Finite Group Databasebeta

← C2×C3×#324 #72172

Subgroups — C2×C3×#324 #72172

Order 384.

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

normal subgroups only

This group has 328 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×C4⋊C4C211yesmaximal, characteristic
2C2×C3×C22⋊C8C212yesmaximal
3C3×#324C214yesmaximal, direct product factor
4C2×#324C311yesmaximal, characteristic, direct product factor, Sylow
5C23×C3×C4C411yescharacteristic
6C23×C3×C4C411yescharacteristic
7C23×C3×C4C2211yescharacteristic
8C22×C3×C8—22no—
9C2×C3×C4⋊C4—22no—
10C2×C3×C4⋊C4—22no—
11C2×C3×C4⋊C4C412yes—
12C2×C3×C4⋊C4C2212yes—
13C3×C22⋊C8—22no—
14C3×C22⋊C8C2214yes—
15C22×C4⋊C4C2×C311yescharacteristic
16C2×C22⋊C8C2×C312yes—
17#324C2×C314yesdirect product factor
18C24×C3C2×C411yescharacteristic
19C22×C3×C4—21no—
20C22×C3×C4—21no—
21C22×C3×C4—22no—
22C22×C3×C4—21no—
23C22×C3×C4—21no—
24C22×C3×C4—42no—
25C22×C3×C4—42no—
26C22×C3×C4C814yescomplemented
27C22×C3×C4C2×C412yes—
28C22×C3×C4C2×C412yes—
29C22×C3×C4C2×C411yescharacteristic
30C22×C3×C4D412yes—
31C22×C3×C4D412yes—
32C22×C3×C4C2311yescharacteristic
33C2×C3×C8—24no—
34C2×C3×C8—44no—
35C3×C4⋊C4—24no—
36C3×C4⋊C4—24no—
37C23×C4C3×C411yescharacteristic
38C23×C4C3×C411yescharacteristic
39C23×C4C22×C311yescharacteristic
40C22×C8—22no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4C3×C412yes—
44C2×C4⋊C4C22×C312yes—
45C22⋊C8—22no—
46C22⋊C8C22×C314yes—
47C23×C3—22no—
48C23×C3—22no—
49C23×C3C22⋊C411yescharacteristic
50C23×C3C22⋊C411yescharacteristic
51C23×C3C2×C812yes—
52C23×C3M1612yes—
53C23×C3C22×C411yescharacteristic
54C2×C3×C4—24no—
55C2×C3×C4—24no—
56C2×C3×C4—24no—
57C2×C3×C4—22no—
58C2×C3×C4—22no—
59C2×C3×C4—22no—
60C2×C3×C4—41no—
61C2×C3×C4—42no—
62C2×C3×C4—41no—
63C2×C3×C4—42no—
64C2×C3×C4—42no—
65C2×C3×C4—42no—
66C2×C3×C4C22⋊C412yes—
67C2×C3×C4C2×C814yescomplemented
68C2×C3×C4C2×D412yes—
69C3×C8—44no—
70C24C2×C3×C411yescharacteristic
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4C3×C814yescomplemented
79C22×C4C2×C3×C412yes—
80C22×C4C2×C3×C412yes—
81C22×C4C2×C3×C411yescharacteristic
82C22×C4C3×D412yes—
83C22×C4C3×D412yes—
84C22×C4C23×C311yescharacteristic
85C2×C8—24no—
86C2×C8—44no—
87C4⋊C4—24no—
88C4⋊C4—24no—
89C22×C3—24no—
90C22×C3—24no—
91C22×C3—21no—
92C22×C3—21no—
93C22×C3—21no—
94C22×C3—21no—
95C22×C3C22⋊C812yes—
96C22×C3C22⋊C812yes—
97C22×C3#9912yes—
98C22×C3#10112yes—
99C22×C3C2×C22⋊C411yescharacteristic
100C22×C3C22×C811yescharacteristic
101C22×C3C2×M1611yescharacteristic
102C3×C4—24no—
103C3×C4—22no—
104C3×C4—22no—
105C3×C4—42no—
106C23—22no—
107C23—22no—
108C23C3×C22⋊C411yescharacteristic
109C23C3×C22⋊C411yescharacteristic
110C23C2×C3×C812yes—
111C23C3×M1612yes—
112C23C22×C3×C411yescharacteristic
113C2×C4—24no—
114C2×C4—24no—
115C2×C4—24no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—41no—
120C2×C4—42no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4C3×C22⋊C412yes—
126C2×C4C2×C3×C814yescomplemented
127C2×C4C2×C3×D412yes—
128C8—44no—
129C2×C3—22no—
130C2×C3—21no—
131C2×C3—21no—
132C2×C3#32414yesdirect product factor
133C2×C3C2×C22⋊C811yescharacteristic
134C2×C3C2×#9911yescharacteristic
135C2×C3C2×#10111yescharacteristic
136C22—21no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—24no—
141C22—24no—
142C22C3×C22⋊C812yes—
143C22C3×C22⋊C812yes—
144C22C3×#9912yes—
145C22C3×#10112yes—
146C22C2×C3×C22⋊C411yescharacteristic
147C22C22×C3×C811yescharacteristic
148C22C2×C3×M1611yescharacteristic
149C4—24no—
150C4—22no—
151C4—22no—
152C4—42no—
153C3C2×#32411yescharacteristic, direct product factor, Sylow
154C2—22no—
155C2—21no—
156C2—21no—
157C2C3×#32414yesdirect product factor
158C2C2×C3×C22⋊C811yescharacteristic
159C2C2×C3×#9911yescharacteristic
160C2C2×C3×#10111yescharacteristic
subgroupG/H
center, socleC23×C3C22⋊C4
commutatorC22C22×C3×C8
Frattini, cosocleC22×C4C23×C3
prime radicalC24×C3C2×C4
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
3. C3×#324C2direct (N × H)
4. C2×#324C3direct (N × H)
17. #324C2×C3direct (N × H)
26. C22×C3×C4C8semidirect
67. C2×C3×C4C2×C8semidirect
78. C22×C4C3×C8semidirect
126. C2×C4C2×C3×C8semidirect
132. C2×C3#324direct (N × H)
153. C3C2×#324direct (N × H)
157. C2C3×#324direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
C2×C3×C22⋊C82
C3×#3242
C2×#3243