Finite Group Databasebeta

← #74005

Subgroups — #74005

Order 384.

816 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×D3×C22⋊C4C212yesmaximal
3C3×#380C211yesmaximal, characteristic, complemented
4#5212C211yesmaximal, characteristic, complemented
5#4989C214yesmaximal, complemented
6#4942C212yesmaximal, complemented
7#4668C214yesmaximal, complemented
8#2315—31nomaximal, Sylow
9C24×D3C2211yescharacteristic
10C23×C3×C4C2211yescharacteristic
11C23×Dic3C2211yescharacteristic
12C22×C4×D3C2214yes—
13C2×C3×C4⋊C4C2212yescomplemented
14C2×C3×C22⋊C4C2212yes—
15C2×#217C2212yes—
16C2×(C2×D3)⋊C4C418yescomplemented
17C2×(C2×D3)⋊C4C2214yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×C4⋊Dic3C2212yescomplemented
21C2×Dic3⋊C4C2214yescomplemented
22D3×C22⋊C4—28no—
23C3×#95C2212yescomplemented
24C3⋊#95C2214yescomplemented
25C3⋊#95C2212yescomplemented
26C22×C22⋊C4—31no—
27C22×C22⋊C4—32no—
28#385—34no—
29#385—34no—
30#380—31no—
31#380—32no—
32#380D311yescharacteristic, complemented
33C24×C3C2311yescharacteristic
34C23×D3—42no—
35C23×D3—42no—
36C23×D3C2×C414yes—
37C23×D3D414yes—
38C23×D3Q814yes—
39C23×D3C2312yes—
40C22×C3×C4—42no—
41C22×C3×C4C2×C414yescomplemented
42C22×C3×C4C2314yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×Dic3—42no—
46C22×Dic3C2×C414yescomplemented
47C22×Dic3C2314yes—
48C22×Dic3C2312yes—
49C2×C4×D3—28no—
50C2×C4×D3—44no—
51C2×C4×D3—44no—
52C3×C4⋊C4—42no—
53C3×C22⋊C4—24no—
54#217—24no—
55(C2×D3)⋊C4—28no—
56(C2×D3)⋊C4C2×C418yescomplemented
57(C2×D3)⋊C4C2×C418yescomplemented
58C4⋊Dic3—42no—
59Dic3⋊C4—44no—
60C25—31no—
61C23×C4—31no—
62C23×C4—34no—
63C23×C4C2×D311yescharacteristic
64C2×C4⋊C4—34no—
65C2×C4⋊C4—32no—
66C2×C4⋊C4C2×D312yescomplemented
67C2×C22⋊C4—38no—
68C2×C22⋊C4—32no—
69C2×C22⋊C4—32no—
70C2×C22⋊C4—32no—
71C2×C22⋊C4—34no—
72C2×C22⋊C4—68no—
73C2×C22⋊C4C2×D312yes—
74#95—34no—
75#95—32no—
76#95C2×D312yescomplemented
77C23×C3—22no—
78C23×C3—22no—
79C23×C3C22×C412yes—
80C23×C3C2×D412yes—
81C23×C3C2×Q812yes—
82C23×C3C2411yescharacteristic
83C22×D3—24no—
84C22×D3—24no—
85C22×D3—44no—
86C22×D3—44no—
87C22×D3—48no—
88C22×D3—48no—
89C22×D3C4⋊C4116yescomplemented
90C22×D3C22×C414yes—
91C22×D3C2×D414yes—
92C22×D3C2×Q814yes—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—24no—
96C2×C3×C4—24no—
97C2×C3×C4—24no—
98C2×C3×C4—44no—
99C2×C3×C4C22×C412yes—
100C2×C3×C4C22×C412yes—
101C2×Dic3—24no—
102C2×Dic3—24no—
103C2×Dic3—24no—
104C2×Dic3—24no—
105C2×Dic3—44no—
106C2×Dic3C22×C414yes—
107C4×D3—416no—
108C24—32no—
109C24—34no—
110C24—34no—
111C24—34no—
112C24—122no—
113C24—122no—
114C24C22×D311yescharacteristic
115C22×C4—32no—
116C22×C4—34no—
117C22×C4—34no—
118C22×C4—42no—
119C22×C4—68no—
120C22×C4—124no—
121C22×C4—124no—
122C22×C4—122no—
123C22×C4C4×D314yescomplemented
124C22×C4C22×D314yes—
125C22×C4C22×D311yescharacteristic
126C22×C4C22×D311yescharacteristic
127C4⋊C4—42no—
128C4⋊C4—124no—
129C4⋊C4—122no—
130C22⋊C4—24no—
131C22⋊C4—38no—
132C22⋊C4—38no—
133C22⋊C4—68no—
134C22⋊C4—64no—
135C22×C3—22no—
136C22×C3—22no—
137C22×C3—24no—
138C22×C3—24no—
139C22×C3C2×C4⋊C414yes—
140C22×C3C23×C411yescharacteristic
141C22×C3C22×D411yescharacteristic
142C22×C3C22×Q811yescharacteristic
143C22×C3H(5,𝔽2)12yes—
144C22×C3H(5,𝔽2)12yes—
145C2×D3—24no—
146C2×D3—24no—
147C2×D3—28no—
148C2×D3—28no—
149C2×D3—416no—
150C2×D3C2×C4⋊C418yes—
151C3×C4—22no—
152C3×C4—22no—
153C3×C4—44no—
154Dic3—24no—
155Dic3—44no—
156C23—22no—
157C23—22no—
158C23—34no—
159C23—34no—
160C23—34no—
161C23—316no—
162C23—64no—
163C23—64no—
164C23—128no—
165C23—124no—
166C23—124no—
167C23—128no—
168C23C2×C4×D312yes—
169C23D3×D412yes—
170C23D3×Q812yes—
171C23C23×D311yescharacteristic
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—34no—
178C2×C4—44no—
179C2×C4—64no—
180C2×C4—64no—
181C2×C4—64no—
182C2×C4—64no—
183C2×C4—1216no—
184C2×C4—124no—
185C2×C4C2×C4×D312yes—
186C2×C4C2×C4×D312yes—
187C2×C3—24no—
188C2×C3C22×C4⋊C411yescharacteristic
189C2×C3#51811yescharacteristic
190C2×C3#51811yescharacteristic
191C2×C3#53412yes—
192C2×C3#54312yes—
193D3—28no—
194C22—24no—
195C22—22no—
196C22—22no—
197C22—24no—
198C22—38no—
199C22—68no—
200C22—64no—
201C22—64no—
202C22—68no—
203C22—1216no—
204C22D3×C4⋊C414yes—
205C22C22×C4×D311yescharacteristic
206C22C2×D3×D411yescharacteristic
207C22#100412yes—
208C22C2×D3×Q811yescharacteristic
209C22#100912yes—
210C4—22no—
211C4—22no—
212C4—44no—
213C4—64no—
214C4—124no—
215C3#231511yescharacteristic, complemented, Sylow
216C2—24no—
217C2—68no—
218C2#548611yescharacteristic
219C2C2×D3×C4⋊C411yescharacteristic
220C2#552111yescharacteristic
221C2#560512yes—
222C2#563412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#380C2
socle, cosocleC23×C3C24
prime radicalC24×D3C22
hypercenter#380D3
hypocenterC3#2315
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C3×#380C2semidirect
4. #5212C2semidirect
5. #4989C2semidirect
6. #4942C2semidirect
7. #4668C2semidirect
13. C2×C3×C4⋊C4C22semidirect
16. C2×(C2×D3)⋊C4C4semidirect
20. C2×C4⋊Dic3C22semidirect
21. C2×Dic3⋊C4C22semidirect
23. C3×#95C22semidirect
24. C3⋊#95C22semidirect
25. C3⋊#95C22semidirect
32. #380D3semidirect
41. C22×C3×C4C2×C4semidirect
46. C22×Dic3C2×C4semidirect
56. (C2×D3)⋊C4C2×C4semidirect
57. (C2×D3)⋊C4C2×C4semidirect
66. C2×C4⋊C4C2×D3semidirect
76. #95C2×D3semidirect
89. C22×D3C4⋊C4semidirect
123. C22×C4C4×D3semidirect
215. C3#2315semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D3)⋊C42
C2×D3×C22⋊C42
C3×#3802
#52122
#49892
#49422
#46682
#23153