Finite Group Databasebeta

← #80413

Subgroups — #80413

Order 384.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5731C211yesmaximal, characteristic, complemented
2#5545C211yesmaximal, characteristic
3C3×#461C211yesmaximal, characteristic, complemented
4#5169C211yesmaximal, characteristic
5#5169C211yesmaximal, characteristic
6#5164C211yesmaximal, characteristic, complemented
7Dic3×SD16C211yesmaximal, characteristic, complemented
8#5083C211yesmaximal, characteristic, complemented
9#5009C211yesmaximal, characteristic, complemented
10#4847C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4817C211yesmaximal, characteristic, complemented
13#4815C211yesmaximal, characteristic, complemented
14#4814C211yesmaximal, characteristic, complemented
15#4788C211yesmaximal, characteristic, complemented
16#3123—31nomaximal, Sylow
17C2×D3×D4C2211yescharacteristic
18C2×C3×SD16C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C3×C4⋊Q8C2211yescharacteristic, complemented
21C3×C4×D4C2211yescharacteristic
22#946C2211yescharacteristic
23#946C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic, complemented
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29C4×#67—21no—
30C2×(C2×D3)⋊C4—21no—
31#898—21no—
32#898—21no—
33#893C2211yescharacteristic
34#892C2211yescharacteristic
35#889—21no—
36#889—21no—
37#67⋊C4—21no—
38D3×C22⋊C4—21no—
39C4×D12C2211yescharacteristic
40#872C2211yescharacteristic
41#870C2211yescharacteristic, complemented
42C3×C4⋊C8C2211yescharacteristic, complemented
43C3×#103C2211yescharacteristic, complemented
44C3×#103C2211yescharacteristic, complemented
45#835C2211yescharacteristic, complemented
46#835C2211yescharacteristic, complemented
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53#818C2211yescharacteristic, complemented
54#817C2211yescharacteristic, complemented
55#815C2211yescharacteristic, complemented
56C8×Dic3C2211yescharacteristic, complemented
57#808C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#541—31no—
61#518—31no—
62#485—31no—
63#485—31no—
64#478—31no—
65#478—31no—
66#461—31no—
67#461D311yescharacteristic, complemented
68#450—31no—
69#450—31no—
70#440—31no—
71#440—31no—
72C4×SD16—31no—
73C4×SD16—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C2×#67—21no—
81C22×Dic3—21no—
82D3×D4—22no—
83D3×D4—22no—
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3C2311yescharacteristic
88C3×SD16—22no—
89C3×SD16—22no—
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C3×C4⋊C4—21no—
93C3×C4⋊C4—21no—
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96C3×C42C2311yescharacteristic
97#217—21no—
98#215—22no—
99#215—22no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4D411yescharacteristic
106(C2×D3)⋊C4D411yescharacteristic
107C4⋊Dic3—21no—
108C4⋊Dic3—21no—
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C22×D4—31no—
125C2×SD16—31no—
126C2×SD16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×SD16C2×D311yescharacteristic
129C4⋊Q8C2×D311yescharacteristic, complemented
130#126—61no—
131#126—61no—
132#125—31no—
133#125—61no—
134#125—61no—
135C22⋊Q8—31no—
136C22⋊Q8—31no—
137C4×Q8—31no—
138C4×Q8—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—61no—
143C4×D4—61no—
144C4×D4C2×D311yescharacteristic
145#117—31no—
146#117—31no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149#107—31no—
150#107—31no—
151#106—31no—
152#106—31no—
153C4⋊C8—31no—
154C4⋊C8C2×D311yescharacteristic, complemented
155#103—31no—
156#103—31no—
157#103C2×D311yescharacteristic, complemented
158#103C2×D311yescharacteristic, complemented
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163C22⋊C8—31no—
164C22⋊C8—31no—
165#97—31no—
166C4×C8—31no—
167C23×C3—21no—
168C22×D3—21no—
169C22×D3—24no—
170C22×D3—81no—
171C22×D3—81no—
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×D4—21no—
178C3×D4#4112yes—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—81no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—24no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—81no—
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194D12—21no—
195D12#4112yes—
196C4×D3—21no—
197C4×D3—21no—
198C4×D3—81no—
199C4×D3—81no—
200C3×C8—21no—
201C3×C8—21no—
202C3⋊C8—21no—
203C3⋊C8—21no—
204C24—61no—
205C2×Q8C22×D311yescharacteristic
206C2×Q8C22×D311yescharacteristic
207C2×D4—31no—
208C2×D4—61no—
209C2×D4—62no—
210C2×D4—62no—
211C2×D4C22×D311yescharacteristic
212C22×C4—21no—
213C22×C4—31no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217SD16—22no—
218SD16—22no—
219SD16—62no—
220SD16—62no—
221C2×C8—31no—
222C2×C8—31no—
223C2×C8C22×D311yescharacteristic
224C2×C8C22×D311yescharacteristic
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—31no—
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—21no—
240C22⋊C4—31no—
241C22⋊C4—31no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C42—31no—
249C42—61no—
250C42—61no—
251C42—61no—
252C42C22×D311yescharacteristic
253C22×C3—22no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3C22×D411yescharacteristic
257C2×D3—21no—
258C2×D3—22no—
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—41no—
262C2×D3—41no—
263C2×D3—84no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4C2×#4111yescharacteristic
270C3×C4C2×#4111yescharacteristic
271Dic3—21no—
272Dic3—21no—
273Dic3—21no—
274Dic3—41no—
275Dic3—41no—
276Dic3—41no—
277C23—21no—
278C23—31no—
279C23—64no—
280C23—61no—
281C23—241no—
282C23—241no—
283Q8—21no—
284Q8—21no—
285Q8—21no—
286Q8—21no—
287D4—21no—
288D4—32no—
289D4—64no—
290D4—61no—
291D4#23712yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—81no—
304C2×C4—241no—
305C2×C4—241no—
306C2×C4—241no—
307C2×C4D3×D411yescharacteristic
308C2×C4D3×D411yescharacteristic
309C2×C4C23×D311yescharacteristic
310C8—21no—
311C8—21no—
312C8—61no—
313C8—61no—
314C2×C3—42no—
315C2×C3#52511yescharacteristic
316C2×C3#57711yescharacteristic
317C2×C3#57711yescharacteristic
318D3—41no—
319D3—42no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—61no—
324C22—61no—
325C22—61no—
326C22—62no—
327C22—121no—
328C22—121no—
329C22—244no—
330C22C2×D3×D411yescharacteristic
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—61no—
337C4—61no—
338C4—61no—
339C4—121no—
340C4—121no—
341C4—121no—
342C4C2×#23711yescharacteristic
343C4D3×#4111yescharacteristic
344C3#312311yescharacteristic, complemented, Sylow
345C2—42no—
346C2—121no—
347C2—122no—
348C2#560611yescharacteristic
349C2#576211yescharacteristic
350C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radicalC2×D3×D4C22
hypercenter#461D3
hypocenterC3#3123
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5731C2semidirect
3. C3×#461C2semidirect
6. #5164C2semidirect
7. Dic3×SD16C2semidirect
8. #5083C2semidirect
9. #5009C2semidirect
10. #4847C2semidirect
11. #4840C2semidirect
12. #4817C2semidirect
13. #4815C2semidirect
14. #4814C2semidirect
15. #4788C2semidirect
20. C3×C4⋊Q8C22semidirect
24. Q8×Dic3C22semidirect
25. Q8×Dic3C22semidirect
41. #870C22semidirect
42. C3×C4⋊C8C22semidirect
43. C3×#103C22semidirect
44. C3×#103C22semidirect
45. #835C22semidirect
46. #835C22semidirect
53. #818C22semidirect
54. #817C22semidirect
55. #815C22semidirect
56. C8×Dic3C22semidirect
57. #808C22semidirect
58. #807C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
67. #461D3semidirect
129. C4⋊Q8C2×D3semidirect
154. C4⋊C8C2×D3semidirect
157. #103C2×D3semidirect
158. #103C2×D3semidirect
344. C3#3123semidirect

Maximal subgroups

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

subgroupindex
#57312
#55452
C3×#4612
#51692
#51642
Dic3×SD162
#50832
#50092
#48472
#48402
#48172
#48152
#48142
#47882
#31233