Finite Group Databasebeta

← C22×#63732 #11858229

Subgroups — C22×#63732 #11858229

Order 1152.

978 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 287 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×C4×#157C211yesmaximal, characteristic
2C22×#3710C212yesmaximal
3C2×#63732C2112yesmaximal, direct product factor
4C23×#157C411yescharacteristic
5C23×#157C411yescharacteristic
6C22×C4×C3⋊D3C2211yescharacteristic
7C2×C4×#157C416yes—
8C2×C4×#157C2216yes—
9C22×C3×C3⋊C8—22no—
10C2×#3710—26no—
11C2×#3710C22112yes—
12#63732C22116yesdirect product factor
13C22×D3×C8—62no—
14C23×C3⋊D3C2×C411yescharacteristic
15C22×#157—24no—
16C22×#157C818yescomplemented
17C22×#157C2×C416yes—
18C22×#157C2×C416yes—
19C22×C32×C4D411yescharacteristic
20C22×C3⋊Dic3Q811yescharacteristic
21C2×C4×C3⋊D3C2×C413yes—
22C2×C4×C3⋊D3D413yes—
23C2×C4×C3⋊D3Q813yes—
24C2×C4×C3⋊D3C2313yes—
25C4×#157C2×C4112yes—
26C4×#157C2314yes—
27C2×C3×C3⋊C8—212no—
28#3710—212no—
29#3710C2318yes—
30C22×C4⋊C8—91nomaximal, Sylow
31C22×C4×D3—62no—
32C22×C3×C8—62no—
33C22×C3⋊C8—22no—
34C2×D3×C8—612no—
35C2×D3×C8—612no—
36C23×C32C4⋊C411yescharacteristic
37C22×C3⋊D3C4⋊C413yes—
38C22×C3⋊D3C2×C814yes—
39C22×C3⋊D3M1614yes—
40C22×C3⋊D3C22×C413yes—
41C2×#157—212no—
42C2×#157C2×C8124yescomplemented
43C2×#157C22×C414yes—
44C2×#157C22×C414yes—
45C2×C32×C4C4⋊C413yes—
46C2×C32×C4C2×D413yes—
47C2×C3⋊Dic3C4⋊C413yes—
48C2×C3⋊Dic3C2×Q813yes—
49C4×C3⋊D3C4⋊C416yes—
50C4×C3⋊D3C22×C413yes—
51C4×C3⋊D3C2×D413yes—
52C4×C3⋊D3C2×Q813yes—
53C4×C3⋊D3C2411yescharacteristic
54C3×C3⋊C8—28no—
55C23×C8—92no—
56C22×C42—91no—
57C2×C4⋊C8—912no—
58C23×D3—62no—
59C22×C3×C4—22no—
60C22×Dic3—62no—
61C2×C4×D3—66no—
62C2×C4×D3—66no—
63C2×C4×D3—66no—
64C2×C4×D3—66no—
65C2×C3×C8—612no—
66C2×C3⋊C8—212no—
67D3×C8—68no—
68D3×C8—624no—
69C22×C32C4⋊C814yescomplemented
70C22×C32C2×C4⋊C413yes—
71C2×C3⋊D3C4⋊C8112yescomplemented
72C2×C3⋊D3C2×C4⋊C413yes—
73C2×C3⋊D3C22×C816yes—
74C2×C3⋊D3C2×M1616yes—
75C2×C3⋊D3C23×C411yescharacteristic
76#157—24no—
77#157C22×C818yescomplemented
78C32×C4C2×C4⋊C413yes—
79C32×C4C22×D411yescharacteristic
80C3⋊Dic3C2×C4⋊C413yes—
81C3⋊Dic3C22×Q811yescharacteristic
82C23×C4—91no—
83C23×C4—91no—
84C23×C4—91no—
85C22×C8—62no—
86C22×C8—912no—
87C22×C8—186no—
88C2×C42—96no—
89C2×C42—96no—
90C4⋊C8—916no—
91C23×C3—22no—
92C22×D3—66no—
93C22×D3—68no—
94C22×D3—68no—
95C22×D3—66no—
96C2×C3×C4—26no—
97C2×C3×C4—26no—
98C2×Dic3—66no—
99C2×Dic3—66no—
100C4×D3—612no—
101C4×D3—66no—
102C4×D3—66no—
103C4×D3—66no—
104C4×D3—62no—
105C3×C8—68no—
106C3⋊C8—28no—
107C2×C32C2×C4⋊C816yescomplemented
108C2×C32C22×C4⋊C411yescharacteristic
109C3⋊D3C2×C4⋊C816yescomplemented
110C3⋊D3C23×C811yescharacteristic
111C3⋊D3C22×M1611yescharacteristic
112C24—91no—
113C22×C4—98no—
114C22×C4—96no—
115C22×C4—91no—
116C22×C4—93no—
117C22×C4—93no—
118C22×C4—93no—
119C22×C4—93no—
120C22×C4—96no—
121C22×C4—184no—
122C22×C4GO+(4,𝔽2)11yescharacteristic
123C2×C8—612no—
124C2×C8—98no—
125C2×C8—1812no—
126C42—94no—
127C42—912no—
128C22×C3—26no—
129C22×C3—28no—
130C2×D3—66no—
131C2×D3—62no—
132C2×D3—624no—
133C2×D3—612no—
134C2×D3—612no—
135C3×C4—22no—
136C3×C4—26no—
137Dic3—66no—
138Dic3—62no—
139C32C22×C4⋊C811yescharacteristic, complemented, Sylow
140C23—93no—
141C23—93no—
142C23—94no—
143C23—94no—
144C23#377311yescharacteristic
145C2×C4—94no—
146C2×C4—924no—
147C2×C4—93no—
148C2×C4—93no—
149C2×C4—91no—
150C2×C4—93no—
151C2×C4—93no—
152C2×C4—93no—
153C2×C4—96no—
154C2×C4—94no—
155C2×C4—1812no—
156C2×C4#377313yes—
157C2×C4C2×GO+(4,𝔽2)13yes—
158C8—68no—
159C2×C3—212no—
160C2×C3—22no—
161D3—62no—
162D3—62no—
163D3—612no—
164C22—96no—
165C22—96no—
166C22—912no—
167C22—91no—
168C22—93no—
169C22#6373214yesdirect product factor
170C22C2×#377313yes—
171C4—98no—
172C4—91no—
173C4—93no—
174C4—184no—
175C4C2×#377313yes—
176C4C22×GO+(4,𝔽2)11yescharacteristic
177C3—22no—
178C2—91no—
179C2—91no—
180C2—96no—
181C2C2×#6373216yesdirect product factor
182C2C22×#377311yescharacteristic
subgroupG/H
center, hypercenterC22×C4GO+(4,𝔽2)
commutatorC3⋊D3C23×C8
FrattiniC4C22×GO+(4,𝔽2)
FittingC22×C32×C4D4
socleC23×C32C4⋊C4
prime radicalC23×C3⋊D3C2×C4
cosocleC4×C3⋊D3C24
hypocenterC32C22×C4⋊C8
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#63732C2direct (N × H)
12. #63732C22direct (N × H)
16. C22×#157C8semidirect
42. C2×#157C2×C8semidirect
69. C22×C32C4⋊C8semidirect
71. C2×C3⋊D3C4⋊C8semidirect
77. #157C22×C8semidirect
107. C2×C32C2×C4⋊C8semidirect
109. C3⋊D3C2×C4⋊C8semidirect
139. C32C22×C4⋊C8semidirect
169. C22#63732direct (N × H)
181. C2C2×#63732direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#1572
C22×#37102
C2×#637322
C22×C4⋊C89