaximal, complemented
6#33943C214yesmaximal, complemented
7#33936C212yesmaximal, complemented
8#33927C212yesmaximal, complemented
9#33784C211yesmaximal, characteristic, complemented
10#33779C212yesmaximal, complemented
11#33777C212yesmaximal, complemented
12#33772C211yesmaximal, characteristic, complemented
13#33677C212yesmaximal, complemented
14#33669C211yesmaximal, characteristic, complemented
15#33665C211yesmaximal, characteristic, complemented
16#33663C212yesmaximal, complemented
17#33498C211yesmaximal, characteristic, complemented
18#3522—21no—
19C2×#556—21no—
20#3396C2211yescharacteristic, complemented
21#3396C2212yescomplemented
22#3393—21no—
23#3393—22no—
24#3393—21no—
25#3393C2212yes—
26#3393C2212yes—
27#3393C2214yes—
28#3391C2211yescharacteristic, complemented
29Q8×Q16C2211yescharacteristic, complemented
30#3380C2211yescharacteristic, complemented
31#3380C2212yescomplemented
32#3362—21no—
33#3361—21no—
34#3361—22no—
35#3247—21no—
36#3247—22no—
37#3169C2212yes—
38#3169C2211yescharacteristic
39#3168C2211yescharacteristic
40#3168C2212yes—
41C2×#495C2212yes—
42C2×#494C2211yescharacteristic
43C2×#464—22no—
44C2×#461—21no—
45C2×C8×Q8C2211yescharacteristic
46C2×C4×Q16C2212yes—
47C2×C4×D8C2211yescharacteristic
48#2251C2212yescomplemented
49#2250C2214yes—
50#2249C2214yes—
51#2248C2212yescomplemented
52#2245C2212yescomplemented
53#2243C2214yes—
54#2242C2214yes—
55#2241C2212yescomplemented
56#2236C2211yescharacteristic
57#2233C2211yescharacteristic
58#2233C2212yes—
59#2229C2214yescomplemented
60#2228C2212yes—
61#2226C2212yes—
62#2225C2214yescomplemented
63#2211C2214yes—
64#2211C2214yescomplemented
65#2210C2212yes—
66#2210C2214yes—
67#2210C2214yes—
68#2210C2214yes—
69#2209C2212yescomplemented
70#2208C2212yescomplemented
71#2208C2214yescomplemented
72#2207C2214yescomplemented
73#2206C2212yes—
74#2188C2212yes—
75#2187C2211yescharacteristic
76#2185C2212yes—
77#2185C2212yes—
78#2185C2212yes—
79#2184C2211yescharacteristic
80#2184C2212yes—
81#2183C2212yescomplemented
82#2183C2214yescomplemented
83Q8×C16C2212yescomplemented
84#2177C2214yes—
85#2176C2214yescomplemented
86#2175C2214yescomplemented
87C4×SD32C2212yes—
88C4×SD32C2214yescomplemented
89C4×SD32C2214yescomplemented
90C4×D16C2212yes—
91#2163C2211yescharacteristic
92#2163C2212yes—
93#2162C2212yes—
94#2162C2211yescharacteristic
95C22×Q16—21no—
96C22×Q16C2312yes—
97C22×SD16—22no—
98C22×D8C2311yescharacteristic
99#559—21no—
100#559—22no—
101Q82—21no—
102#556—22no—
103#556—22no—
104#556—24no—
105#556—41no—
106#556—41no—
107#556—42no—
108#554—21no—
109#554—22no—
110#550—21no—
111#548—41no—
112#547—41no—
113#547—42no—
114#542—41no—
115#542—42no—
116#532—21no—
117#532—22no—
118C2×C4⋊Q8—21no—
119#529—21no—
120#529—22no—
121C2×#126—42no—
122C2×#124—22no—
123C2×C22⋊Q8—41no—
124#520—41no—
125C2×C4×Q8—22no—
126C2×C4×Q8D411yescharacteristic
127C2×C4×D4—21no—
128C2×#117—41no—
129C2×Q32C2314yes—
130C2×SD32C2312yes—
131C2×SD32C2314yes—
132C2×SD32C2314yes—
133C2×SD32C2314yes—
134C2×D16C2312yes—
135C2×D16C2314yes—
136#500C2312yes—
137#500C2311yescharacteristic
138#499C2311yescharacteristic
139#499C2312yes—
140#495—22no—
141#495—22no—
142#495C2314yes—
143#495C2314yes—
144#495C2311yescharacteristic
145#495C2312yes—
146#494—21no—
147#494—21no—
148#494C2312yes—
149#494C2312yes—
150#493C2311yescharacteristic
151#493C2312yes—
152#488—21no—
153#488—22no—
154#486—21no—
155#486—22no—
156#478—22no—
157#478—22no—
158#478—22no—
159#477—21no—
160#477—22no—
161#476—21no—
162#476—22no—
163#467C2311yescharacteristic
164#466—22no—
165#466—21no—
166#466C2311yescharacteristic
167#466C2312yes—
168#465—22no—
169#465—21no—
170#464—24no—
171#464—24no—
172#464—24no—
173#464—24no—
174#463—22no—
175#463—22no—
176#463—22no—
177#461—22no—
178#461—22no—
179#461—24no—
180#460—21no—
181#460—22no—
182#459—21no—
183#459—22no—
184C8×Q8D412yes—
185C8×Q8D411yescharacteristic
186C8×Q8D411yescharacteristic
187C8×Q8C2311yescharacteristic
188C8×Q8C2312yes—
189C8×Q8C2311yescharacteristic
190C4×Q16—22no—
191C4×Q16—22no—
192C4×Q16C2311yescharacteristic
193C4×Q16C2312yes—
194C4×Q16C2314yes—
195C4×Q16C2314yes—
196C4×SD16—21no—
197C4×SD16—22no—
198C4×D8—21no—
199C4×D8—22no—
200C4×D8—21no—
201C4×D8—21no—
202C4×D8C2311yescharacteristic
203C4×D8C2312yes—
204C4×D8C2312yes—
205C4×D8C2312yes—
206C4×D8C2312yes—
207C4×D8C2312yes—
208C4×D8C2312yes—
209C2×#107C2311yescharacteristic
210C2×C4⋊C8D411yescharacteristic
211C2×C4⋊C8D412yes—
212C2×#103—22no—
213C2×#102—21no—
214C2×C4×C8C2311yescharacteristic
215C2×C4×C8C2312yes—
216#367C2312yes—
217#367C2314yes—
218#366C2312yes—
219C4⋊C16C2314yes—
220C4⋊C16C2312yes—
221C4⋊C16C2312yes—
222C4⋊C16C2314yes—
223#358C2312yes—
224#358C2314yes—
225#358C2314yes—
226#358C2314yes—
227#358C2314yes—
228#357C2312yes—
229#357C2312yes—
230#357C2312yes—
231#357C2314yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2311yescharacteristic
235C22⋊C16C2312yes—
236#346C2314yes—
237#346C2312yes—
238C4×C16C2312yes—
239C4×C16C2314yes—
240C2×#41—41no—
241C2×#41—42no—
242C2×#41—41no—
243C22×Q8—22no—
244C22×Q8D811yescharacteristic
245C22×D4—21no—
246C23×C4—41no—
247C2×Q16—22no—
248C2×Q16—22no—
249C2×Q16—22no—
250C2×Q16—44no—
251C2×Q16—82no—
252C2×Q16C2414yes—
253C2×Q16C2411yescharacteristic
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—21no—
257C2×SD16—22no—
258C2×SD16—84no—
259C2×D8—21no—
260C2×D8—21no—
261C2×D8—22no—
262C2×D8—42no—
263C2×D8C2412yes—
264C2×D8C2411yescharacteristic
265C2×D8C2412yes—
266C22×C8C2×D412yes—
267C22×C8C2×D411yescharacteristic
268C22×C8C2411yescharacteristic
269C4⋊Q8—21no—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—41no—
274C4⋊Q8—42no—
275C4⋊Q8—41no—
276C4⋊Q8—41no—
277#127—21no—
278#127—22no—
279#126—44no—
280#126—44no—
281#126—44no—
282#126—44no—
283#126—41no—
284#126—42no—
285#125—21no—
286#125—22no—
287#124—24no—
288#124—24no—
289#124—21no—
290#124—22no—
291#124—41no—
292#124—42no—
293#124—42no—
294#124—42no—
295#124—41no—
296#124—42no—
297#123—42no—
298#123—41no—
299#123—42no—
300#123—42no—
301#123—42no—
302C22⋊Q8—21no—
303C22⋊Q8—42no—
304C22⋊Q8—42no—
305C22⋊Q8—44no—
306C22⋊Q8—41no—
307C22⋊Q8—42no—
308C22⋊Q8—42no—
309C22⋊Q8—42no—
310C22⋊Q8—42no—
311#121—21no—
312#121—22no—
313#121—41no—
314#121—42no—
315#121—41no—
316#121—42no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—21no—
320C4×Q8—22no—
321C4×Q8—41no—
322C4×Q8—42no—
323C4×Q8—42no—
324C4×Q8D811yescharacteristic
325C4×Q8D811yescharacteristic
326C4×Q8D812yes—
327C4×Q8C2×D411yescharacteristic
328C4×Q8C2×D412yes—
329C4×Q8C2×D411yescharacteristic
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—21no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—41no—
338C4×D4—42no—
339C4×D4—41no—
340C4×D4—42no—
341C4×D4—41no—
342C4×D4—41no—
343C4×D4—41no—
344C4×D4—42no—
345#117—42no—
346#117—42no—
347#117—44no—
348#117—41no—
349#117—42no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—41no—
354C2×C4⋊C4—41no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4D811yescharacteristic
357C2×C4⋊C4D812yes—
358C2×C22⋊C4—42no—
359C2×C22⋊C4—41no—
360C2×C22⋊C4—42no—
361C2×C42—42no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364Q32—44no—
365SD32—44no—
366SD32—44no—
367SD32—44no—
368SD32—42no—
369D16—42no—
370D16—44no—
371C2×C16C2412yes—
372C2×C16C2414yes—
373C2×C16C2412yes—
374#107—21no—
375#107—21no—
376#107—22no—
377#107C2412yes—
378#107C2411yescharacteristic
379#107C2411yescharacteristic
380#107C2412yes—
381#106—21no—
382#106—22no—
383C4⋊C8C2×D411yescharacteristic
384C4⋊C8C2×D412yes—
385C4⋊C8C2×D411yescharacteristic
386C4⋊C8C2×D412yes—
387C4⋊C8C2×D412yes—
388C4⋊C8C2×D412yes—
389C4⋊C8C2×D412yes—
390#103—24no—
391#103—24no—
392#103—21no—
393#103—21no—
394#103—22no—
395#102—22no—
396#102—22no—
397#102—22no—
398#102—21no—
399#102—21no—
400#102—22no—
401#102—22no—
402#102—22no—
403#102—22no—
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D412yes—
407C4×C8C2×D412yes—
408C4×C8C2411yescharacteristic
409C4×C8C2412yes—
410C4×C8C2411yescharacteristic
411C4×C8C2412yes—
412C24—41no—
413#41—84no—
414#41—164no—
415#41—162no—
416C2×Q8—21no—
417C2×Q8—24no—
418C2×Q8—42no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—41no—
424C2×Q8C2×D811yescharacteristic
425C2×Q8C2×D811yescharacteristic
426C2×Q8C2×D812yes—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—41no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—42no—
438C2×D4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—84no—
449C22×C4C2×D812yes—
450C22×C4C2×D811yescharacteristic
451C22×C4C2×D811yescharacteristic
452C22×C4C2×D812yes—
453C22×C4C22×D411yescharacteristic
454Q16—24no—
455Q16—22no—
456Q16—24no—
457Q16—44no—
458Q16—82no—
459Q16—82no—
460SD16—82no—
461SD16—81no—
462SD16—84no—
463SD16—84no—
464D8—22no—
465D8—22no—
466D8—24no—
467D8—22no—
468D8—42no—
469D8—82no—
470D8—81no—
471C2×C8—22no—
472C2×C8—42no—
473C2×C8—41no—
474C2×C8C22×D411yescharacteristic
475C2×C8C22×D411yescharacteristic
476C2×C8C22×D411yescharacteristic
477C2×C8C22×D412yes—
478C2×C8C22×D412yes—
479C2×C8C2511yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—41no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—41no—
500C4⋊C4C2×D811yescharacteristic
501C4⋊C4C2×D812yes—
502C4⋊C4C2×D811yescharacteristic
503C4⋊C4C2×D812yes—
504C4⋊C4C2×D812yes—
505C4⋊C4C2×D812yes—
506C4⋊C4C2×D812yes—
507C22⋊C4—41no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—41no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—41no—
518C22⋊C4—42no—
519C22⋊C4—41no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—41no—
525C42—44no—
526C42—44no—
527C42—42no—
528C42—41no—
529C42C2×D811yescharacteristic
530C42C2×D812yes—
531C42C2×D811yescharacteristic
532C42C2×D812yes—
533C42C22×D411yescharacteristic
534C42C22×D411yescharacteristic
535C42C22×D412yes—
536C42C22×D412yes—
537C16—22no—
538C16—24no—
539C16—22no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23C22×D811yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—42no—
550Q8—41no—
551Q8—44no—
552Q8—44no—
553Q8—41no—
554Q8—84no—
555Q8—162no—
556Q8—161no—
557D4—41no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—41no—
562D4—42no—
563D4—82no—
564D4—84no—
565D4—82no—
566D4—84no—
567D4—82no—
568D4—162no—
569D4—161no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—82no—
588C2×C4—162no—
589C2×C4—162no—
590C2×C4—162no—
591C2×C4C22×D812yes—
592C2×C4C22×D812yes—
593C2×C4C22×D812yes—
594C2×C4C22×D811yescharacteristic
595C2×C4C22×D812yes—
596C2×C4C22×D811yescharacteristic
597C2×C4C22×D811yescharacteristic
598C2×C4C22×D811yescharacteristic
599C2×C4C22×D811yescharacteristic
600C2×C4C23×D411yescharacteristic
601C8—22no—
602C8—21no—
603C8—22no—
604C8—21no—
605C8—21no—
606C8#58512yes—
607C22—21no—
608C22—21no—
609C22—82no—
610C22—82no—
611C22—81no—
612C22—82no—
613C22—81no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—81no—
618C22—164no—
619C22#341612yes—
620C22C23×D811yescharacteristic
621C4—21no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—21no—
626C4—22no—
627C4—21no—
628C4—21no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—81no—
633C4—84no—
634C4—81no—
635C4#341412yes—
636C4#348711yescharacteristic
637C4#358511yescharacteristic
638C2—22no—
639C2—162no—
640C2—161no—
641C2—162no—
642C2#6188011yescharacteristic
643C2C2×#341411yescharacteristic
644C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62363C2semidirect
2. C2×#3393C2semidirect
3. #33968C2semidirect
4. #33964C2semidirect
5. #33961C2semidirect
6. #33943C2semidirect
7. #33936C2semidirect
8. #33927C2semidirect
9. #33784C2semidirect
10. #33779C2semidirect
11. #33777C2semidirect
12. #33772C2semidirect
13. #33677C2semidirect
14. #33669C2semidirect
15. #33665C2semidirect
16. #33663C2semidirect
17. #33498C2semidirect
20. #3396C22semidirect
21. #3396C22semidirect
28. #3391C22semidirect
29. Q8×Q16C22semidirect
30. #3380C22semidirect
31. #3380C22semidirect
48. #2251C22semidirect
51. #2248C22semidirect
52. #2245C22semidirect
55. #2241C22semidirect
59. #2229C22semidirect
62. #2225C22semidirect
64. #2211C22semidirect
69. #2209C22semidirect
70. #2208C22semidirect
71. #2208C22semidirect
72. #2207C22semidirect
81. #2183C22semidirect
82. #2183C22semidirect
83. Q8×C16C22semidirect
85. #2176C22semidirect
86. #2175C22semidirect
88. C4×SD32C22semidirect
89. C4×SD32C22semidirect

Maximal subgroups

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

subgroupindex
#623632
C2×#33932
#339682
#339642
#339612
#339432
#339362
#339272
#337842
#337792
#337772
#337722
#336772
#336692
#336652
#336632
#334982