Finite Group Databasebeta

← C2×C3×#323 #72171

Subgroups — C2×C3×#323 #72171

Order 384.

456 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 136 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×C3×C22⋊C4C211yesmaximal, characteristic
2C2×C3×C22⋊C8C212yesmaximal, complemented
3C3×#323C214yesmaximal, direct product factor
4C2×#323C311yesmaximal, characteristic, direct product factor, Sylow
5C25×C3C411yescharacteristic
6C23×C3×C4C411yescharacteristic
7C23×C3×C4C2211yescharacteristic
8C22×C3×C8—22no—
9C2×C3×C22⋊C4—22no—
10C2×C3×C22⋊C4—22no—
11C2×C3×C22⋊C4C412yes—
12C2×C3×C22⋊C4C2212yes—
13C3×C22⋊C8—22no—
14C3×C22⋊C8C2214yescomplemented
15C22×C22⋊C4C2×C311yescharacteristic
16C2×C22⋊C8C2×C312yescomplemented
17#323C2×C314yesdirect product factor
18C24×C3—21no—
19C24×C3—21no—
20C24×C3—22no—
21C24×C3—42no—
22C24×C3—42no—
23C24×C3C814yescomplemented
24C24×C3C2×C411yescharacteristic
25C24×C3C2×C412yes—
26C22×C3×C4—21no—
27C22×C3×C4—21no—
28C22×C3×C4—42no—
29C22×C3×C4—42no—
30C22×C3×C4C2×C412yes—
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4D412yes—
33C22×C3×C4D412yes—
34C22×C3×C4C2311yescharacteristic
35C2×C3×C8—24no—
36C2×C3×C8—44no—
37C3×C22⋊C4—24no—
38C3×C22⋊C4—24no—
39C25C3×C411yescharacteristic
40C23×C4C3×C411yescharacteristic
41C23×C4C22×C311yescharacteristic
42C22×C8—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4C3×C412yes—
46C2×C22⋊C4C22×C312yes—
47C22⋊C8—22no—
48C22⋊C8C22×C314yescomplemented
49C23×C3—22no—
50C23×C3—22no—
51C23×C3—24no—
52C23×C3—24no—
53C23×C3—22no—
54C23×C3—22no—
55C23×C3—48no—
56C23×C3—42no—
57C23×C3—42no—
58C23×C3—48no—
59C23×C3—42no—
60C23×C3—42no—
61C23×C3—42no—
62C23×C3—42no—
63C23×C3C22⋊C411yescharacteristic
64C23×C3C22⋊C411yescharacteristic
65C23×C3C2×C812yes—
66C23×C3C2×C814yescomplemented
67C23×C3M1612yes—
68C23×C3C22×C411yescharacteristic
69C2×C3×C4—24no—
70C2×C3×C4—22no—
71C2×C3×C4—41no—
72C2×C3×C4—42no—
73C2×C3×C4—41no—
74C2×C3×C4—42no—
75C2×C3×C4—42no—
76C2×C3×C4—42no—
77C2×C3×C4C22⋊C412yes—
78C2×C3×C4C2×D412yes—
79C3×C8—44no—
80C24—21no—
81C24—21no—
82C24—22no—
83C24—42no—
84C24—42no—
85C24C3×C814yescomplemented
86C24C2×C3×C411yescharacteristic
87C24C2×C3×C412yes—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4C2×C3×C412yes—
93C22×C4C2×C3×C411yescharacteristic
94C22×C4C3×D412yes—
95C22×C4C3×D412yes—
96C22×C4C23×C311yescharacteristic
97C2×C8—24no—
98C2×C8—44no—
99C22⋊C4—24no—
100C22⋊C4—24no—
101C22×C3—24no—
102C22×C3—24no—
103C22×C3—21no—
104C22×C3—21no—
105C22×C3—21no—
106C22×C3—21no—
107C22×C3—24no—
108C22×C3—44no—
109C22×C3—48no—
110C22×C3—44no—
111C22×C3—48no—
112C22×C3—42no—
113C22×C3—42no—
114C22×C3C22⋊C812yes—
115C22×C3C22⋊C812yes—
116C22×C3#9912yes—
117C22×C3#10012yes—
118C22×C3C2×C22⋊C411yescharacteristic
119C22×C3C22×C811yescharacteristic
120C22×C3C2×M1611yescharacteristic
121C3×C4—22no—
122C3×C4—22no—
123C3×C4—42no—
124C23—22no—
125C23—22no—
126C23—24no—
127C23—24no—
128C23—22no—
129C23—22no—
130C23—48no—
131C23—42no—
132C23—42no—
133C23—42no—
134C23—42no—
135C23—48no—
136C23—42no—
137C23—42no—
138C23C3×C22⋊C411yescharacteristic
139C23C3×C22⋊C411yescharacteristic
140C23C2×C3×C812yes—
141C23C2×C3×C814yescomplemented
142C23C3×M1612yes—
143C23C22×C3×C411yescharacteristic
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4C3×C22⋊C412yes—
153C2×C4C2×C3×D412yes—
154C8—44no—
155C2×C3—22no—
156C2×C3—21no—
157C2×C3—21no—
158C2×C3—44no—
159C2×C3#32314yesdirect product factor
160C2×C3C2×C22⋊C811yescharacteristic
161C2×C3C2×#9911yescharacteristic
162C2×C3C2×#10011yescharacteristic
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—24no—
168C22—24no—
169C22—24no—
170C22—44no—
171C22—48no—
172C22—44no—
173C22—42no—
174C22—42no—
175C22—48no—
176C22C3×C22⋊C812yes—
177C22C3×C22⋊C812yes—
178C22C3×#9912yes—
179C22C3×#10012yes—
180C22C2×C3×C22⋊C411yescharacteristic
181C22C22×C3×C811yescharacteristic
182C22C2×C3×M1611yescharacteristic
183C4—22no—
184C4—22no—
185C4—42no—
186C3C2×#32311yescharacteristic, direct product factor, Sylow
187C2—22no—
188C2—21no—
189C2—21no—
190C2—44no—
191C2C3×#32314yesdirect product factor
192C2C2×C3×C22⋊C811yescharacteristic
193C2C2×C3×#9911yescharacteristic
194C2C2×C3×#10011yescharacteristic
subgroupG/H
center, socleC23×C3C22⋊C4
commutatorC22C22×C3×C8
Frattini, cosocleC22×C4C23×C3
prime radicalC25×C3C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C22⋊C8C2semidirect
3. C3×#323C2direct (N × H)
4. C2×#323C3direct (N × H)
14. C3×C22⋊C8C22semidirect
16. C2×C22⋊C8C2×C3semidirect
17. #323C2×C3direct (N × H)
23. C24×C3C8semidirect
48. C22⋊C8C22×C3semidirect
66. C23×C3C2×C8semidirect
85. C24C3×C8semidirect
141. C23C2×C3×C8semidirect
159. C2×C3#323direct (N × H)
186. C3C2×#323direct (N × H)
191. C2C3×#323direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊C42
C2×C3×C22⋊C82
C3×#3232
C2×#3233