Finite Group Databasebeta

← #11920978

Subgroups — #11920978

Order 1152.

360 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 49 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⋊(C3⋊GL(2,𝔽3))C212yesmaximal, complemented
2C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#73022—33nomaximal
4#70939—31nomaximal
5C2×C3⋊GL(2,𝔽3)—22no—
6C2×#3745—41nomaximal
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C4⋊GL(2,𝔽3)—36no—
9C3×C8×Q8D311yescharacteristic, complemented
10#5027—32no—
11#4695—31no—
12C8×SL(2,𝔽3)D313yescomplemented
13#4485—32no—
14#4458—31no—
15C2×C4⋊(C3⋊D3)—42no—
16C2×C3×SL(2,𝔽3)D411yescharacteristic
17C3⋊GL(2,𝔽3)—42no—
18C2×C32×C8—41no—
19#3745—44no—
20#1668—91noSylow
21C2×GL(2,𝔽3)—66no—
22C3×C4×Q8C2×D311yescharacteristic
23C2×#215—62no—
24C2×D24—61no—
25C2×D24—61no—
26C2×D24—123no—
27#874—32no—
28C4×SL(2,𝔽3)C2×D313yes—
29C3×C4⋊C8—31no—
30C3×C4×C8—31no—
31#809—62no—
32C4⋊(C3⋊C8)—32no—
33C22×C3⋊D3—82no—
34C2×C32×C4—41no—
35C4⋊(C3⋊D3)—44no—
36C4⋊(C3⋊D3)—82no—
37C3×SL(2,𝔽3)D811yescharacteristic, complemented
38C32×C8—42no—
39#493—91no—
40#460—92no—
41C8×Q8C3⋊D311yescharacteristic, complemented
42#335—91no—
43#331—92no—
44C2×C3×Q8D1211yescharacteristic
45C2×D12—62no—
46C2×D12—62no—
47C2×D12—62no—
48C2×D12—126no—
49C2×SL(2,𝔽3)D1213yes—
50GL(2,𝔽3)—126no—
51C2×C3×C8—31no—
52C2×C3×C8—43no—
53C2×C3×C8S411yescharacteristic
54C3×C4⋊C4—31no—
55C3×C42—31no—
56#215—122no—
57#215—122no—
58C2×C3⋊C8—62no—
59D24—1212no—
60D24—122no—
61D24—122no—
62C22×C32—41no—
63C2×C3⋊D3—84no—
64C2×C3⋊D3—162no—
65C2×C3⋊D3—162no—
66C32×C4—41no—
67C32×C4—41no—
68C2×SD16—182no—
69C2×D8—181no—
70C2×D8—181no—
71#127—92no—
72C4×Q8C2×C3⋊D311yescharacteristic
73C4⋊C8—31no—
74C4⋊C8—92no—
75#102—182no—
76C4×C8—31no—
77C22×D3—122no—
78C22×D3—246no—
79C3×Q8—31no—
80C3×Q8D2411yescharacteristic, complemented
81C2×C3×C4—31no—
82C2×C3×C4—31no—
83C2×C3×C4—43no—
84C2×C3×C4C2×S411yescharacteristic
85D12—122no—
86D12—122no—
87D12—1212no—
88D12—122no—
89D12—122no—
90D12—246no—
91D12—242no—
92SL(2,𝔽3)D2413yescomplemented
93C3×C8—46no—
94C3×C8—61no—
95C3×C8GL(2,𝔽3)12yescomplemented
96C3⋊C8—122no—
97C2×C32—41no—
98C2×C32—41no—
99C2×C32—41no—
100C3⋊D3—164no—
101C2×Q8C4⋊(C3⋊D3)11yescharacteristic
102C2×D4—182no—
103C2×D4—182no—
104C2×D4—182no—
105SD16—362no—
106SD16—362no—
107D8—362no—
108D8—362no—
109C2×C8—31no—
110C2×C8—182no—
111C2×C8C3⋊S411yescharacteristic
112C4⋊C4—31no—
113C42—31no—
114C22×C3—43no—
115C22×C3C4⋊S411yescharacteristic
116C2×D3—2412no—
117C2×D3—242no—
118C2×D3—242no—
119C2×D3—242no—
120C2×D3—486no—
121C2×D3—486no—
122C3×C4—31no—
123C3×C4—31no—
124C3×C4—43no—
125C3×C4—43no—
126C3×C4—61no—
127C3×C4C2×GL(2,𝔽3)11yescharacteristic
128C3×C4#98611yescharacteristic
129C32—41noSylow
130C23—362no—
131Q8—31no—
132Q8#374511yescharacteristic, complemented
133D4—362no—
134D4—362no—
135D4—362no—
136D4—362no—
137D4—722no—
138C2×C4—31no—
139C2×C4—31no—
140C2×C4C2×C3⋊S411yescharacteristic
141C8—61no—
142C8—362no—
143C8C3⋊GL(2,𝔽3)12yescomplemented
144C2×C3—43no—
145C2×C3—43no—
146C2×C3—43no—
147C2×C3C4⋊GL(2,𝔽3)11yescharacteristic
148C2×C3#540211yescharacteristic
149C2×C3#540711yescharacteristic
150D3—4812no—
151D3—482no—
152C22—722no—
153C22—722no—
154C22—722no—
155C22C4⋊(C3⋊S4)11yescharacteristic
156C4—31no—
157C4—31no—
158C4—61no—
159C4C2×C3⋊GL(2,𝔽3)11yescharacteristic
160C4#6426711yescharacteristic
161C3—43no—
162C3#7302211yescharacteristic, complemented
163C2—1442no—
164C2C4⋊(C3⋊GL(2,𝔽3))11yescharacteristic
165C2#1059363611yescharacteristic
166C2#1059364111yescharacteristic
subgroupG/H
centerC22C4⋊(C3⋊S4)
commutator, cosocleC3×C4×SL(2,𝔽3)C22
FrattiniC2×C4C2×C3⋊S4
FittingC3×C8×Q8D3
socleC22×C3C4⋊S4
hypercenterC2×C8C3⋊S4
hypocenterC3×SL(2,𝔽3)D8
LCSG ▷ C3×C4×SL(2,𝔽3) ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
2. C3×C8×SL(2,𝔽3)C2semidirect
9. C3×C8×Q8D3semidirect
12. C8×SL(2,𝔽3)D3semidirect
37. C3×SL(2,𝔽3)D8semidirect
41. C8×Q8C3⋊D3semidirect
80. C3×Q8D24semidirect
92. SL(2,𝔽3)D24semidirect
95. C3×C8GL(2,𝔽3)semidirect
132. Q8#3745semidirect
143. C8C3⋊GL(2,𝔽3)semidirect
162. C3#73022semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊GL(2,𝔽3))2
C3×C8×SL(2,𝔽3)2
#730223
#709393
C2×#37454