Finite Group Databasebeta

← #86713

Subgroups — #86713

Order 384.

1658 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1009C211yesmaximal, characteristic, complemented
2C22×#239C211yesmaximal, characteristic, complemented
3C3×#548C211yesmaximal, characteristic, complemented
4#5834C211yesmaximal, characteristic, complemented
5#5817C211yesmaximal, characteristic, complemented
6#5676C213yesmaximal, complemented
7#5674C213yesmaximal, complemented
8#5630C213yesmaximal, complemented
9#5629C213yesmaximal, complemented
10#5609C213yesmaximal, complemented
11D3×#121C213yesmaximal, complemented
12#5577C211yesmaximal, characteristic, complemented
13#5574C211yesmaximal, characteristic, complemented
14D4×D12C213yesmaximal, complemented
15#5547C213yesmaximal, complemented
16#3488—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18C22×C3×Q8C2211yescharacteristic, complemented
19#1009—28no—
20C2×#239—28no—
21C2×#239C2211yescharacteristic
22C2×#239C2213yescomplemented
23C2×#239C2213yescomplemented
24C2×#239C2211yescharacteristic, complemented
25C2×#239C2211yescharacteristic, complemented
26C2×D3×D4C2213yescomplemented
27C2×D3×D4C2213yescomplemented
28C2×D3×D4C2216yescomplemented
29C2×#235C2213yescomplemented
30C22×D12C2213yescomplemented
31C22×D12C2213yescomplemented
32C22×C4×D3C2213yescomplemented
33C3×#124C2213yescomplemented
34C3×C22⋊Q8C2213yescomplemented
35C3×#121C2213yescomplemented
36C3×C4×Q8C2211yescharacteristic, complemented
37C3×C4×D4C2213yescomplemented
38#947C2213yescomplemented
39#946C2213yescomplemented
40Q8×Dic3C2211yescharacteristic, complemented
41#940C2213yescomplemented
42#939C2213yescomplemented
43#938C2213yescomplemented
44#937C2213yescomplemented
45#930C2213yescomplemented
46#930C2213yescomplemented
47C4×#67C2213yescomplemented
48C4×#67C2213yescomplemented
49#896C2213yescomplemented
50#895C2216yescomplemented
51#895C2213yescomplemented
52#895C2213yescomplemented
53#894C2213yescomplemented
54#893C2213yescomplemented
55D3×C4⋊C4C2213yescomplemented
56#884C2213yescomplemented
57#884C2216yescomplemented
58#884C2216yescomplemented
59#883C2213yescomplemented
60#882C2216yescomplemented
61#882C2216yescomplemented
62#882C2213yescomplemented
63D3×C22⋊C4C2216yescomplemented
64#878C2211yescharacteristic, complemented
65#874C2213yescomplemented
66C4×D12C2213yescomplemented
67C4×D12C2213yescomplemented
68#872C2213yescomplemented
69C2×H(5,𝔽2)—31no—
70C22×#41—31no—
71#550—31no—
72#550—31no—
73#548—31no—
74#548D311yescharacteristic, complemented
75#546—33no—
76#546—33no—
77D42—33no—
78D42—33no—
79#537—33no—
80#535—33no—
81#525—33no—
82C2×#121—33no—
83#520—31no—
84C23×D3C2313yes—
85C23×D3C2316yes—
86C3×#41—24no—
87C2×C3×Q8—24no—
88C2×C3×Q8C2313yes—
89C2×C3×Q8C2311yescharacteristic
90C2×C3×D4C2313yes—
91C2×C3×D4C2313yes—
92C22×C3×C4C2313yes—
93C22×C3×C4C2313yes—
94C2×#67C2313yes—
95C2×#67C2313yes—
96C2×#67C2313yes—
97C2×#67C2313yes—
98C2×#67C2316yes—
99C22×Dic3C2311yescharacteristic
100#239—24no—
101#239—24no—
102#239—212no—
103#239—212no—
104#239D418yescomplemented
105D3×D4—212no—
106D3×D4—212no—
107D3×D4—412no—
108#235—212no—
109C2×D12—212no—
110C2×D12—212no—
111C2×D12C2313yes—
112C2×D12C2313yes—
113C2×D12C2316yes—
114C2×D12C2313yes—
115C2×D12C2313yes—
116C2×D12C2316yes—
117C2×D12C2313yes—
118C2×C4×D3—212no—
119C2×C4×D3C2313yes—
120C2×C4×D3C2313yes—
121C2×C4×D3C2313yes—
122C2×C4×D3C2313yes—
123C2×C4×D3C2316yes—
124C2×C4×D3C2313yes—
125C2×Dic6C2311yescharacteristic
126C3×C4⋊C4C2313yes—
127C3×C4⋊C4C2313yes—
128C3×C22⋊C4C2311yescharacteristic
129C3×C22⋊C4C2316yes—
130C3×C22⋊C4C2313yes—
131C3×C42C2313yes—
132#217C2311yescharacteristic
133#217C2313yes—
134(C2×D3)⋊C4C2313yes—
135(C2×D3)⋊C4C2313yes—
136(C2×D3)⋊C4C2313yes—
137(C2×D3)⋊C4C2313yes—
138(C2×D3)⋊C4C2316yes—
139(C2×D3)⋊C4C2313yes—
140(C2×D3)⋊C4C2313yes—
141C4⋊Dic3C2311yescharacteristic
142C4⋊Dic3C2313yes—
143Dic3⋊C4C2311yescharacteristic
144Dic3⋊C4C2313yes—
145Dic3⋊C4C2311yescharacteristic
146Dic3⋊C4C2313yes—
147C4×Dic3C2313yes—
148H(5,𝔽2)—68no—
149C2×#41—31no—
150C2×#41—33no—
151C2×#41—33no—
152C2×#41—31no—
153C2×#41—31no—
154C2×#41—33no—
155C2×#41—68no—
156C2×#41C2×D311yescharacteristic, complemented
157C22×Q8C2×D311yescharacteristic, complemented
158C22×D4—33no—
159C22×D4—33no—
160C22×D4—36no—
161C22×D4—33no—
162C22×D4—33no—
163C23×C4—33no—
164#127—33no—
165#127—33no—
166#124—33no—
167#124C2×D313yescomplemented
168#123—33no—
169#123—33no—
170C22⋊Q8—33no—
171C22⋊Q8—33no—
172C22⋊Q8—31no—
173C22⋊Q8C2×D313yescomplemented
174#121—33no—
175#121—33no—
176#121—33no—
177#121—33no—
178#121—33no—
179#121—36no—
180#121—33no—
181#121—36no—
182#121—33no—
183#121—36no—
184#121C2×D313yescomplemented
185#120—33no—
186#120—33no—
187#120—36no—
188#120—36no—
189C4×Q8—31no—
190C4×Q8C2×D311yescharacteristic, complemented
191C4×D4—33no—
192C4×D4—33no—
193C4×D4—33no—
194C4×D4—33no—
195C4×D4—33no—
196C4×D4C2×D313yescomplemented
197#117—33no—
198C2×C4⋊C4—33no—
199C2×C22⋊C4—36no—
200C23×C3C2413yes—
201C23×C3C2411yescharacteristic
202C22×D3—26no—
203C22×D3—26no—
204C22×D3—212no—
205C22×D3—43no—
206C22×D3—46no—
207C22×D3—46no—
208C22×D3C2413yes—
209C22×D3C2413yes—
210C22×D3C2413yes—
211C3×Q8—26no—
212C3×Q8C2×D414yescomplemented
213C3×D4—26no—
214C3×D4—26no—
215C2×C3×C4—26no—
216C2×C3×C4—26no—
217C2×C3×C4C2411yescharacteristic
218C2×C3×C4C2413yes—
219C2×C3×C4C2413yes—
220C2×C3×C4C2411yescharacteristic
221C2×C3×C4C2413yes—
222#67—26no—
223#67—26no—
224#67—26no—
225#67—46no—
226#67—43no—
227C2×Dic3—22no—
228C2×Dic3C2411yescharacteristic
229C2×Dic3C2413yes—
230C2×Dic3C2411yescharacteristic
231C2×Dic3C2411yescharacteristic
232C2×Dic3C2411yescharacteristic
233D12—26no—
234D12—212no—
235D12—26no—
236D12—26no—
237D12—43no—
238D12—46no—
239D12C2×D4112yescomplemented
240C4×D3—26no—
241C4×D3—26no—
242C4×D3—26no—
243C4×D3—26no—
244C4×D3—46no—
245C4×D3C2×D4112yescomplemented
246Dic6—22no—
247C24—33no—
248C24—36no—
249#41—24no—
250#41—38no—
251#41—612no—
252#41—612no—
253#41—612no—
254#41—64no—
255#41—64no—
256C2×Q8—24no—
257C2×Q8—31no—
258C2×Q8C22×D313yes—
259C2×Q8C22×D311yescharacteristic
260C2×D4—33no—
261C2×D4—33no—
262C2×D4—33no—
263C2×D4—33no—
264C2×D4—36no—
265C2×D4—33no—
266C2×D4—36no—
267C2×D4—33no—
268C2×D4—33no—
269C2×D4—33no—
270C2×D4—33no—
271C2×D4—36no—
272C2×D4—612no—
273C2×D4—612no—
274C2×D4—612no—
275C2×D4—612no—
276C2×D4—1212no—
277C2×D4C22×D313yes—
278C2×D4C22×D313yes—
279C22×C4—33no—
280C22×C4—31no—
281C22×C4—33no—
282C22×C4—33no—
283C22×C4—33no—
284C22×C4—33no—
285C22×C4—36no—
286C22×C4—612no—
287C22×C4C22×D313yes—
288C22×C4C22×D313yes—
289C4⋊C4—33no—
290C4⋊C4—33no—
291C4⋊C4—31no—
292C4⋊C4—31no—
293C4⋊C4—31no—
294C4⋊C4—33no—
295C4⋊C4C22×D313yes—
296C4⋊C4C22×D313yes—
297C22⋊C4—33no—
298C22⋊C4—31no—
299C22⋊C4—33no—
300C22⋊C4—36no—
301C22⋊C4—33no—
302C22⋊C4—33no—
303C22⋊C4—33no—
304C22⋊C4—33no—
305C22⋊C4—33no—
306C22⋊C4C22×D316yes—
307C22⋊C4C22×D311yescharacteristic
308C22⋊C4C22×D313yes—
309C42—33no—
310C42C22×D313yes—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—23no—
314C22×C3—23no—
315C22×C3—23no—
316C22×C3H(5,𝔽2)12yescomplemented
317C22×C3C2511yescharacteristic
318C2×D3—23no—
319C2×D3—23no—
320C2×D3—23no—
321C2×D3—23no—
322C2×D3—23no—
323C2×D3—23no—
324C2×D3—23no—
325C2×D3—23no—
326C2×D3—412no—
327C2×D3—812no—
328C2×D3C22×D416yes—
329C3×C4—21no—
330C3×C4—23no—
331C3×C4—23no—
332C3×C4—21no—
333C3×C4C22×D416yes—
334Dic3—21no—
335Dic3—21no—
336Dic3—23no—
337Dic3—21no—
338Dic3C22×D412yes—
339C23—33no—
340C23—33no—
341C23—33no—
342C23—66no—
343C23—612no—
344C23—66no—
345C23—123no—
346C23—126no—
347C23—126no—
348C23C23×D313yes—
349C23C23×D311yescharacteristic
350Q8—26no—
351Q8—62no—
352Q8D3×D414yescomplemented
353D4—26no—
354D4—26no—
355D4—312no—
356D4—66no—
357D4—612no—
358D4—66no—
359D4—66no—
360D4—66no—
361D4—66no—
362D4—66no—
363D4—126no—
364D4—123no—
365D4—126no—
366D4—123no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—31no—
370C2×C4—312no—
371C2×C4—31no—
372C2×C4—33no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—66no—
376C2×C4—66no—
377C2×C4—62no—
378C2×C4—66no—
379C2×C4—66no—
380C2×C4—126no—
381C2×C4C23×D311yescharacteristic
382C2×C4C23×D313yes—
383C2×C4C23×D313yes—
384C2×C4C23×D311yescharacteristic
385C2×C4C23×D313yes—
386C2×C3—22no—
387C2×C3—43no—
388C2×C3C23×D411yescharacteristic
389C2×C3C2×H(5,𝔽2)11yescharacteristic
390C2×C3#58511yescharacteristic
391D3—26no—
392D3—43no—
393D3—43no—
394C22—21no—
395C22—21no—
396C22—23no—
397C22—23no—
398C22—23no—
399C22—36no—
400C22—63no—
401C22—63no—
402C22—63no—
403C22—63no—
404C22—63no—
405C22—63no—
406C22—63no—
407C22—63no—
408C22—1212no—
409C22—2412no—
410C22#100912yescomplemented
411C22C24×D311yescharacteristic
412C4—21no—
413C4—23no—
414C4—23no—
415C4—21no—
416C4—32no—
417C4—61no—
418C4—61no—
419C4—63no—
420C4—61no—
421C4C2×D3×D416yes—
422C3#348811yescharacteristic, complemented, Sylow
423C2—22no—
424C2—43no—
425C2—66no—
426C2—123no—
427C2—123no—
428C2C22×D3×D411yescharacteristic
429C2C2×#100911yescharacteristic
430C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#548C2
hypercenter#548D3
hypocenterC3#3488
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. C22×#239C2semidirect
3. C3×#548C2semidirect
4. #5834C2semidirect
5. #5817C2semidirect
6. #5676C2semidirect
7. #5674C2semidirect
8. #5630C2semidirect
9. #5629C2semidirect
10. #5609C2semidirect
11. D3×#121C2semidirect
12. #5577C2semidirect
13. #5574C2semidirect
14. D4×D12C2semidirect
15. #5547C2semidirect
17. C2×C3×#41C22semidirect
18. C22×C3×Q8C22semidirect
22. C2×#239C22semidirect
23. C2×#239C22semidirect
24. C2×#239C22semidirect
25. C2×#239C22semidirect
26. C2×D3×D4C22semidirect
27. C2×D3×D4C22semidirect
28. C2×D3×D4C22semidirect
29. C2×#235C22semidirect
30. C22×D12C22semidirect
31. C22×D12C22semidirect
32. C22×C4×D3C22semidirect
33. C3×#124C22semidirect
34. C3×C22⋊Q8C22semidirect
35. C3×#121C22semidirect
36. C3×C4×Q8C22semidirect
37. C3×C4×D4C22semidirect
38. #947C22semidirect
39. #946C22semidirect
40. Q8×Dic3C22semidirect
41. #940C22semidirect
42. #939C22semidirect
43. #938C22semidirect
44. #937C22semidirect
45. #930C22semidirect
46. #930C22semidirect
47. C4×#67C22semidirect
48. C4×#67C22semidirect
49. #896C22semidirect
50. #895C22semidirect
51. #895C22semidirect
52. #895C22semidirect
53. #894C22semidirect
54. #893C22semidirect
55. D3×C4⋊C4C22semidirect
56. #884C22semidirect
57. #884C22semidirect
58. #884C22semidirect
59. #883C22semidirect
60. #882C22semidirect
61. #882C22semidirect
62. #882C22semidirect
63. D3×C22⋊C4C22semidirect
64. #878C22semidirect
65. #874C22semidirect
66. C4×D12C22semidirect
67. C4×D12C22semidirect
68. #872C22semidirect
74. #548D3semidirect
104. #239D4semidirect
156. C2×#41C2×D3semidirect
157. C22×Q8C2×D3semidirect
167. #124C2×D3semidirect
173. C22⋊Q8C2×D3semidirect
184. #121C2×D3semidirect
190. C4×Q8C2×D3semidirect
196. C4×D4C2×D3semidirect
212. C3×Q8C2×D4semidirect
239. D12C2×D4semidirect
245. C4×D3C2×D4semidirect
316. C22×C3H(5,𝔽2)semidirect
352. Q8D3×D4semidirect
410. C22#1009semidirect
422. C3#3488semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C22×#2392
C3×#5482
#58342
#58172
#56762
#56742
#56302
#56292
#56092
D3×#1212
#55772
#55742
D4×D122
#55472
#34883