Finite Group Databasebeta

← #7624667

Subgroups — #7624667

Order 512.

1233 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62472C211yesmaximal, characteristic, complemented
2#61427C211yesmaximal, characteristic
3#33952C212yesmaximal, complemented
4#33936C212yesmaximal, complemented
5#33924C212yesmaximal, complemented
6#33911C214yesmaximal, complemented
7#33898C212yesmaximal, complemented
8#33881C214yesmaximal, complemented
9#33818C212yesmaximal, complemented
10#33815C211yesmaximal, characteristic, complemented
11#33804C212yesmaximal, complemented
12#33796C211yesmaximal, characteristic, complemented
13#33676C212yesmaximal, complemented
14#33661C211yesmaximal, characteristic, complemented
15#33629C212yesmaximal, complemented
16C2×#2226C211yesmaximal, characteristic, complemented
17#33503C211yesmaximal, characteristic, complemented
18#3503—21no—
19#3475—21no—
20#3393C2211yescharacteristic
21Q8×Q16—21no—
22Q8×Q16C2212yescomplemented
23#3380C2211yescharacteristic
24#3360—21no—
25#3341C2211yescharacteristic
26#3338—22no—
27#3338—21no—
28#3338C2214yes—
29#3338C2212yescomplemented
30#3336C2212yes—
31#3303—21no—
32#3296—22no—
33#3283C2211yescharacteristic
34#3280C2212yes—
35#3230—21no—
36#3230—22no—
37#3170C2211yescharacteristic
38#3170C2211yescharacteristic
39#3170C2212yescomplemented
40#3168C2212yes—
41C2×#500C2211yescharacteristic
42C2×#500C2212yescomplemented
43#3102—22no—
44#3101—21no—
45C8×#41C2211yescharacteristic
46C4×#135C2212yes—
47C2×C4×D8C2211yescharacteristic
48#2255C2214yescomplemented
49#2252C2212yescomplemented
50#2243C2214yes—
51#2241C2212yescomplemented
52#2237C2214yescomplemented
53#2236C2214yes—
54#2235C2212yes—
55#2235C2214yescomplemented
56#2232C2214yes—
57#2229C2214yescomplemented
58#2227C2212yescomplemented
59#2227C2212yescomplemented
60#2227C2212yescomplemented
61#2227C2212yescomplemented
62#2226C2211yescharacteristic
63#2226C2212yes—
64#2226C2212yes—
65#2226C2212yes—
66#2226C2212yes—
67#2226C2211yescharacteristic
68#2225C2214yescomplemented
69#2224C2214yes—
70#2220C2214yes—
71#2219C2214yescomplemented
72#2216C2214yes—
73#2211C2214yes—
74#2209C2212yescomplemented
75#2209C2212yescomplemented
76#2208C2211yescharacteristic
77#2208C2211yescharacteristic
78#2189C2214yes—
79#2187C2212yes—
80#2183C2212yescomplemented
81#2176C2214yescomplemented
82#2175C2214yes—
83C4×SD32C2212yes—
84C4×SD32C2212yes—
85#2169C2212yescomplemented
86#2168C2214yes—
87#2163C2212yescomplemented
88#2162C2211yescharacteristic
89#2159C2214yescomplemented
90C2×#367C2212yes—
91#2150C2212yescomplemented
92C2×C4⋊C16C2211yescharacteristic
93#2139C2214yes—
94C2×#357C2212yes—
95C2×#135C2312yes—
96C22×D8C2311yescharacteristic
97Q82—22no—
98#557—41no—
99#556—21no—
100#555—21no—
101#554—24no—
102#554—22no—
103#554—21no—
104#553—41no—
105#552—22no—
106#552—41no—
107#551—42no—
108D4×Q8—42no—
109D4×Q8—41no—
110#547—21no—
111#546—22no—
112#544—41no—
113#544—42no—
114#533—21no—
115#533—21no—
116#533—22no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—22no—
119#529—22no—
120#529—42no—
121#525—41no—
122#520—41no—
123C4×#41—22no—
124C4×#41D411yescharacteristic
125C2×C4×D4—21no—
126C2×#117—41no—
127C2×Q32C2314yes—
128C2×SD32C2312yes—
129C2×SD32C2312yes—
130C2×M32C2314yes—
131C22×C16C2312yes—
132#500—21no—
133#500—22no—
134#500—21no—
135#500—22no—
136#500C2314yes—
137#500C2312yes—
138#500C2311yescharacteristic
139#500C2312yes—
140#500C2311yescharacteristic
141#499C2312yes—
142#495C2312yes—
143#494C2311yescharacteristic
144#494C2311yescharacteristic
145#488—22no—
146#487—21no—
147#487—21no—
148#487—22no—
149#485—22no—
150#484—24no—
151#484—24no—
152#484—24no—
153#483—22no—
154#482—22no—
155#479—22no—
156#478—22no—
157#477—22no—
158#477—24no—
159#477—24no—
160#477—22no—
161#476—21no—
162#476—21no—
163#467—21no—
164#467—22no—
165#467—21no—
166#467C2312yes—
167#467C2312yes—
168#467C2312yes—
169#467C2314yes—
170#466C2312yes—
171#465—22no—
172#464—22no—
173#463—22no—
174#461—21no—
175#461—21no—
176#452—22no—
177#450—22no—
178#449—22no—
179C8×Q8D411yescharacteristic
180C8×Q8C2311yescharacteristic
181C4×Q16—21no—
182C4×Q16—22no—
183C4×Q16—21no—
184C4×Q16C2312yes—
185C4×Q16C2314yes—
186C4×Q16C2312yes—
187C4×Q16C2312yes—
188C4×SD16—22no—
189C4×SD16—22no—
190C4×SD16—22no—
191C4×D8—21no—
192C4×D8—21no—
193C4×D8C2311yescharacteristic
194C4×D8C2311yescharacteristic
195C4×D8C2312yes—
196C4×D8C2311yescharacteristic
197C4×D8C2311yescharacteristic
198C8×D4D412yes—
199C8×D4D411yescharacteristic
200C8×D4C2312yes—
201C8×D4C2311yescharacteristic
202#431D411yescharacteristic
203#431D412yes—
204#427C2312yes—
205C2×#107—21no—
206C2×#107—22no—
207C2×#107C2311yescharacteristic
208C2×#107C2312yes—
209C2×#107C2314yes—
210#416—22no—
211C2×#102—21no—
212C2×C4×C8C2311yescharacteristic
213C2×C4×C8C2312yes—
214#367C2314yes—
215#367C2314yes—
216#367C2312yes—
217#367C2312yes—
218#366C2314yes—
219#366C2314yes—
220#366C2314yes—
221C4⋊C16C2312yes—
222C4⋊C16C2312yes—
223C4⋊C16C2311yescharacteristic
224C4⋊C16C2314yes—
225C4⋊C16C2311yescharacteristic
226#358C2314yes—
227#358C2314yes—
228#358C2312yes—
229#358C2312yes—
230#358C2314yes—
231#358C2314yes—
232#357C2314yes—
233#357C2314yes—
234#357C2312yes—
235#357C2312yes—
236C22⋊C16C2312yes—
237C22⋊C16C2314yes—
238#346C2314yes—
239C4×C16C2312yes—
240C2×#41—22no—
241C2×#41D811yescharacteristic
242C22×Q8—41no—
243C22×Q8—42no—
244C22×D4—21no—
245C23×C4—41no—
246#135—24no—
247#135—42no—
248C2×Q16—21no—
249C2×Q16—21no—
250C2×Q16C2412yes—
251C2×Q16C2412yes—
252C2×SD16—22no—
253C2×SD16—22no—
254C2×D8—22no—
255C2×D8—21no—
256C2×D8—41no—
257C2×D8C2411yescharacteristic
258C2×D8C2411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D412yes—
261C22×C8C2411yescharacteristic
262C22×C8C2412yes—
263C4⋊Q8—22no—
264C4⋊Q8—21no—
265C4⋊Q8—22no—
266C4⋊Q8—24no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—42no—
271C4⋊Q8—44no—
272C4⋊Q8—42no—
273C4⋊Q8—41no—
274C4⋊Q8—42no—
275C4⋊Q8—41no—
276#126—42no—
277#126—42no—
278#126—42no—
279#125—22no—
280#125—42no—
281#125—42no—
282#125—44no—
283#125—42no—
284#125—41no—
285#125—41no—
286#124—22no—
287#124—42no—
288#123—42no—
289#123—42no—
290#123—42no—
291#123—42no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—24no—
296C22⋊Q8—41no—
297C22⋊Q8—41no—
298C22⋊Q8—41no—
299C22⋊Q8—42no—
300C22⋊Q8—41no—
301C22⋊Q8—42no—
302C22⋊Q8—44no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305#121—22no—
306#121—41no—
307#121—41no—
308#121—42no—
309#120—42no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—24no—
314C4×Q8—42no—
315C4×Q8—41no—
316C4×Q8—42no—
317C4×Q8—41no—
318C4×Q8D811yescharacteristic
319C4×Q8C2×D411yescharacteristic
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—22no—
325C4×D4—44no—
326C4×D4—42no—
327C4×D4—42no—
328C4×D4—42no—
329C4×D4—42no—
330C4×D4—41no—
331C4×D4—41no—
332C4×D4D812yes—
333C4×D4D811yescharacteristic
334C4×D4C2×D412yes—
335C4×D4C2×D411yescharacteristic
336#117—22no—
337#117—41no—
338#117—42no—
339#117—41no—
340#117—41no—
341#117—41no—
342#117—42no—
343#117—42no—
344#117—42no—
345#117D811yescharacteristic
346#117D812yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4—41no—
354C2×C22⋊C4—41no—
355C2×C22⋊C4—42no—
356C2×C42—42no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359Q32—44no—
360SD32—24no—
361SD32—42no—
362M32—44no—
363C2×C16—42no—
364C2×C16C2412yes—
365C2×C16C2412yes—
366C2×C16C2414yes—
367#107—22no—
368#107—21no—
369#107—22no—
370#107—24no—
371#107—21no—
372#107—24no—
373#107—22no—
374#107—21no—
375#107C2414yes—
376#107C2412yes—
377#107C2412yes—
378#107C2411yescharacteristic
379#107C2411yescharacteristic
380#106—22no—
381#106—22no—
382C4⋊C8C2×D411yescharacteristic
383C4⋊C8C2×D412yes—
384C4⋊C8C2×D411yescharacteristic
385C4⋊C8C2×D412yes—
386#103—22no—
387#103—22no—
388#103—22no—
389#103—22no—
390#103—24no—
391#103—22no—
392#103—22no—
393#103—24no—
394#102—21no—
395#102—21no—
396#102—21no—
397#102—21no—
398#102—22no—
399#102—22no—
400C22⋊C8C2×D412yes—
401C22⋊C8C2×D412yes—
402C22⋊C8C2×D412yes—
403C4×C8C2×D411yescharacteristic
404C4×C8C2×D412yes—
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D412yes—
407C4×C8C2411yescharacteristic
408C4×C8C2412yes—
409C4×C8C2411yescharacteristic
410C24—41no—
411#41—24no—
412#41—44no—
413#41—82no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—42no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—164no—
422C2×Q8—162no—
423C2×Q8C2×D811yescharacteristic
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—41no—
430C2×D4—81no—
431C2×D4C2×D812yes—
432C2×D4C2×D811yescharacteristic
433C22×C4—41no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—84no—
443C22×C4C2×D812yes—
444C22×C4C2×D811yescharacteristic
445C22×C4C2×D812yes—
446C22×C4C2×D811yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449Q16—22no—
450Q16—22no—
451Q16—22no—
452Q16—22no—
453Q16—42no—
454Q16—81no—
455SD16—44no—
456SD16—82no—
457D8—21no—
458D8—21no—
459D8—21no—
460D8—42no—
461D82−1+412yes—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—42no—
467C2×C8—41no—
468C2×C8C22×D411yescharacteristic
469C2×C8C22×D412yes—
470C2×C8C22×D411yescharacteristic
471C2×C8C22×D411yescharacteristic
472C2×C8C22×D412yes—
473C2×C8C2511yescharacteristic
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—41no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—41no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—41no—
501C4⋊C4C2×D811yescharacteristic
502C4⋊C4C2×D811yescharacteristic
503C4⋊C4C2×D812yes—
504C4⋊C4C2×D812yes—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4C2×D812yes—
519C22⋊C4C2×D812yes—
520C22⋊C4C2×D812yes—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42C2×D812yes—
527C42C2×D812yes—
528C42C2×D811yescharacteristic
529C42C2×D811yescharacteristic
530C42C22×D411yescharacteristic
531C42C22×D411yescharacteristic
532C42C22×D412yes—
533C16—22no—
534C16—24no—
535C16—22no—
536C23—41no—
537C23—41no—
538C23—82no—
539C23—82no—
540C23—161no—
541C23C22×D812yes—
542C23C22×D811yescharacteristic
543Q8—22no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—84no—
549Q8—82no—
550Q8—161no—
551Q8—164no—
552Q8—162no—
553D4—24no—
554D4—22no—
555D4—22no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—82no—
560D4—84no—
561D4—162no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—162no—
583C2×C4—162no—
584C2×C4—164no—
585C2×C4#51014yes—
586C2×C4C22×D812yes—
587C2×C4C22×D812yes—
588C2×C4C22×D812yes—
589C2×C4C22×D811yescharacteristic
590C2×C4C22×D811yescharacteristic
591C2×C4C22×D811yescharacteristic
592C2×C4C22×D811yescharacteristic
593C2×C4C22×D811yescharacteristic
594C2×C4C23×D411yescharacteristic
595C8—21no—
596C8—22no—
597C8—21no—
598C8—21no—
599C8—22no—
600C8C2×2−1+411yescharacteristic
601C8#58511yescharacteristic
602C22—21no—
603C22—21no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—164no—
614C22C2×#51012yes—
615C22C23×D811yescharacteristic
616C4—22no—
617C4—21no—
618C4—22no—
619C4—21no—
620C4—22no—
621C4—21no—
622C4—82no—
623C4—82no—
624C4—81no—
625C4—81no—
626C4—84no—
627C4—82no—
628C4—82no—
629C4C2×#51012yes—
630C4C2×#51012yes—
631C4#348911yescharacteristic
632C4#358611yescharacteristic
633C2—22no—
634C2—42no—
635C2—82no—
636C2—161no—
637C2#6209511yescharacteristic
638C2C22×#51011yescharacteristic
639C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#61427C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62472C2semidirect
3. #33952C2semidirect
4. #33936C2semidirect
5. #33924C2semidirect
6. #33911C2semidirect
7. #33898C2semidirect
8. #33881C2semidirect
9. #33818C2semidirect
10. #33815C2semidirect
11. #33804C2semidirect
12. #33796C2semidirect
13. #33676C2semidirect
14. #33661C2semidirect
15. #33629C2semidirect
16. C2×#2226C2semidirect
17. #33503C2semidirect
22. Q8×Q16C22semidirect
29. #3338C22semidirect
39. #3170C22semidirect
42. C2×#500C22semidirect
48. #2255C22semidirect
49. #2252C22semidirect
51. #2241C22semidirect
52. #2237C22semidirect
55. #2235C22semidirect
57. #2229C22semidirect
58. #2227C22semidirect
59. #2227C22semidirect
60. #2227C22semidirect
61. #2227C22semidirect
68. #2225C22semidirect
71. #2219C22semidirect
74. #2209C22semidirect
75. #2209C22semidirect
80. #2183C22semidirect
81. #2176C22semidirect
85. #2169C22semidirect
87. #2163C22semidirect
89. #2159C22semidirect
91. #2150C22semidirect

Maximal subgroups

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

subgroupindex
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#614272
#339522
#339362
#339242
#339112
#338982
#338812
#338182
#338152
#338042
#337962
#336762
#336612
#336292
C2×#22262
#335032