Finite Group Databasebeta

← C3×#1478 #72193

Subgroups — C3×#1478 #72193

Order 384.

312 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 124 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×#431C211yesmaximal, characteristic, complemented
3C3×#431C211yesmaximal, characteristic, complemented
4C3×D4⋊C8C214yesmaximal, complemented
5#1478C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×D4C411yescharacteristic
7C23×C3×C4—21no—
8C3×C4×D4—21no—
9C3×C4×D4—21no—
10C3×C4×D4C412yes—
11C3×C4×D4C2212yes—
12C2×C3×C4⋊C4C411yescharacteristic
13C2×C3×C22⋊C4—21no—
14C2×C3×C42C2211yescharacteristic
15C3×C4⋊C8—21no—
16C3×C4⋊C8C2212yescomplemented
17C3×C22⋊C8—21no—
18C3×C22⋊C8—21no—
19C3×C4×C8—21no—
20C3×C4×C8C2212yescomplemented
21C2×C4×D4C2×C311yescharacteristic
22#431C2×C311yescharacteristic, complemented
23#431C2×C311yescharacteristic, complemented
24D4⋊C8C2×C314yescomplemented
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—44no—
34C22×C3×C4C2×C411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4D411yescharacteristic
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—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4C3×C412yes—
52C4×D4C22×C312yes—
53C2×C4⋊C4C3×C411yescharacteristic
54C2×C22⋊C4—21no—
55C2×C42C22×C311yescharacteristic
56C4⋊C8—21no—
57C4⋊C8C22×C312yescomplemented
58C22⋊C8—21no—
59C22⋊C8—21no—
60C4×C8—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×D4C2×C814yes—
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—41no—
73C2×C3×C4—44no—
74C2×C3×C4—42no—
75C2×C3×C4—44no—
76C2×C3×C4C22⋊C411yescharacteristic
77C2×C3×C4C22⋊C411yescharacteristic
78C2×C3×C4C22⋊C411yescharacteristic
79C2×C3×C4C2×C812yes—
80C2×C3×C4M1612yes—
81C2×C3×C4C22×C411yescharacteristic
82C2×C3×C4C2×D411yescharacteristic
83C2×C3×C4C2×D411yescharacteristic
84C3×C8—42no—
85C3×C8—42no—
86C24—21no—
87C2×D4—21no—
88C2×D4—22no—
89C2×D4C3×C814yescomplemented
90C2×D4C2×C3×C412yes—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—44no—
95C22×C4C2×C3×C411yescharacteristic
96C22×C4C3×D411yescharacteristic
97C22×C4C3×D411yescharacteristic
98C2×C8—22no—
99C2×C8—22no—
100C4⋊C4—21no—
101C4⋊C4C2×C3×C412yes—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C42C2×C3×C411yescharacteristic
105C42C3×D411yescharacteristic
106C42C3×D411yescharacteristic
107C42C23×C311yescharacteristic
108C22×C3—21no—
109C22×C3—21no—
110C22×C3—24no—
111C22×C3—48no—
112C22×C3—42no—
113C22×C3—42no—
114C22×C3C22⋊C812yes—
115C22×C3C2×C22⋊C411yescharacteristic
116C3×C4—21no—
117C3×C4—21no—
118C3×C4—21no—
119C3×C4—42no—
120C3×C4C22⋊C812yes—
121C3×C4C22×C811yescharacteristic
122C3×C4C2×M1611yescharacteristic
123C3×C4AGL(1,ℤ/8)12yes—
124C23—24no—
125C23—22no—
126C23—42no—
127C23—42no—
128C23C3×C22⋊C411yescharacteristic
129D4—24no—
130D4—22no—
131D4C2×C3×C814yes—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—44no—
136C2×C4—42no—
137C2×C4—44no—
138C2×C4C3×C22⋊C411yescharacteristic
139C2×C4C3×C22⋊C411yescharacteristic
140C2×C4C3×C22⋊C411yescharacteristic
141C2×C4C2×C3×C812yes—
142C2×C4C3×M1612yes—
143C2×C4C22×C3×C411yescharacteristic
144C2×C4C2×C3×D411yescharacteristic
145C2×C4C2×C3×D411yescharacteristic
146C8—42no—
147C8—42no—
148C2×C3—22no—
149C2×C3—44no—
150C2×C3C2×C22⋊C811yescharacteristic
151C2×C3#41811yescharacteristic
152C2×C3#42111yescharacteristic
153C22—21no—
154C22—21no—
155C22—24no—
156C22—48no—
157C22—42no—
158C22—42no—
159C22C3×C22⋊C812yes—
160C22C2×C3×C22⋊C411yescharacteristic
161C4—21no—
162C4—21no—
163C4—21no—
164C4—42no—
165C4C3×C22⋊C812yes—
166C4C22×C3×C811yescharacteristic
167C4C2×C3×M1611yescharacteristic
168C4C3×AGL(1,ℤ/8)12yes—
169C3#147811yescharacteristic, direct product factor, Sylow
170C2—22no—
171C2—44no—
172C2C2×C3×C22⋊C811yescharacteristic
173C2C3×#41811yescharacteristic
174C2C3×#42111yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D4
commutatorC4C22×C3×C8
Frattini, cosocleC42C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC22×C3×D4C4
LCSG ▷ 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×#431C2semidirect
3. C3×#431C2semidirect
4. C3×D4⋊C8C2semidirect
5. #1478C3direct (N × H)
16. C3×C4⋊C8C22semidirect
20. C3×C4×C8C22semidirect
22. #431C2×C3semidirect
23. #431C2×C3semidirect
24. D4⋊C8C2×C3semidirect
28. C2×C3×D4C8semidirect
57. C4⋊C8C22×C3semidirect
61. C4×C8C22×C3semidirect
89. C2×D4C3×C8semidirect
169. C3#1478direct (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×#4312
C3×D4⋊C82
#14783