Finite Group Databasebeta

← #10952003

Subgroups — #10952003

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
1D8×#67C211yesmaximal, characteristic, complemented
2#80708C211yesmaximal, characteristic, complemented
3C3×#2184C211yesmaximal, characteristic, complemented
4C2×#4562C211yesmaximal, characteristic, complemented
5#71469C211yesmaximal, characteristic, complemented
6#71467C211yesmaximal, characteristic, complemented
7#71462C211yesmaximal, characteristic, complemented
8Dic3×D16C211yesmaximal, characteristic, complemented
9#71071C211yesmaximal, characteristic, complemented
10#69199C211yesmaximal, characteristic, complemented
11D3×#357C211yesmaximal, characteristic, complemented
12#69183C211yesmaximal, characteristic, complemented
13#69178C211yesmaximal, characteristic, complemented
14#67⋊C16C211yesmaximal, characteristic, complemented
15#69095C211yesmaximal, characteristic, complemented
16#33861—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18D4×#67—21no—
19C2×D3×D8C2211yescharacteristic
20#5611—21no—
21C2×C3×D16C2211yescharacteristic
22C2×C3×D16C2211yescharacteristic, complemented
23C3×#466C2211yescharacteristic, complemented
24#5217—21no—
25#5157C2211yescharacteristic
26#5157C2211yescharacteristic, complemented
27#5156—21no—
28#5151C2211yescharacteristic
29#5150—21no—
30Dic3×D8C2211yescharacteristic
31Dic3×D8C2211yescharacteristic, complemented
32C2×#826C2211yescharacteristic, complemented
33#5115C2211yescharacteristic
34C8×#67C2211yescharacteristic
35C2×#818C2211yescharacteristic, complemented
36C22×C3⋊C16C2211yescharacteristic, complemented
37#5033—21no—
38C2×D3×C16C2211yescharacteristic, complemented
39D3×#107C2211yescharacteristic, complemented
40#4874C2211yescharacteristic, complemented
41#4774—21no—
42#4759—21no—
43C3×#357C2211yescharacteristic
44C3×#357C2211yescharacteristic, complemented
45C3×C22⋊C16C2211yescharacteristic, complemented
46#4562C2211yescharacteristic
47#4562C2211yescharacteristic
48#4562C2211yescharacteristic, complemented
49#4562C2211yescharacteristic, complemented
50#4562C2211yescharacteristic, complemented
51#4509C2211yescharacteristic, complemented
52#4509C2211yescharacteristic, complemented
53(C2×D3)⋊C16C2211yescharacteristic, complemented
54#4504C2211yescharacteristic, complemented
55#4504C2211yescharacteristic, complemented
56Dic3⋊C16C2211yescharacteristic, complemented
57Dic3×C16C2211yescharacteristic, complemented
58#4491C2211yescharacteristic, 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#2215—31no—
68#2206—31no—
69#2184—31no—
70#2184D311yescharacteristic, complemented
71C4×D16—31no—
72D4×C16—31no—
73C2×#357—31no—
74C2×#357—31no—
75C22×C3×D4—21no—
76C22×#67—41no—
77C2×#237—41no—
78C2×D3×D4—21no—
79C2×C3×D8—22no—
80C2×C3×D8—21no—
81C2×C3×D8—41no—
82C2×C3×D8C2311yescharacteristic
83C2×C3×D8C2311yescharacteristic
84C2×C3×D8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#121—21no—
87#953—41no—
88#940—21no—
89#939—41no—
90#938—21no—
91#938—21no—
92#937—41no—
93#935—41no—
94D4×Dic3—21no—
95D4×Dic3—21no—
96C2×#213—21no—
97C2×#213—21no—
98#930—21no—
99C4×#67D411yescharacteristic
100C2×C4⋊Dic3—21no—
101D3×D8—22no—
102D3×D8—41no—
103C2×D24C2311yescharacteristic
104C2×D3×C8C2311yescharacteristic
105D3×C4⋊C4—21no—
106#888—21no—
107#883—41no—
108#878—41no—
109C3×D16—22no—
110C3×D16—41no—
111C2×C3×C16C2311yescharacteristic
112C2×C3×C16C2311yescharacteristic
113C3×#107C2311yescharacteristic
114C3×#102—21no—
115#832—21no—
116#832—21no—
117#832—21no—
118#832—21no—
119#830D411yescharacteristic
120#826—41no—
121#821—21no—
122(C2×D3)⋊C8D411yescharacteristic
123#818—21no—
124#818C2311yescharacteristic
125#818C2311yescharacteristic
126#818C2311yescharacteristic
127Dic3⋊C8D411yescharacteristic
128C8×Dic3C2311yescharacteristic
129C2×C3⋊C16—41no—
130C2×C3⋊C16C2311yescharacteristic
131C2×C3⋊C16C2311yescharacteristic
132#807—21no—
133#807—21no—
134D3×C16—41no—
135C22×D8—31no—
136C22×D8C2×D311yescharacteristic
137#547—61no—
138D42—61no—
139C2×D16—31no—
140C2×D16C2×D311yescharacteristic
141C2×D16C2×D311yescharacteristic, complemented
142C22×C16—31no—
143C22×C16—31no—
144#500—31no—
145#493—31no—
146#480—61no—
147#480—61no—
148#474—61no—
149#466—31no—
150#466—31no—
151#466—31no—
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#366—31no—
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
174#357C2×D311yescharacteristic, complemented
175C22⋊C16—31no—
176C22⋊C16C2×D311yescharacteristic, complemented
177C4×C16—31no—
178C24×C3—41no—
179C23×D3—41no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—42no—
184C2×C3×D4—41no—
185C2×C3×D4—41no—
186C2×C3×D4—81no—
187C22×C3×C4C2×D411yescharacteristic
188C2×#67—41no—
189C2×#67—41no—
190C2×#67—41no—
191C2×#67—41no—
192C2×#67—41no—
193C2×#67—41no—
194C2×#67—84no—
195C2×#67D811yescharacteristic
196C22×Dic3—41no—
197C22×Dic3—41no—
198C22×Dic3—41no—
199#237—84no—
200D3×D4—42no—
201D3×D4—81no—
202C2×D12—21no—
203C2×C4×D3—41no—
204C2×C4×D3C2×D411yescharacteristic
205C2×Dic6—41no—
206C3×D8—21no—
207C3×D8—21no—
208C3×D8—21no—
209C3×D8—21no—
210C3×D8—21no—
211C3×D8—42no—
212C3×D8#4112yescomplemented
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8C2×D411yescharacteristic
216C2×C3×C8C2411yescharacteristic
217C3×C4⋊C4—21no—
218C3×C22⋊C4—41no—
219#217—41no—
220#217—41no—
221#217—41no—
222#217—41no—
223#217D811yescharacteristic
224#213—42no—
225#213—81no—
226(C2×D3)⋊C4—41no—
227(C2×D3)⋊C4D811yescharacteristic
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—41no—
232Dic3⋊C4—41no—
233Dic3⋊C4—41no—
234Dic3⋊C4D811yescharacteristic
235C4×Dic3C2×D411yescharacteristic
236C2×C3⋊C8C2×D411yescharacteristic
237C2×C3⋊C8C2×D411yescharacteristic
238D24—21no—
239D24—21no—
240D3×C8—21no—
241D3×C8—21no—
242C3×C16—21no—
243C3×C16—21no—
244C3⋊C16—21no—
245C3⋊C16—21no—
246C2×#41—121no—
247C22×D4—21no—
248C22×D4—61no—
249C22×D4—121no—
250C2×D8—22no—
251C2×D8—21no—
252C2×D8—31no—
253C2×D8—41no—
254C2×D8—61no—
255C2×D8—61no—
256C2×D8—62no—
257C2×D8—121no—
258C2×D8C22×D311yescharacteristic
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—61no—
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—22no—
281D16—41no—
282D16—121no—
283C2×C16—31no—
284C2×C16—31no—
285C2×C16—121no—
286C2×C16—121no—
287C2×C16C22×D311yescharacteristic
288C2×C16C22×D311yescharacteristic
289#107—31no—
290#107—31no—
291#107—31no—
292#107—61no—
293#107—61no—
294#107—61no—
295#107C22×D311yescharacteristic
296C4⋊C8—31no—
297#102—21no—
298#102—61no—
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×C3—41no—
309C23×C3—82no—
310C23×C3—82no—
311C23×C3—161no—
312C23×C3C2×D811yescharacteristic
313C22×D3—41no—
314C22×D3—82no—
315C22×D3—82no—
316C22×D3—161no—
317C22×D3C2×D811yescharacteristic
318C3×D4—22no—
319C3×D4—41no—
320C3×D4—41no—
321C3×D4—41no—
322C3×D4—41no—
323C3×D4—41no—
324C3×D4—82no—
325C3×D4—82no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4C2×D811yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331#67—82no—
332#67—82no—
333#67—82no—
334#67—82no—
335#67—82no—
336#67—161no—
337#67D1614yescomplemented
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—82no—
342C2×Dic3—82no—
343C2×Dic3—82no—
344C2×Dic3C2×D811yescharacteristic
345C2×Dic3C2×D811yescharacteristic
346C2×Dic3C2×D811yescharacteristic
347D12—41no—
348D12—41no—
349C4×D3—21no—
350C4×D3—21no—
351C4×D3—82no—
352Dic6—82no—
353C3×C8—21no—
354C3×C8C2×#4111yescharacteristic
355C3×C8H(5,𝔽2)11yescharacteristic
356C3⋊C8—21no—
357C3⋊C8—21no—
358C24—41no—
359C24—121no—
360#41—244no—
361C2×Q8—121no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—31no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—61no—
370C2×D4—81no—
371C2×D4—121no—
372C2×D4—121no—
373C2×D4—121no—
374C2×D4—122no—
375C2×D4—121no—
376C2×D4—121no—
377C2×D4—121no—
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—21no—
391D8—42no—
392D8—61no—
393D8—61no—
394D8—122no—
395D8—241no—
396D8#23712yescomplemented
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—31no—
407C4⋊C4—61no—
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—121no—
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—42no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—164no—
438C22×C3C2×D1612yes—
439C22×C3C22×D811yescharacteristic
440C2×D3—21no—
441C2×D3—21no—
442C2×D3—81no—
443C2×D3—81no—
444C2×D3—81no—
445C2×D3—164no—
446C2×D3C2×D1612yes—
447C3×C4—21no—
448C3×C4—81no—
449C3×C4#54611yescharacteristic
450C3×C4#57711yescharacteristic
451Dic3—21no—
452Dic3—21no—
453Dic3—81no—
454Dic3—81no—
455Dic3—81no—
456Dic3C2×D1612yes—
457C23—31no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—82no—
462C23—82no—
463C23—121no—
464C23—161no—
465C23—242no—
466C23—242no—
467C23—481no—
468C23D3×D811yescharacteristic
469Q8—242no—
470D4—22no—
471D4—34no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—82no—
478D4—82no—
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—61no—
494C2×C4—61no—
495C2×C4—121no—
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—
507C8C2×#23711yescharacteristic
508C8#100411yescharacteristic
509C2×C3—22no—
510C2×C3—82no—
511C2×C3—161no—
512C2×C3—161no—
513C2×C3#329411yescharacteristic
514C2×C3C22×D1611yescharacteristic
515C2×C3#341611yescharacteristic
516D3—22no—
517D3—161no—
518C22—21no—
519C22—21no—
520C22—32no—
521C22—42no—
522C22—61no—
523C22—61no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—164no—
533C22—241no—
534C22—241no—
535C22—241no—
536C22—484no—
537C22D3×D1612yes—
538C22C2×D3×D811yescharacteristic
539C4—21no—
540C4—32no—
541C4—61no—
542C4—61no—
543C4—81no—
544C4—241no—
545C4—241no—
546C4—241no—
547C4#559111yescharacteristic
548C4#577511yescharacteristic
549C3#3386111yescharacteristic, complemented, Sylow
550C2—22no—
551C2—62no—
552C2—82no—
553C2—161no—
554C2—161no—
555C2—481no—
556C2#8001411yescharacteristic
557C2C2×D3×D1611yescharacteristic
558C2#8203711yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2184C2
socleC22×C3C22×D8
hypercenter#2184D3
hypocenterC3#33861
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2184

Complements

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

NH ≅ G/Ntype
1. D8×#67C2semidirect
2. #80708C2semidirect
3. C3×#2184C2semidirect
4. C2×#4562C2semidirect
5. #71469C2semidirect
6. #71467C2semidirect
7. #71462C2semidirect
8. Dic3×D16C2semidirect
9. #71071C2semidirect
10. #69199C2semidirect
11. D3×#357C2semidirect
12. #69183C2semidirect
13. #69178C2semidirect
14. #67⋊C16C2semidirect
15. #69095C2semidirect
22. C2×C3×D16C22semidirect
23. C3×#466C22semidirect
26. #5157C22semidirect
31. Dic3×D8C22semidirect
32. C2×#826C22semidirect
35. C2×#818C22semidirect
36. C22×C3⋊C16C22semidirect
38. C2×D3×C16C22semidirect
39. D3×#107C22semidirect
40. #4874C22semidirect
44. C3×#357C22semidirect
45. C3×C22⋊C16C22semidirect
48. #4562C22semidirect
49. #4562C22semidirect
50. #4562C22semidirect
51. #4509C22semidirect
52. #4509C22semidirect
53. (C2×D3)⋊C16C22semidirect
54. #4504C22semidirect
55. #4504C22semidirect
56. Dic3⋊C16C22semidirect
57. Dic3×C16C22semidirect
58. #4491C22semidirect
59. #4489C22semidirect
70. #2184D3semidirect
141. C2×D16C2×D3semidirect
152. #466C2×D3semidirect
174. #357C2×D3semidirect
176. C22⋊C16C2×D3semidirect
212. C3×D8#41semidirect
337. #67D16semidirect
396. D8#237semidirect
549. C3#33861semidirect

Maximal subgroups

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

subgroupindex
D8×#672
#807082
C3×#21842
C2×#45622
#714692
#714672
#714622
Dic3×D162
#710712
#691992
D3×#3572
#691832
#691782
#67⋊C162
#690952
#338613