Finite Group Databasebeta

← #78690

Subgroups — #78690

Order 384.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5964C211yesmaximal, characteristic, complemented
2#5828C211yesmaximal, characteristic, complemented
3#5524C212yesmaximal, complemented
4C3×#410C211yesmaximal, characteristic, complemented
5#5006C212yesmaximal, complemented
6#4742C214yesmaximal, complemented
7#4741C214yesmaximal, complemented
8#3025—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10#1010—22no—
11#1009—22no—
12D3×#41—24no—
13#1007—21no—
14#1004—21no—
15#1004C2214yes—
16C2×#235—22no—
17C2×#235C2212yes—
18C22×C4×D3C2211yescharacteristic, complemented
19C3×#117C2212yescomplemented
20#946—21no—
21#938—21no—
22#937C2212yescomplemented
23#935C2212yescomplemented
24C4×#67—22no—
25#926C2211yescharacteristic, complemented
26#897—22no—
27#895—22no—
28#886C2214yescomplemented
29#882C2214yescomplemented
30C4×D12—24no—
31C3×#99C2214yescomplemented
32#806C2218yescomplemented
33#585—31no—
34#525—31no—
35#525—32no—
36#458—34no—
37B(4,𝔽2)—34no—
38#410—32no—
39#410D311yescharacteristic, complemented
40C23×D3C2311yescharacteristic
41C3×#41—22no—
42C3×#41—22no—
43C2×C3×Q8D411yescharacteristic, complemented
44C2×C3×D4D411yescharacteristic, complemented
45C2×C3×D4C2312yes—
46C22×C3×C4D412yescomplemented
47C22×C3×C4C2311yescharacteristic
48C2×#67—24no—
49C2×#67—22no—
50C2×#67C2314yes—
51C22×Dic3C2311yescharacteristic
52#239—24no—
53D3×Q8—24no—
54#237—24no—
55#237—28no—
56D3×D4—24no—
57D3×D4—28no—
58#235—24no—
59#235—24no—
60#235—22no—
61#235—24no—
62#235—22no—
63#235—22no—
64#235—22no—
65C2×D12—22no—
66C2×D12—24no—
67C2×D12D412yescomplemented
68C2×C4×D3—24no—
69C2×C4×D3—22no—
70C2×C4×D3—22no—
71C2×C4×D3—21no—
72C2×C4×D3—21no—
73C2×C4×D3D414yescomplemented
74C2×Dic6—22no—
75C2×Dic6D412yescomplemented
76C3×C4⋊C4—22no—
77C3×C22⋊C4C2314yes—
78C3×C42—22no—
79#217—22no—
80#217C2312yes—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—24no—
84C4⋊Dic3—21no—
85C4⋊Dic3—24no—
86Dic3⋊C4—22no—
87C4×Dic3—21no—
882−1+4—62no—
892−1+4—61no—
90H(5,𝔽2)—34no—
91H(5,𝔽2)—62no—
92H(5,𝔽2)—61no—
93C2×#41—32no—
94C2×#41—64no—
95C2×#41—62no—
96C2×#41C2×D311yescharacteristic
97C23×C4—31no—
98#123—32no—
99#123—34no—
100C22⋊Q8—62no—
101C22⋊Q8—61no—
102#121—62no—
103#121—61no—
104#120—32no—
105#120—34no—
106C4×D4—64no—
107C4×D4—62no—
108#117—31no—
109#117C2×D312yescomplemented
110#99—38no—
111#99C2×D314yescomplemented
112C23×C3C2×D412yes—
113C23×C3C2411yescharacteristic
114C22×D3—22no—
115C22×D3—21no—
116C22×D3—24no—
117C22×D3—42no—
118C22×D3C2×D414yes—
119C3×Q8—22no—
120C3×D4—24no—
121C3×D4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—24no—
125C2×C3×C4C2×D412yes—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D412yes—
129C2×C3×C4C2×D412yes—
130#67—28no—
131#67—24no—
132#67—24no—
133#67—44no—
134C2×Dic3—24no—
135C2×Dic3—22no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3C2×D414yes—
139D12—24no—
140D12—22no—
141D12—22no—
142D12—48no—
143C4×D3—28no—
144C4×D3—24no—
145C4×D3—24no—
146C4×D3—41no—
147C4×D3—42no—
148C4×D3—41no—
149Dic6—24no—
150Dic6—22no—
151Dic6—22no—
152C24—31no—
153#41—22no—
154#41—22no—
155#41—64no—
156#41—64no—
157#41—68no—
158#41—64no—
159#41—64no—
160#41—62no—
161#41—62no—
162#41—64no—
163#41—62no—
164#41—62no—
165C2×Q8—32no—
166C2×Q8—62no—
167C2×Q8—64no—
168C2×Q8D1211yescharacteristic, complemented
169C2×D4—34no—
170C2×D4—32no—
171C2×D4—62no—
172C2×D4—64no—
173C2×D4—68no—
174C2×D4—64no—
175C2×D4—64no—
176C2×D4—62no—
177C2×D4D1211yescharacteristic, complemented
178C2×D4C22×D312yes—
179C22×C4—34no—
180C22×C4—31no—
181C22×C4—64no—
182C22×C4—61no—
183C22×C4—61no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4D1212yescomplemented
187C22×C4C22×D311yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—64no—
190C4⋊C4—61no—
191C4⋊C4—62no—
192C22⋊C4—32no—
193C22⋊C4—64no—
194C22⋊C4—64no—
195C22⋊C4—62no—
196C22⋊C4—62no—
197C22⋊C4C22×D314yes—
198C42—22no—
199C42—61no—
200C22×C3—22no—
201C22×C3—41no—
202C22×C3—42no—
203C22×C3C22×D411yescharacteristic
204C22×C3C22×D412yes—
205C2×D3—24no—
206C2×D3—22no—
207C2×D3—24no—
208C2×D3—41no—
209C2×D3—42no—
210C2×D3—41no—
211C2×D3—44no—
212C2×D3—84no—
213C3×C4—22no—
214C3×C4—44no—
215C3×C4#12011yescharacteristic
216C3×C4#12012yes—
217C3×C4#12011yescharacteristic
218Dic3—24no—
219Dic3—42no—
220Dic3—41no—
221C23—34no—
222C23—64no—
223C23—61no—
224C23—62no—
225C23—122no—
226C23C2×D1212yes—
227C23C23×D311yescharacteristic
228Q8—22no—
229Q8—64no—
230Q8—62no—
231Q8—62no—
232D4—24no—
233D4—22no—
234D4—64no—
235D4—68no—
236D4—64no—
237D4—64no—
238D4—62no—
239D4—62no—
240D4—128no—
241D4—124no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—34no—
246C2×C4—68no—
247C2×C4—64no—
248C2×C4—64no—
249C2×C4—64no—
250C2×C4—62no—
251C2×C4—61no—
252C2×C4—62no—
253C2×C4—121no—
254C2×C4—122no—
255C2×C4—121no—
256C2×C4C2×D1212yes—
257C2×C4C2×D1211yescharacteristic
258C2×C4C2×D1211yescharacteristic
259C2×C4D3×D412yes—
260C2×C4D3×D412yes—
261C2×C3—21no—
262C2×C3—22no—
263C2×C3—42no—
264C2×C3C2×#12011yescharacteristic
265D3—42no—
266D3—44no—
267C22—22no—
268C22—41no—
269C22—42no—
270C22—64no—
271C22—62no—
272C22—64no—
273C22—124no—
274C22—122no—
275C22—121no—
276C22—121no—
277C22—244no—
278C22C22×D1211yescharacteristic
279C22C2×D3×D412yes—
280C4—22no—
281C4—44no—
282C4—64no—
283C4—122no—
284C4—121no—
285C4#88211yescharacteristic
286C4#88212yes—
287C4#88211yescharacteristic
288C3#302511yescharacteristic, complemented, Sylow
289C2—22no—
290C2—21no—
291C2—42no—
292C2—122no—
293C2—124no—
294C2C2×#88211yescharacteristic
subgroupG/H
centerC4#882
commutator, cosocleC23×C3C24
FrattiniC23C23×D3
FittingC3×#410C2
socleC2×C3C2×#120
hypercenter#410D3
hypocenterC3#3025
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. #5828C2semidirect
3. #5524C2semidirect
4. C3×#410C2semidirect
5. #5006C2semidirect
6. #4742C2semidirect
7. #4741C2semidirect
18. C22×C4×D3C22semidirect
19. C3×#117C22semidirect
22. #937C22semidirect
23. #935C22semidirect
25. #926C22semidirect
28. #886C22semidirect
29. #882C22semidirect
31. C3×#99C22semidirect
32. #806C22semidirect
39. #410D3semidirect
43. C2×C3×Q8D4semidirect
44. C2×C3×D4D4semidirect
46. C22×C3×C4D4semidirect
67. C2×D12D4semidirect
73. C2×C4×D3D4semidirect
75. C2×Dic6D4semidirect
109. #117C2×D3semidirect
111. #99C2×D3semidirect
168. C2×Q8D12semidirect
177. C2×D4D12semidirect
186. C22×C4D12semidirect
288. C3#3025semidirect

Maximal subgroups

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

subgroupindex
#59642
#58282
#55242
C3×#4102
#50062
#47422
#47412
#30253