Finite Group Databasebeta

← #10952005

Subgroups — #10952005

Order 768.

654 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#80729C211yesmaximal, characteristic, complemented
2#80708C211yesmaximal, characteristic, complemented
3C3×#2215C211yesmaximal, characteristic, complemented
4#71467C211yesmaximal, characteristic, complemented
5#71462C211yesmaximal, characteristic, complemented
6C2×#4509C211yesmaximal, characteristic, complemented
7C16×#67C211yesmaximal, characteristic, complemented
8#71075C211yesmaximal, characteristic, complemented
9#71068C211yesmaximal, characteristic, complemented
10#69333C211yesmaximal, characteristic, complemented
11#69330C211yesmaximal, characteristic, complemented
12#69195C211yesmaximal, characteristic, complemented
13D3×#357C211yesmaximal, characteristic, complemented
14#69183C211yesmaximal, characteristic, complemented
15#69174C211yesmaximal, characteristic, complemented
16#33861—31nomaximal, Sylow
17C2×D3×D8C2211yescharacteristic
18C22×D24C2211yescharacteristic
19#5611—21no—
20#5609—21no—
21C2×C3×D16C2211yescharacteristic, complemented
22C22×C3×C16C2211yescharacteristic, complemented
23C3×#466C2211yescharacteristic
24C3×#466C2211yescharacteristic, complemented
25#5157C2211yescharacteristic, complemented
26#5151C2211yescharacteristic
27Dic3×D8C2211yescharacteristic, complemented
28C2×#826C2211yescharacteristic
29C2×#826C2211yescharacteristic, complemented
30C8×#67C2211yescharacteristic
31C2×#818C2211yescharacteristic, complemented
32#5036—21no—
33#5033—21no—
34C2×D3×C16C2211yescharacteristic, complemented
35#4883C2211yescharacteristic
36D3×#107C2211yescharacteristic, complemented
37#4874C2211yescharacteristic, complemented
38#4872C2211yescharacteristic
39#4786—21no—
40#4774—21no—
41#4773—21no—
42#4759—21no—
43C3×#366C2211yescharacteristic, complemented
44C3×#357C2211yescharacteristic, complemented
45C3×#357C2211yescharacteristic, complemented
46#4562C2211yescharacteristic, complemented
47#4562C2211yescharacteristic, complemented
48#4549C2211yescharacteristic, complemented
49#4509C2211yescharacteristic
50#4509C2211yescharacteristic
51#4509C2211yescharacteristic, complemented
52#4509C2211yescharacteristic, complemented
53(C2×D3)⋊C16C2211yescharacteristic, complemented
54Dic3⋊C16C2211yescharacteristic, complemented
55Dic3×C16C2211yescharacteristic, complemented
56#4491C2211yescharacteristic
57#4491C2211yescharacteristic, complemented
58#4489C2211yescharacteristic, complemented
59#4489C2211yescharacteristic, complemented
60#3334—31no—
61D4×D8—31no—
62#2240—31no—
63#2232—31no—
64#2232—31no—
65#2224—31no—
66#2215—31no—
67#2215D311yescharacteristic, complemented
68#2206—31no—
69#2184—31no—
70#2184—31no—
71C4×D16—31no—
72D4×C16—31no—
73C2×#357—31no—
74C2×#357—31no—
75C2×#237—41no—
76C2×D3×D4—21no—
77C2×D3×D4—41no—
78C22×D12—21no—
79C2×C3×D8C2311yescharacteristic
80C2×C3×D8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×#121—21no—
83C3×#121—21no—
84#940—21no—
85#938—21no—
86#937—41no—
87#935—41no—
88D4×Dic3—21no—
89C2×#213—21no—
90C2×#213—21no—
91C4×#67D411yescharacteristic
92C2×C4⋊Dic3—21no—
93D3×D8—22no—
94D3×D8—41no—
95C2×D24—22no—
96C2×D24—21no—
97C2×D24—41no—
98C2×D24C2311yescharacteristic
99C2×D24C2311yescharacteristic
100C2×D3×C8C2311yescharacteristic
101#895—21no—
102#893—21no—
103D3×C4⋊C4—21no—
104#888—21no—
105#884—41no—
106#883—41no—
107#882—41no—
108#878—41no—
109C3×D16—41no—
110C2×C3×C16—41no—
111C2×C3×C16C2311yescharacteristic
112C2×C3×C16C2311yescharacteristic
113C3×#107C2311yescharacteristic
114C3×#107C2311yescharacteristic
115C3×#102—21no—
116C3×#102—21no—
117#832—21no—
118#832—21no—
119#830D411yescharacteristic
120#826—22no—
121#826—41no—
122(C2×D3)⋊C8D411yescharacteristic
123#818—21no—
124#818C2311yescharacteristic
125#818C2311yescharacteristic
126Dic3⋊C8D411yescharacteristic
127C8×Dic3C2311yescharacteristic
128C2×C3⋊C16C2311yescharacteristic
129C2×C3⋊C16C2311yescharacteristic
130#809—21no—
131#809—21no—
132#807—21no—
133#807—21no—
134D3×C16—41no—
135C22×D8—31no—
136C22×D8—31no—
137#547—61no—
138D42—61no—
139C2×D16—31no—
140C2×D16—31no—
141C2×D16C2×D311yescharacteristic, complemented
142C22×C16—31no—
143C22×C16C2×D311yescharacteristic, complemented
144#500—31no—
145#493—31no—
146#480—61no—
147#480—61no—
148#474—61no—
149#466—31no—
150#466—31no—
151#466C2×D311yescharacteristic
152#466C2×D311yescharacteristic, complemented
153#459—61no—
154#447—61no—
155#447—61no—
156C4×D8—31no—
157C4×D8—31no—
158C8×D4—31no—
159C2×#107—31no—
160C2×#107—31no—
161#366—31no—
162#366—31no—
163#366C2×D311yescharacteristic, complemented
164C4⋊C16—31no—
165#357—31no—
166#357—31no—
167#357—31no—
168#357—31no—
169#357—31no—
170#357—31no—
171#357—31no—
172#357—31no—
173#357C2×D311yescharacteristic, complemented
174#357C2×D311yescharacteristic, complemented
175C22⋊C16—31no—
176C22⋊C16—31no—
177C4×C16—31no—
178C23×D3—41no—
179C23×D3—41no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—41no—
183C2×C3×D4—41no—
184C22×C3×C4C2×D411yescharacteristic
185C2×#67—41no—
186C2×#67—41no—
187C2×#67—41no—
188C2×#67—41no—
189C2×#67D811yescharacteristic
190C22×Dic3—41no—
191C22×Dic3—41no—
192#237—84no—
193D3×D4—42no—
194D3×D4—81no—
195D3×D4—84no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—42no—
199C2×D12—41no—
200C2×D12—41no—
201C2×D12—81no—
202C2×C4×D3—41no—
203C2×C4×D3—41no—
204C2×C4×D3C2×D411yescharacteristic
205C2×Dic6—41no—
206C3×D8—21no—
207C3×D8—21no—
208C3×D8—21no—
209C3×D8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8C2×D411yescharacteristic
213C2×C3×C8C2411yescharacteristic
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C22⋊C4—41no—
217C3×C22⋊C4—41no—
218#217—41no—
219#217—41no—
220#217D811yescharacteristic
221#213—42no—
222#213—81no—
223(C2×D3)⋊C4—41no—
224(C2×D3)⋊C4—41no—
225(C2×D3)⋊C4D811yescharacteristic
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—41no—
229Dic3⋊C4—41no—
230Dic3⋊C4—41no—
231Dic3⋊C4D811yescharacteristic
232C4×Dic3C2×D411yescharacteristic
233C2×C3⋊C8C2×D411yescharacteristic
234C2×C3⋊C8C2×D411yescharacteristic
235D24—21no—
236D24—21no—
237D24—21no—
238D24—42no—
239D24#4112yescomplemented
240D3×C8—21no—
241D3×C8—21no—
242C3×C16—21no—
243C3×C16—21no—
244C3⋊C16—21no—
245C3⋊C16—21no—
246C2×#41—121no—
247C22×D4—61no—
248C22×D4—61no—
249C22×D4—121no—
250C2×D8—31no—
251C2×D8—31no—
252C2×D8—62no—
253C2×D8—61no—
254C2×D8—61no—
255C2×D8—62no—
256C2×D8—61no—
257C2×D8—121no—
258C2×D8—121no—
259C2×D8C22×D311yescharacteristic
260C2×D8C22×D311yescharacteristic
261C22×C8—31no—
262C22×C8C22×D311yescharacteristic
263C4⋊Q8—61no—
264#127—61no—
265#123—121no—
266#123—121no—
267C22⋊Q8—121no—
268#121—21no—
269#121—21no—
270#121—61no—
271#121—61no—
272#121—121no—
273#120—121no—
274#120—121no—
275C4×D4—31no—
276C4×D4—61no—
277C4×D4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280D16—41no—
281D16—62no—
282D16—121no—
283C2×C16—31no—
284C2×C16—31no—
285C2×C16—41no—
286C2×C16—121no—
287C2×C16C22×D311yescharacteristic
288C2×C16C22×D311yescharacteristic
289#107—31no—
290#107—31no—
291#107—61no—
292#107—61no—
293#107—61no—
294#107C22×D311yescharacteristic
295#107C22×D311yescharacteristic
296C4⋊C8—31no—
297#102—21no—
298#102—21no—
299#102—61no—
300#102—61no—
301#102—61no—
302#102—61no—
303C22⋊C8—31no—
304C22⋊C8—31no—
305C4×C8—31no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3C2×D811yescharacteristic
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—82no—
312C22×D3—82no—
313C22×D3—82no—
314C22×D3—82no—
315C22×D3—161no—
316C22×D3—161no—
317C22×D3C2×D811yescharacteristic
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—41no—
321C3×D4—41no—
322C3×D4—82no—
323C3×D4—82no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4C2×D811yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330#67—82no—
331#67—82no—
332#67—82no—
333#67—161no—
334#67D1614yescomplemented
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—82no—
338C2×Dic3—82no—
339C2×Dic3C2×D811yescharacteristic
340C2×Dic3C2×D811yescharacteristic
341C2×Dic3C2×D811yescharacteristic
342D12—22no—
343D12—41no—
344D12—41no—
345D12—41no—
346D12—82no—
347D12—82no—
348C4×D3—21no—
349C4×D3—21no—
350C4×D3—82no—
351C4×D3—82no—
352Dic6—82no—
353C3×C8—21no—
354C3×C8C2×#4111yescharacteristic
355C3×C8H(5,𝔽2)11yescharacteristic
356C3⋊C8—21no—
357C3⋊C8—21no—
358C24—121no—
359C24—121no—
360#41—244no—
361C2×Q8—121no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—31no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—122no—
370C2×D4—121no—
371C2×D4—121no—
372C2×D4—121no—
373C2×D4—121no—
374C2×D4—122no—
375C2×D4—121no—
376C2×D4—121no—
377C2×D4—241no—
378C2×D4—241no—
379C2×D4—244no—
380C22×C4—31no—
381C22×C4—121no—
382C22×C4—121no—
383C22×C4—121no—
384C22×C4—121no—
385C22×C4D3×D411yescharacteristic
386D8—21no—
387D8—21no—
388D8—21no—
389D8—21no—
390D8—32no—
391D8—61no—
392D8—61no—
393D8—61no—
394D8—122no—
395D8—122no—
396D8—241no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—31no—
400C2×C8—31no—
401C2×C8—61no—
402C2×C8—61no—
403C2×C8D3×D411yescharacteristic
404C2×C8C23×D311yescharacteristic
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—31no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—121no—
411C4⋊C4—121no—
412C4⋊C4—121no—
413C22⋊C4—31no—
414C22⋊C4—31no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—121no—
418C22⋊C4—121no—
419C22⋊C4—121no—
420C22⋊C4—121no—
421C42—31no—
422C16—21no—
423C16—21no—
424C16—61no—
425C16—61no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3C2×D1612yes—
435C22×C3C22×D811yescharacteristic
436C2×D3—21no—
437C2×D3—21no—
438C2×D3—42no—
439C2×D3—81no—
440C2×D3—81no—
441C2×D3—81no—
442C2×D3—81no—
443C2×D3—81no—
444C2×D3—164no—
445C2×D3—164no—
446C2×D3C2×D1612yes—
447C3×C4—21no—
448C3×C4—81no—
449C3×C4—81no—
450C3×C4#54611yescharacteristic
451C3×C4#57711yescharacteristic
452Dic3—21no—
453Dic3—21no—
454Dic3—81no—
455Dic3—81no—
456Dic3C2×D1612yes—
457C23—31no—
458C23—41no—
459C23—41no—
460C23—121no—
461C23—121no—
462C23—242no—
463C23—242no—
464C23—242no—
465C23—242no—
466C23—481no—
467C23—481no—
468C23D3×D811yescharacteristic
469Q8—242no—
470D4—34no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—62no—
476D4—82no—
477D4—82no—
478D4—121no—
479D4—121no—
480D4—121no—
481D4—242no—
482D4—242no—
483D4—242no—
484D4—242no—
485D4—242no—
486D4—481no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—31no—
490C2×C4—31no—
491C2×C4—31no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—242no—
499C2×C4—242no—
500C2×C4—242no—
501C2×C4—242no—
502C2×C4D3×D811yescharacteristic
503C2×C4C2×D3×D411yescharacteristic
504C8—21no—
505C8—61no—
506C8—61no—
507C8#100411yescharacteristic
508C8D3×#4111yescharacteristic
509C2×C3—22no—
510C2×C3—161no—
511C2×C3—161no—
512C2×C3#329411yescharacteristic
513C2×C3C22×D1611yescharacteristic
514C2×C3#341611yescharacteristic
515D3—22no—
516D3—82no—
517D3—161no—
518C22—21no—
519C22—21no—
520C22—32no—
521C22—61no—
522C22—61no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—122no—
530C22—241no—
531C22—241no—
532C22—241no—
533C22—241no—
534C22—241no—
535C22—484no—
536C22—484no—
537C22D3×D1612yes—
538C22C2×D3×D811yescharacteristic
539C4—21no—
540C4—32no—
541C4—61no—
542C4—61no—
543C4—81no—
544C4—81no—
545C4—241no—
546C4—241no—
547C4#561011yescharacteristic
548C4#577011yescharacteristic
549C3#3386111yescharacteristic, complemented, Sylow
550C2—22no—
551C2—62no—
552C2—161no—
553C2—161no—
554C2—242no—
555C2—481no—
556C2#8073111yescharacteristic
557C2C2×D3×D1611yescharacteristic
558C2#8203211yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2215C2
socleC22×C3C22×D8
hypercenter#2215D3
hypocenterC3#33861
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2215

Complements

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

NH ≅ G/Ntype
1. #80729C2semidirect
2. #80708C2semidirect
3. C3×#2215C2semidirect
4. #71467C2semidirect
5. #71462C2semidirect
6. C2×#4509C2semidirect
7. C16×#67C2semidirect
8. #71075C2semidirect
9. #71068C2semidirect
10. #69333C2semidirect
11. #69330C2semidirect
12. #69195C2semidirect
13. D3×#357C2semidirect
14. #69183C2semidirect
15. #69174C2semidirect
21. C2×C3×D16C22semidirect
22. C22×C3×C16C22semidirect
24. C3×#466C22semidirect
25. #5157C22semidirect
27. Dic3×D8C22semidirect
29. C2×#826C22semidirect
31. C2×#818C22semidirect
34. C2×D3×C16C22semidirect
36. D3×#107C22semidirect
37. #4874C22semidirect
43. C3×#366C22semidirect
44. C3×#357C22semidirect
45. C3×#357C22semidirect
46. #4562C22semidirect
47. #4562C22semidirect
48. #4549C22semidirect
51. #4509C22semidirect
52. #4509C22semidirect
53. (C2×D3)⋊C16C22semidirect
54. Dic3⋊C16C22semidirect
55. Dic3×C16C22semidirect
57. #4491C22semidirect
58. #4489C22semidirect
59. #4489C22semidirect
67. #2215D3semidirect
141. C2×D16C2×D3semidirect
143. C22×C16C2×D3semidirect
152. #466C2×D3semidirect
163. #366C2×D3semidirect
173. #357C2×D3semidirect
174. #357C2×D3semidirect
239. D24#41semidirect
334. #67D16semidirect
549. C3#33861semidirect

Maximal subgroups

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

subgroupindex
#807292
#807082
C3×#22152
#714672
#714622
C2×#45092
C16×#672
#710752
#710682
#693332
#693302
#691952
D3×#3572
#691832
#691742
#338613