Finite Group Databasebeta

← #7624433

Subgroups — #7624433

Order 512.

1223 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62364C211yesmaximal, characteristic, complemented
2C2×#3380C211yesmaximal, characteristic, complemented
3#33955C214yesmaximal, complemented
4#33951C212yesmaximal, complemented
5#33948C212yesmaximal, complemented
6#33943C214yesmaximal, complemented
7#33939C212yesmaximal, complemented
8#33927C212yesmaximal, complemented
9#33804C211yesmaximal, characteristic, complemented
10#33802C212yesmaximal, complemented
11#33798C212yesmaximal, complemented
12#33795C211yesmaximal, characteristic, complemented
13#33674C212yesmaximal
14#33670C211yesmaximal, characteristic, complemented
15#33665C211yesmaximal, characteristic
16#33661C212yesmaximal, complemented
17#33498C211yesmaximal, characteristic, complemented
18#3523—21no—
19C2×#554—21no—
20#3395C2211yescharacteristic, complemented
21#3393C2211yescharacteristic, complemented
22#3393C2212yescomplemented
23#3383C2211yescharacteristic, complemented
24#3383C2212yescomplemented
25#3380—21no—
26#3380—21no—
27#3380—22no—
28#3380C2212yes—
29#3380C2212yes—
30#3380C2214yes—
31Q8×D8C2211yescharacteristic, complemented
32#3364—21no—
33#3364—22no—
34#3359—21no—
35#3248—22no—
36#3248—21no—
37#3170C2211yescharacteristic
38#3170C2212yes—
39#3168C2212yes—
40#3168C2211yescharacteristic
41C2×#500C2211yescharacteristic
42C2×#499C2212yes—
43C2×#478—22no—
44C2×#476—21no—
45C2×C8×Q8C2211yescharacteristic
46C2×C4×D8C2212yes—
47C2×C4×D8C2211yescharacteristic
48#2255C2214yescomplemented
49#2255C2212yescomplemented
50#2254C2212yescomplemented
51#2253C2214yescomplemented
52#2245C2214yescomplemented
53#2243C2212yescomplemented
54#2242C2214yescomplemented
55#2241C2212yescomplemented
56#2236C2212yes—
57#2236C2211yescharacteristic
58#2235C2212yes—
59#2235C2212yes—
60#2235C2212yes—
61#2233C2211yescharacteristic
62#2232C2212yes—
63#2229C2214yescomplemented
64#2229C2214yescomplemented
65#2229C2214yescomplemented
66#2228C2212yes—
67#2228C2214yes—
68#2228C2214yescomplemented
69#2227C2212yescomplemented
70#2226C2214yes—
71#2226C2212yescomplemented
72#2225C2214yescomplemented
73#2224C2212yes—
74#2210C2212yes—
75#2210C2214yes—
76#2208C2214yes—
77#2208C2212yes—
78#2187C2211yescharacteristic
79#2187C2212yes—
80#2184C2211yescharacteristic
81#2183C2212yescomplemented
82#2183C2214yescomplemented
83Q8×C16C2212yescomplemented
84#2177C2214yes—
85#2175C2214yes—
86#2175C2214yes—
87C4×SD32C2214yes—
88C4×SD32C2214yes—
89C4×SD32C2212yes—
90C4×D16C2212yes—
91#2163C2212yes—
92#2163C2211yescharacteristic
93#2162C2211yescharacteristic
94#2162C2212yes—
95C22×D8C2311yescharacteristic
96#557—21no—
97#557—22no—
98#557—21no—
99#556—21no—
100#556—22no—
101#554—22no—
102#554—22no—
103#554—24no—
104#554—41no—
105#554—41no—
106#554—42no—
107#552—41no—
108#552—42no—
109D4×Q8—21no—
110D4×Q8—41no—
111#541—42no—
112#541—41no—
113#533—22no—
114#533—21no—
115C2×C4⋊Q8—21no—
116#529—22no—
117#529—21no—
118C2×#125—22no—
119C2×C22⋊Q8—41no—
120C2×C22⋊Q8—42no—
121#519—41no—
122C2×C4×Q8D411yescharacteristic
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C22×C4⋊C4—41no—
126C2×SD32C2312yes—
127C2×SD32C2314yes—
128C2×D16C2312yes—
129#500—21no—
130#500—21no—
131#500C2311yescharacteristic
132#500C2312yes—
133#500C2312yes—
134#500C2312yes—
135#499—22no—
136#499—22no—
137#499C2314yes—
138#499C2314yes—
139#499C2312yes—
140#499C2311yescharacteristic
141#495C2312yes—
142#495C2311yescharacteristic
143#494C2311yescharacteristic
144#494C2312yes—
145#488—22no—
146#488—21no—
147#487—22no—
148#487—21no—
149#479—22no—
150#479—22no—
151#479—22no—
152#478—24no—
153#478—24no—
154#478—24no—
155#478—24no—
156#476—22no—
157#476—22no—
158#476—24no—
159#475—21no—
160#475—22no—
161#474—21no—
162#474—22no—
163#467—22no—
164#467—21no—
165#467C2312yes—
166#467C2311yescharacteristic
167#466C2311yescharacteristic
168#465—22no—
169#465—21no—
170#464—22no—
171#464—22no—
172#464—22no—
173#462—21no—
174#462—22no—
175#461—21no—
176#461—22no—
177C8×Q8D411yescharacteristic
178C8×Q8D412yes—
179C8×Q8D411yescharacteristic
180C8×Q8C2311yescharacteristic
181C8×Q8C2311yescharacteristic
182C8×Q8C2312yes—
183C4×Q16—22no—
184C4×Q16—21no—
185C4×Q16C2312yes—
186C4×Q16C2312yes—
187C4×Q16C2312yes—
188C4×Q16C2311yescharacteristic
189C4×Q16C2312yes—
190C4×SD16—22no—
191C4×SD16—21no—
192C4×D8—22no—
193C4×D8—21no—
194C4×D8—22no—
195C4×D8—21no—
196C4×D8C2311yescharacteristic
197C4×D8C2312yes—
198C4×D8C2312yes—
199C4×D8C2314yes—
200C4×D8C2312yes—
201C4×D8C2314yes—
202C2×#107—21no—
203C2×#107C2311yescharacteristic
204C2×#107C2312yes—
205C2×#106—22no—
206C2×C4⋊C8D412yes—
207C2×C4⋊C8D411yescharacteristic
208C2×#102—21no—
209C2×#102—22no—
210C2×C4×C8C2312yes—
211C2×C4×C8C2311yescharacteristic
212#367C2314yes—
213#367C2314yes—
214#367C2312yes—
215#367C2314yes—
216#366C2312yes—
217#366C2314yes—
218#366C2314yes—
219C4⋊C16C2314yes—
220C4⋊C16C2312yes—
221C4⋊C16C2312yes—
222C4⋊C16C2314yes—
223#358C2314yes—
224#358C2314yes—
225#358C2312yes—
226#358C2314yes—
227#358C2314yes—
228#357C2312yes—
229#357C2312yes—
230#357C2312yes—
231#357C2314yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2311yescharacteristic
235C22⋊C16C2312yes—
236#346C2312yes—
237#346C2314yes—
238C4×C16C2312yes—
239C4×C16C2314yes—
240C22×Q8—41no—
241C22×Q8—41no—
242C22×Q8D811yescharacteristic
243C22×D4—21no—
244C23×C4—41no—
245C23×C4—42no—
246C2×Q16—22no—
247C2×Q16—21no—
248C2×Q16C2412yes—
249C2×Q16C2411yescharacteristic
250C2×SD16—22no—
251C2×SD16—21no—
252C2×D8—24no—
253C2×D8—21no—
254C2×D8C2411yescharacteristic
255C2×D8C2412yes—
256C22×C8C2×D412yes—
257C22×C8C2×D411yescharacteristic
258C22×C8C2411yescharacteristic
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—21no—
263C4⋊Q8—22no—
264C4⋊Q8—21no—
265C4⋊Q8—41no—
266C4⋊Q8—42no—
267C4⋊Q8—41no—
268C4⋊Q8—41no—
269#126—42no—
270#126—42no—
271#126—42no—
272#126—42no—
273#126—41no—
274#125—24no—
275#125—24no—
276#125—22no—
277#125—21no—
278#125—42no—
279#125—42no—
280#125—42no—
281#125—41no—
282#125—42no—
283#125—42no—
284#125—42no—
285#125—42no—
286#125—41no—
287#124—22no—
288#124—21no—
289#123—41no—
290#123—42no—
291C22⋊Q8—22no—
292C22⋊Q8—21no—
293C22⋊Q8—42no—
294C22⋊Q8—42no—
295C22⋊Q8—44no—
296C22⋊Q8—41no—
297C22⋊Q8—42no—
298C22⋊Q8—41no—
299C22⋊Q8—42no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C22⋊Q8—44no—
304C22⋊Q8—41no—
305C22⋊Q8—42no—
306#121—21no—
307C4×Q8—22no—
308C4×Q8—21no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—41no—
313C4×Q8—41no—
314C4×Q8—42no—
315C4×Q8—42no—
316C4×Q8—41no—
317C4×Q8D811yescharacteristic
318C4×Q8D812yes—
319C4×Q8D811yescharacteristic
320C4×Q8C2×D411yescharacteristic
321C4×Q8C2×D412yes—
322C4×Q8C2×D411yescharacteristic
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—22no—
328C4×D4—21no—
329C4×D4—41no—
330C4×D4—42no—
331C4×D4—42no—
332C4×D4—41no—
333C4×D4—42no—
334C4×D4—41no—
335#117—41no—
336#117—42no—
337#117—41no—
338#117—42no—
339#117—42no—
340#117—41no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—42no—
344C2×C4⋊C4—41no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—42no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4D811yescharacteristic
350C2×C4⋊C4D812yes—
351C2×C22⋊C4—41no—
352C2×C22⋊C4—42no—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355SD32—24no—
356SD32—28no—
357D16—24no—
358C2×C16C2412yes—
359C2×C16C2414yes—
360C2×C16C2412yes—
361#107—22no—
362#107—22no—
363#107—21no—
364#107—22no—
365#107—22no—
366#107—21no—
367#107C2411yescharacteristic
368#107C2411yescharacteristic
369#107C2412yes—
370#107C2412yes—
371#107C2414yes—
372#106—24no—
373#106—24no—
374#106—22no—
375#106—21no—
376C4⋊C8C2×D411yescharacteristic
377C4⋊C8C2×D412yes—
378C4⋊C8C2×D411yescharacteristic
379C4⋊C8C2×D412yes—
380C4⋊C8C2×D412yes—
381C4⋊C8C2×D412yes—
382C4⋊C8C2×D412yes—
383#103—22no—
384#103—21no—
385#103—22no—
386#103—22no—
387#103—22no—
388#103—22no—
389#103—21no—
390#102—24no—
391#102—24no—
392#102—22no—
393#102—22no—
394#102—22no—
395#102—21no—
396#102—21no—
397C4×C8C2×D412yes—
398C4×C8C2×D411yescharacteristic
399C4×C8C2×D412yes—
400C4×C8C2×D411yescharacteristic
401C4×C8C2411yescharacteristic
402C4×C8C2412yes—
403C4×C8C2412yes—
404C4×C8C2411yescharacteristic
405C24—41no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—41no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—84no—
414C2×Q8—162no—
415C2×Q8C2×D811yescharacteristic
416C2×Q8C2×D811yescharacteristic
417C2×Q8C2×D812yes—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—42no—
422C2×D4—41no—
423C2×D4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—84no—
433C22×C4—164no—
434C22×C4C2×D811yescharacteristic
435C22×C4C2×D812yes—
436C22×C4C2×D811yescharacteristic
437C22×C4C2×D812yes—
438C22×C4C22×D411yescharacteristic
439Q16—22no—
440Q16—24no—
441Q16—82no—
442Q16—81no—
443SD16—82no—
444SD16—81no—
445D8—24no—
446D8—22no—
447D8—22no—
448D82−1+414yes—
449C2×C8—22no—
450C2×C8—42no—
451C2×C8—41no—
452C2×C8C22×D411yescharacteristic
453C2×C8C22×D412yes—
454C2×C8C22×D411yescharacteristic
455C2×C8C22×D412yes—
456C2×C8C22×D411yescharacteristic
457C2×C8C2511yescharacteristic
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—41no—
473C4⋊C4—42no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4C2×D811yescharacteristic
489C4⋊C4C2×D811yescharacteristic
490C4⋊C4C2×D812yes—
491C4⋊C4C2×D812yes—
492C4⋊C4C2×D812yes—
493C4⋊C4C2×D812yes—
494C4⋊C4C2×D812yes—
495C22⋊C4—42no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—42no—
504C22⋊C4—41no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—41no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—41no—
513C42C2×D812yes—
514C42C2×D811yescharacteristic
515C42C2×D812yes—
516C42C2×D811yescharacteristic
517C42C22×D412yes—
518C42C22×D411yescharacteristic
519C42C22×D412yes—
520C42C22×D411yescharacteristic
521C16—22no—
522C16—22no—
523C16—24no—
524C23—44no—
525C23—42no—
526C23—41no—
527C23—82no—
528C23—82no—
529C23C22×D811yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—42no—
533Q8—42no—
534Q8—41no—
535Q8—41no—
536Q8—42no—
537Q8—84no—
538Q8—82no—
539Q8—162no—
540Q8—162no—
541D4—24no—
542D4—44no—
543D4—42no—
544D4—41no—
545D4—41no—
546D4—82no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—82no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—82no—
566C2×C4—164no—
567C2×C4—164no—
568C2×C4C22×D811yescharacteristic
569C2×C4C22×D812yes—
570C2×C4C22×D812yes—
571C2×C4C22×D811yescharacteristic
572C2×C4C22×D812yes—
573C2×C4C22×D811yescharacteristic
574C2×C4C22×D812yes—
575C2×C4C22×D811yescharacteristic
576C2×C4C22×D811yescharacteristic
577C2×C4C23×D411yescharacteristic
578C8—22no—
579C8—21no—
580C8—22no—
581C8—21no—
582C8—21no—
583C8C2×2−1+412yes—
584C22—21no—
585C22—21no—
586C22—44no—
587C22—88no—
588C22—82no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22#341612yes—
594C22C23×D811yescharacteristic
595C4—21no—
596C4—21no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—22no—
601C4—21no—
602C4—21no—
603C4—82no—
604C4—81no—
605C4—82no—
606C4—82no—
607C4—81no—
608C4—84no—
609C4—81no—
610C4#341412yes—
611C4#348611yescharacteristic
612C4#358611yescharacteristic
613C2—22no—
614C2—84no—
615C2—161no—
616C2#6181711yescharacteristic
617C2C2×#341411yescharacteristic
618C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#2184C22
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. #62364C2semidirect
2. C2×#3380C2semidirect
3. #33955C2semidirect
4. #33951C2semidirect
5. #33948C2semidirect
6. #33943C2semidirect
7. #33939C2semidirect
8. #33927C2semidirect
9. #33804C2semidirect
10. #33802C2semidirect
11. #33798C2semidirect
12. #33795C2semidirect
14. #33670C2semidirect
16. #33661C2semidirect
17. #33498C2semidirect
20. #3395C22semidirect
21. #3393C22semidirect
22. #3393C22semidirect
23. #3383C22semidirect
24. #3383C22semidirect
31. Q8×D8C22semidirect
48. #2255C22semidirect
49. #2255C22semidirect
50. #2254C22semidirect
51. #2253C22semidirect
52. #2245C22semidirect
53. #2243C22semidirect
54. #2242C22semidirect
55. #2241C22semidirect
63. #2229C22semidirect
64. #2229C22semidirect
65. #2229C22semidirect
68. #2228C22semidirect
69. #2227C22semidirect
71. #2226C22semidirect
72. #2225C22semidirect
81. #2183C22semidirect
82. #2183C22semidirect
83. Q8×C16C22semidirect

Maximal subgroups

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

subgroupindex
#623642
C2×#33802
#339552
#339512
#339482
#339432
#339392
#339272
#338042
#338022
#337982
#337952
#336742
#336702
#336652
#336612
#334982