Finite Group Databasebeta

← #7625747

Subgroups — #7625747

Order 512.

10574 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3486C211yesmaximal, characteristic
2C2×C4×2−1+4C211yesmaximal, characteristic, complemented
3C22×#519C211yesmaximal, characteristic, complemented
4#34312C218yesmaximal, complemented
5#34311C218yesmaximal, complemented
6#34293C218yesmaximal, complemented
7#34291C218yesmaximal, complemented
8#34266C216yesmaximal, complemented
9#34257C211yesmaximal, characteristic, complemented
10#34249C211yesmaximal, characteristic, complemented
11#34201C216yesmaximal, complemented
12#34173C211yesmaximal, characteristic, complemented
13C4×#536C216yesmaximal, complemented
14#34065C216yesmaximal, complemented
15C4×#519C211yesmaximal, characteristic, complemented
16C22×2−1+4C2211yescharacteristic
17C24×Q8C2211yescharacteristic
18#3486C4132yescomplemented
19C2×D4×Q8C2218yes—
20C2×#548C2218yes—
21C2×#536C2216yes—
22C2×#525C2216yes—
23C4×2−1+4—216no—
24C2×#520C2216yes—
25C2×#520C2214yescomplemented
26C2×#519—216no—
27C2×#519C2218yes—
28C2×#519C2211yescharacteristic
29C2×#519C2211yescharacteristic
30C2×#519C2218yes—
31C2×#519C2211yescharacteristic
32C2×#519C2214yescomplemented
33C2×C4×#41C2216yes—
34C2×C4×#41C2214yescomplemented
35C22×C4×Q8C2213yes—
36C22×C4×Q8C2211yescharacteristic
37C22×C4×Q8C2213yes—
38C22×C4×Q8C2211yescharacteristic
39C22×C4×Q8C2214yescomplemented
40C22×#117C2213yes—
41C22×#117C2213yes—
42#2367C22124yescomplemented
43#2365C22124yescomplemented
44#2363C22124yes—
45#2362C22124yes—
46#2362C22124yescomplemented
47#2360C22124yescomplemented
48#2354C22124yes—
49#2352C22124yes—
50#2351C2218yescomplemented
51Q8×C4⋊C4C2218yescomplemented
52#2341C2218yescomplemented
53Q8×C22⋊C4C2218yes—
54Q8×C22⋊C4C2218yes—
55Q8×C22⋊C4C2218yescomplemented
56#2327C22112yescomplemented
57#2326C22112yescomplemented
58#2324C22112yes—
59#2324C22112yes—
60#2323C22112yes—
61#2323C22112yes—
62#2323C22124yescomplemented
63#2322C22112yes—
64#2322C22112yes—
65#2321C2214yes—
66#2321C22112yes—
67#2317C2216yes—
68#2311C2216yes—
69#2310C2216yes—
70C4×C4⋊Q8C22112yescomplemented
71C4×#124C22112yescomplemented
72C4×C22⋊Q8C22124yescomplemented
73C4×#123C22124yes—
74#2297C2216yes—
75#2296C2216yes—
76#2293C22124yes—
77#2290C22124yes—
78#2280C2216yes—
79#2276C2218yescomplemented
80#2274C2216yes—
81C42×Q8C2218yescomplemented
82C4×#117C2216yes—
83C2×2−1+4—28no—
84C2×2−1+4—28no—
85C2×2−1+4C2×C4116yes—
86C22×#41C2316yes—
87C22×#41C2314yes—
88C23×Q8—28no—
89C23×Q8—28no—
90C23×Q8C2×C4116yescomplemented
91C23×Q8C2313yes—
92C23×Q8C2318yes—
93C23×Q8C2314yes—
94C23×Q8C2311yescharacteristic
95C24×C4C2313yes—
96D4×Q8C2×C41128yescomplemented
97#548C2×C41128yescomplemented
98#536C2×C4196yescomplemented
99C2×C4⋊Q8C23112yes—
100C2×#124C23112yes—
101#525C2×C4196yescomplemented
102C2×#123C23112yes—
103C2×C22⋊Q8C23112yes—
104C2×C22⋊Q8C23124yes—
105C2×#121C23112yes—
106C2×#120C2314yes—
107#520—232no—
108#520—248no—
109#519—28no—
110#519—28no—
111#519—264no—
112#519—264no—
113#519—232no—
114#519C2×C4116yescomplemented
115C4×#41—232no—
116C4×#41—248no—
117C2×C4×Q8—28no—
118C2×C4×Q8—232no—
119C2×C4×Q8—224no—
120C2×C4×Q8—224no—
121C2×C4×Q8D4116yescomplemented
122C2×C4×Q8C2316yes—
123C2×C4×Q8C23112yes—
124C2×C4×Q8C2316yes—
125C2×C4×Q8C23112yes—
126C2×C4×Q8C2318yes—
127C2×C4×Q8C2314yes—
128C2×C4×Q8C2314yes—
129C2×C4×Q8C2318yes—
130C2×C4×Q8C23124yes—
131C2×C4×Q8C23112yes—
132C2×C4×Q8C2318yes—
133C2×C4×Q8C2314yes—
134C2×C4×Q8C2314yes—
135C2×C4×D4C2314yes—
136C2×C4×D4C23112yes—
137C2×C4×D4C2316yes—
138C2×C4×D4C2316yes—
139C2×C4×D4C23112yes—
140C2×C4×D4C23124yes—
141C2×#117—224no—
142C2×#117—224no—
143C2×#117C2316yes—
144C2×#117C2316yes—
145C2×#117C23112yes—
146C2×#117C23112yes—
147C2×#117C23112yes—
148C2×#117C23124yes—
149C2×#117C23112yes—
150C2×#117C2316yes—
151C22×C4⋊C4C2313yes—
152C22×C4⋊C4C2313yes—
153C22×C4⋊C4C2313yes—
154C22×C4⋊C4C23112yes—
155C22×C4⋊C4C2313yes—
156C22×C22⋊C4C2311yescharacteristic
157C22×C22⋊C4C2313yes—
158C22×C42C2313yes—
159C22×C42C2316yes—
160C22×C42C23112yes—
161C22×C42C2313yes—
162#391C23112yes—
163#391C23112yes—
164#391C23124yes—
165#391C23112yes—
166#391C23112yes—
167#390C23112yes—
168#390C2318yes—
169#390C23124yes—
170#390C23112yes—
171#389C23124yes—
172#389C23124yes—
173#389C23124yes—
174#389C23112yes—
175#389C23112yes—
176#388C2318yes—
177#388C2318yes—
178#388C23124yes—
179#388C23124yes—
180#388C23124yes—
181#388C23124yes—
182#388C23148yes—
183#387C23124yes—
184#387C23124yes—
185#387C23148yes—
186#386C23112yes—
187#386C2314yes—
188#386C2318yes—
189#386C23124yes—
190#385C23112yes—
191#385C2314yes—
192#385C23124yes—
193#385C23124yes—
194#384C23112yes—
195#382C23112yes—
196#381C23112yes—
197#380C23124yes—
198#380C23112yes—
199#380C23112yes—
200#379C2314yes—
201C4×C4⋊C4C23148yes—
202C4×C4⋊C4C23112yes—
203C4×C4⋊C4C23112yes—
204C4×C4⋊C4C23124yes—
205C4×C4⋊C4C23124yes—
206C4×C4⋊C4C23124yes—
207C4×C22⋊C4C2314yes—
208C4×C22⋊C4C23112yes—
209C4×C22⋊C4C23124yes—
210C4×C22⋊C4C23124yes—
211C4×C22⋊C4C23148yes—
212#376C23112yes—
213C43C23112yes—
214C25C2411yescharacteristic
2152−1+4—2128no—
216C2×#41—216no—
217C2×#41—216no—
218C2×#41—224no—
219C2×#41—224no—
220C2×#41C22×C4148yes—
221C2×#41C22×C4132yes—
222C22×Q8—2128no—
223C22×Q8—216no—
224C22×Q8—232no—
225C22×Q8—232no—
226C22×Q8—216no—
227C22×Q8—212no—
228C22×Q8—212no—
229C22×Q8C22×C4124yes—
230C22×Q8C22×C4164yes—
231C22×Q8C22×C4132yes—
232C22×Q8C22×C418yes—
233C22×Q8C2×D418yes—
234C22×Q8#4118yes—
235C22×Q8C24112yes—
236C22×Q8C2416yes—
237C22×Q8C2414yes—
238C22×D4C2414yes—
239C22×D4C2416yes—
240C22×D4C2418yes—
241C23×C4—212no—
242C23×C4—212no—
243C23×C4C22×C4124yes—
244C23×C4C2413yes—
245C23×C4C24112yes—
246C23×C4C24112yes—
247C23×C4C2413yes—
248C23×C4C2411yescharacteristic
249C23×C4C24112yes—
250C23×C4C2414yes—
251C23×C4C2413yes—
252C23×C4C2411yescharacteristic
253C23×C4C2413yes—
254C4⋊Q8C22×C4196yes—
255#124C22×C4196yes—
256#123C22×C4196yes—
257C22⋊Q8C22×C4196yes—
258C22⋊Q8C22×C41192yes—
259#121C22×C4196yes—
260#120C22×C4132yes—
261C4×Q8—232no—
262C4×Q8—216no—
263C4×Q8—216no—
264C4×Q8—232no—
265C4×Q8—296no—
266C4×Q8—248no—
267C4×Q8—224no—
268C4×Q8—248no—
269C4×Q8—248no—
270C4×Q8—224no—
271C4×Q8C22×C4164yes—
272C4×Q8C2×D4132yes—
273C4×Q8C2×D4132yes—
274C4×D4—216no—
275C4×D4—248no—
276C4×D4—248no—
277C4×D4—224no—
278C4×D4—224no—
279C4×D4C22×C41192yes—
280#117—224no—
281#117—224no—
282#117—296no—
283#117—248no—
284#117—248no—
285#117—248no—
286#117—248no—
287#117C22×C4148yes—
288C2×C4⋊C4—212no—
289C2×C4⋊C4—212no—
290C2×C4⋊C4—212no—
291C2×C4⋊C4—248no—
292C2×C4⋊C4C2×D4124yes—
293C2×C4⋊C4C2416yes—
294C2×C4⋊C4C24112yes—
295C2×C4⋊C4C24112yes—
296C2×C4⋊C4C2416yes—
297C2×C4⋊C4C24112yes—
298C2×C4⋊C4C2416yes—
299C2×C4⋊C4C2416yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C24112yes—
302C2×C4⋊C4C24124yes—
303C2×C4⋊C4C2418yes—
304C2×C4⋊C4C24112yes—
305C2×C4⋊C4C24124yes—
306C2×C4⋊C4C24112yes—
307C2×C22⋊C4—212no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4C2412yes—
310C2×C22⋊C4C24112yes—
311C2×C22⋊C4C2412yes—
312C2×C22⋊C4C2414yes—
313C2×C22⋊C4C2414yes—
314C2×C22⋊C4C24112yes—
315C2×C22⋊C4C2414yes—
316C2×C22⋊C4C2418yes—
317C2×C22⋊C4C2418yes—
318C2×C22⋊C4C24124yes—
319C2×C22⋊C4C2418yes—
320C2×C22⋊C4C2414yes—
321C2×C22⋊C4C2418yes—
322C2×C22⋊C4C24124yes—
323C2×C42—224no—
324C2×C42—212no—
325C2×C42—248no—
326C2×C42C2×D4124yes—
327C2×C42C24112yes—
328C2×C42C24112yes—
329C2×C42C24124yes—
330C2×C42C24112yes—
331C2×C42C24112yes—
332C2×C42C2414yes—
333C2×C42C2418yes—
334C2×C42C24124yes—
335C2×C42C24112yes—
336C2×C42C2416yes—
337C2×C42C2416yes—
338#95C2418yes—
339#95C24124yes—
340#95C24124yes—
341#95C2418yes—
342#95C24124yes—
343#95C24124yes—
344#95C24124yes—
345#95C2418yes—
346C24—22no—
347C24—22no—
348C24—42no—
349C24—42no—
350C24C23×C414yes—
351C24C2512yes—
352C24C2514yes—
353C24C2511yescharacteristic
354#41—2128no—
355#41—2192no—
356C2×Q8—2256no—
357C2×Q8—2128no—
358C2×Q8—296no—
359C2×Q8—224no—
360C2×Q8—212no—
361C2×Q8—224no—
362C2×Q8—212no—
363C2×Q8C4×D4164yes—
364C2×Q8C23×C4148yes—
365C2×Q8C23×C4124yes—
366C2×Q8C23×C4116yes—
367C2×Q8C22×D4116yes—
368C2×Q8C2×#41116yes—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—212no—
372C2×D4—212no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4C23×C4116yes—
376C2×D4C23×C4124yes—
377C2×D4C23×C4132yes—
378C22×C4—26no—
379C22×C4—26no—
380C22×C4—296no—
381C22×C4—224no—
382C22×C4—224no—
383C22×C4—224no—
384C22×C4—224no—
385C22×C4—26no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—224no—
389C22×C4—42no—
390C22×C4—46no—
391C22×C4—48no—
392C22×C4—42no—
393C22×C4—46no—
394C22×C4—424no—
395C22×C4C23×C4148yes—
396C22×C4C23×C4148yes—
397C22×C4C23×C4112yes—
398C22×C4C23×C4112yes—
399C22×C4C22×D4112yes—
400C22×C4C22×D414yes—
401C22×C4C22×D4112yes—
402C22×C4C2×#41112yes—
403C22×C4C2518yes—
404C22×C4C2514yes—
405C22×C4C2514yes—
406C22×C4C2512yes—
407C22×C4C2516yes—
408C22×C4C2518yes—
409C22×C4C2516yes—
410C22×C4C2514yes—
411C22×C4C2511yescharacteristic
412C22×C4C2516yes—
413C22×C4C2511yescharacteristic
414C22×C4C2516yes—
415C4⋊C4—224no—
416C4⋊C4—212no—
417C4⋊C4—224no—
418C4⋊C4—212no—
419C4⋊C4—212no—
420C4⋊C4—224no—
421C4⋊C4—28no—
422C4⋊C4—212no—
423C4⋊C4—224no—
424C4⋊C4—48no—
425C4⋊C4—424no—
426C4⋊C4C23×C4148yes—
427C4⋊C4C23×C4196yes—
428C4⋊C4C22×D4148yes—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—224no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—224no—
435C22⋊C4—28no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—424no—
439C22⋊C4—48no—
440C22⋊C4C23×C4116yes—
441C22⋊C4C23×C4132yes—
442C22⋊C4C23×C4196yes—
443C42—28no—
444C42—224no—
445C42—224no—
446C42—248no—
447C42—224no—
448C42—224no—
449C42—48no—
450C42—424no—
451C42C23×C4148yes—
452C42C22×D4124yes—
453C42C22×D4124yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—28no—
463C23—28no—
464C23—22no—
465C23—22no—
466C23—28no—
467C23—48no—
468C23—44no—
469C23—48no—
470C23—44no—
471C23C24×C418yes—
472C23C24×C414yes—
473C23C24×C412yes—
474C23C23×D412yes—
475C23C22×#4112yes—
476C23C2611yescharacteristic
477Q8—296no—
478Q8—248no—
479Q8C2×C4×D4164yes—
480D4—232no—
481D4—248no—
482D4—432no—
483C2×C4—24no—
484C2×C4—26no—
485C2×C4—24no—
486C2×C4—26no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—212no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—224no—
493C2×C4—296no—
494C2×C4—296no—
495C2×C4—24no—
496C2×C4—26no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—212no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—44no—
508C2×C4—412no—
509C2×C4—416no—
510C2×C4—412no—
511C2×C4—44no—
512C2×C4—824no—
513C2×C4C2×C4×D4148yes—
514C2×C4C24×C4116yes—
515C2×C4C24×C4112yes—
516C2×C4C24×C418yes—
517C2×C4C24×C4112yes—
518C2×C4C23×D4112yes—
519C2×C4C23×D412yes—
520C2×C4C23×D412yes—
521C2×C4C23×D4112yes—
522C2×C4C22×#41112yes—
523C22—22no—
524C22—24no—
525C22—22no—
526C22—24no—
527C22—24no—
528C22—28no—
529C22—24no—
530C22—216no—
531C22—24no—
532C22—416no—
533C22—44no—
534C22—42no—
535C22—48no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—42no—
540C22C22×C4×D414yes—
541C22C25×C411yescharacteristic
542C22C24×D411yescharacteristic
543C22C23×#4111yescharacteristic
544C222−1+612yes—
545C222−1+612yes—
546C4—22no—
547C4—22no—
548C4—212no—
549C4—28no—
550C4—212no—
551C4—216no—
552C4—44no—
553C4—46no—
554C4—48no—
555C4—42no—
556C4—44no—
557C4C22×C4×D4124yes—
558C2—24no—
559C2—48no—
560C2—44no—
561C2C23×C4×D411yescharacteristic
562C2#6265312yes—
563C2#6284811yescharacteristic
564C2#6285012yes—
565C2#6285911yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC23C26
commutatorC22C25×C4
prime radicalC2×#3486C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×2−1+4C2semidirect
3. C22×#519C2semidirect
4. #34312C2semidirect
5. #34311C2semidirect
6. #34293C2semidirect
7. #34291C2semidirect
8. #34266C2semidirect
9. #34257C2semidirect
10. #34249C2semidirect
11. #34201C2semidirect
12. #34173C2semidirect
13. C4×#536C2semidirect
14. #34065C2semidirect
15. C4×#519C2semidirect
18. #3486C4semidirect
25. C2×#520C22semidirect
32. C2×#519C22semidirect
34. C2×C4×#41C22semidirect
39. C22×C4×Q8C22semidirect
42. #2367C22semidirect
43. #2365C22semidirect
46. #2362C22semidirect
47. #2360C22semidirect
50. #2351C22semidirect
51. Q8×C4⋊C4C22semidirect
52. #2341C22semidirect
55. Q8×C22⋊C4C22semidirect
56. #2327C22semidirect
57. #2326C22semidirect
62. #2323C22semidirect
70. C4×C4⋊Q8C22semidirect
71. C4×#124C22semidirect
72. C4×C22⋊Q8C22semidirect
79. #2276C22semidirect
81. C42×Q8C22semidirect
90. C23×Q8C2×C4semidirect
96. D4×Q8C2×C4semidirect
97. #548C2×C4semidirect
98. #536C2×C4semidirect
101. #525C2×C4semidirect
114. #519C2×C4semidirect
121. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#34862
C2×C4×2−1+42
C22×#5192
#343122
#343112
#342932
#342912
#342662
#342572
#342492
#342012
#341732
C4×#5362
#340652
C4×#5192