Finite Group Databasebeta

← #41×#67 #87272

Subgroups — #41×#67 #87272

Order 384.

1648 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
1C22×C3×#41C211yesmaximal, characteristic, complemented
2C2×D3×#41C211yesmaximal, characteristic, complemented
3#5834C211yesmaximal, characteristic, complemented
4#5832C213yesmaximal, complemented
5#5831C211yesmaximal, characteristic, complemented
6#5829C213yesmaximal, complemented
7#5828C213yesmaximal, complemented
8Dic3×#41C211yesmaximal, characteristic, complemented
9#5824C213yesmaximal, complemented
10#5817C211yesmaximal, characteristic, complemented
11Q8×#67C211yesmaximal, characteristic, complemented
12#5806C213yesmaximal, complemented
13D4×#67C213yesmaximal, complemented
14#5791C213yesmaximal, complemented
15C2×C4×#67C213yesmaximal, complemented
16D4×#41—31nomaximal, Sylow
17C2×C3×#41—28no—
18C2×C3×#41C2211yescharacteristic
19C2×C3×#41C2213yescomplemented
20C2×C3×#41C2213yescomplemented
21C2×C3×#41C2211yescharacteristic, complemented
22C22×C3×Q8C2211yescharacteristic, complemented
23C22×C3×D4C2213yescomplemented
24C23×C3×C4C2213yescomplemented
25C22×#67C2213yes—
26D3×#41—28no—
27C2×#239C2211yescharacteristic, complemented
28C2×D3×Q8C2211yescharacteristic, complemented
29C2×#237C2213yescomplemented
30C2×D3×D4C2213yescomplemented
31C2×#235C2213yescomplemented
32C22×C4×D3C2213yescomplemented
33#953C2216yescomplemented
34C2×#217C2213yescomplemented
35#947C2213yescomplemented
36#946C2213yescomplemented
37#946C2213yescomplemented
38Q8×Dic3C2211yescharacteristic, complemented
39Q8×Dic3C2211yescharacteristic, complemented
40#944C2213yescomplemented
41#940C2213yescomplemented
42#939C2216yescomplemented
43#939C2216yescomplemented
44#938C2213yescomplemented
45#938C2213yescomplemented
46#937C2216yescomplemented
47#936C2213yescomplemented
48#935C2216yescomplemented
49D4×Dic3C2213yescomplemented
50D4×Dic3C2213yescomplemented
51#930C2213yescomplemented
52#930C2213yescomplemented
53#929C2216yescomplemented
54C4×#67C2211yescharacteristic
55C4×#67C2213yes—
56C4×#67C2213yescomplemented
57C4×#67C2213yescomplemented
58C4×#67C2213yescomplemented
59C4×#67C2213yescomplemented
60C4×#67C2213yescomplemented
61C4×#67C2213yescomplemented
62C4×#67C2216yescomplemented
63C2×(C2×D3)⋊C4C2213yescomplemented
64#926C2213yescomplemented
65#924C2213yescomplemented
66#924C2213yescomplemented
67C2×Dic3⋊C4C2213yescomplemented
68C2×C4×Dic3C2213yescomplemented
69C22×#41—31no—
70C22×#41D311yescharacteristic, complemented
71#550—31no—
72D4×Q8—31no—
73#548—31no—
74#548—31no—
75#547—33no—
76#546—33no—
77#546—33no—
78D42—33no—
79#532—33no—
80#525—33no—
81#525—33no—
82C4×#41—31no—
83C2×C4×D4—33no—
84C24×C3C2313yes—
85C23×D3C2313yes—
86C3×#41—24no—
87C3×#41—212no—
88C3×#41—212no—
89C3×#41D418yescomplemented
90C2×C3×Q8—24no—
91C2×C3×Q8C2313yes—
92C2×C3×Q8C2311yescharacteristic
93C2×C3×D4—212no—
94C2×C3×D4C2316yes—
95C2×C3×D4C2313yes—
96C2×C3×D4C2313yes—
97C22×C3×C4—212no—
98C22×C3×C4C2313yes—
99C22×C3×C4C2313yes—
100C22×C3×C4C2313yes—
101C22×C3×C4C2311yescharacteristic
102C22×C3×C4C2313yes—
103C22×C3×C4C2313yes—
104C2×#67—212no—
105C2×#67C2313yes—
106C2×#67C2313yes—
107C2×#67C2313yes—
108C2×#67C2316yes—
109C2×#67C2316yes—
110C2×#67C2311yescharacteristic
111C22×Dic3C2313yes—
112C22×Dic3C2313yes—
113C22×Dic3C2313yes—
114#239—24no—
115D3×Q8—24no—
116#237—212no—
117D3×D4—212no—
118#235—212no—
119C2×D12C2313yes—
120C2×C4×D3—212no—
121C2×C4×D3C2311yescharacteristic
122C2×C4×D3C2313yes—
123C2×C4×D3C2313yes—
124C2×C4×D3C2313yes—
125C2×Dic6C2313yes—
126#217C2313yes—
127#217C2316yes—
128#217C2316yes—
129#217C2311yescharacteristic
130#217C2313yes—
131#217C2313yes—
132(C2×D3)⋊C4C2311yescharacteristic
133(C2×D3)⋊C4C2313yes—
134(C2×D3)⋊C4C2313yes—
135(C2×D3)⋊C4C2313yes—
136(C2×D3)⋊C4C2316yes—
137C4⋊Dic3C2313yes—
138C4⋊Dic3C2313yes—
139Dic3⋊C4C2311yescharacteristic
140Dic3⋊C4C2313yes—
141Dic3⋊C4C2313yes—
142Dic3⋊C4C2313yes—
143Dic3⋊C4C2316yes—
144C4×Dic3C2311yescharacteristic
145C4×Dic3C2313yes—
146C4×Dic3C2313yes—
147C4×Dic3C2313yes—
148C2×#41—28no—
149C2×#41—31no—
150C2×#41—33no—
151C2×#41—33no—
152C2×#41—68no—
153C2×#41C2×D311yescharacteristic
154C2×#41C2×D313yescomplemented
155C2×#41C2×D313yescomplemented
156C2×#41C2×D311yescharacteristic, complemented
157C22×Q8—31no—
158C22×Q8C2×D311yescharacteristic, complemented
159C22×D4—33no—
160C22×D4—33no—
161C22×D4C2×D313yescomplemented
162C23×C4—33no—
163C23×C4C2×D313yescomplemented
164C4⋊Q8—33no—
165#127—33no—
166#124—33no—
167#124—33no—
168#123—36no—
169#123—36no—
170C22⋊Q8—33no—
171C22⋊Q8—33no—
172C22⋊Q8—33no—
173C22⋊Q8—33no—
174#121—33no—
175#121—33no—
176#121—36no—
177#121—36no—
178#121—33no—
179#121—33no—
180#120—36no—
181#120—36no—
182C4×Q8—31no—
183C4×Q8—31no—
184C4×D4—33no—
185C4×D4—33no—
186C4×D4—31no—
187C4×D4—33no—
188C4×D4—33no—
189C4×D4—33no—
190C4×D4—33no—
191C4×D4—33no—
192C4×D4—33no—
193C4×D4—33no—
194C4×D4—36no—
195#117—33no—
196C2×C4⋊C4—33no—
197C2×C22⋊C4—33no—
198C2×C22⋊C4—33no—
199C2×C42—33no—
200C23×C3—26no—
201C23×C3—26no—
202C23×C3—43no—
203C23×C3C2413yes—
204C23×C3C2413yes—
205C23×C3C2411yescharacteristic
206C22×D3—26no—
207C22×D3—26no—
208C22×D3—43no—
209C22×D3C2413yes—
210C22×D3C2411yescharacteristic
211C3×Q8—26no—
212C3×Q8C2×D414yescomplemented
213C3×D4—212no—
214C3×D4—26no—
215C3×D4C2×D4112yescomplemented
216C2×C3×C4—26no—
217C2×C3×C4—22no—
218C2×C3×C4—26no—
219C2×C3×C4—26no—
220C2×C3×C4—26no—
221C2×C3×C4C2×D4112yescomplemented
222C2×C3×C4C2411yescharacteristic
223C2×C3×C4C2413yes—
224C2×C3×C4C2413yes—
225C2×C3×C4C2411yescharacteristic
226#67—212no—
227#67—26no—
228#67—26no—
229#67—26no—
230#67—46no—
231#67#4114yesdirect product factor
232C2×Dic3—26no—
233C2×Dic3—26no—
234C2×Dic3—43no—
235C2×Dic3C2411yescharacteristic
236C2×Dic3C2413yes—
237C2×Dic3C2413yes—
238C2×Dic3C2413yes—
239C2×Dic3C2411yescharacteristic
240C2×Dic3C2411yescharacteristic
241D12—26no—
242C4×D3—22no—
243C4×D3—26no—
244C4×D3—26no—
245C4×D3—26no—
246Dic6—26no—
247C24—33no—
248C24C22×D313yes—
249#41—24no—
250#41—212no—
251#41—212no—
252#41—64no—
253#41—612no—
254#41—612no—
255#41#6718yesdirect product factor
256C2×Q8—24no—
257C2×Q8—33no—
258C2×Q8—64no—
259C2×Q8C22×D313yes—
260C2×Q8C22×D311yescharacteristic
261C2×D4—212no—
262C2×D4—31no—
263C2×D4—33no—
264C2×D4—33no—
265C2×D4—33no—
266C2×D4—36no—
267C2×D4—36no—
268C2×D4—33no—
269C2×D4—612no—
270C2×D4—612no—
271C2×D4C22×D316yes—
272C2×D4C22×D313yes—
273C2×D4C22×D313yes—
274C22×C4—212no—
275C22×C4—33no—
276C22×C4—33no—
277C22×C4—33no—
278C22×C4—33no—
279C22×C4—33no—
280C22×C4—31no—
281C22×C4—33no—
282C22×C4—612no—
283C22×C4C22×D313yes—
284C22×C4C22×D313yes—
285C22×C4C22×D313yes—
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D313yes—
288C22×C4C22×D313yes—
289C4⋊C4—31no—
290C4⋊C4—33no—
291C4⋊C4—33no—
292C4⋊C4—33no—
293C4⋊C4—33no—
294C4⋊C4—33no—
295C4⋊C4—36no—
296C22⋊C4—31no—
297C22⋊C4—33no—
298C22⋊C4—33no—
299C22⋊C4—33no—
300C22⋊C4—36no—
301C22⋊C4—36no—
302C22⋊C4—31no—
303C22⋊C4—33no—
304C22⋊C4—33no—
305C22⋊C4—33no—
306C22⋊C4—36no—
307C42—33no—
308C42—33no—
309C42—31no—
310C42—33no—
311C22×C3—21no—
312C22×C3—23no—
313C22×C3—23no—
314C22×C3—21no—
315C22×C3—23no—
316C22×C3—23no—
317C22×C3—23no—
318C22×C3—412no—
319C22×C3C22×D416yes—
320C22×C3C2×#4112yescomplemented
321C22×C3C2511yescharacteristic
322C2×D3—21no—
323C2×D3—21no—
324C2×D3—23no—
325C2×D3—23no—
326C2×D3—23no—
327C2×D3—412no—
328C2×D3C2×#4112yescomplemented
329C3×C4—21no—
330C3×C4—23no—
331C3×C4C22×D416yes—
332C3×C4C22×D412yes—
333Dic3—23no—
334Dic3—21no—
335Dic3—23no—
336Dic3—21no—
337Dic3—23no—
338Dic3C2×#4112yescomplemented
339C23—26no—
340C23—26no—
341C23—31no—
342C23—33no—
343C23—43no—
344C23—66no—
345C23—66no—
346C23—123no—
347C23C23×D313yes—
348C23C23×D311yescharacteristic
349C23C23×D313yes—
350Q8—26no—
351Q8—66no—
352Q8C2×#6714yescomplemented
353D4—212no—
354D4—26no—
355D4—34no—
356D4—66no—
357D4—66no—
358D4—66no—
359D4—612no—
360D4—66no—
361D4—126no—
362D4C2×#67112yescomplemented
363C2×C4—26no—
364C2×C4—22no—
365C2×C4—26no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—33no—
371C2×C4—33no—
372C2×C4—31no—
373C2×C4—33no—
374C2×C4—66no—
375C2×C4—66no—
376C2×C4—62no—
377C2×C4—66no—
378C2×C4—66no—
379C2×C4—66no—
380C2×C4—123no—
381C2×C4C2×#67112yescomplemented
382C2×C4C23×D311yescharacteristic
383C2×C4C23×D313yes—
384C2×C4C23×D313yes—
385C2×C4C23×D311yescharacteristic
386C2×C3—26no—
387C2×C3—22no—
388C2×C3—43no—
389C2×C3C23×D411yescharacteristic
390C2×C3C22×#4111yescharacteristic
391C2×C3#58511yescharacteristic
392D3—22no—
393D3—43no—
394C22—23no—
395C22—23no—
396C22—21no—
397C22—21no—
398C22—23no—
399C22—23no—
400C22—23no—
401C22—32no—
402C22—412no—
403C22—61no—
404C22—61no—
405C22—63no—
406C22—63no—
407C22—63no—
408C22—1212no—
409C22D3×#4112yescomplemented
410C22C22×#6716yes—
411C22C24×D311yescharacteristic
412C4—21no—
413C4—23no—
414C4—32no—
415C4—63no—
416C4—61no—
417C4—63no—
418C4—61no—
419C4—63no—
420C4C22×#6712yes—
421C4C22×#6716yes—
422C3D4×#4111yescharacteristic, complemented, Sylow
423C2—22no—
424C2—26no—
425C2—43no—
426C2—62no—
427C2—123no—
428C2C2×D3×#4111yescharacteristic
429C2#596411yescharacteristic
430C2C23×#6711yescharacteristic
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC22×C3×#41C2
hypercenterC22×#41D3
hypocenterC3D4×#41
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41C2semidirect
2. C2×D3×#41C2semidirect
3. #5834C2semidirect
4. #5832C2semidirect
5. #5831C2semidirect
6. #5829C2semidirect
7. #5828C2semidirect
8. Dic3×#41C2semidirect
9. #5824C2semidirect
10. #5817C2semidirect
11. Q8×#67C2semidirect
12. #5806C2semidirect
13. D4×#67C2semidirect
14. #5791C2semidirect
15. C2×C4×#67C2semidirect
19. C2×C3×#41C22semidirect
20. C2×C3×#41C22semidirect
21. C2×C3×#41C22semidirect
22. C22×C3×Q8C22semidirect
23. C22×C3×D4C22semidirect
24. C23×C3×C4C22semidirect
27. C2×#239C22semidirect
28. C2×D3×Q8C22semidirect
29. C2×#237C22semidirect
30. C2×D3×D4C22semidirect
31. C2×#235C22semidirect
32. C22×C4×D3C22semidirect
33. #953C22semidirect
34. C2×#217C22semidirect
35. #947C22semidirect
36. #946C22semidirect
37. #946C22semidirect
38. Q8×Dic3C22semidirect
39. Q8×Dic3C22semidirect
40. #944C22semidirect
41. #940C22semidirect
42. #939C22semidirect
43. #939C22semidirect
44. #938C22semidirect
45. #938C22semidirect
46. #937C22semidirect
47. #936C22semidirect
48. #935C22semidirect
49. D4×Dic3C22semidirect
50. D4×Dic3C22semidirect
51. #930C22semidirect
52. #930C22semidirect
53. #929C22semidirect
56. C4×#67C22semidirect
57. C4×#67C22semidirect
58. C4×#67C22semidirect
59. C4×#67C22semidirect
60. C4×#67C22semidirect
61. C4×#67C22semidirect
62. C4×#67C22semidirect
63. C2×(C2×D3)⋊C4C22semidirect
64. #926C22semidirect
65. #924C22semidirect
66. #924C22semidirect
67. C2×Dic3⋊C4C22semidirect
68. C2×C4×Dic3C22semidirect
70. C22×#41D3semidirect
89. C3×#41D4semidirect
154. C2×#41C2×D3semidirect
155. C2×#41C2×D3semidirect
156. C2×#41C2×D3semidirect
158. C22×Q8C2×D3semidirect
161. C22×D4C2×D3semidirect
163. C23×C4C2×D3semidirect
212. C3×Q8C2×D4semidirect
215. C3×D4C2×D4semidirect
221. C2×C3×C4C2×D4semidirect
231. #67#41direct (N × H)
255. #41#67direct (N × H)
320. C22×C3C2×#41semidirect
328. C2×D3C2×#41semidirect
338. Dic3C2×#41semidirect
352. Q8C2×#67semidirect
362. D4C2×#67semidirect
381. C2×C4C2×#67semidirect
409. C22D3×#41semidirect
422. C3D4×#41semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#412
C2×D3×#412
#58342
#58322
#58312
#58292
#58282
Dic3×#412
#58242
#58172
Q8×#672
#58062
D4×#672
#57912
C2×C4×#672
D4×#413