Finite Group Databasebeta

← C3×#1490 #72205

Subgroups — C3×#1490 #72205

Order 384.

316 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 104 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
2C3×#432C211yesmaximal, characteristic, complemented
3C3×C4×M16C211yesmaximal, characteristic, complemented
4C3×D4⋊C8C214yesmaximal, complemented
5#1490C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×D4C411yescharacteristic
7C23×C3×C4—21no—
8C2×C3×M16—21no—
9C3×C4×D4—21no—
10C3×C4×D4—21no—
11C3×C4×D4C412yes—
12C3×C4×D4C2212yes—
13C2×C3×C4⋊C4C411yescharacteristic
14C2×C3×C22⋊C4—21no—
15C2×C3×C42C2211yescharacteristic
16C3×C4⋊C8C2212yescomplemented
17C3×C22⋊C8—21no—
18C3×#97—21no—
19C3×#97—21no—
20C3×C4×C8C2212yescomplemented
21C2×C4×D4C2×C311yescharacteristic
22#432C2×C311yescharacteristic, complemented
23C4×M16C2×C311yescharacteristic, complemented
24D4⋊C8C2×C314yescomplemented
25C24×C3—21no—
26C2×C3×D4—24no—
27C2×C3×D4—21no—
28C2×C3×D4C2×C412yes—
29C22×C3×C4—21no—
30C22×C3×C4—22no—
31C22×C3×C4—22no—
32C22×C3×C4—44no—
33C22×C3×C4C2×C411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C3×M16—24no—
37C2×C3×C8—22no—
38C2×C3×C8—22no—
39C3×C4⋊C4—21no—
40C3×C4⋊C4C2×C412yes—
41C3×C22⋊C4—22no—
42C3×C22⋊C4—22no—
43C3×C42C2×C411yescharacteristic
44C3×C42D411yescharacteristic
45C3×C42D411yescharacteristic
46C3×C42C2311yescharacteristic
47C22×D4C3×C411yescharacteristic
48C23×C4—21no—
49C2×M16—21no—
50C4×D4—21no—
51C4×D4—21no—
52C4×D4C3×C412yes—
53C4×D4C22×C312yes—
54C2×C4⋊C4C3×C411yescharacteristic
55C2×C22⋊C4—21no—
56C2×C42C22×C311yescharacteristic
57C4⋊C8C22×C312yescomplemented
58C22⋊C8—21no—
59#97—21no—
60#97—21no—
61C4×C8C22×C312yescomplemented
62C23×C3—24no—
63C23×C3—22no—
64C23×C3—42no—
65C23×C3—42no—
66C23×C3C22⋊C411yescharacteristic
67C3×D4—24no—
68C3×D4—22no—
69C3×D4M1614yescomplemented
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—22no—
74C2×C3×C4—44no—
75C2×C3×C4—42no—
76C2×C3×C4—44no—
77C2×C3×C4C22⋊C411yescharacteristic
78C2×C3×C4C22⋊C411yescharacteristic
79C2×C3×C4C22⋊C411yescharacteristic
80C2×C3×C4C22×C411yescharacteristic
81C2×C3×C4C2×D411yescharacteristic
82C2×C3×C4C2×D411yescharacteristic
83C3×C8—24no—
84C3×C8—42no—
85C24—21no—
86C2×D4—24no—
87C2×D4—21no—
88C2×D4C2×C3×C412yes—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—44no—
93C22×C4C2×C3×C411yescharacteristic
94C22×C4C3×D411yescharacteristic
95C22×C4C3×D411yescharacteristic
96M16—24no—
97C2×C8—22no—
98C2×C8—22no—
99C4⋊C4—21no—
100C4⋊C4C2×C3×C412yes—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C42C2×C3×C411yescharacteristic
104C42C3×D411yescharacteristic
105C42C3×D411yescharacteristic
106C42C23×C311yescharacteristic
107C22×C3—21no—
108C22×C3—21no—
109C22×C3—24no—
110C22×C3—48no—
111C22×C3—42no—
112C22×C3—42no—
113C22×C3#10412yes—
114C22×C3C2×C22⋊C411yescharacteristic
115C3×C4—21no—
116C3×C4—21no—
117C3×C4—21no—
118C3×C4—21no—
119C3×C4—42no—
120C3×C4C2×M1612yes—
121C3×C4AGL(1,ℤ/8)12yes—
122C23—24no—
123C23—22no—
124C23—42no—
125C23—42no—
126C23C3×C22⋊C411yescharacteristic
127D4—24no—
128D4—22no—
129D4C3×M1614yescomplemented
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—44no—
135C2×C4—42no—
136C2×C4—44no—
137C2×C4C3×C22⋊C411yescharacteristic
138C2×C4C3×C22⋊C411yescharacteristic
139C2×C4C3×C22⋊C411yescharacteristic
140C2×C4C22×C3×C411yescharacteristic
141C2×C4C2×C3×D411yescharacteristic
142C2×C4C2×C3×D411yescharacteristic
143C8—24no—
144C8—42no—
145C2×C3—22no—
146C2×C3—44no—
147C2×C3#40711yescharacteristic
148C2×C3#41811yescharacteristic
149C2×C3C2×#10411yescharacteristic
150C22—21no—
151C22—21no—
152C22—24no—
153C22—48no—
154C22—42no—
155C22—42no—
156C22C3×#10412yes—
157C22C2×C3×C22⋊C411yescharacteristic
158C4—21no—
159C4—21no—
160C4—21no—
161C4—21no—
162C4—42no—
163C4C2×C3×M1612yes—
164C4C3×AGL(1,ℤ/8)12yes—
165C3#149011yescharacteristic, direct product factor, Sylow
166C2—22no—
167C2—44no—
168C2C3×#40711yescharacteristic
169C2C3×#41811yescharacteristic
170C2C2×C3×#10411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC42C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC22×C3×D4C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#432C2semidirect
3. C3×C4×M16C2semidirect
4. C3×D4⋊C8C2semidirect
5. #1490C3direct (N × H)
16. C3×C4⋊C8C22semidirect
20. C3×C4×C8C22semidirect
22. #432C2×C3semidirect
23. C4×M16C2×C3semidirect
24. D4⋊C8C2×C3semidirect
57. C4⋊C8C22×C3semidirect
61. C4×C8C22×C3semidirect
69. C3×D4M16semidirect
129. D4C3×M16semidirect
165. C3#1490direct (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
C3×#4322
C3×C4×M162
C3×D4⋊C82
#14903