Finite Group Databasebeta

← #74007

Subgroups — #74007

Order 384.

1252 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 1252 conjugacy classes of subgroups — too many to draw. Only its 253 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
1C23×D12C211yesmaximal, characteristic
2C2×#893C212yesmaximal, complemented
3C2×D3×C22⋊C4C212yesmaximal, complemented
4C3×#380C211yesmaximal, characteristic, complemented
5C4×#217C211yesmaximal, characteristic, complemented
6#4988C212yesmaximal, complemented
7#4955C212yesmaximal, complemented
8#4665C214yesmaximal, complemented
9D4×C22⋊C4—31nomaximal, Sylow
10C24×D3C2212yes—
11C23×C3×C4C2211yescharacteristic
12C22×D12—28no—
13C22×D12C418yescomplemented
14C22×D12C2212yes—
15C22×D12C2211yescharacteristic
16C22×D12C2211yescharacteristic
17C22×C4×D3C2214yescomplemented
18C2×C3×C4⋊C4C2212yescomplemented
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×(C2×D3)⋊C4C2214yescomplemented
23C2×(C2×D3)⋊C4C2214yescomplemented
24C2×(C2×D3)⋊C4C2214yescomplemented
25C2×C4×Dic3C2212yescomplemented
26#893—28no—
27D3×C22⋊C4—28no—
28C3×#95C2212yescomplemented
29C3⋊#95C2212yescomplemented
30C23×D4—31no—
31C2×C4×D4—32no—
32C22×C22⋊C4—32no—
33#390—32no—
34#386—34no—
35#380D311yescharacteristic, complemented
36#379—32no—
37C4×C22⋊C4—31no—
38C24×C3C2311yescharacteristic
39C23×D3—24no—
40C23×D3—24no—
41C23×D3—28no—
42C23×D3—44no—
43C23×D3C2×C418yescomplemented
44C23×D3C2312yes—
45C23×D3C2312yes—
46C22×C3×C4—24no—
47C22×C3×C4C2×C414yescomplemented
48C22×C3×C4C2314yes—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C2×D12—28no—
54C2×D12—24no—
55C2×D12—232no—
56C2×D12C2×C414yes—
57C2×D12C2×C418yescomplemented
58C2×D12C2×C414yescomplemented
59C2×D12D418yes—
60C2×C4×D3—28no—
61C2×C4×D3—28no—
62C2×C4×D3—44no—
63C3×C4⋊C4—24no—
64C3×C22⋊C4—24no—
65#217—22no—
66#217D414yescomplemented
67(C2×D3)⋊C4—28no—
68(C2×D3)⋊C4—28no—
69(C2×D3)⋊C4—44no—
70C4×Dic3—24no—
71C25—32no—
72C22×D4—38no—
73C22×D4—31no—
74C22×D4—32no—
75C22×D4—31no—
76C22×D4—68no—
77C23×C4—34no—
78C23×C4C2×D311yescharacteristic
79C4×D4—68no—
80C2×C4⋊C4C2×D312yescomplemented
81C2×C22⋊C4—31no—
82C2×C22⋊C4—31no—
83C2×C22⋊C4—34no—
84C2×C22⋊C4—34no—
85C2×C22⋊C4—34no—
86C2×C22⋊C4—68no—
87C2×C22⋊C4C2×D312yes—
88C2×C42—32no—
89#95—32no—
90#95C2×D312yescomplemented
91C23×C3—22no—
92C23×C3—22no—
93C23×C3C22×C412yes—
94C23×C3C2×D412yes—
95C23×C3#4112yes—
96C23×C3C2411yescharacteristic
97C22×D3—22no—
98C22×D3—22no—
99C22×D3—24no—
100C22×D3—216no—
101C22×D3—22no—
102C22×D3—24no—
103C22×D3—22no—
104C22×D3—24no—
105C22×D3—216no—
106C22×D3—48no—
107C22×D3—416no—
108C22×D3—416no—
109C22×D3C22×C414yes—
110C22×D3C22×C414yes—
111C22×D3C2×D418yes—
112C2×C3×C4—22no—
113C2×C3×C4—28no—
114C2×C3×C4—24no—
115C2×C3×C4—24no—
116C2×C3×C4—24no—
117C2×C3×C4C22×C412yes—
118C2×C3×C4C22×C412yes—
119C2×C3×C4C2×D414yes—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3C2×D414yescomplemented
126D12—216no—
127D12—28no—
128D12C22⋊C4116yescomplemented
129C4×D3—416no—
130C24—32no—
131C24—32no—
132C24—38no—
133C24—68no—
134C24—64no—
135C24—64no—
136C24—124no—
137C24C22×D311yescharacteristic
138C2×D4—34no—
139C2×D4—38no—
140C2×D4—34no—
141C2×D4—38no—
142C2×D4—632no—
143C2×D4—68no—
144C2×D4—64no—
145C22×C4—24no—
146C22×C4—32no—
147C22×C4—32no—
148C22×C4—68no—
149C22×C4—68no—
150C22×C4—124no—
151C22×C4C4×D314yescomplemented
152C22×C4C22×D314yes—
153C22×C4C22×D311yescharacteristic
154C22×C4C22×D311yescharacteristic
155C4⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—34no—
158C22⋊C4—68no—
159C22⋊C4—68no—
160C22⋊C4—62no—
161C22⋊C4—124no—
162C42—64no—
163C22×C3—22no—
164C22×C3—22no—
165C22×C3—24no—
166C22×C3—24no—
167C22×C3C4×D414yescomplemented
168C22×C3C23×C411yescharacteristic
169C22×C3C22×D412yes—
170C22×C3C22×D411yescharacteristic
171C22×C3C2×#4111yescharacteristic
172C22×C3H(5,𝔽2)12yes—
173C2×D3—24no—
174C2×D3—24no—
175C2×D3—28no—
176C2×D3—28no—
177C2×D3—24no—
178C2×D3—24no—
179C2×D3—24no—
180C2×D3—432no—
181C2×D3—44no—
182C2×D3—44no—
183C2×D3C2×C22⋊C418yescomplemented
184C3×C4—22no—
185C3×C4—44no—
186C3×C4C2×C22⋊C414yescomplemented
187Dic3—24no—
188Dic3—42no—
189C23—22no—
190C23—22no—
191C23—34no—
192C23—38no—
193C23—34no—
194C23—616no—
195C23—62no—
196C23—62no—
197C23—64no—
198C23—616no—
199C23—62no—
200C23—64no—
201C23—62no—
202C23—64no—
203C23—128no—
204C23—1216no—
205C23—1216no—
206C23C2×C4×D312yes—
207C23D3×D412yes—
208C23#23912yes—
209C23C23×D311yescharacteristic
210D4—316no—
211D4—616no—
212D4—68no—
213C2×C4—22no—
214C2×C4—28no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—34no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—1216no—
225C2×C4C2×C4×D312yes—
226C2×C4C2×C4×D312yes—
227C2×C4D3×D414yes—
228C2×C3—24no—
229C2×C3C22×C22⋊C411yescharacteristic
230C2×C3C2×C4×D411yescharacteristic
231C2×C3#51811yescharacteristic
232C2×C3D4212yes—
233C2×C3#54612yes—
234D3—28no—
235D3—44no—
236C22—22no—
237C22—22no—
238C22—24no—
239C22—24no—
240C22—38no—
241C22—64no—
242C22—64no—
243C22—68no—
244C22—68no—
245C22—64no—
246C22—64no—
247C22—64no—
248C22—1232no—
249C22—124no—
250C22—124no—
251C22#89314yescomplemented
252C22C22×C4×D311yescharacteristic
253C22C2×D3×D412yes—
254C22C2×D3×D411yescharacteristic
255C22C2×#23911yescharacteristic
256C22#100912yes—
257C4—22no—
258C4—44no—
259C4—64no—
260C4—122no—
261C4D3×C22⋊C414yescomplemented
262C3D4×C22⋊C411yescharacteristic, complemented, Sylow
263C2—24no—
264C2—68no—
265C2—124no—
266C2C2×D3×C22⋊C411yescharacteristic
267C2C2×#89311yescharacteristic
268C2#552111yescharacteristic
269C2#560912yes—
270C2#563012yes—
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#380C2
socle, cosocleC23×C3C24
prime radicalC23×D12C2
hypercenter#380D3
hypocenterC3D4×C22⋊C4
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#893C2semidirect
3. C2×D3×C22⋊C4C2semidirect
4. C3×#380C2semidirect
5. C4×#217C2semidirect
6. #4988C2semidirect
7. #4955C2semidirect
8. #4665C2semidirect
13. C22×D12C4semidirect
17. C22×C4×D3C22semidirect
18. C2×C3×C4⋊C4C22semidirect
22. C2×(C2×D3)⋊C4C22semidirect
23. C2×(C2×D3)⋊C4C22semidirect
24. C2×(C2×D3)⋊C4C22semidirect
25. C2×C4×Dic3C22semidirect
28. C3×#95C22semidirect
29. C3⋊#95C22semidirect
35. #380D3semidirect
43. C23×D3C2×C4semidirect
47. C22×C3×C4C2×C4semidirect
57. C2×D12C2×C4semidirect
58. C2×D12C2×C4semidirect
66. #217D4semidirect
80. C2×C4⋊C4C2×D3semidirect
90. #95C2×D3semidirect
125. C2×Dic3C2×D4semidirect
128. D12C22⋊C4semidirect
151. C22×C4C4×D3semidirect
167. C22×C3C4×D4semidirect
183. C2×D3C2×C22⋊C4semidirect
186. C3×C4C2×C22⋊C4semidirect
251. C22#893semidirect
261. C4D3×C22⋊C4semidirect
262. C3D4×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×D122
C2×#8932
C2×D3×C22⋊C42
C3×#3802
C4×#2172
#49882
#49552
#46652
D4×C22⋊C43