Finite Group Databasebeta

← #82936

Subgroups — #82936

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
2C3×#525C211yesmaximal, characteristic, complemented
3#5791C212yesmaximal, complemented
4#5240C211yesmaximal, characteristic, complemented
5#5053C214yesmaximal, complemented
6#5032C214yesmaximal, complemented
7#5006C212yesmaximal, complemented
8#3025—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C23×C3×C4C2211yescharacteristic, complemented
11#1010—22no—
12#1009—22no—
13D3×#41—24no—
14#1007—21no—
15#1004—21no—
16#1004C2214yes—
17C2×#235—22no—
18C2×#235C2212yes—
19C3×#123C2212yescomplemented
20C3×C22⋊Q8—21no—
21C3×#121—21no—
22C3×#120C2212yescomplemented
23C3×C4×D4—22no—
24C3×#117C2211yescharacteristic, complemented
25#953C2214yescomplemented
26#930—22no—
27#929C2214yescomplemented
28C4×#67—24no—
29#926C2212yescomplemented
30#924—22no—
31#834C2214yescomplemented
32#806C2218yescomplemented
33#585—31no—
34#525—32no—
35#525D311yescharacteristic, complemented
36#458—34no—
37B(4,𝔽2)—34no—
38#410—31no—
39#410—32no—
40C24×C3C2311yescharacteristic
41C3×#41—22no—
42C3×#41—22no—
43C2×C3×Q8D411yescharacteristic, complemented
44C2×C3×D4—22no—
45C2×C3×D4D411yescharacteristic, complemented
46C2×C3×D4C2312yes—
47C22×C3×C4—22no—
48C22×C3×C4—22no—
49C22×C3×C4—21no—
50C22×C3×C4—21no—
51C22×C3×C4D412yescomplemented
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C2×#67—24no—
55C2×#67—24no—
56C2×#67C2314yes—
57#239—24no—
58D3×Q8—24no—
59#237—24no—
60#237—28no—
61D3×D4—24no—
62D3×D4—28no—
63#235—24no—
64#235—24no—
65#235—22no—
66#235—24no—
67#235—22no—
68#235—22no—
69#235—22no—
70C2×D12—22no—
71C2×D12D412yescomplemented
72C2×C4×D3—24no—
73C2×C4×D3D414yescomplemented
74C2×Dic6—22no—
75C2×Dic6D412yescomplemented
76C3×C4⋊C4—21no—
77C3×C4⋊C4—22no—
78C3×C22⋊C4—22no—
79C3×C22⋊C4—22no—
80C3×C22⋊C4C2312yes—
81C3×C42—21no—
82#217—24no—
83#217C2314yes—
84(C2×D3)⋊C4—24no—
85C4⋊Dic3—22no—
86Dic3⋊C4—24no—
87C4×Dic3—22no—
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×C4C2×D311yescharacteristic, complemented
98#123—34no—
99#123C2×D312yescomplemented
100C22⋊Q8—21no—
101C22⋊Q8—62no—
102#121—21no—
103#121—62no—
104#120—34no—
105#120C2×D312yescomplemented
106C4×D4—22no—
107C4×D4—64no—
108#117—32no—
109#117C2×D311yescharacteristic, complemented
110#99—34no—
111#99—38no—
112C23×C3—22no—
113C23×C3—21no—
114C23×C3—42no—
115C23×C3C2×D412yes—
116C23×C3C2411yescharacteristic
117C22×D3—24no—
118C22×D3C2×D414yes—
119C3×Q8—22no—
120C3×D4—24no—
121C3×D4—22no—
122C3×D4—44no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—41no—
129C2×C3×C4—42no—
130C2×C3×C4—41no—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136#67—28no—
137#67—24no—
138#67—24no—
139#67—48no—
140C2×Dic3—24no—
141C2×Dic3—24no—
142C2×Dic3C2×D414yes—
143D12—24no—
144D12—22no—
145D12—22no—
146C4×D3—28no—
147C4×D3—24no—
148C4×D3—24no—
149Dic6—24no—
150Dic6—22no—
151Dic6—22no—
152C24C22×D311yescharacteristic
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—62no—
163#41—62no—
164#41—64no—
165C2×Q8—32no—
166C2×Q8—62no—
167C2×Q8—64no—
168C2×Q8#6711yescharacteristic, complemented
169C2×D4—22no—
170C2×D4—34no—
171C2×D4—32no—
172C2×D4—62no—
173C2×D4—68no—
174C2×D4—64no—
175C2×D4—64no—
176C2×D4—64no—
177C2×D4#6711yescharacteristic, complemented
178C2×D4C22×D312yes—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—34no—
184C22×C4—64no—
185C22×C4#6712yescomplemented
186C22×C4C22×D311yescharacteristic
187C22×C4C22×D311yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—22no—
190C4⋊C4—64no—
191C4⋊C4—62no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—34no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4C22×D312yes—
198C42—21no—
199C42—62no—
200C22×C3—24no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3—41no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3C22×D412yes—
209C22×C3C22×D411yescharacteristic
210C2×D3—24no—
211C2×D3—44no—
212C2×D3—84no—
213C3×C4—22no—
214C3×C4—42no—
215C3×C4—41no—
216C3×C4#12011yescharacteristic
217C3×C4#12012yes—
218C3×C4#12011yescharacteristic
219Dic3—24no—
220Dic3—44no—
221C23—22no—
222C23—21no—
223C23—34no—
224C23—42no—
225C23—64no—
226C23C2×#6712yes—
227C23C23×D311yescharacteristic
228Q8—22no—
229Q8—64no—
230Q8—62no—
231Q8—62no—
232D4—24no—
233D4—22no—
234D4—44no—
235D4—64no—
236D4—68no—
237D4—64no—
238D4—64no—
239D4—62no—
240D4—62no—
241D4—128no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—34no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—68no—
252C2×C4—64no—
253C2×C4—64no—
254C2×C4—64no—
255C2×C4—64no—
256C2×C4D3×D412yes—
257C2×C4D3×D412yes—
258C2×C4C2×#6712yes—
259C2×C4C2×#6711yescharacteristic
260C2×C4C2×#6711yescharacteristic
261C2×C3—22no—
262C2×C3—21no—
263C2×C3—42no—
264C2×C3—42no—
265C2×C3C2×#12011yescharacteristic
266D3—44no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—64no—
276C22—124no—
277C22—244no—
278C22C2×D3×D412yes—
279C22C22×#6711yescharacteristic
280C4—22no—
281C4—42no—
282C4—41no—
283C4—64no—
284C4—124no—
285C4#93711yescharacteristic
286C4#93712yes—
287C4#93711yescharacteristic
288C3#302511yescharacteristic, complemented, Sylow
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—124no—
294C2C2×#93711yescharacteristic
subgroupG/H
centerC4#937
commutator, cosocleC23×C3C24
FrattiniC23C23×D3
FittingC3×#525C2
socleC2×C3C2×#120
hypercenter#525D3
hypocenterC3#3025
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. C3×#525C2semidirect
3. #5791C2semidirect
4. #5240C2semidirect
5. #5053C2semidirect
6. #5032C2semidirect
7. #5006C2semidirect
10. C23×C3×C4C22semidirect
19. C3×#123C22semidirect
22. C3×#120C22semidirect
24. C3×#117C22semidirect
25. #953C22semidirect
27. #929C22semidirect
29. #926C22semidirect
31. #834C22semidirect
32. #806C22semidirect
35. #525D3semidirect
43. C2×C3×Q8D4semidirect
45. C2×C3×D4D4semidirect
51. C22×C3×C4D4semidirect
71. C2×D12D4semidirect
73. C2×C4×D3D4semidirect
75. C2×Dic6D4semidirect
97. C23×C4C2×D3semidirect
99. #123C2×D3semidirect
105. #120C2×D3semidirect
109. #117C2×D3semidirect
168. C2×Q8#67semidirect
177. C2×D4#67semidirect
185. C22×C4#67semidirect
288. C3#3025semidirect

Maximal subgroups

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

subgroupindex
#59642
C3×#5252
#57912
#52402
#50532
#50322
#50062
#30253