Finite Group Databasebeta

← #7623940

Subgroups — #7623940

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61373C211yesmaximal, characteristic, complemented
2#61370C211yesmaximal, characteristic, complemented
3#33956C214yesmaximal, complemented
4#33953C212yesmaximal, complemented
5#33952C214yesmaximal, complemented
6#33951C212yesmaximal, complemented
7#33948C212yesmaximal, complemented
8Q8×D16C212yesmaximal, complemented
9#33731C211yesmaximal, characteristic, complemented
10#33729C212yesmaximal, complemented
11#33633C212yesmaximal, complemented
12#33631C212yesmaximal, complemented
13#33626C211yesmaximal, characteristic, complemented
14C2×#2224C211yesmaximal, characteristic, complemented
15#33476C211yesmaximal, characteristic, complemented
16#33454C212yesmaximal
17C4×#509C211yesmaximal, characteristic
18#3503—21no—
19#3501—21no—
20C2×#509C2211yescharacteristic
21#3389—22no—
22#3386—22no—
23#3385—22no—
24#3384—22no—
25#3383C2212yescomplemented
26Q8×Q16C2212yescomplemented
27#3380C2212yes—
28#3380C2214yes—
29#3380C2212yescomplemented
30Q8×D8C2212yes—
31Q8×D8C2212yescomplemented
32#3164C2212yes—
33#3164C2212yes—
34#3160C2211yescharacteristic
35C2×#500C2211yescharacteristic
36#3083—22no—
37#3081—22no—
38#3079—21no—
39#3077—21no—
40#2960C2211yescharacteristic
41#2951C2212yes—
42#2948C2212yes—
43C4×#135C2211yescharacteristic
44C2×C4×D8C2211yescharacteristic
45#2255C2214yescomplemented
46#2255C2214yescomplemented
47#2255C2214yescomplemented
48#2254C2212yescomplemented
49#2253C2214yescomplemented
50#2252C2214yescomplemented
51#2252C2212yescomplemented
52#2229C2212yescomplemented
53#2229C2214yescomplemented
54#2228C2214yes—
55#2228C2214yes—
56#2228C2212yescomplemented
57#2228C2214yescomplemented
58#2227C2211yescharacteristic, complemented
59#2227C2214yescomplemented
60#2226C2214yes—
61#2226C2211yescharacteristic, complemented
62#2225C2211yescharacteristic, complemented
63#2224C2212yes—
64#2224C2212yes—
65#2224C2214yes—
66#2224C2214yes—
67#2224C2211yescharacteristic, complemented
68#2224C2214yescomplemented
69#2183C2212yescomplemented
70#2183C2214yescomplemented
71Q8×C16C2212yescomplemented
72#2177C2214yes—
73#2177C2212yes—
74#2175C2212yes—
75#2175C2214yes—
76C4×SD32C2214yes—
77C4×SD32C2212yes—
78C4×D16C2212yes—
79C4×D16C2214yes—
80#2159C2212yes—
81#2159C2211yescharacteristic
82#2159C2212yes—
83#2158C2212yes—
84C2×#366C2212yes—
85#2151C2212yes—
86C2×C4⋊C16C2211yescharacteristic
87#2140C2211yescharacteristic
88#2139C2211yescharacteristic
89#2138C2212yes—
90C2×#357C2212yes—
91#2108C2212yes—
92C4×M32C2211yescharacteristic
93C2×#135C2311yescharacteristic
94C22×D8C2311yescharacteristic
95Q82—22no—
96#557—22no—
97#554—22no—
98#554—24no—
99#554—22no—
100#554—41no—
101#554—42no—
102#554—42no—
103D4×Q8—22no—
104D4×Q8—22no—
105D4×Q8—41no—
106D4×Q8—42no—
107#544—22no—
108#544—22no—
109#544—42no—
110#543—42no—
111#533—21no—
112#533—41no—
113#533D411yescharacteristic
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—41no—
116C2×C22⋊Q8—42no—
117#520—22no—
118#518—22no—
119C4×#41—21no—
120C2×C4×D4—21no—
121#509Q818yescomplemented
122C2×SD32C2312yes—
123C2×D16C2312yes—
124C2×M32C2311yescharacteristic
125C2×M32C2311yescharacteristic
126C22×C16C2312yes—
127#501—24no—
128#501—24no—
129#500—21no—
130#500—21no—
131#500C2311yescharacteristic
132#500C2314yes—
133#500C2314yes—
134#500C2312yes—
135#500C2311yescharacteristic
136#500C2312yes—
137#499—21no—
138#499C2312yes—
139#499C2314yes—
140#499C2314yes—
141#498—21no—
142#479—24no—
143#478—24no—
144#478—24no—
145#478—24no—
146#478—24no—
147#477—22no—
148#477—24no—
149#476—22no—
150#476—22no—
151#476—24no—
152#475—22no—
153#474—22no—
154#474—22no—
155#474—24no—
156#474—24no—
157#446D412yes—
158#446D412yes—
159C8×Q8C2312yes—
160C8×Q8C2312yes—
161#442—24no—
162#440—24no—
163C4×Q16C2311yescharacteristic
164C4×Q16C2314yes—
165C4×Q16C2311yescharacteristic
166C4×SD16—21no—
167C4×SD16—21no—
168C4×D8—21no—
169C4×D8—21no—
170C4×D8C2312yes—
171C4×D8C2314yes—
172C4×D8C2311yescharacteristic
173C4×D8C2314yes—
174C4×D8C2312yes—
175C4×D8C2314yes—
176C4×D8C2311yescharacteristic
177#428—22no—
178#428—22no—
179#427—21no—
180#427C2312yes—
181#427C2311yescharacteristic
182C2×#107—21no—
183C2×#107C2312yes—
184C2×#107C2311yescharacteristic
185#424D412yes—
186#423D411yescharacteristic
187#418—22no—
188#417—22no—
189#416—21no—
190C2×#102—21no—
191#405C2312yes—
192C2×C4×C8C2311yescharacteristic
193#367C2312yes—
194#367C2312yes—
195#367C2312yes—
196#367C2314yes—
197#367C2314yes—
198#366C2314yes—
199#366C2314yes—
200#366C2312yes—
201#366C2312yes—
202#366C2314yes—
203#366C2312yes—
204#366C2314yes—
205C4⋊C16C2311yescharacteristic
206C4⋊C16C2311yescharacteristic
207C4⋊C16C2312yes—
208C4⋊C16C2314yes—
209C4⋊C16C2314yes—
210#358C2312yes—
211#358C2312yes—
212#358C2312yes—
213#357C2314yes—
214#357C2314yes—
215#357C2312yes—
216#357C2312yes—
217#357C2312yes—
218#357C2312yes—
219#357C2312yes—
220#346C2312yes—
221#346C2314yes—
222C4×C16C2312yes—
223C4×C16C2314yes—
224C2×#41—21no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8—41no—
228C22×D4—21no—
229C23×C4—41no—
230#135—22no—
231#135C2×Q814yescomplemented
232C2×Q16C2411yescharacteristic
233C2×SD16—21no—
234C2×D8—21no—
235C2×D8—22no—
236C2×D8C2×Q814yescomplemented
237C2×D8C2412yes—
238C2×D8C2411yescharacteristic
239C2×M16C2×D412yes—
240C22×C8C2×D411yescharacteristic
241C22×C8C2411yescharacteristic
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—42no—
249C4⋊Q8—42no—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C4⋊Q8—44no—
253C4⋊Q8—41no—
254C4⋊Q8—41no—
255C4⋊Q8D811yescharacteristic
256C4⋊Q8D811yescharacteristic
257#125—22no—
258#125—24no—
259#125—24no—
260#125—42no—
261#125—42no—
262#125—44no—
263#125D812yes—
264C22⋊Q8—42no—
265C22⋊Q8—42no—
266C22⋊Q8—44no—
267C22⋊Q8—41no—
268C22⋊Q8—41no—
269C22⋊Q8—42no—
270C22⋊Q8—44no—
271C22⋊Q8—42no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C22⋊Q8D812yes—
280C22⋊Q8D812yes—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—42no—
286C4×Q8—41no—
287C4×Q8—41no—
288C4×Q8C2×D412yes—
289C4×Q8C2×D412yes—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—41no—
298C4×D4—41no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—44no—
302C4×D4—41no—
303C4×D4—41no—
304#117—21no—
305#117—22no—
306#117—42no—
307#117—41no—
308#117C2×D412yes—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—41no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C22⋊C4—41no—
317C2×C22⋊C4—44no—
318C2×C42—41no—
319C2×C42C2×D411yescharacteristic
320SD32C2×Q818yescomplemented
321D16C2×Q818yescomplemented
322M32—22no—
323M32C2×Q814yescomplemented
324C2×C16—24no—
325C2×C16C2412yes—
326C2×C16C2412yes—
327C2×C16C2412yes—
328C2×C16C2412yes—
329#107—22no—
330#107—22no—
331#107—21no—
332#107—22no—
333#107—21no—
334#107—21no—
335#107—24no—
336#107—24no—
337#107C2414yes—
338#107C2411yescharacteristic
339#107C2411yescharacteristic
340#107C2412yes—
341#107C2412yes—
342#107C2412yes—
343#106—22no—
344#106—21no—
345#106—22no—
346#106—24no—
347#106—24no—
348C4⋊C8C2×D412yes—
349C4⋊C8C2×D412yes—
350C4⋊C8C2×D414yes—
351C4⋊C8C2×D414yes—
352#103—21no—
353#103—21no—
354#103—24no—
355#102—22no—
356#102—22no—
357#102—21no—
358#102—21no—
359#102—24no—
360#102—24no—
361#102—24no—
362#97C2×D414yes—
363C4×C8C2×D411yescharacteristic
364C4×C8C2×D411yescharacteristic
365C4×C8C2411yescharacteristic
366C4×C8C2414yes—
367C4×C8C2411yescharacteristic
368C24—41no—
369#41—24no—
370#41—42no—
371C2×Q8—21no—
372C2×Q8—41no—
373C2×Q8—42no—
374C2×Q8—41no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8—162no—
379C2×Q8—164no—
380C2×Q8—164no—
381C2×Q8C2×D812yes—
382C2×Q8C2×D812yes—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—162no—
400C22×C4C2×D811yescharacteristic
401C22×C4C2×D811yescharacteristic
402C22×C4C22×D411yescharacteristic
403Q16—21no—
404Q16C22×Q812yes—
405SD16—24no—
406D8—24no—
407D8—22no—
408D8—21no—
409D8C22×Q812yes—
410D8C22×Q814yes—
411M16—24no—
412C2×C8—22no—
413C2×C8C22×D411yescharacteristic
414C2×C8C22×D411yescharacteristic
415C2×C8C22×D414yes—
416C2×C8C22×D411yescharacteristic
417C2×C8C22×Q812yes—
418C2×C82−1+412yes—
419C2×C8C2511yescharacteristic
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—24no—
425C4⋊C4—21no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—41no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4C2×D812yes—
444C4⋊C4C2×D812yes—
445C4⋊C4C2×D814yes—
446C4⋊C4C2×D812yes—
447C4⋊C4C2×D812yes—
448C4⋊C4C2×D814yes—
449C22⋊C4—42no—
450C22⋊C4—41no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—44no—
458C22⋊C4C2×D814yes—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—42no—
463C42—44no—
464C42C2×D811yescharacteristic
465C42C2×D811yescharacteristic
466C42C22×D411yescharacteristic
467C42C22×D411yescharacteristic
468C42C22×D414yes—
469C16—22no—
470C16—22no—
471C16—22no—
472C16C22×Q814yes—
473C23—41no—
474C23—42no—
475C23—44no—
476C23—161no—
477C23—161no—
478C23C22×D811yescharacteristic
479Q8—24no—
480Q8—22no—
481Q8—42no—
482Q8—41no—
483Q8—82no—
484Q8—84no—
485Q8—161no—
486Q8—162no—
487Q8—164no—
488Q8—164no—
489D4—24no—
490D4—22no—
491D4—44no—
492D4—44no—
493D4—42no—
494D4—41no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4—82no—
508C2×C4—164no—
509C2×C4—162no—
510C2×C4—164no—
511C2×C4—164no—
512C2×C4—161no—
513C2×C4—162no—
514C2×C4D4×Q812yes—
515C2×C4C22×D811yescharacteristic
516C2×C4C22×D814yes—
517C2×C4C22×D811yescharacteristic
518C2×C4C22×D811yescharacteristic
519C2×C4C22×D814yes—
520C2×C4C22×D811yescharacteristic
521C2×C4C22×D811yescharacteristic
522C2×C4#57812yes—
523C2×C4C23×D411yescharacteristic
524C8—21no—
525C8—21no—
526C8—24no—
527C8D4×Q812yes—
528C8C23×Q811yescharacteristic
529C8C2×2−1+411yescharacteristic
530C22—21no—
531C22—21no—
532C22—42no—
533C22—44no—
534C22—81no—
535C22—82no—
536C22—81no—
537C22—82no—
538C22—164no—
539C22Q8×D812yes—
540C22C23×D811yescharacteristic
541C4—21no—
542C4—24no—
543C4—21no—
544C4—21no—
545C4—24no—
546C4—42no—
547C4—82no—
548C4—81no—
549C4—82no—
550C4—84no—
551C4—81no—
552C4—82no—
553C4Q8×D812yes—
554C4#341512yes—
555C4C2×D4×Q811yescharacteristic
556C4C2×#57811yescharacteristic
557C2—22no—
558C2—82no—
559C2—84no—
560C2C2×Q8×D811yescharacteristic
561C2C2×#341511yescharacteristic
562C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC2×#509C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #533 ◁ #2960 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #61373C2semidirect
2. #61370C2semidirect
3. #33956C2semidirect
4. #33953C2semidirect
5. #33952C2semidirect
6. #33951C2semidirect
7. #33948C2semidirect
8. Q8×D16C2semidirect
9. #33731C2semidirect
10. #33729C2semidirect
11. #33633C2semidirect
12. #33631C2semidirect
13. #33626C2semidirect
14. C2×#2224C2semidirect
15. #33476C2semidirect
25. #3383C22semidirect
26. Q8×Q16C22semidirect
29. #3380C22semidirect
31. Q8×D8C22semidirect
45. #2255C22semidirect
46. #2255C22semidirect
47. #2255C22semidirect
48. #2254C22semidirect
49. #2253C22semidirect
50. #2252C22semidirect
51. #2252C22semidirect
52. #2229C22semidirect
53. #2229C22semidirect
56. #2228C22semidirect
57. #2228C22semidirect
58. #2227C22semidirect
59. #2227C22semidirect
61. #2226C22semidirect
62. #2225C22semidirect
67. #2224C22semidirect
68. #2224C22semidirect
69. #2183C22semidirect
70. #2183C22semidirect
71. Q8×C16C22semidirect
121. #509Q8semidirect
231. #135C2×Q8semidirect
236. C2×D8C2×Q8semidirect
320. SD32C2×Q8semidirect
321. D16C2×Q8semidirect
323. M32C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#613732
#613702
#339562
#339532
#339522
#339512
#339482
Q8×D162
#337312
#337292
#336332
#336312
#336262
C2×#22242
#334762
#334542
C4×#5092