Finite Group Databasebeta

← #7624500

Subgroups — #7624500

Order 512.

1277 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62364C211yesmaximal, characteristic, complemented
2C2×#3396C211yesmaximal, characteristic, complemented
3#33968C214yesmaximal, complemented
4#33967C212yesmaximal, complemented
5#33961C212yesmaximal, complemented
6#33940C212yesmaximal, complemented
7#33938C214yesmaximal, complemented
8#33932C212yesmaximal, complemented
9#33783C211yesmaximal, characteristic
10#33781C212yesmaximal, complemented
11#33777C212yesmaximal, complemented
12#33774C211yesmaximal, characteristic, complemented
13#33674C212yesmaximal, complemented
14#33673C211yesmaximal, characteristic, complemented
15#33668C211yesmaximal, characteristic, complemented
16#33661C212yesmaximal, complemented
17#33498C211yesmaximal, characteristic, complemented
18#3523—21no—
19C2×#559—21no—
20#3396—21no—
21#3396—21no—
22#3396—22no—
23#3396C2212yes—
24#3396C2214yes—
25#3396C2212yes—
26#3395C2211yescharacteristic, complemented
27#3393C2211yescharacteristic, complemented
28#3393C2212yescomplemented
29#3383C2211yescharacteristic, complemented
30#3383C2212yescomplemented
31Q8×D8C2211yescharacteristic, complemented
32#3364—21no—
33#3364—22no—
34#3359—21no—
35#3248—22no—
36#3248—21no—
37#3170C2212yes—
38#3170C2211yescharacteristic
39#3168C2211yescharacteristic
40#3168C2212yes—
41C2×#495C2212yes—
42C2×#493C2211yescharacteristic
43C2×#460—21no—
44C2×#463—22no—
45C2×C8×Q8C2211yescharacteristic
46C2×C4×Q16C2211yescharacteristic
47C2×C4×D8C2212yes—
48#2251C2214yescomplemented
49#2250C2212yes—
50#2249C2214yescomplemented
51#2248C2212yes—
52#2244C2212yes—
53#2243C2214yescomplemented
54#2242C2214yescomplemented
55#2240C2212yes—
56#2235C2211yescharacteristic
57#2235C2212yes—
58#2232C2211yescharacteristic
59#2229C2212yescomplemented
60#2228C2214yescomplemented
61#2226C2214yescomplemented
62#2225C2212yescomplemented
63#2211C2212yescomplemented
64#2211C2214yescomplemented
65#2211C2214yescomplemented
66#2211C2214yescomplemented
67#2210C2214yescomplemented
68#2210C2214yescomplemented
69#2209C2212yescomplemented
70#2208C2214yescomplemented
71#2207C2212yes—
72#2207C2214yes—
73#2206C2212yes—
74#2188C2211yescharacteristic
75#2188C2212yes—
76#2187C2212yes—
77#2187C2212yes—
78#2187C2212yes—
79#2185C2211yescharacteristic
80#2184C2212yes—
81#2183C2214yescomplemented
82#2183C2212yescomplemented
83Q8×C16C2212yescomplemented
84#2177C2214yescomplemented
85#2176C2214yescomplemented
86#2175C2214yescomplemented
87C4×Q32C2212yescomplemented
88C4×SD32C2214yescomplemented
89C4×SD32C2214yescomplemented
90C4×SD32C2212yescomplemented
91#2163C2211yescharacteristic
92#2163C2212yes—
93#2162C2212yes—
94#2162C2211yescharacteristic
95C22×Q16C2311yescharacteristic
96C22×SD16—22no—
97C22×D8—21no—
98C22×D8C2312yes—
99#559—22no—
100#559—22no—
101#559—24no—
102#559—41no—
103#559—41no—
104#559—42no—
105#557—21no—
106#557—22no—
107#557—21no—
108#556—21no—
109#556—22no—
110#552—41no—
111#552—42no—
112D4×Q8—21no—
113D4×Q8—41no—
114#541—42no—
115#541—41no—
116#533—22no—
117#533—21no—
118C2×#127—21no—
119#529—22no—
120#529—21no—
121C2×#124—22no—
122C2×#124—41no—
123C2×#123—42no—
124#519—41no—
125C2×C4×Q8—21no—
126C2×C4×Q8D411yescharacteristic
127C2×C4×D4—22no—
128C2×C4×D4—41no—
129C2×Q32C2312yes—
130C2×Q32C2314yes—
131C2×SD32C2312yes—
132C2×SD32C2314yes—
133C2×SD32C2314yes—
134C2×SD32C2314yes—
135C2×D16C2314yes—
136#500C2311yescharacteristic
137#500C2312yes—
138#499C2312yes—
139#499C2311yescharacteristic
140#495—22no—
141#495—22no—
142#495C2312yes—
143#495C2311yescharacteristic
144#495C2314yes—
145#495C2314yes—
146#494C2312yes—
147#494C2311yescharacteristic
148#493—21no—
149#493—21no—
150#493C2312yes—
151#493C2312yes—
152#488—22no—
153#488—21no—
154#487—22no—
155#487—21no—
156#479—22no—
157#479—22no—
158#479—22no—
159#475—21no—
160#475—22no—
161#474—21no—
162#474—22no—
163#467—21no—
164#467—22no—
165#467C2311yescharacteristic
166#467C2312yes—
167#466C2311yescharacteristic
168#465—21no—
169#465—22no—
170#464—22no—
171#464—22no—
172#464—22no—
173#463—24no—
174#463—24no—
175#463—24no—
176#463—24no—
177#462—21no—
178#462—22no—
179#461—21no—
180#461—22no—
181#460—22no—
182#460—22no—
183#460—24no—
184C8×Q8D411yescharacteristic
185C8×Q8D412yes—
186C8×Q8D411yescharacteristic
187C8×Q8C2311yescharacteristic
188C8×Q8C2311yescharacteristic
189C8×Q8C2312yes—
190C4×Q16—22no—
191C4×Q16—21no—
192C4×Q16—21no—
193C4×Q16—21no—
194C4×Q16C2312yes—
195C4×Q16C2312yes—
196C4×Q16C2312yes—
197C4×Q16C2311yescharacteristic
198C4×Q16C2312yes—
199C4×Q16C2312yes—
200C4×Q16C2312yes—
201C4×SD16—22no—
202C4×SD16—21no—
203C4×D8—22no—
204C4×D8—22no—
205C4×D8C2311yescharacteristic
206C4×D8C2312yes—
207C4×D8C2314yes—
208C4×D8C2314yes—
209C2×#107C2311yescharacteristic
210C2×C4⋊C8D412yes—
211C2×C4⋊C8D411yescharacteristic
212C2×#103—21no—
213C2×#102—22no—
214C2×C4×C8C2311yescharacteristic
215C2×C4×C8C2312yes—
216#367C2312yes—
217#367C2314yes—
218#366C2312yes—
219C4⋊C16C2312yes—
220C4⋊C16C2314yes—
221C4⋊C16C2312yes—
222C4⋊C16C2314yes—
223#358C2312yes—
224#358C2314yes—
225#358C2312yes—
226#358C2312yes—
227#358C2314yes—
228#357C2312yes—
229#357C2314yes—
230#357C2314yes—
231#357C2314yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2312yes—
235C22⋊C16C2311yescharacteristic
236#346C2314yes—
237#346C2312yes—
238C4×C16C2314yes—
239C4×C16C2312yes—
240C22×Q8—21no—
241C22×Q8—41no—
242C22×Q8D811yescharacteristic
243C22×D4—22no—
244C22×D4—41no—
245C22×D4—41no—
246C23×C4—42no—
247C2×Q16—21no—
248C2×Q16—22no—
249C2×Q16—21no—
250C2×Q16—42no—
251C2×Q16C2412yes—
252C2×Q16C2411yescharacteristic
253C2×Q16C2412yes—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—22no—
257C2×SD16—21no—
258C2×SD16—84no—
259C2×D8—22no—
260C2×D8—22no—
261C2×D8—22no—
262C2×D8—44no—
263C2×D8—82no—
264C2×D8C2414yes—
265C2×D8C2411yescharacteristic
266C22×C8C2×D411yescharacteristic
267C22×C8C2×D412yes—
268C22×C8C2411yescharacteristic
269C4⋊Q8—22no—
270C4⋊Q8—21no—
271C4⋊Q8—22no—
272C4⋊Q8—21no—
273C4⋊Q8—41no—
274C4⋊Q8—42no—
275#127—22no—
276#127—22no—
277#127—41no—
278#127—41no—
279#126—42no—
280#126—42no—
281#126—42no—
282#126—42no—
283#126—41no—
284#125—22no—
285#125—21no—
286#125—42no—
287#125—42no—
288#125—42no—
289#125—41no—
290#125—42no—
291#125—42no—
292#125—41no—
293#124—24no—
294#124—24no—
295#124—22no—
296#124—21no—
297#124—42no—
298#124—42no—
299#124—44no—
300#124—42no—
301#124—42no—
302#123—42no—
303#123—41no—
304#123—44no—
305#123—44no—
306#123—44no—
307#123—44no—
308C22⋊Q8—22no—
309C22⋊Q8—21no—
310C22⋊Q8—41no—
311C22⋊Q8—42no—
312C22⋊Q8—41no—
313C22⋊Q8—42no—
314C22⋊Q8—41no—
315C22⋊Q8—42no—
316#121—21no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—21no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—41no—
325C4×Q8—41no—
326C4×Q8—42no—
327C4×Q8—41no—
328C4×Q8—41no—
329C4×Q8—42no—
330C4×Q8—41no—
331C4×Q8D811yescharacteristic
332C4×Q8D812yes—
333C4×Q8D811yescharacteristic
334C4×Q8C2×D411yescharacteristic
335C4×Q8C2×D412yes—
336C4×Q8C2×D411yescharacteristic
337C4×D4—24no—
338C4×D4—24no—
339C4×D4—22no—
340C4×D4—21no—
341C4×D4—41no—
342C4×D4—42no—
343C4×D4—42no—
344C4×D4—42no—
345C4×D4—44no—
346C4×D4—42no—
347C4×D4—42no—
348#117—42no—
349#117—41no—
350#117—41no—
351#117—42no—
352#117—42no—
353#117—41no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4D811yescharacteristic
357C2×C4⋊C4D812yes—
358C2×C22⋊C4—42no—
359C2×C22⋊C4—42no—
360C2×C22⋊C4—42no—
361C2×C42—41no—
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364Q32—44no—
365Q32—42no—
366SD32—44no—
367SD32—42no—
368SD32—44no—
369SD32—44no—
370D16—44no—
371C2×C16C2412yes—
372C2×C16C2412yes—
373C2×C16C2414yes—
374#107—21no—
375#107—22no—
376#107—21no—
377#107C2411yescharacteristic
378#107C2411yescharacteristic
379#107C2412yes—
380#107C2412yes—
381#106—22no—
382#106—21no—
383C4⋊C8C2×D411yescharacteristic
384C4⋊C8C2×D411yescharacteristic
385C4⋊C8C2×D412yes—
386C4⋊C8C2×D412yes—
387C4⋊C8C2×D412yes—
388C4⋊C8C2×D412yes—
389C4⋊C8C2×D412yes—
390#103—22no—
391#103—22no—
392#103—22no—
393#103—21no—
394#103—22no—
395#103—22no—
396#103—22no—
397#103—22no—
398#103—21no—
399#102—24no—
400#102—24no—
401#102—22no—
402#102—21no—
403#102—21no—
404C4×C8C2×D412yes—
405C4×C8C2×D412yes—
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2412yes—
409C4×C8C2411yescharacteristic
410C4×C8C2412yes—
411C4×C8C2411yescharacteristic
412C24—42no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—84no—
423C2×Q8C2×D811yescharacteristic
424C2×Q8C2×D811yescharacteristic
425C2×Q8C2×D812yes—
426C2×D4—21no—
427C2×D4—24no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—41no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—84no—
437C2×D4—162no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—164no—
445C22×C4C2×D811yescharacteristic
446C22×C4C2×D812yes—
447C22×C4C2×D811yescharacteristic
448C22×C4C2×D812yes—
449C22×C4C22×D411yescharacteristic
450Q16—22no—
451Q16—24no—
452Q16—22no—
453Q16—22no—
454Q16—42no—
455Q16—81no—
456Q16—82no—
457SD16—81no—
458SD16—82no—
459SD16—84no—
460SD16—84no—
461D8—22no—
462D8—24no—
463D8—24no—
464D8—44no—
465D8—82no—
466D8—82no—
467C2×C8—22no—
468C2×C8—41no—
469C2×C8—42no—
470C2×C8C22×D411yescharacteristic
471C2×C8C22×D412yes—
472C2×C8C22×D412yes—
473C2×C8C22×D411yescharacteristic
474C2×C8C22×D411yescharacteristic
475C2×C8C2511yescharacteristic
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—41no—
491C4⋊C4—42no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
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—42no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—41no—
518C22⋊C4—42no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—41no—
529C42C2×D812yes—
530C42C2×D811yescharacteristic
531C42C2×D812yes—
532C42C2×D811yescharacteristic
533C42C22×D412yes—
534C42C22×D412yes—
535C42C22×D411yescharacteristic
536C42C22×D411yescharacteristic
537C16—24no—
538C16—22no—
539C16—22no—
540C23—41no—
541C23—44no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23C22×D811yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—42no—
549Q8—42no—
550Q8—41no—
551Q8—42no—
552Q8—42no—
553Q8—41no—
554Q8—42no—
555Q8—82no—
556Q8—84no—
557Q8—82no—
558D4—44no—
559D4—41no—
560D4—44no—
561D4—41no—
562D4—84no—
563D4—84no—
564D4—82no—
565D4—84no—
566D4—162no—
567D4—162no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—164no—
586C2×C4—164no—
587C2×C4C22×D812yes—
588C2×C4C22×D811yescharacteristic
589C2×C4C22×D811yescharacteristic
590C2×C4C22×D812yes—
591C2×C4C22×D811yescharacteristic
592C2×C4C22×D812yes—
593C2×C4C22×D812yes—
594C2×C4C22×D811yescharacteristic
595C2×C4C22×D811yescharacteristic
596C2×C4C23×D411yescharacteristic
597C8—21no—
598C8—21no—
599C8—22no—
600C8—22no—
601C8—21no—
602C8#58512yes—
603C22—21no—
604C22—21no—
605C22—81no—
606C22—84no—
607C22—84no—
608C22—81no—
609C22—84no—
610C22—81no—
611C22—168no—
612C22#341612yes—
613C22C23×D811yescharacteristic
614C4—21no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—82no—
623C4—81no—
624C4—82no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—82no—
629C4#341412yes—
630C4#348711yescharacteristic
631C4#358511yescharacteristic
632C2—22no—
633C2—161no—
634C2—164no—
635C2#6188011yescharacteristic
636C2C2×#341411yescharacteristic
637C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#33783C2
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×#3396C2semidirect
3. #33968C2semidirect
4. #33967C2semidirect
5. #33961C2semidirect
6. #33940C2semidirect
7. #33938C2semidirect
8. #33932C2semidirect
10. #33781C2semidirect
11. #33777C2semidirect
12. #33774C2semidirect
13. #33674C2semidirect
14. #33673C2semidirect
15. #33668C2semidirect
16. #33661C2semidirect
17. #33498C2semidirect
26. #3395C22semidirect
27. #3393C22semidirect
28. #3393C22semidirect
29. #3383C22semidirect
30. #3383C22semidirect
31. Q8×D8C22semidirect
48. #2251C22semidirect
50. #2249C22semidirect
53. #2243C22semidirect
54. #2242C22semidirect
59. #2229C22semidirect
60. #2228C22semidirect
61. #2226C22semidirect
62. #2225C22semidirect
63. #2211C22semidirect
64. #2211C22semidirect
65. #2211C22semidirect
66. #2211C22semidirect
67. #2210C22semidirect
68. #2210C22semidirect
69. #2209C22semidirect
70. #2208C22semidirect
81. #2183C22semidirect
82. #2183C22semidirect
83. Q8×C16C22semidirect
84. #2177C22semidirect
85. #2176C22semidirect
86. #2175C22semidirect
87. C4×Q32C22semidirect
88. C4×SD32C22semidirect
89. C4×SD32C22semidirect
90. C4×SD32C22semidirect

Maximal subgroups

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

subgroupindex
#623642
C2×#33962
#339682
#339672
#339612
#339402
#339382
#339322
#337832
#337812
#337772
#337742
#336742
#336732
#336682
#336612
#334982