Finite Group Databasebeta

← C2×C3×D4⋊C8 #72189

Subgroups — C2×C3×D4⋊C8 #72189

Order 384.

344 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C2×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×C4⋊C8C211yesmaximal, characteristic, complemented
3C2×C3×C4×C8C211yesmaximal, characteristic, complemented
4C3×D4⋊C8C214yesmaximal, direct product factor
5C2×D4⋊C8C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×D4C411yescharacteristic
7C23×C3×C4—21no—
8C22×C3×C8—21no—
9C22×C3×C8—21no—
10C3×C4×D4—21no—
11C3×C4×D4—21no—
12C3×C4×D4C412yes—
13C3×C4×D4C2212yes—
14C2×C3×C4⋊C4C411yescharacteristic
15C2×C3×C22⋊C4—21no—
16C2×C3×C42C2211yescharacteristic
17C3×C4⋊C8—21no—
18C3×C4⋊C8C2212yescomplemented
19C3×C4×C8—21no—
20C3×C4×C8C2212yescomplemented
21C2×C4×D4C2×C311yescharacteristic
22C2×C4⋊C8C2×C311yescharacteristic, complemented
23C2×C4×C8C2×C311yescharacteristic, complemented
24D4⋊C8C2×C314yesdirect product factor
25C24×C3—21no—
26C2×C3×D4—21no—
27C2×C3×D4—22no—
28C2×C3×D4C814yescomplemented
29C2×C3×D4C2×C412yes—
30C22×C3×C4—21no—
31C22×C3×C4—22no—
32C22×C3×C4—22no—
33C22×C3×C4—42no—
34C22×C3×C4—42no—
35C22×C3×C4C2×C411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4D411yescharacteristic
38C2×C3×C8—24no—
39C2×C3×C8—22no—
40C2×C3×C8—22no—
41C2×C3×C8—42no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4C2×C412yes—
44C3×C22⋊C4—22no—
45C3×C22⋊C4—22no—
46C3×C42C2×C411yescharacteristic
47C3×C42D411yescharacteristic
48C3×C42D411yescharacteristic
49C3×C42C2311yescharacteristic
50C22×D4C3×C411yescharacteristic
51C23×C4—21no—
52C22×C8—21no—
53C22×C8—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4C3×C412yes—
57C4×D4C22×C312yes—
58C2×C4⋊C4C3×C411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C42C22×C311yescharacteristic
61C4⋊C8—21no—
62C4⋊C8C22×C312yescomplemented
63C4×C8—21no—
64C4×C8C22×C312yescomplemented
65C23×C3—24no—
66C23×C3—22no—
67C23×C3—42no—
68C23×C3—42no—
69C23×C3C22⋊C411yescharacteristic
70C3×D4—24no—
71C3×D4—22no—
72C3×D4C2×C814yescomplemented
73C2×C3×C4—22no—
74C2×C3×C4—22no—
75C2×C3×C4—42no—
76C2×C3×C4—44no—
77C2×C3×C4—42no—
78C2×C3×C4—42no—
79C2×C3×C4C22⋊C411yescharacteristic
80C2×C3×C4C22⋊C411yescharacteristic
81C2×C3×C4C22⋊C411yescharacteristic
82C2×C3×C4C2×C812yes—
83C2×C3×C4M1612yes—
84C2×C3×C4D812yes—
85C2×C3×C4SD1612yes—
86C2×C3×C4C22×C411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88C2×C3×C4C2×D411yescharacteristic
89C3×C8—24no—
90C3×C8—42no—
91C24—21no—
92C2×D4—21no—
93C2×D4—22no—
94C2×D4C3×C814yescomplemented
95C2×D4C2×C3×C412yes—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4C2×C3×C411yescharacteristic
102C22×C4C3×D411yescharacteristic
103C22×C4C3×D411yescharacteristic
104C2×C8—24no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—42no—
108C4⋊C4—21no—
109C4⋊C4C2×C3×C412yes—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42C2×C3×C411yescharacteristic
113C42C3×D411yescharacteristic
114C42C3×D411yescharacteristic
115C42C23×C311yescharacteristic
116C22×C3—24no—
117C22×C3—48no—
118C22×C3—42no—
119C22×C3—42no—
120C22×C3C22⋊C812yes—
121C22×C3#10212yes—
122C22×C3#10412yes—
123C22×C3C2×C22⋊C411yescharacteristic
124C3×C4—21no—
125C3×C4—21no—
126C3×C4—42no—
127C3×C4C22⋊C812yes—
128C3×C4#10212yes—
129C3×C4C22×C811yescharacteristic
130C3×C4C2×M1611yescharacteristic
131C3×C4C2×D811yescharacteristic
132C3×C4C2×SD1611yescharacteristic
133C23—24no—
134C23—22no—
135C23—42no—
136C23—42no—
137C23C3×C22⋊C411yescharacteristic
138D4—24no—
139D4—22no—
140D4C2×C3×C814yescomplemented
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—44no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4C3×C22⋊C411yescharacteristic
148C2×C4C3×C22⋊C411yescharacteristic
149C2×C4C3×C22⋊C411yescharacteristic
150C2×C4C2×C3×C812yes—
151C2×C4C3×M1612yes—
152C2×C4C3×D812yes—
153C2×C4C3×SD1612yes—
154C2×C4C22×C3×C411yescharacteristic
155C2×C4C2×C3×D411yescharacteristic
156C2×C4C2×C3×D411yescharacteristic
157C8—24no—
158C8—42no—
159C2×C3—44no—
160C2×C3D4⋊C814yesdirect product factor
161C2×C3C2×C22⋊C811yescharacteristic
162C2×C3C2×#10211yescharacteristic
163C2×C3C2×#10411yescharacteristic
164C22—24no—
165C22—42no—
166C22—42no—
167C22—48no—
168C22C3×C22⋊C812yes—
169C22C3×#10212yes—
170C22C3×#10412yes—
171C22C2×C3×C22⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—42no—
175C4C3×C22⋊C812yes—
176C4C3×#10212yes—
177C4C22×C3×C811yescharacteristic
178C4C2×C3×M1611yescharacteristic
179C4C2×C3×D811yescharacteristic
180C4C2×C3×SD1611yescharacteristic
181C3C2×D4⋊C811yescharacteristic, direct product factor, Sylow
182C2—44no—
183C2C3×D4⋊C814yesdirect product factor
184C2C2×C3×C22⋊C811yescharacteristic
185C2C2×C3×#10211yescharacteristic
186C2C2×C3×#10411yescharacteristic
subgroupG/H
centerC22×C3×C4D4
commutatorC4C22×C3×C8
Frattini, cosocleC42C23×C3
socleC23×C3C22⋊C4
prime radicalC22×C3×D4C4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊C8C2semidirect
3. C2×C3×C4×C8C2semidirect
4. C3×D4⋊C8C2direct (N × H)
5. C2×D4⋊C8C3direct (N × H)
18. C3×C4⋊C8C22semidirect
20. C3×C4×C8C22semidirect
22. C2×C4⋊C8C2×C3semidirect
23. C2×C4×C8C2×C3semidirect
24. D4⋊C8C2×C3direct (N × H)
28. C2×C3×D4C8semidirect
62. C4⋊C8C22×C3semidirect
64. C4×C8C22×C3semidirect
72. C3×D4C2×C8semidirect
94. C2×D4C3×C8semidirect
140. D4C2×C3×C8semidirect
160. C2×C3D4⋊C8direct (N × H)
181. C3C2×D4⋊C8direct (N × H)
183. C2C3×D4⋊C8direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×C3×C4⋊C82
C2×C3×C4×C82
C3×D4⋊C82
C2×D4⋊C83