Finite Group Databasebeta

← #71572

Subgroups — #71572

Order 384.

252 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 85 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
2#4926C212yesmaximal, complemented
3#4528C212yesmaximal, complemented
4#4525C212yesmaximal
5#1464—31nomaximal, 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
15#830C2214yes—
16C4⋊(C3⋊C8)—22no—
17#803—22no—
18C2×#117D311yescharacteristic
19#432—32no—
20#324—32no—
21#323—32no—
22C24×C3C2×C411yescharacteristic
23C22×C3×C4—22no—
24C22×C3×C4—21no—
25C22×C3×C4—21no—
26C22×C3×C4—22no—
27C22×C3×C4C2×C411yescharacteristic
28C22×C3×C4C2×C411yescharacteristic
29C22×C3×C4C2×C411yescharacteristic
30C22×C3×C4C2×C411yescharacteristic
31C22×C3×C4C2×C411yescharacteristic
32C22×C3×C4C2311yescharacteristic
33C3×C4⋊C4—22no—
34C3×C4⋊C4—22no—
35C3×C22⋊C4—22no—
36C3×C22⋊C4—22no—
37C3×C42—21no—
38C3×C42—21no—
39C3×C42D412yes—
40C3×C42D412yes—
41C2×C3⋊C8—24no—
42C23×C4Dic311yescharacteristic
43#117—22no—
44#117—22no—
45C2×C4⋊C4Dic311yescharacteristic
46C2×C4⋊C4C2×D311yescharacteristic
47C2×C22⋊C4Dic311yescharacteristic
48C2×C22⋊C4C2×D311yescharacteristic
49C2×C42Dic311yescharacteristic
50C2×C42C2×D311yescharacteristic
51C4⋊C8—62no—
52C22⋊C8—34no—
53#97—62no—
54C23×C3—21no—
55C23×C3—21no—
56C23×C3—41no—
57C23×C3—41no—
58C23×C3M1612yes—
59C23×C3C22×C411yescharacteristic
60C2×C3×C4—24no—
61C2×C3×C4—24no—
62C2×C3×C4—21no—
63C2×C3×C4—21no—
64C2×C3×C4—21no—
65C2×C3×C4—21no—
66C2×C3×C4—21no—
67C2×C3×C4—21no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4C22⋊C411yescharacteristic
73C2×C3×C4C22⋊C411yescharacteristic
74C2×C3×C4C22⋊C412yes—
75C2×C3×C4M1612yes—
76C2×C3×C4C2×D412yes—
77C3⋊C8—44no—
78C24C2×Dic311yescharacteristic
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4C2×Dic311yescharacteristic
84C22×C4C2×Dic311yescharacteristic
85C22×C4C2×Dic311yescharacteristic
86C22×C4C2×Dic311yescharacteristic
87C22×C4C2×Dic311yescharacteristic
88C22×C4C22×D311yescharacteristic
89C2×C8—64no—
90C4⋊C4—22no—
91C4⋊C4—22no—
92C22⋊C4—22no—
93C22⋊C4—22no—
94C42—21no—
95C42—21no—
96C42#6712yes—
97C42#6712yes—
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×C3C2×M1611yescharacteristic
107C22×C3C2×M1611yescharacteristic
108C3×C4—21no—
109C3×C4—21no—
110C3×C4—22no—
111C3×C4—22no—
112C3×C4—22no—
113C3×C4—41no—
114C3×C4—41no—
115C23—21no—
116C23—21no—
117C23—41no—
118C23—41no—
119C23#20812yes—
120C23C22×Dic311yescharacteristic
121C2×C4—24no—
122C2×C4—24no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4#20812yes—
134C2×C4#21711yescharacteristic
135C2×C4#21711yescharacteristic
136C2×C4#21712yes—
137C2×C4C2×#6712yes—
138C8—124no—
139C2×C3—21no—
140C2×C3—21no—
141C2×C3—42no—
142C2×C3#40711yescharacteristic
143C2×C3#41011yescharacteristic
144C2×C3#41311yescharacteristic
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22C2×#20811yescharacteristic
153C22C2×#20811yescharacteristic
154C22C2×#21711yescharacteristic
155C4—22no—
156C4—21no—
157C4—21no—
158C4—22no—
159C4—22no—
160C4—41no—
161C4—41no—
162C3#146411yescharacteristic, complemented, Sylow
163C2—21no—
164C2—21no—
165C2—42no—
166C2#520711yescharacteristic
167C2#523711yescharacteristic
168C2#524011yescharacteristic
subgroupG/H
centerC22C2×#217
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC2×C3×#117C2
socleC22×C3C2×C22⋊C4
prime radicalC24×C3C2×C4
cosocleC22×C3×C4C23
hypercenterC2×#117D3
hypocenterC3#1464
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #4926C2semidirect
3. #4528C2semidirect
162. C3#1464semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1172
#49262
#45282
#45252
#14643