Finite Group Databasebeta

← #7625753

Subgroups — #7625753

Order 512.

11050 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3487C211yesmaximal, characteristic
2C2×C4×2−1+4C211yesmaximal, characteristic, complemented
3C22×#520C211yesmaximal, characteristic, complemented
4#34313C218yesmaximal, complemented
5#34312C218yesmaximal, complemented
6#34295C218yesmaximal, complemented
7#34294C218yesmaximal, complemented
8#34265C216yesmaximal, complemented
9#34257C211yesmaximal, characteristic, complemented
10#34249C211yesmaximal, characteristic, complemented
11#34200C216yesmaximal, complemented
12#34174C211yesmaximal, characteristic, complemented
13C4×#535C216yesmaximal, complemented
14#34065C216yesmaximal, complemented
15C4×#519C211yesmaximal, characteristic, complemented
16C22×2−1+4C2211yescharacteristic
17C23×#41C2211yescharacteristic
18#3487C4132yescomplemented
19C2×#550C2218yes—
20C2×#548C2218yes—
21C2×#535C2216yes—
22C2×#525C2216yes—
23C4×2−1+4—216no—
24C2×#520—216no—
25C2×#520C2218yes—
26C2×#520C2216yes—
27C2×#520C2218yes—
28C2×#520C2214yescomplemented
29C2×#519C2211yescharacteristic
30C2×#519C2211yescharacteristic
31C2×#519C2211yescharacteristic
32C2×#519C2214yescomplemented
33C2×C4×#41C2216yes—
34C2×C4×#41C2214yescomplemented
35C22×C4×Q8C2211yescharacteristic
36C22×C4×Q8C2211yescharacteristic
37C22×C4×Q8C2214yescomplemented
38C22×C4×D4C2213yes—
39C22×C4×D4C2213yes—
40C22×#117C2213yes—
41C23×C4⋊C4C2213yes—
42#2367C22124yescomplemented
43#2366C22124yescomplemented
44#2363C22124yescomplemented
45#2361C22124yescomplemented
46#2361C22124yescomplemented
47#2360C22124yescomplemented
48#2352C22124yescomplemented
49#2351C2218yescomplemented
50Q8×C4⋊C4C2218yescomplemented
51D4×C4⋊C4C22124yescomplemented
52#2341C2218yes—
53#2341C2218yes—
54#2341C2218yescomplemented
55Q8×C22⋊C4C2218yescomplemented
56#2328C22112yescomplemented
57#2326C22112yescomplemented
58#2324C22124yescomplemented
59#2324C22112yescomplemented
60#2324C22112yescomplemented
61#2323C22112yescomplemented
62#2323C22112yescomplemented
63#2322C2214yes—
64#2322C22112yes—
65#2321C22112yes—
66#2321C22112yes—
67#2317C2216yes—
68#2311C2216yes—
69#2309C2216yes—
70C4×#127C22112yescomplemented
71C4×#124C22112yescomplemented
72C4×#121C22124yescomplemented
73C4×#120C22124yes—
74#2297C2216yes—
75#2296C2216yes—
76#2293C22124yescomplemented
77#2290C22124yescomplemented
78#2280C2216yes—
79#2276C2218yescomplemented
80#2274C2216yes—
81C42×Q8C2218yescomplemented
82C4×#117C2216yes—
83C2×2−1+4—28no—
84C2×2−1+4—28no—
85C2×2−1+4C2×C4116yescomplemented
86C22×#41—28no—
87C22×#41—28no—
88C22×#41C2×C4116yescomplemented
89C22×#41C2318yes—
90C22×#41C2316yes—
91C22×#41C2314yes—
92C23×Q8C2314yes—
93C23×Q8C2311yescharacteristic
94C23×D4C2313yes—
95C24×C4C2313yes—
96C24×C4C2316yes—
97#550C2×C41128yescomplemented
98#548C2×C41128yescomplemented
99#535C2×C4196yescomplemented
100C2×#127C23112yes—
101C2×#124C23112yes—
102#525C2×C4196yescomplemented
103C2×#123C2314yes—
104C2×C22⋊Q8C23112yes—
105C2×#121C23112yes—
106C2×#121C23124yes—
107C2×#120C23112yes—
108#520—264no—
109#520—264no—
110#520—232no—
111#520—248no—
112#519—28no—
113#519—28no—
114#519—232no—
115#519C2×C4116yescomplemented
116C4×#41—232no—
117C4×#41—248no—
118C2×C4×Q8—28no—
119C2×C4×Q8—232no—
120C2×C4×Q8D4116yescomplemented
121C2×C4×Q8C2316yes—
122C2×C4×Q8C2316yes—
123C2×C4×Q8C2318yes—
124C2×C4×Q8C2318yes—
125C2×C4×Q8C2314yes—
126C2×C4×Q8C2314yes—
127C2×C4×Q8C23112yes—
128C2×C4×Q8C2318yes—
129C2×C4×Q8C2314yes—
130C2×C4×Q8C2314yes—
131C2×C4×D4—224no—
132C2×C4×D4—224no—
133C2×C4×D4C2316yes—
134C2×C4×D4C23112yes—
135C2×C4×D4C2314yes—
136C2×C4×D4C23112yes—
137C2×C4×D4C2316yes—
138C2×C4×D4C23112yes—
139C2×C4×D4C23112yes—
140C2×C4×D4C23124yes—
141C2×C4×D4C23148yes—
142C2×C4×D4C23124yes—
143C2×#117—224no—
144C2×#117C2316yes—
145C2×#117C2316yes—
146C2×#117C23112yes—
147C2×#117C23112yes—
148C2×#117C23112yes—
149C2×#117C23112yes—
150C2×#117C2316yes—
151C22×C4⋊C4—224no—
152C22×C4⋊C4C2313yes—
153C22×C4⋊C4C2313yes—
154C22×C4⋊C4C2311yescharacteristic
155C22×C4⋊C4C23112yes—
156C22×C4⋊C4C23124yes—
157C22×C4⋊C4C2313yes—
158C22×C22⋊C4C2313yes—
159C22×C22⋊C4C2313yes—
160C22×C42C2313yes—
161C22×C42C23112yes—
162C22×C42C2313yes—
163#391C23112yes—
164#391C23112yes—
165#390C23124yes—
166#390C23112yes—
167#390C23112yes—
168#390C23124yes—
169#390C23112yes—
170#390C23124yes—
171#390C23112yes—
172#389C23124yes—
173#389C2318yes—
174#389C23124yes—
175#389C23112yes—
176#389C23112yes—
177#388C23124yes—
178#388C23124yes—
179#388C23124yes—
180#388C23124yes—
181#388C2318yes—
182#388C23124yes—
183#387C2318yes—
184#387C23124yes—
185#386C23112yes—
186#386C2314yes—
187#386C23124yes—
188#386C23124yes—
189#386C23148yes—
190#385C23112yes—
191#385C2314yes—
192#385C2318yes—
193#385C23124yes—
194#385C23148yes—
195#384C23112yes—
196#382C23112yes—
197#381C2314yes—
198#380C23124yes—
199#380C23112yes—
200#380C23112yes—
201#379C23112yes—
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—
214C25C2413yes—
2152−1+4—2128no—
216C2×#41—2128no—
217C2×#41—216no—
218C2×#41—232no—
219C2×#41—232no—
220C2×#41—216no—
221C2×#41—224no—
222C2×#41—224no—
223C2×#41C22×C4148yes—
224C2×#41C22×C4164yes—
225C2×#41C22×C4132yescomplemented
226C22×Q8—216no—
227C22×Q8—216no—
228C22×Q8C22×C418yes—
229C22×Q8C22×C4132yescomplemented
230C22×Q8C2×D418yes—
231C22×Q8#4118yes—
232C22×Q8C2416yes—
233C22×Q8C2414yes—
234C22×D4—212no—
235C22×D4—212no—
236C22×D4C22×C4124yes—
237C22×D4C2414yes—
238C22×D4C24112yes—
239C22×D4C2416yes—
240C22×D4C24124yes—
241C23×C4—212no—
242C23×C4—212no—
243C23×C4—224no—
244C23×C4—812no—
245C23×C4C22×C4124yes—
246C23×C4C2411yescharacteristic
247C23×C4C24112yes—
248C23×C4C24112yes—
249C23×C4C2411yescharacteristic
250C23×C4C2413yes—
251C23×C4C24112yes—
252C23×C4C24124yes—
253C23×C4C2414yes—
254C23×C4C2411yescharacteristic
255C23×C4C2413yes—
256C23×C4C2413yes—
257#127C22×C4196yescomplemented
258#124C22×C4196yescomplemented
259#123C22×C4132yes—
260C22⋊Q8C22×C4196yescomplemented
261#121C22×C4196yescomplemented
262#121C22×C41192yescomplemented
263#120C22×C4196yes—
264C4×Q8—216no—
265C4×Q8—232no—
266C4×Q8—216no—
267C4×Q8—232no—
268C4×Q8—248no—
269C4×Q8—224no—
270C4×Q8—224no—
271C4×Q8C22×C4164yescomplemented
272C4×Q8C2×D4132yes—
273C4×Q8C2×D4132yes—
274C4×D4—216no—
275C4×D4—248no—
276C4×D4—296no—
277C4×D4—248no—
278C4×D4—248no—
279C4×D4—248no—
280C4×D4—224no—
281C4×D4—224no—
282C4×D4—848no—
283C4×D4C22×C41192yescomplemented
284#117—224no—
285#117—224no—
286#117—248no—
287#117—248no—
288#117—248no—
289#117—248no—
290#117C22×C4148yes—
291C2×C4⋊C4—212no—
292C2×C4⋊C4—212no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—296no—
295C2×C4⋊C4—248no—
296C2×C4⋊C4C2×D4124yes—
297C2×C4⋊C4C24112yes—
298C2×C4⋊C4C24112yes—
299C2×C4⋊C4C2416yes—
300C2×C4⋊C4C24112yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2416yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C24112yes—
306C2×C4⋊C4C24124yes—
307C2×C4⋊C4C2418yes—
308C2×C4⋊C4C24112yes—
309C2×C4⋊C4C2418yes—
310C2×C4⋊C4C24112yes—
311C2×C22⋊C4—212no—
312C2×C22⋊C4—212no—
313C2×C22⋊C4C2416yes—
314C2×C22⋊C4C24112yes—
315C2×C22⋊C4C2416yes—
316C2×C22⋊C4C2414yes—
317C2×C22⋊C4C24112yes—
318C2×C22⋊C4C2414yes—
319C2×C22⋊C4C2414yes—
320C2×C22⋊C4C24124yes—
321C2×C22⋊C4C24124yes—
322C2×C22⋊C4C24124yes—
323C2×C22⋊C4C2418yes—
324C2×C22⋊C4C2414yes—
325C2×C22⋊C4C2418yes—
326C2×C22⋊C4C24124yes—
327C2×C42—212no—
328C2×C42—248no—
329C2×C42C2×D4124yes—
330C2×C42C24112yes—
331C2×C42C24112yes—
332C2×C42C24112yes—
333C2×C42C24112yes—
334C2×C42C2414yes—
335C2×C42C24124yes—
336C2×C42C2418yes—
337C2×C42C24112yes—
338C2×C42C2416yes—
339C2×C42C2416yes—
340#95C2418yes—
341#95C24124yes—
342#95C2418yes—
343#95C2418yes—
344#95C24124yes—
345#95C2418yes—
346#95C24124yes—
347#95C2418yes—
348C24—26no—
349C24—26no—
350C24—46no—
351C24—46no—
352C24C23×C4112yes—
353C24C2516yes—
354C24C2514yes—
355C24C2511yescharacteristic
356#41—2256no—
357#41—2128no—
358#41—2192no—
359C2×Q8—2128no—
360C2×Q8—212no—
361C2×Q8—212no—
362C2×Q8C4×D4164yescomplemented
363C2×Q8C23×C4124yes—
364C2×Q8C23×C4116yes—
365C2×Q8C22×D4116yes—
366C2×Q8C2×#41116yes—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—296no—
370C2×D4—224no—
371C2×D4—212no—
372C2×D4—224no—
373C2×D4—212no—
374C2×D4—424no—
375C2×D4—424no—
376C2×D4C23×C4116yes—
377C2×D4C23×C4148yes—
378C2×D4C23×C4124yes—
379C2×D4C23×C4196yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—296no—
383C22×C4—224no—
384C22×C4—224no—
385C22×C4—224no—
386C22×C4—224no—
387C22×C4—22no—
388C22×C4—26no—
389C22×C4—28no—
390C22×C4—224no—
391C22×C4—248no—
392C22×C4—46no—
393C22×C4—46no—
394C22×C4—42no—
395C22×C4—48no—
396C22×C4—42no—
397C22×C4—424no—
398C22×C4—448no—
399C22×C4—848no—
400C22×C4C23×C4148yes—
401C22×C4C23×C4148yes—
402C22×C4C23×C414yes—
403C22×C4C23×C4112yes—
404C22×C4C22×D414yes—
405C22×C4C22×D4112yes—
406C22×C4C22×D4112yes—
407C22×C4C2×#41112yes—
408C22×C4C2518yes—
409C22×C4C2512yes—
410C22×C4C2514yes—
411C22×C4C2516yes—
412C22×C4C2514yes—
413C22×C4C2518yes—
414C22×C4C2514yes—
415C22×C4C2512yes—
416C22×C4C2511yescharacteristic
417C22×C4C2516yes—
418C22×C4C2511yescharacteristic
419C22×C4C2516yes—
420C4⋊C4—224no—
421C4⋊C4—224no—
422C4⋊C4—224no—
423C4⋊C4—212no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—28no—
427C4⋊C4—212no—
428C4⋊C4—224no—
429C4⋊C4—48no—
430C4⋊C4—424no—
431C4⋊C4C23×C4148yes—
432C4⋊C4C23×C4132yes—
433C4⋊C4C22×D4148yes—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—224no—
437C22⋊C4—212no—
438C22⋊C4—212no—
439C22⋊C4—224no—
440C22⋊C4—28no—
441C22⋊C4—48no—
442C22⋊C4—424no—
443C22⋊C4—424no—
444C22⋊C4—424no—
445C22⋊C4C23×C4116yes—
446C22⋊C4C23×C4132yes—
447C22⋊C4C23×C4196yes—
448C42—28no—
449C42—224no—
450C42—224no—
451C42—224no—
452C42—224no—
453C42—48no—
454C42—424no—
455C42C23×C4148yes—
456C42C22×D4124yes—
457C42C22×D4124yes—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—26no—
462C23—24no—
463C23—26no—
464C23—28no—
465C23—212no—
466C23—212no—
467C23—28no—
468C23—224no—
469C23—424no—
470C23—412no—
471C23—412no—
472C23—424no—
473C23C24×C4112yes—
474C23C24×C418yes—
475C23C24×C412yes—
476C23C23×D412yes—
477C23C22×#4112yes—
478C23C2611yescharacteristic
479Q8—248no—
480Q8C2×C4×D4164yes—
481D4—232no—
482D4—296no—
483D4—248no—
484D4—496no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—212no—
491C2×C4—22no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—296no—
496C2×C4—296no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—26no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—212no—
505C2×C4—28no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—22no—
509C2×C4—412no—
510C2×C4—44no—
511C2×C4—416no—
512C2×C4—44no—
513C2×C4—412no—
514C2×C4—824no—
515C2×C4—848no—
516C2×C4C2×C4×D4148yes—
517C2×C4C24×C4116yes—
518C2×C4C24×C418yes—
519C2×C4C24×C414yes—
520C2×C4C24×C4112yes—
521C2×C4C23×D412yes—
522C2×C4C23×D4112yes—
523C2×C4C23×D412yes—
524C2×C4C23×D4112yes—
525C2×C4C22×#41112yes—
526C22—22no—
527C22—24no—
528C22—22no—
529C22—24no—
530C22—28no—
531C22—212no—
532C22—216no—
533C22—224no—
534C22—448no—
535C22—44no—
536C22—46no—
537C22—48no—
538C22—412no—
539C22—46no—
540C22—44no—
541C22C22×C4×D414yes—
542C22C25×C411yescharacteristic
543C22C24×D411yescharacteristic
544C22C23×#4111yescharacteristic
545C22H(7,𝔽2)12yes—
546C222−1+612yes—
547C4—22no—
548C4—22no—
549C4—212no—
550C4—24no—
551C4—28no—
552C4—216no—
553C4—42no—
554C4—44no—
555C4—48no—
556C4—46no—
557C4—44no—
558C4C22×C4×D4124yes—
559C2—24no—
560C2—48no—
561C2—412no—
562C2C23×C4×D411yescharacteristic
563C2#6265412yes—
564C2#6284811yescharacteristic
565C2#6285112yes—
566C2#6285211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC23C26
commutatorC22C25×C4
prime radicalC2×#3487C2
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×#520C2semidirect
4. #34313C2semidirect
5. #34312C2semidirect
6. #34295C2semidirect
7. #34294C2semidirect
8. #34265C2semidirect
9. #34257C2semidirect
10. #34249C2semidirect
11. #34200C2semidirect
12. #34174C2semidirect
13. C4×#535C2semidirect
14. #34065C2semidirect
15. C4×#519C2semidirect
18. #3487C4semidirect
28. C2×#520C22semidirect
32. C2×#519C22semidirect
34. C2×C4×#41C22semidirect
37. C22×C4×Q8C22semidirect
42. #2367C22semidirect
43. #2366C22semidirect
44. #2363C22semidirect
45. #2361C22semidirect
46. #2361C22semidirect
47. #2360C22semidirect
48. #2352C22semidirect
49. #2351C22semidirect
50. Q8×C4⋊C4C22semidirect
51. D4×C4⋊C4C22semidirect
54. #2341C22semidirect
55. Q8×C22⋊C4C22semidirect
56. #2328C22semidirect
57. #2326C22semidirect
58. #2324C22semidirect
59. #2324C22semidirect
60. #2324C22semidirect
61. #2323C22semidirect
62. #2323C22semidirect
70. C4×#127C22semidirect
71. C4×#124C22semidirect
72. C4×#121C22semidirect
76. #2293C22semidirect
77. #2290C22semidirect
79. #2276C22semidirect
81. C42×Q8C22semidirect
85. C2×2−1+4C2×C4semidirect
88. C22×#41C2×C4semidirect
97. #550C2×C4semidirect
98. #548C2×C4semidirect
99. #535C2×C4semidirect
102. #525C2×C4semidirect
115. #519C2×C4semidirect
120. C2×C4×Q8D4semidirect
225. C2×#41C22×C4semidirect
229. C22×Q8C22×C4semidirect
257. #127C22×C4semidirect
258. #124C22×C4semidirect
260. C22⋊Q8C22×C4semidirect
261. #121C22×C4semidirect
262. #121C22×C4semidirect
271. C4×Q8C22×C4semidirect
283. C4×D4C22×C4semidirect
362. C2×Q8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#34872
C2×C4×2−1+42
C22×#5202
#343132
#343122
#342952
#342942
#342652
#342572
#342492
#342002
#341742
C4×#5352
#340652
C4×#5192