Finite Group Databasebeta

← #11922979

Subgroups — #11922979

Order 1152.

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

normal subgroups only

This group has 746 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
1C3×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊GL(2,𝔽3))C213yesmaximal, complemented
3C4×C3⋊GL(2,𝔽3)C213yesmaximal, complemented
4#85430—33nomaximal
5#83703—31nomaximal
6C2×#3832—41nomaximal
7C2×C3⋊GL(2,𝔽3)C2213yes—
8C3×C4×SL(2,𝔽3)C2213yes—
9C3⋊#859C2211yescharacteristic
10C3×Q82D311yescharacteristic, complemented
11#5577—31no—
12Q8×SL(2,𝔽3)D313yescomplemented
13C4⋊GL(2,𝔽3)—39no—
14C4×GL(2,𝔽3)—39no—
15#5088—33no—
16#5084—33no—
17#5027—33no—
18C4×#215—33no—
19Q8×C3⋊C8—31no—
20C2×C32×Q8—41no—
21#3832—48no—
22C2×C4⋊(C3⋊D3)—43no—
23C2×C4×C3⋊D3—43no—
24C2×C3×SL(2,𝔽3)C2311yescharacteristic
25C3⋊GL(2,𝔽3)—23no—
26#3362—91noSylow
27C2×#239—61no—
28C2×#239—123no—
29C2×GL(2,𝔽3)—39no—
30C3×C4⋊Q8—33no—
31C3×C4×Q8—31no—
32C3×C4×Q8C2×D313yes—
33C2×#215—33no—
34#895—63no—
35#874—33no—
36C4×D12—33no—
37C4×SL(2,𝔽3)C2×D319yes—
38#859—33no—
39#835—31no—
40#809—33no—
41#809—36no—
42#808—33no—
43C4⋊(C3⋊C8)—33no—
44C4×C3⋊C8—33no—
45C22×C3⋊D3—43no—
46C32×Q8—44no—
47C2×C32×C4—43no—
48C2×C3⋊Dic3—41no—
49C4⋊(C3⋊D3)—412no—
50C4×C3⋊D3—412no—
51C3×SL(2,𝔽3)#4111yescharacteristic, complemented
52Q82C3⋊D311yescharacteristic, complemented
53#550—91no—
54#486—93no—
55#476—93no—
56#460—93no—
57C8×Q8—91no—
58C4×SD16—93no—
59C2×C3×Q8—43no—
60C2×C3×Q8—61no—
61C2×C3×Q8S411yescharacteristic
62C2×C3×Q8C22×D311yescharacteristic
63#239—1224no—
64#239—124no—
65C2×D12—33no—
66C2×D12—63no—
67C2×D12—63no—
68C2×D12—129no—
69C2×C4×D3—63no—
70C2×C4×D3—129no—
71C2×SL(2,𝔽3)C22×D313yes—
72GL(2,𝔽3)—69no—
73C3×C4⋊C4—33no—
74C3×C4⋊C4—33no—
75C3×C42—33no—
76#215—63no—
77#215—63no—
78(C2×D3)⋊C4—63no—
79C4⋊Dic3—31no—
80C2×C3⋊C8—31no—
81C2×C3⋊C8—33no—
82C22×C32—41no—
83C2×C3⋊D3—46no—
84C2×C3⋊D3—83no—
85C2×C3⋊D3—83no—
86C32×C4—46no—
87C3⋊Dic3—42no—
88C2×#41—181no—
89C2×SD16—93no—
90C4⋊Q8—33no—
91#127—93no—
92#121—183no—
93C4×Q8—31no—
94C4×Q8C2×C3⋊D313yes—
95C4×D4—93no—
96#106—93no—
97C4⋊C8—93no—
98#103—91no—
99#102—93no—
100#102—96no—
101C4×C8—93no—
102C22×D3—63no—
103C22×D3—129no—
104C3×Q8—31no—
105C3×Q8—412no—
106C3×Q8—241no—
107C3×Q8GL(2,𝔽3)14yescomplemented
108C3×Q8#23911yescharacteristic, complemented
109C2×C3×C4—31no—
110C2×C3×C4—33no—
111C2×C3×C4—49no—
112C2×C3×C4C2×S413yes—
113C2×Dic3—61no—
114C2×Dic3—123no—
115D12—63no—
116D12—63no—
117D12—1236no—
118D12—126no—
119D12—243no—
120C4×D3—1236no—
121C4×D3—126no—
122SL(2,𝔽3)#23913yescomplemented
123C3⋊C8—61no—
124C3⋊C8—63no—
125C2×C32—41no—
126C2×C32—41no—
127C2×C32—41no—
128C3⋊D3—86no—
129#41—364no—
130C2×Q8—61no—
131C2×Q8C3⋊S411yescharacteristic
132C2×Q8C22×C3⋊D311yescharacteristic
133C2×D4—93no—
134C2×D4—183no—
135C2×D4—183no—
136C22×C4—183no—
137SD16—183no—
138SD16—183no—
139C2×C8—91no—
140C2×C8—93no—
141C4⋊C4—33no—
142C4⋊C4—33no—
143C4⋊C4—91no—
144C22⋊C4—183no—
145C42—33no—
146C22×C3—43no—
147C22×C3C22×S411yescharacteristic
148C2×D3—1218no—
149C2×D3—123no—
150C2×D3—123no—
151C2×D3—123no—
152C2×D3—249no—
153C2×D3—249no—
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—418no—
157C3×C4—63no—
158C3×C4C2×GL(2,𝔽3)16yes—
159Dic3—126no—
160Dic3—121no—
161C32—41noSylow
162C23—183no—
163Q8—31no—
164Q8—241no—
165Q8C3⋊GL(2,𝔽3)14yescomplemented
166Q8#383211yescharacteristic, complemented
167D4—183no—
168D4—183no—
169D4—366no—
170D4—723no—
171C2×C4—31no—
172C2×C4—33no—
173C2×C4—181no—
174C2×C4—366no—
175C2×C4C2×C3⋊S413yes—
176C8—183no—
177C8—181no—
178C2×C3—43no—
179C2×C3—43no—
180C2×C3—43no—
181C2×C3C22×GL(2,𝔽3)11yescharacteristic
182C2×C3#591911yescharacteristic
183C2×C3#592511yescharacteristic
184D3—2418no—
185D3—243no—
186C22—363no—
187C22—363no—
188C22—363no—
189C22C22×C3⋊S411yescharacteristic
190C4—31no—
191C4—31no—
192C4—63no—
193C4—361no—
194C4C2×C3⋊GL(2,𝔽3)16yes—
195C3—43no—
196C3#8543011yescharacteristic, complemented
197C2—723no—
198C2C22×C3⋊GL(2,𝔽3)11yescharacteristic
199C2#1059658211yescharacteristic
200C2#1059658811yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×Q82D3
socleC22×C3C22×S4
hypercenterC2×Q8C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
2. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C4×C3⋊GL(2,𝔽3)C2semidirect
10. C3×Q82D3semidirect
12. Q8×SL(2,𝔽3)D3semidirect
51. C3×SL(2,𝔽3)#41semidirect
52. Q82C3⋊D3semidirect
107. C3×Q8GL(2,𝔽3)semidirect
108. C3×Q8#239semidirect
122. SL(2,𝔽3)#239semidirect
165. Q8C3⋊GL(2,𝔽3)semidirect
166. Q8#3832semidirect
196. C3#85430semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
C4⋊(C3⋊GL(2,𝔽3))2
C4×C3⋊GL(2,𝔽3)2
#854303
#837033
C2×#38324