Finite Group Databasebeta

← C3×#1458 #72173

Subgroups — C3×#1458 #72173

Order 384.

264 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 120 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×#117C211yesmaximal, characteristic
2C3×#431C212yesmaximal, complemented
3C3×#324C212yesmaximal, complemented
4C3×#323C212yesmaximal
5#1458C311yesmaximal, characteristic, direct product factor, Sylow
6C23×C3×C4C411yescharacteristic
7C3×#117—22no—
8C3×#117—22no—
9C2×C3×C4⋊C4C411yescharacteristic
10C2×C3×C4⋊C4C2211yescharacteristic
11C2×C3×C22⋊C4C411yescharacteristic
12C2×C3×C22⋊C4C2211yescharacteristic
13C2×C3×C42C411yescharacteristic
14C2×C3×C42C2211yescharacteristic
15C3×C4⋊C8—22no—
16C3×C22⋊C8C2214yes—
17C3×C4×C8—22no—
18C2×#117C2×C311yescharacteristic
19#431C2×C312yescomplemented
20#324C2×C312yescomplemented
21#323C2×C312yes—
22C24×C3C2×C411yescharacteristic
23C22×C3×C4—21no—
24C22×C3×C4—21no—
25C22×C3×C4—22no—
26C22×C3×C4C814yescomplemented
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4C2×C411yescharacteristic
29C22×C3×C4C2×C411yescharacteristic
30C22×C3×C4C2×C411yescharacteristic
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4C2311yescharacteristic
33C2×C3×C8—24no—
34C3×C4⋊C4—22no—
35C3×C4⋊C4—22no—
36C3×C22⋊C4—22no—
37C3×C22⋊C4—22no—
38C3×C42—21no—
39C3×C42—21no—
40C3×C42D412yes—
41C3×C42D412yes—
42C23×C4C3×C411yescharacteristic
43#117—22no—
44#117—22no—
45C2×C4⋊C4C3×C411yescharacteristic
46C2×C4⋊C4C22×C311yescharacteristic
47C2×C22⋊C4C3×C411yescharacteristic
48C2×C22⋊C4C22×C311yescharacteristic
49C2×C42C3×C411yescharacteristic
50C2×C42C22×C311yescharacteristic
51C4⋊C8—22no—
52C22⋊C8C22×C314yes—
53C4×C8—22no—
54C23×C3—21no—
55C23×C3—21no—
56C23×C3—41no—
57C23×C3—41no—
58C23×C3C2×C812yes—
59C23×C3C22×C411yescharacteristic
60C2×C3×C4—21no—
61C2×C3×C4—21no—
62C2×C3×C4—24no—
63C2×C3×C4—24no—
64C2×C3×C4—21no—
65C2×C3×C4—21no—
66C2×C3×C4—41no—
67C2×C3×C4—41no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4C22⋊C412yes—
71C2×C3×C4C22⋊C411yescharacteristic
72C2×C3×C4C22⋊C411yescharacteristic
73C2×C3×C4C2×C812yes—
74C2×C3×C4C2×C812yes—
75C2×C3×C4M1612yes—
76C2×C3×C4C2×D412yes—
77C3×C8—44no—
78C24C2×C3×C411yescharacteristic
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4C3×C814yescomplemented
83C22×C4C2×C3×C411yescharacteristic
84C22×C4C2×C3×C411yescharacteristic
85C22×C4C2×C3×C411yescharacteristic
86C22×C4C2×C3×C411yescharacteristic
87C22×C4C2×C3×C411yescharacteristic
88C22×C4C23×C311yescharacteristic
89C2×C8—24no—
90C4⋊C4—22no—
91C4⋊C4—22no—
92C22⋊C4—22no—
93C22⋊C4—22no—
94C42—21no—
95C42—21no—
96C42C3×D412yes—
97C42C3×D412yes—
98C22×C3—21no—
99C22×C3—21no—
100C22×C3—22no—
101C22×C3—42no—
102C22×C3—42no—
103C22×C3—41no—
104C22×C3—41no—
105C22×C3C2×C22⋊C411yescharacteristic
106C22×C3C22×C811yescharacteristic
107C22×C3C2×M1611yescharacteristic
108C3×C4—22no—
109C3×C4—22no—
110C3×C4—22no—
111C3×C4—41no—
112C3×C4—41no—
113C3×C4C22⋊C812yes—
114C3×C4C22⋊C812yes—
115C23—21no—
116C23—21no—
117C23—41no—
118C23—41no—
119C23C2×C3×C812yes—
120C23C22×C3×C411yescharacteristic
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—24no—
124C2×C4—24no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4C3×C22⋊C412yes—
132C2×C4C3×C22⋊C411yescharacteristic
133C2×C4C3×C22⋊C411yescharacteristic
134C2×C4C2×C3×C812yes—
135C2×C4C2×C3×C812yes—
136C2×C4C3×M1612yes—
137C2×C4C2×C3×D412yes—
138C8—44no—
139C2×C3—21no—
140C2×C3—21no—
141C2×C3—42no—
142C2×C3C2×C22⋊C811yescharacteristic
143C2×C3#41011yescharacteristic
144C2×C3#41311yescharacteristic
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22C2×C3×C22⋊C411yescharacteristic
153C22C22×C3×C811yescharacteristic
154C22C2×C3×M1611yescharacteristic
155C4—22no—
156C4—22no—
157C4—22no—
158C4—41no—
159C4—41no—
160C4C3×C22⋊C812yes—
161C4C3×C22⋊C812yes—
162C3#145811yescharacteristic, direct product factor, Sylow
163C2—21no—
164C2—21no—
165C2—42no—
166C2C2×C3×C22⋊C811yescharacteristic
167C2C3×#41011yescharacteristic
168C2C3×#41311yescharacteristic
subgroupG/H
centerC2×C3×C4C22⋊C4
commutatorC22C22×C3×C8
Frattini, cosocleC22×C4C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC24×C3C2×C4
LCSG ▷ C22 ▷ 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×#324C2semidirect
5. #1458C3direct (N × H)
19. #431C2×C3semidirect
20. #324C2×C3semidirect
26. C22×C3×C4C8semidirect
82. C22×C4C3×C8semidirect
162. C3#1458direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1172
C3×#4312
C3×#3242
C3×#3232
#14583