Finite Group Databasebeta

← #11771212

Subgroups — #11771212

Order 1152.

230 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C4×C4⋊#157C211yesmaximal, characteristic
2#10593729C211yesmaximal, characteristic
3#10593727C211yesmaximal, characteristic
4C2×C4⋊#157—21no—
5C2×C4×#157—21no—
6C2×#3788—21no—
7C42×C3⋊D3C2211yescharacteristic
8#63781C412yes—
9#63779—21no—
10#63779C412yescomplemented
11#63778—21no—
12C4⋊#622C412yescomplemented
13C4×#622—21no—
14C22×#157—22no—
15C2×C4×C3⋊D3C2×C411yescharacteristic
16C2×C4×C3⋊D3D411yescharacteristic
17C2×C4×C3⋊D3D411yescharacteristic
18C2×#622—22no—
19C2×#622—22no—
20C4⋊#157—24no—
21C4×#157—41no—
22C4×#157—41no—
23#3788—41no—
24#3788—41no—
25C32×C42Q811yescharacteristic
26C4×C3⋊Dic3C2×C411yescharacteristic
27C4×C3⋊Dic3C2×C411yescharacteristic
28C4×C3⋊Dic3D411yescharacteristic
29#1303—91nomaximal, Sylow
30C42×D3—121no—
31C22×C3⋊D3C22⋊C411yescharacteristic
32C2×#157—24no—
33C2×#157—42no—
34C2×#157—42no—
35C2×C32×C4C4⋊C411yescharacteristic
36C2×C32×C4C4⋊C411yescharacteristic
37C2×C32×C4C4⋊C411yescharacteristic
38C2×C3⋊Dic3C4211yescharacteristic
39C2×C3⋊Dic3C22⋊C411yescharacteristic
40C2×C3⋊Dic3C22⋊C411yescharacteristic
41C4×C3⋊D3—22no—
42C4×C3⋊D3—41no—
43C4×C3⋊D3D811yescharacteristic
44C4×C3⋊D3SD1611yescharacteristic
45C4×C3⋊D3SD1611yescharacteristic
46C4×C3⋊D3Q1611yescharacteristic
47#622—42no—
48#622—42no—
49#431—91no—
50#423—91no—
51C4×C4⋊C4—91no—
52C2×C4×D3—121no—
53C2×C4×D3—121no—
54C2×C4×D3—121no—
55C3×C42—41no—
56C4×Dic3—121no—
57C4×Dic3—121no—
58C4×Dic3—121no—
59C22×C32#9511yescharacteristic
60C2×C3⋊D3—21no—
61C2×C3⋊D3—21no—
62C2×C3⋊D3#10211yescharacteristic
63C2×C3⋊D3#10311yescharacteristic
64#157—44no—
65C32×C4—21no—
66C32×C4—21no—
67C32×C4#10611yescharacteristic
68C32×C4#10711yescharacteristic
69C3⋊Dic3—21no—
70C3⋊Dic3#10211yescharacteristic
71C3⋊Dic3#10311yescharacteristic
72C3⋊Dic3#10412yes—
73C2×M16—181no—
74C2×C4⋊C4—181no—
75C2×C42—91no—
76C2×C42—181no—
77C4⋊C8—92no—
78C4⋊C8—92no—
79C4⋊C8—181no—
80C22⋊C8—181no—
81C4×C8—181no—
82#95—92no—
83C22×D3—121no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×Dic3—121no—
88C2×Dic3—121no—
89C2×Dic3—121no—
90C4×D3—121no—
91C4×D3—121no—
92C4×D3—121no—
93C4×D3—121no—
94C4×D3—124no—
95C4×D3—124no—
96C2×C32#33911yescharacteristic
97C2×C32#34011yescharacteristic
98C2×C32#34411yescharacteristic
99C3⋊D3—21no—
100C3⋊D3—21no—
101C22×C4—91no—
102C22×C4—91no—
103C22×C4—91no—
104C22×C4—182no—
105M16—361no—
106M16—361no—
107C2×C8—182no—
108C2×C8—182no—
109C4⋊C4—184no—
110C42—91no—
111C42—91no—
112C42—91no—
113C42—361no—
114C42—361no—
115C42PSU(3,𝔽2)11yescharacteristic
116C22×C3—41no—
117C2×D3—121no—
118C2×D3—121no—
119C2×D3—122no—
120C2×D3—122no—
121C3×C4—42no—
122C3×C4—42no—
123C3×C4—41no—
124C3×C4—41no—
125Dic3—121no—
126Dic3—121no—
127Dic3—122no—
128Dic3—122no—
129C32#130311yescharacteristic, complemented, Sylow
130C23—91no—
131C2×C4—91no—
132C2×C4—91no—
133C2×C4—91no—
134C2×C4—91no—
135C2×C4—91no—
136C2×C4—91no—
137C2×C4—91no—
138C2×C4—182no—
139C2×C4—184no—
140C2×C4—361no—
141C2×C4—362no—
142C2×C4—362no—
143C2×C4#377711yescharacteristic
144C2×C4#377711yescharacteristic
145C2×C4#377711yescharacteristic
146C8—362no—
147C8—362no—
148C2×C3—41no—
149C2×C3—41no—
150C2×C3—41no—
151D3—122no—
152D3—122no—
153C22—91no—
154C22—91no—
155C22—181no—
156C22—181no—
157C22#6374711yescharacteristic
158C4—21no—
159C4—21no—
160C4—92no—
161C4—91no—
162C4—91no—
163C4—181no—
164C4—364no—
165C4#6374511yescharacteristic
166C4#6374611yescharacteristic
167C3—41no—
168C2—181no—
169C2—181no—
170C2#1059009211yescharacteristic
171C2#1059009311yescharacteristic
172C2#1059009511yescharacteristic
subgroupG/H
centerC22#63747
commutatorC2×C3⋊Dic3C42
Frattini, hypercenterC42PSU(3,𝔽2)
FittingC32×C42Q8
socleC22×C32#95
prime radicalC22×C3⋊D3C22⋊C4
cosocleC42×C3⋊D3C22
hypocenterC32#1303
LCSG ▷ C2×C3⋊Dic3 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
10. #63779C4semidirect
12. C4⋊#622C4semidirect
129. C32#1303semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#1572
#105937292
#105937272
#13039