Finite Group Databasebeta

← #11675825

Subgroups — #11675825

Order 768.

1818 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86280C212yesmaximal, complemented
2C3×#3160C211yesmaximal, characteristic, complemented
3#83350C212yesmaximal, complemented
4#81953C212yesmaximal, complemented
5#81952C212yesmaximal, complemented
6#81931C214yesmaximal, complemented
7#81912C214yesmaximal, complemented
8#81905C214yesmaximal, complemented
9#79197C214yesmaximal, complemented
10#79173C212yesmaximal, complemented
11#79167C212yesmaximal
12C2×C4×D24C211yesmaximal, characteristic
13#78445C211yesmaximal, characteristic, complemented
14#61370—31nomaximal, Sylow
15C3×#533C2212yes—
16C22×D24C2211yescharacteristic
17#5730C2214yescomplemented
18#5727C2214yescomplemented
19#5690C2218yescomplemented
20#5634—24no—
21D3×C22⋊Q8—24no—
22#5579—24no—
23#5579—22no—
24Q8×D12—22no—
25#5521C2214yes—
26#5519C2214yes—
27C2×C4×D12C2212yes—
28#5470C2211yescharacteristic
29C3×#500C2211yescharacteristic, complemented
30C3×#500C2211yescharacteristic, complemented
31C3×#499C2212yescomplemented
32C3×#499C2212yescomplemented
33C3×#498C2212yescomplemented
34C3×#427C2212yes—
35C3×#427C2212yes—
36C2×C3×C4×C8C2211yescharacteristic
37C2×#821C2212yes—
38C2×#818C2211yescharacteristic
39#5102C2212yes—
40#5101C2212yes—
41#5087C2214yescomplemented
42#5084C2214yescomplemented
43#5066C2218yescomplemented
44#5001C2214yes—
45#4995C2214yes—
46#4924C2211yescharacteristic
47#4891C2218yescomplemented
48#4890C2218yescomplemented
49#4878C2214yescomplemented
50#4874C2214yescomplemented
51#4872C2214yes—
52#4872C2214yes—
53#4871C2218yescomplemented
54#4870C2218yescomplemented
55#4869C2218yes—
56#4856C2218yescomplemented
57C4×D24C2212yes—
58C4×D24C2212yes—
59C4×D24C2212yes—
60C4×D24C2212yes—
61#4679C2212yescomplemented
62#4679C2212yescomplemented
63C8×Dic6C2212yescomplemented
64C8×Dic6C2212yescomplemented
65#3501—32no—
66#3389—34no—
67#3384—34no—
68#3380—32no—
69#3380—34no—
70Q8×D8—32no—
71#3164—32no—
72#3160D311yescharacteristic, complemented
73#3081—34no—
74#3077—32no—
75#2960—31no—
76#2948—32no—
77C2×C4×D8—31no—
78C2×D3×Q8—24no—
79C22×D12C2312yes—
80C22×C4×D3—22no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4⋊Q8C2312yes—
84C3×C4⋊Q8C2312yes—
85C3×#125C2314yes—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×#117C2314yes—
91C2×C3×C42C2311yescharacteristic
92#946—24no—
93#946—28no—
94C2×(C2×D3)⋊C4—22no—
95#926C2312yes—
96C2×C4⋊Dic3C2312yes—
97#924D412yes—
98#924D412yes—
99C2×#208C2312yes—
100C2×D24Q818yescomplemented
101C2×D24C2312yes—
102C2×D24C2312yes—
103#897—24no—
104#897—28no—
105#896—28no—
106#896—28no—
107#896—28no—
108#893C2318yes—
109#893C2318yes—
110D3×C4⋊C4—24no—
111D3×C4⋊C4—28no—
112#890C2318yes—
113#888C2318yes—
114D3×C22⋊C4—28no—
115#878—28no—
116C4×D12—22no—
117C4×D12—22no—
118C4×D12—24no—
119C4×D12—24no—
120C4×D12C2314yes—
121C4×D12C2314yes—
122#870—24no—
123#870D412yes—
124#869—24no—
125#869D411yescharacteristic
126#869D411yescharacteristic
127C4×Dic6C2312yes—
128C4×Dic6C2312yes—
129C3×#107C2312yes—
130C3×#107C2312yes—
131C3×#107C2312yes—
132C3×#107C2312yes—
133C3×#106C2314yes—
134C3×#106C2314yes—
135C3×C4×C8C2311yescharacteristic
136C3×C4×C8C2311yescharacteristic
137C3×C4×C8C2311yescharacteristic
138C3×C4×C8C2311yescharacteristic
139#821C2314yes—
140#821C2314yes—
141#818C2312yes—
142#818C2312yes—
143#815C2314yes—
144Dic3⋊C8C2314yes—
145Dic3⋊C8C2314yes—
146C8×Dic3C2314yes—
147#809C2318yes—
148#809C2318yes—
149#808C2318yes—
150#807C2318yes—
151C4⋊(C3⋊C8)C2312yes—
152C4⋊(C3⋊C8)C2312yes—
153C22×D8—31no—
154#554—34no—
155#554—38no—
156#554—64no—
157#554—62no—
158D4×Q8—34no—
159D4×Q8—62no—
160#544—34no—
161#543—64no—
162#533—31no—
163#533C2×D312yes—
164C2×C22⋊Q8—64no—
165#518—34no—
166C2×C4×D4—32no—
167#501—38no—
168#500—34no—
169#500C2×D311yescharacteristic, complemented
170#500C2×D311yescharacteristic, complemented
171#499—34no—
172#499C2×D312yescomplemented
173#499C2×D312yescomplemented
174#498C2×D312yescomplemented
175#478—38no—
176#478—38no—
177#478—38no—
178#476—34no—
179#476—38no—
180#474—34no—
181#474—38no—
182#446—32no—
183#446—32no—
184C8×Q8—32no—
185C8×Q8—32no—
186#442—38no—
187C4×D8—34no—
188C4×D8—34no—
189C4×D8—32no—
190C4×D8—32no—
191C4×D8—32no—
192C4×D8—32no—
193#428—34no—
194#427C2×D312yes—
195#427C2×D312yes—
196C2×#107—31no—
197#424—32no—
198#423—31no—
199#418—34no—
200C2×#102—32no—
201#405—32no—
202C2×C4×C8C2×D311yescharacteristic
203C23×D3—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—22no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—22no—
210D3×Q8—88no—
211C2×D12—22no—
212C2×D12—24no—
213C2×D12C2×Q818yescomplemented
214C2×D12C2414yes—
215C2×C4×D3—24no—
216C2×C4×D3—24no—
217C2×C4×D3—28no—
218C2×C4×D3—28no—
219C2×C4×D3—84no—
220C2×Dic6—28no—
221C2×Dic6C2×D412yes—
222C2×Dic6C2×D412yes—
223C2×C3×C8—22no—
224C2×C3×C8C2×Q814yescomplemented
225C2×C3×C8C2411yescharacteristic
226C2×C3×C8C2411yescharacteristic
227C2×C3×C8C2411yescharacteristic
228C2×C3×C8C2411yescharacteristic
229C3×C4⋊C4—24no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4C2414yes—
234C3×C4⋊C4C2414yes—
235C3×C22⋊C4—24no—
236C3×C42—24no—
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42C2411yescharacteristic
240C3×C42C2411yescharacteristic
241#217C2×D414yes—
242(C2×D3)⋊C4—24no—
243(C2×D3)⋊C4—24no—
244(C2×D3)⋊C4—28no—
245(C2×D3)⋊C4—28no—
246C4⋊Dic3—22no—
247C4⋊Dic3—28no—
248C4⋊Dic3C2×D412yes—
249C4⋊Dic3C2×D412yes—
250C4⋊Dic3C2×D412yes—
251C4⋊Dic3C2×D412yes—
252C4⋊Dic3C2414yes—
253Dic3⋊C4—28no—
254Dic3⋊C4—28no—
255Dic3⋊C4C2×D414yes—
256Dic3⋊C4C2×D414yes—
257C4×Dic3C2414yes—
258#208—24no—
259C2×C3⋊C8C2414yes—
260D24C2×Q818yescomplemented
261D24C2×Q818yescomplemented
262C22×Q8—64no—
263C22×D4—32no—
264C23×C4—62no—
265C2×D8—38no—
266C2×D8—32no—
267C2×D8—32no—
268C2×M16—32no—
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C4⋊Q8—38no—
272C4⋊Q8—31no—
273C4⋊Q8—31no—
274C4⋊Q8—64no—
275C4⋊Q8C22×D312yes—
276C4⋊Q8C22×D312yes—
277#125—38no—
278#125—32no—
279#125—64no—
280#125C22×D314yes—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—32no—
284C22⋊Q8—32no—
285C22⋊Q8—68no—
286C22⋊Q8—68no—
287C22⋊Q8—68no—
288C22⋊Q8—68no—
289C22⋊Q8—64no—
290C22⋊Q8—68no—
291C22⋊Q8—64no—
292C22⋊Q8—68no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—32no—
296C4×Q8—32no—
297C4×D4—38no—
298C4×D4—38no—
299C4×D4—34no—
300C4×D4—34no—
301C4×D4—64no—
302C4×D4—64no—
303C4×D4—62no—
304C4×D4—62no—
305#117—32no—
306#117C22×D314yes—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—68no—
309C2×C4⋊C4—64no—
310C2×C22⋊C4—68no—
311C2×C22⋊C4—62no—
312C2×C42C22×D311yescharacteristic
313#107—32no—
314#107—32no—
315#107—38no—
316#107C22×D312yes—
317#107C22×D312yes—
318#107C22×D312yes—
319#107C22×D312yes—
320#106—38no—
321#106C22×D314yes—
322#106C22×D314yes—
323C4⋊C8—34no—
324C4⋊C8—34no—
325C4⋊C8—32no—
326C4⋊C8—32no—
327#102—34no—
328#102—34no—
329#102—38no—
330#102—38no—
331#97—34no—
332C4×C8—34no—
333C4×C8C22×D311yescharacteristic
334C4×C8C22×D311yescharacteristic
335C4×C8C22×D311yescharacteristic
336C4×C8C22×D311yescharacteristic
337C23×C3C22×D411yescharacteristic
338C22×D3—24no—
339C22×D3—28no—
340C22×D3—82no—
341C22×D3—82no—
342C3×Q8—44no—
343C3×Q8—82no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—41no—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×Q812yes—
354C2×C3×C42−1+412yes—
355C2×C3×C4C2511yescharacteristic
356C2×Dic3—24no—
357C2×Dic3—82no—
358C2×Dic3C22×D414yes—
359C2×Dic3C22×D414yes—
360D12—28no—
361D12—24no—
362D12C22×Q818yes—
363C4×D3—48no—
364C4×D3—88no—
365C4×D3—88no—
366C4×D3—84no—
367Dic6—24no—
368Dic6—42no—
369Dic6—88no—
370C3×C8—21no—
371C3×C8—21no—
372C3×C8C22×Q812yes—
373C3×C8C22×Q812yes—
374C3⋊C8—24no—
375C24—62no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—32no—
379C2×Q8—32no—
380C2×Q8—68no—
381C2×Q8—248no—
382C2×D4—38no—
383C2×D4—34no—
384C2×D4—62no—
385C2×D4—64no—
386C22×C4—68no—
387C22×C4—68no—
388C22×C4—62no—
389C22×C4—64no—
390C22×C4—64no—
391C22×C4—244no—
392C22×C4D3×D411yescharacteristic
393C22×C4D3×D411yescharacteristic
394C22×C4C23×D311yescharacteristic
395D8—38no—
396D8—38no—
397M16—64no—
398C2×C8—22no—
399C2×C8—34no—
400C2×C8D3×Q814yescomplemented
401C2×C8C23×D311yescharacteristic
402C2×C8C23×D311yescharacteristic
403C2×C8C23×D311yescharacteristic
404C2×C8C23×D311yescharacteristic
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—34no—
410C4⋊C4—32no—
411C4⋊C4—32no—
412C4⋊C4—32no—
413C4⋊C4—34no—
414C4⋊C4—32no—
415C4⋊C4—34no—
416C4⋊C4—68no—
417C4⋊C4—68no—
418C4⋊C4—68no—
419C4⋊C4—62no—
420C4⋊C4C23×D314yes—
421C4⋊C4C23×D314yes—
422C22⋊C4—24no—
423C22⋊C4—34no—
424C22⋊C4—68no—
425C22⋊C4—68no—
426C22⋊C4—64no—
427C22⋊C4—64no—
428C42—24no—
429C42—34no—
430C42D3×D411yescharacteristic
431C42D3×D411yescharacteristic
432C42C23×D311yescharacteristic
433C42C23×D311yescharacteristic
434C22×C3—21no—
435C22×C3—21no—
436C22×C3D4×Q812yes—
437C22×C3C23×D411yescharacteristic
438C2×D3—28no—
439C2×D3—44no—
440C2×D3—44no—
441C2×D3—88no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—44no—
446C3×C4—44no—
447C3×C4D4×Q812yes—
448C3×C4#57612yes—
449C3×C4C23×Q811yescharacteristic
450C3×C4C2×2−1+411yescharacteristic
451Dic3—24no—
452Dic3—24no—
453Dic3—44no—
454C23—64no—
455C23—68no—
456C23—242no—
457C23—242no—
458C23C2×D3×D411yescharacteristic
459Q8—44no—
460Q8—64no—
461Q8—82no—
462Q8—122no—
463Q8—248no—
464D4—38no—
465D4—64no—
466D4—68no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—34no—
471C2×C4—34no—
472C2×C4—41no—
473C2×C4—64no—
474C2×C4—128no—
475C2×C4—244no—
476C2×C4—248no—
477C2×C4—248no—
478C2×C4—242no—
479C2×C4C2×D3×D411yescharacteristic
480C2×C4C2×D3×D411yescharacteristic
481C2×C4C2×D3×D411yescharacteristic
482C2×C4C2×D3×D411yescharacteristic
483C2×C4C2×D3×D411yescharacteristic
484C2×C4C2×D3×Q812yes—
485C2×C4#100712yes—
486C2×C4C24×D311yescharacteristic
487C8—21no—
488C8—21no—
489C8—64no—
490C8C2×D3×Q812yes—
491C8C2×D3×Q812yes—
492C2×C3—22no—
493C2×C3C2×D4×Q811yescharacteristic
494C2×C3C2×#57611yescharacteristic
495C2×C3#358411yescharacteristic
496D3—48no—
497C22—21no—
498C22—21no—
499C22—68no—
500C22—124no—
501C22—124no—
502C22—248no—
503C22#572712yes—
504C22C22×D3×D411yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—44no—
509C4—44no—
510C4—64no—
511C4—64no—
512C4—124no—
513C4#572712yes—
514C4#576912yes—
515C4C22×D3×Q811yescharacteristic
516C4C2×#100711yescharacteristic
517C3#6137011yescharacteristic, complemented, Sylow
518C2—22no—
519C2—128no—
520C2C2×#572711yescharacteristic
521C2C2×#576911yescharacteristic
522C2#8718711yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3160C2
socleC22×C3C23×D4
prime radicalC22×D24C22
hypercenter#3160D3
hypocenterC3#61370
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
1. #86280C2semidirect
2. C3×#3160C2semidirect
3. #83350C2semidirect
4. #81953C2semidirect
5. #81952C2semidirect
6. #81931C2semidirect
7. #81912C2semidirect
8. #81905C2semidirect
9. #79197C2semidirect
10. #79173C2semidirect
13. #78445C2semidirect
17. #5730C22semidirect
18. #5727C22semidirect
19. #5690C22semidirect
29. C3×#500C22semidirect
30. C3×#500C22semidirect
31. C3×#499C22semidirect
32. C3×#499C22semidirect
33. C3×#498C22semidirect
41. #5087C22semidirect
42. #5084C22semidirect
43. #5066C22semidirect
47. #4891C22semidirect
48. #4890C22semidirect
49. #4878C22semidirect
50. #4874C22semidirect
53. #4871C22semidirect
54. #4870C22semidirect
56. #4856C22semidirect
61. #4679C22semidirect
62. #4679C22semidirect
63. C8×Dic6C22semidirect
64. C8×Dic6C22semidirect
72. #3160D3semidirect
100. C2×D24Q8semidirect
169. #500C2×D3semidirect
170. #500C2×D3semidirect
172. #499C2×D3semidirect
173. #499C2×D3semidirect
174. #498C2×D3semidirect
213. C2×D12C2×Q8semidirect
224. C2×C3×C8C2×Q8semidirect
260. D24C2×Q8semidirect
261. D24C2×Q8semidirect
400. C2×C8D3×Q8semidirect
517. C3#61370semidirect

Maximal subgroups

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

subgroupindex
#862802
C3×#31602
#833502
#819532
#819522
#819312
#819122
#819052
#791972
#791732
#791672
C2×C4×D242
#784452
#613703