Finite Group Databasebeta

← C3×#1826 #72541

Subgroups — C3×#1826 #72541

Order 384.

306 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 108 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×#407C211yesmaximal, characteristic
3C3×C4×C22⋊C4C211yesmaximal, characteristic
4C3×#339C214yesmaximal, complemented
5#1826C311yesmaximal, characteristic, direct product factor, Sylow
6C23×C3×C4C2211yescharacteristic
7C2×C3×M16C2212yes—
8C3×#117—21no—
9C3×#117—21no—
10C3×#117C412yescomplemented
11C3×#117C2212yes—
12C2×C3×C4⋊C4C411yescharacteristic, complemented
13C2×C3×C4⋊C4C411yescharacteristic, complemented
14C2×C3×C22⋊C4—21no—
15C2×C3×C22⋊C4—21no—
16C2×C3×C42—21no—
17C2×C3×C42C2212yes—
18C3×C22⋊C8—21no—
19C3×C22⋊C8—21no—
20C3×#95—21no—
21C2×#117C2×C311yescharacteristic
22#407C2×C311yescharacteristic
23C4×C22⋊C4C2×C311yescharacteristic
24#339C2×C314yescomplemented
25C24×C3D411yescharacteristic
26C22×C3×C4—24no—
27C22×C3×C4—21no—
28C22×C3×C4—21no—
29C22×C3×C4—22no—
30C22×C3×C4C2×C411yescharacteristic
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4D411yescharacteristic
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4C2311yescharacteristic
36C3×M16—24no—
37C2×C3×C8—22no—
38C3×C4⋊C4—21no—
39C3×C4⋊C4—21no—
40C3×C4⋊C4C2×C412yes—
41C3×C4⋊C4C2×C412yes—
42C3×C22⋊C4—24no—
43C3×C22⋊C4—22no—
44C3×C22⋊C4—22no—
45C3×C42—22no—
46C3×C42—22no—
47C3×C42—24no—
48C23×C4C22×C311yescharacteristic
49C2×M16C22×C312yes—
50#117—21no—
51#117—21no—
52#117C3×C412yescomplemented
53#117C22×C312yes—
54C2×C4⋊C4C3×C411yescharacteristic, complemented
55C2×C4⋊C4C3×C411yescharacteristic, complemented
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C42—21no—
59C2×C42C22×C312yes—
60C22⋊C8—21no—
61C22⋊C8—21no—
62#95—21no—
63C23×C3—22no—
64C23×C3—22no—
65C23×C3—41no—
66C23×C3C22⋊C411yescharacteristic
67C23×C3C22⋊C411yescharacteristic
68C23×C3C2×D411yescharacteristic
69C2×C3×C4—22no—
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—22no—
74C2×C3×C4—24no—
75C2×C3×C4—22no—
76C2×C3×C4—22no—
77C2×C3×C4—42no—
78C2×C3×C4—42no—
79C2×C3×C4—42no—
80C2×C3×C4—42no—
81C2×C3×C4C22⋊C411yescharacteristic
82C2×C3×C4C22⋊C411yescharacteristic
83C2×C3×C4C22×C411yescharacteristic
84C2×C3×C4C2×D411yescharacteristic
85C2×C3×C4#4114yes—
86C3×C8—42no—
87C24C3×D411yescharacteristic
88C22×C4—24no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4C2×C3×C411yescharacteristic
93C22×C4C2×C3×C411yescharacteristic
94C22×C4C3×D411yescharacteristic
95C22×C4C3×D411yescharacteristic
96C22×C4C3×D411yescharacteristic
97C22×C4C23×C311yescharacteristic
98M16—24no—
99C2×C8—22no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4C2×C3×C412yes—
103C4⋊C4C2×C3×C412yes—
104C22⋊C4—24no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C42—22no—
108C42—22no—
109C42—24no—
110C22×C3—21no—
111C22×C3—21no—
112C22×C3—21no—
113C22×C3—21no—
114C22×C3—21no—
115C22×C3—21no—
116C22×C3—21no—
117C22×C3—44no—
118C22×C3#10412yes—
119C22×C3C2×C22⋊C411yescharacteristic
120C22×C3#12312yes—
121C3×C4—21no—
122C3×C4—21no—
123C3×C4—44no—
124C3×C4—42no—
125C3×C4—42no—
126C3×C4#11712yes—
127C3×C4#12312yes—
128C23—22no—
129C23—22no—
130C23—41no—
131C23C3×C22⋊C411yescharacteristic
132C23C3×C22⋊C411yescharacteristic
133C23C2×C3×D411yescharacteristic
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—24no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4C3×C22⋊C411yescharacteristic
147C2×C4C3×C22⋊C411yescharacteristic
148C2×C4C22×C3×C411yescharacteristic
149C2×C4C2×C3×D411yescharacteristic
150C2×C4C3×#4114yes—
151C8—42no—
152C2×C3—22no—
153C2×C3—22no—
154C2×C3—41no—
155C2×C3#38111yescharacteristic
156C2×C3C2×#10411yescharacteristic
157C2×C3#42111yescharacteristic
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—44no—
166C22C3×#10412yes—
167C22C2×C3×C22⋊C411yescharacteristic
168C22C3×#12312yes—
169C4—21no—
170C4—21no—
171C4—44no—
172C4—42no—
173C4—42no—
174C4C3×#11712yes—
175C4C3×#12312yes—
176C3#182611yescharacteristic, direct product factor, Sylow
177C2—22no—
178C2—22no—
179C2—41no—
180C2C3×#38111yescharacteristic
181C2C2×C3×#10411yescharacteristic
182C2C3×#42111yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC24×C3D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#339C2semidirect
5. #1826C3direct (N × H)
10. C3×#117C4semidirect
12. C2×C3×C4⋊C4C4semidirect
13. C2×C3×C4⋊C4C4semidirect
24. #339C2×C3semidirect
52. #117C3×C4semidirect
54. C2×C4⋊C4C3×C4semidirect
55. C2×C4⋊C4C3×C4semidirect
176. C3#1826direct (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×#4072
C3×C4×C22⋊C42
C3×#3392
#18263