Finite Group Databasebeta

← #71539

Subgroups — #71539

Order 384.

440 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×C3×C22⋊C4C211yesmaximal, characteristic
2#5207C212yesmaximal, complemented
3#4525C214yesmaximal, complemented
4#1462—31nomaximal, Sylow
5C25×C3C411yescharacteristic
6C23×C3×C4C411yescharacteristic
7C23×C3×C4C2211yescharacteristic
8C2×C3×C22⋊C4—22no—
9C2×C3×C22⋊C4—22no—
10C2×C3×C22⋊C4C412yes—
11C2×C3×C22⋊C4C2212yes—
12C2×#208—22no—
13#830—22no—
14#830C2214yescomplemented
15C22×C22⋊C4D311yescharacteristic
16#407—32no—
17#323—34no—
18C24×C3—21no—
19C24×C3—21no—
20C24×C3—24no—
21C24×C3—42no—
22C24×C3—42no—
23C24×C3C2×C411yescharacteristic
24C24×C3C2×C412yes—
25C22×C3×C4—21no—
26C22×C3×C4—21no—
27C22×C3×C4—42no—
28C22×C3×C4—42no—
29C22×C3×C4C2×C412yes—
30C22×C3×C4C2×C411yescharacteristic
31C22×C3×C4D412yes—
32C22×C3×C4D412yes—
33C22×C3×C4C2311yescharacteristic
34C3×C22⋊C4—24no—
35C3×C22⋊C4—24no—
36#208—44no—
37C2×C3⋊C8—24no—
38C25Dic311yescharacteristic
39C23×C4Dic311yescharacteristic
40C23×C4C2×D311yescharacteristic
41C2×M16—62no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4Dic312yes—
45C2×C22⋊C4C2×D312yes—
46C22⋊C8—34no—
47C22⋊C8—62no—
48C23×C3—22no—
49C23×C3—22no—
50C23×C3—22no—
51C23×C3—24no—
52C23×C3—24no—
53C23×C3—22no—
54C23×C3—22no—
55C23×C3—42no—
56C23×C3—42no—
57C23×C3—416no—
58C23×C3—42no—
59C23×C3—42no—
60C23×C3—42no—
61C23×C3—42no—
62C23×C3C22⋊C411yescharacteristic
63C23×C3C22⋊C411yescharacteristic
64C23×C3M1614yescomplemented
65C23×C3C22×C411yescharacteristic
66C2×C3×C4—24no—
67C2×C3×C4—22no—
68C2×C3×C4—41no—
69C2×C3×C4—42no—
70C2×C3×C4—41no—
71C2×C3×C4—42no—
72C2×C3×C4—42no—
73C2×C3×C4—42no—
74C2×C3×C4C22⋊C412yes—
75C2×C3×C4C2×D412yes—
76C3⋊C8—44no—
77C24—24no—
78C24—21no—
79C24—21no—
80C24—42no—
81C24—42no—
82C24C2×Dic311yescharacteristic
83C24C2×Dic312yes—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4C2×Dic312yes—
89C22×C4C2×Dic311yescharacteristic
90C22×C4#6712yes—
91C22×C4#6712yes—
92C22×C4C22×D311yescharacteristic
93M16—124no—
94C2×C8—64no—
95C22⋊C4—24no—
96C22⋊C4—24no—
97C22×C3—28no—
98C22×C3—21no—
99C22×C3—21no—
100C22×C3—21no—
101C22×C3—21no—
102C22×C3—21no—
103C22×C3—21no—
104C22×C3—24no—
105C22×C3—44no—
106C22×C3—48no—
107C22×C3—48no—
108C22×C3—44no—
109C22×C3—42no—
110C22×C3—42no—
111C22×C3#9912yes—
112C22×C3#10012yes—
113C22×C3C2×C22⋊C411yescharacteristic
114C22×C3C2×M1612yes—
115C3×C4—22no—
116C3×C4—22no—
117C3×C4—42no—
118C23—22no—
119C23—22no—
120C23—22no—
121C23—24no—
122C23—24no—
123C23—22no—
124C23—22no—
125C23—42no—
126C23—42no—
127C23—416no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23—42no—
132C23#20814yescomplemented
133C23#21711yescharacteristic
134C23#21711yescharacteristic
135C23C22×Dic311yescharacteristic
136C2×C4—24no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4#21712yes—
145C2×C4C2×#6712yes—
146C8—124no—
147C2×C3—22no—
148C2×C3—22no—
149C2×C3—22no—
150C2×C3—44no—
151C2×C3#40711yescharacteristic
152C2×C3C2×#9911yescharacteristic
153C2×C3C2×#10011yescharacteristic
154C22—28no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—24no—
162C22—44no—
163C22—48no—
164C22—48no—
165C22—44no—
166C22—42no—
167C22—42no—
168C22#83312yes—
169C22#83412yes—
170C22C2×#20812yes—
171C22C2×#21711yescharacteristic
172C4—22no—
173C4—22no—
174C4—42no—
175C3#146211yescharacteristic, complemented, Sylow
176C2—22no—
177C2—22no—
178C2—22no—
179C2—44no—
180C2#520711yescharacteristic
181C2C2×#83311yescharacteristic
182C2C2×#83411yescharacteristic
subgroupG/H
centerC22C2×#217
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC22×C3×C22⋊C4C2
socleC22×C3C2×C22⋊C4
prime radicalC25×C3C4
cosocleC22×C3×C4C23
hypercenterC22×C22⋊C4D3
hypocenterC3#1462
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #5207C2semidirect
3. #4525C2semidirect
14. #830C22semidirect
64. C23×C3M16semidirect
132. C23#208semidirect
175. C3#1462semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊C42
#52072
#45252
#14623