Finite Group Databasebeta

← #10952016

Subgroups — #10952016

Order 768.

622 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#80753C211yesmaximal, characteristic, complemented
2#80729C211yesmaximal, characteristic
3C3×#2217C211yesmaximal, characteristic, complemented
4#71485C211yesmaximal, characteristic, complemented
5#71475C211yesmaximal, characteristic, complemented
6C2×#4509C211yesmaximal, characteristic, complemented
7C16×#67C211yesmaximal, characteristic, complemented
8#71085C211yesmaximal, characteristic, complemented
9#71069C211yesmaximal, characteristic, complemented
10#69353C211yesmaximal, characteristic, complemented
11#69350C211yesmaximal, characteristic, complemented
12#69228C211yesmaximal, characteristic, complemented
13#69206C211yesmaximal, characteristic, complemented
14D3×#357C211yesmaximal, characteristic, complemented
15#69183C211yesmaximal, characteristic, complemented
16#33862—31nomaximal, Sylow
17C2×D3×D8C2211yescharacteristic
18C22×D24C2211yescharacteristic
19#5635—21no—
20#5609—21no—
21C2×C3×SD32C2211yescharacteristic, complemented
22C22×C3×C16C2211yescharacteristic, complemented
23C3×#467C2211yescharacteristic, complemented
24C3×#466C2211yescharacteristic
25#5188C2211yescharacteristic, complemented
26Dic3×Q16C2211yescharacteristic, complemented
27C2×#828C2211yescharacteristic
28C2×#828C2211yescharacteristic, complemented
29#5151C2211yescharacteristic
30C8×#67C2211yescharacteristic
31C2×#818C2211yescharacteristic, complemented
32#5044—21no—
33#5036—21no—
34C2×D3×C16C2211yescharacteristic, complemented
35#4883C2211yescharacteristic
36D3×#107C2211yescharacteristic, complemented
37#4874C2211yescharacteristic, complemented
38#4872C2211yescharacteristic
39#4811—21no—
40#4793—21no—
41#4786—21no—
42#4773—21no—
43C3×#367C2211yescharacteristic, complemented
44C3×#358C2211yescharacteristic, 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#4490C2211yescharacteristic, complemented
59#4490C2211yescharacteristic, complemented
60#3338—31no—
61D4×D8—31no—
62#2240—31no—
63#2233—31no—
64#2233—31no—
65#2226—31no—
66#2217—31no—
67#2217D311yescharacteristic, complemented
68#2207—31no—
69#2187—31no—
70#2187—31no—
71C4×SD32—31no—
72D4×C16—31no—
73C2×#357—31no—
74C2×#357—31no—
75C2×D3×D4—21no—
76C2×D3×D4—41no—
77C22×D12—21no—
78C2×C3×Q16C2311yescharacteristic
79C2×C3×D8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C22⋊Q8—21no—
82C3×#121—21no—
83#946—21no—
84Q8×Dic3—21no—
85#940—21no—
86#937—41no—
87C2×#213—21no—
88C2×#213—21no—
89C4×#67D411yescharacteristic
90C2×C4⋊Dic3—21no—
91D3×D8—22no—
92D3×D8—41no—
93C2×D24—22no—
94C2×D24—21no—
95C2×D24—41no—
96C2×D24C2311yescharacteristic
97C2×D24C2311yescharacteristic
98C2×D3×C8C2311yescharacteristic
99#897—41no—
100#895—21no—
101#893—21no—
102D3×C4⋊C4—21no—
103#889—41no—
104#888—21no—
105#884—41no—
106#882—41no—
107#67⋊C4—41no—
108#878—41no—
109C3×SD32—41no—
110C2×C3×C16—41no—
111C2×C3×C16C2311yescharacteristic
112C2×C3×C16C2311yescharacteristic
113C3×#107C2311yescharacteristic
114C3×#107C2311yescharacteristic
115C3×#103—21no—
116C3×#102—21no—
117#835—21no—
118#835—21no—
119#830D411yescharacteristic
120#828—22no—
121#828—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#554—61no—
138D42—61no—
139C2×SD32—31no—
140C2×SD32—31no—
141C2×SD32C2×D311yescharacteristic, complemented
142C22×C16—31no—
143C22×C16C2×D311yescharacteristic, complemented
144#500—31no—
145#493—31no—
146#484—61no—
147#484—61no—
148#477—61no—
149#467—31no—
150#467C2×D311yescharacteristic, complemented
151#466—31no—
152#466C2×D311yescharacteristic
153#459—61no—
154#447—61no—
155#447—61no—
156C4×Q16—31no—
157C4×D8—31no—
158C8×D4—31no—
159C2×#107—31no—
160C2×#107—31no—
161#367—31no—
162#367—31no—
163#367C2×D311yescharacteristic, complemented
164C4⋊C16—31no—
165#358C2×D311yescharacteristic, complemented
166#357—31no—
167#357—31no—
168#357—31no—
169#357—31no—
170#357—31no—
171#357—31no—
172#357—31no—
173#357—31no—
174#357C2×D311yescharacteristic, complemented
175C22⋊C16—31no—
176C22⋊C16—31no—
177C4×C16—31no—
178C23×D3—41no—
179C23×D3—41no—
180C2×C3×Q8—21no—
181C2×C3×D4—21no—
182C2×C3×D4—41no—
183C22×C3×C4C2×D411yescharacteristic
184C2×#67—41no—
185C2×#67—41no—
186C2×#67—41no—
187C2×#67D811yescharacteristic
188C22×Dic3—41no—
189D3×D4—42no—
190D3×D4—81no—
191D3×D4—84no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—42no—
195C2×D12—41no—
196C2×D12—41no—
197C2×D12—81no—
198C2×C4×D3—41no—
199C2×C4×D3—41no—
200C2×C4×D3C2×D411yescharacteristic
201C2×Dic6—41no—
202C3×Q16—21no—
203C3×Q16—21no—
204C3×D8—21no—
205C3×D8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8C2×D411yescharacteristic
209C2×C3×C8C2411yescharacteristic
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—41no—
213C3×C22⋊C4—41no—
214C3×C22⋊C4—41no—
215#217D811yescharacteristic
216#213—42no—
217#213—81no—
218(C2×D3)⋊C4—41no—
219(C2×D3)⋊C4—41no—
220(C2×D3)⋊C4—41no—
221(C2×D3)⋊C4D811yescharacteristic
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—41no—
225C4⋊Dic3—41no—
226Dic3⋊C4—41no—
227Dic3⋊C4—41no—
228Dic3⋊C4—41no—
229Dic3⋊C4D811yescharacteristic
230C4×Dic3—41no—
231C4×Dic3C2×D411yescharacteristic
232C2×C3⋊C8C2×D411yescharacteristic
233C2×C3⋊C8C2×D411yescharacteristic
234D24—21no—
235D24—21no—
236D24—21no—
237D24—42no—
238D24#4112yes—
239D3×C8—21no—
240D3×C8—21no—
241C3×C16—21no—
242C3×C16—21no—
243C3⋊C16—21no—
244C3⋊C16—21no—
245C22×D4—61no—
246C22×D4—61no—
247C22×D4—121no—
248C2×Q16C22×D311yescharacteristic
249C2×D8—31no—
250C2×D8—31no—
251C2×D8—62no—
252C2×D8—61no—
253C2×D8—61no—
254C2×D8—62no—
255C2×D8—61no—
256C2×D8—121no—
257C2×D8—121no—
258C2×D8C22×D311yescharacteristic
259C22×C8—31no—
260C22×C8C22×D311yescharacteristic
261C4⋊Q8—61no—
262#127—61no—
263#125—121no—
264C22⋊Q8—21no—
265C22⋊Q8—61no—
266C22⋊Q8—121no—
267C22⋊Q8—121no—
268#121—21no—
269#121—61no—
270#121—121no—
271#120—121no—
272#120—121no—
273C4×Q8—61no—
274C4×D4—31no—
275C4×D4—61no—
276C4×D4—121no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279SD32—41no—
280SD32—62no—
281SD32—121no—
282C2×C16—31no—
283C2×C16—31no—
284C2×C16—41no—
285C2×C16—121no—
286C2×C16C22×D311yescharacteristic
287C2×C16C22×D311yescharacteristic
288#107—31no—
289#107—31no—
290#107—61no—
291#107—61no—
292#107—61no—
293#107C22×D311yescharacteristic
294#107C22×D311yescharacteristic
295C4⋊C8—31no—
296#103—21no—
297#103—61no—
298#103—61no—
299#102—21no—
300#102—61no—
301#102—61no—
302C22⋊C8—31no—
303C22⋊C8—31no—
304C4×C8—31no—
305C23×C3—41no—
306C23×C3C2×D811yescharacteristic
307C22×D3—41no—
308C22×D3—41no—
309C22×D3—82no—
310C22×D3—82no—
311C22×D3—82no—
312C22×D3—82no—
313C22×D3—161no—
314C22×D3—161no—
315C22×D3C2×D811yescharacteristic
316C3×Q8—41no—
317C3×Q8—41no—
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—82no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4C2×D811yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328#67—82no—
329#67—82no—
330#67—161no—
331#67SD3214yescomplemented
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—161no—
335C2×Dic3C2×D811yescharacteristic
336C2×Dic3C2×D811yescharacteristic
337C2×Dic3C2×D811yescharacteristic
338D12—22no—
339D12—41no—
340D12—41no—
341D12—41no—
342D12—82no—
343D12—82no—
344C4×D3—21no—
345C4×D3—21no—
346C4×D3—82no—
347C4×D3—161no—
348Dic6—161no—
349C3×C8—21no—
350C3×C8C2×#4111yescharacteristic
351C3×C8H(5,𝔽2)11yescharacteristic
352C3⋊C8—21no—
353C3⋊C8—21no—
354C24—121no—
355C24—121no—
356C2×Q8—21no—
357C2×Q8—121no—
358C2×D4—21no—
359C2×D4—31no—
360C2×D4—41no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—122no—
364C2×D4—121no—
365C2×D4—121no—
366C2×D4—121no—
367C2×D4—121no—
368C2×D4—122no—
369C2×D4—121no—
370C2×D4—241no—
371C2×D4—241no—
372C2×D4—244no—
373C22×C4—31no—
374C22×C4—121no—
375C22×C4—121no—
376C22×C4—121no—
377C22×C4D3×D411yescharacteristic
378Q16—21no—
379Q16—21no—
380D8—21no—
381D8—21no—
382D8—32no—
383D8—61no—
384D8—61no—
385D8—61no—
386D8—122no—
387D8—122no—
388D8—241no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—31no—
392C2×C8—31no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8D3×D411yescharacteristic
396C2×C8C23×D311yescharacteristic
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—31no—
400C4⋊C4—41no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—121no—
404C4⋊C4—121no—
405C4⋊C4—121no—
406C4⋊C4—121no—
407C4⋊C4—121no—
408C22⋊C4—31no—
409C22⋊C4—31no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—121no—
413C22⋊C4—121no—
414C22⋊C4—121no—
415C42—31no—
416C42—121no—
417C16—21no—
418C16—21no—
419C16—61no—
420C16—61no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3C2×SD3212yes—
427C22×C3C22×D811yescharacteristic
428C2×D3—21no—
429C2×D3—21no—
430C2×D3—42no—
431C2×D3—81no—
432C2×D3—81no—
433C2×D3—81no—
434C2×D3—81no—
435C2×D3—81no—
436C2×D3—164no—
437C2×D3—164no—
438C2×D3C2×SD3212yes—
439C3×C4—21no—
440C3×C4—81no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4#54611yescharacteristic
444C3×C4#57711yescharacteristic
445Dic3—21no—
446Dic3—21no—
447Dic3—81no—
448Dic3—81no—
449Dic3C2×SD3212yes—
450C23—31no—
451C23—41no—
452C23—121no—
453C23—121no—
454C23—242no—
455C23—242no—
456C23—242no—
457C23—242no—
458C23—481no—
459C23—481no—
460C23D3×D811yescharacteristic
461Q8—41no—
462Q8—41no—
463Q8—481no—
464D4—34no—
465D4—41no—
466D4—41no—
467D4—62no—
468D4—82no—
469D4—121no—
470D4—121no—
471D4—121no—
472D4—242no—
473D4—242no—
474D4—242no—
475D4—242no—
476D4—481no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—31no—
480C2×C4—31no—
481C2×C4—31no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—242no—
490C2×C4—481no—
491C2×C4—481no—
492C2×C4D3×D811yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C8—21no—
495C8—61no—
496C8—61no—
497C8#100411yescharacteristic
498C8D3×#4111yescharacteristic
499C2×C3—22no—
500C2×C3—161no—
501C2×C3#329411yescharacteristic
502C2×C3C22×SD3211yescharacteristic
503C2×C3#341511yescharacteristic
504D3—22no—
505D3—82no—
506D3—161no—
507C22—21no—
508C22—21no—
509C22—32no—
510C22—61no—
511C22—61no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—122no—
516C22—241no—
517C22—241no—
518C22—241no—
519C22—241no—
520C22—241no—
521C22—484no—
522C22—484no—
523C22D3×SD3212yes—
524C22C2×D3×D811yescharacteristic
525C4—21no—
526C4—32no—
527C4—61no—
528C4—61no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—241no—
533C4—241no—
534C4#561011yescharacteristic
535C4#577011yescharacteristic
536C3#3386211yescharacteristic, complemented, Sylow
537C2—22no—
538C2—62no—
539C2—161no—
540C2—242no—
541C2—481no—
542C2#8073111yescharacteristic
543C2C2×D3×SD3211yescharacteristic
544C2#8203111yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2217C2
socleC22×C3C22×D8
prime radical#80729C2
hypercenter#2217D3
hypocenterC3#33862
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. #80753C2semidirect
3. C3×#2217C2semidirect
4. #71485C2semidirect
5. #71475C2semidirect
6. C2×#4509C2semidirect
7. C16×#67C2semidirect
8. #71085C2semidirect
9. #71069C2semidirect
10. #69353C2semidirect
11. #69350C2semidirect
12. #69228C2semidirect
13. #69206C2semidirect
14. D3×#357C2semidirect
15. #69183C2semidirect
21. C2×C3×SD32C22semidirect
22. C22×C3×C16C22semidirect
23. C3×#467C22semidirect
25. #5188C22semidirect
26. Dic3×Q16C22semidirect
28. C2×#828C22semidirect
31. C2×#818C22semidirect
34. C2×D3×C16C22semidirect
36. D3×#107C22semidirect
37. #4874C22semidirect
43. C3×#367C22semidirect
44. C3×#358C22semidirect
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. #4490C22semidirect
59. #4490C22semidirect
67. #2217D3semidirect
141. C2×SD32C2×D3semidirect
143. C22×C16C2×D3semidirect
150. #467C2×D3semidirect
163. #367C2×D3semidirect
165. #358C2×D3semidirect
174. #357C2×D3semidirect
331. #67SD32semidirect
536. C3#33862semidirect

Maximal subgroups

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

subgroupindex
#807532
#807292
C3×#22172
#714852
#714752
C2×#45092
C16×#672
#710852
#710692
#693532
#693502
#692282
#692062
D3×#3572
#691832
#338623