Finite Group Databasebeta

← #11920974

Subgroups — #11920974

Order 1152.

318 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 51 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)C211yesmaximal, characteristic
2C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(SL(2,𝔽3)⋊C8)C211yesmaximal, characteristic, complemented
4#73018—33nomaximal
5#70933—31nomaximal
6C2×C3⋊GL(2,𝔽3)C411yescharacteristic
7C2×#3743—41nomaximal
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3⋊#859C411yescharacteristic
10C4×GL(2,𝔽3)—33no—
11C3×C8×Q8D311yescharacteristic, complemented
12C4×#215—31no—
13#4688—31no—
14C8×SL(2,𝔽3)D313yescomplemented
15SL(2,𝔽3)⋊C8—33no—
16#4535—31no—
17#4483—31no—
18#4480—31no—
19#4455—31no—
20C2×C4×C3⋊D3—41no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C3⋊GL(2,𝔽3)—21no—
23C2×C32×C8—41no—
24C2×C3⋊(C3⋊C8)—41no—
25#3743—44no—
26#1583—91noSylow
27C2×GL(2,𝔽3)—33no—
28C3×C4×Q8C2×D311yescharacteristic
29C2×#215—31no—
30C2×ΓL(1,𝔽25)—61no—
31C2×ΓL(1,𝔽25)—123no—
32C4×D12—31no—
33C4×SL(2,𝔽3)C2×D313yes—
34#859—33no—
35C3×C4⋊C8—31no—
36C3×C4×C8—31no—
37#835—31no—
38(C2×D3)⋊C8—61no—
39#809—31no—
40#808—31no—
41C4⋊(C3⋊C8)—31no—
42C4×C3⋊C8—31no—
43C4×C3⋊C8—31no—
44C22×C3⋊D3—41no—
45C2×C32×C4—41no—
46C2×C3⋊Dic3—41no—
47C4×C3⋊D3—42no—
48C4×C3⋊D3—42no—
49C3×SL(2,𝔽3)M1611yescharacteristic, complemented
50C32×C8—42no—
51C3⋊(C3⋊C8)—42no—
52C8×Q8C3⋊D311yescharacteristic, complemented
53C4×SD16—91no—
54#436—91no—
55#334—91no—
56Q8⋊C8—91no—
57D4⋊C8—91no—
58#322—91no—
59C2×C3×Q8C4×D311yescharacteristic
60C2×D12—31no—
61C2×C4×D3—61no—
62C2×C4×D3—123no—
63C2×SL(2,𝔽3)C4×D313yes—
64GL(2,𝔽3)—63no—
65C2×C3×C8—31no—
66C2×C3×C8—43no—
67C2×C3×C8S411yescharacteristic
68C3×C4⋊C4—31no—
69C3×C42—31no—
70#215—61no—
71#215—61no—
72(C2×D3)⋊C4—61no—
73C4⋊Dic3—31no—
74C2×C3⋊C8—31no—
75C2×C3⋊C8—31no—
76C2×C3⋊C8—61no—
77C2×C3⋊C8—61no—
78C2×C3⋊C8—123no—
79ΓL(1,𝔽25)—1212no—
80ΓL(1,𝔽25)—122no—
81C22×C32—41no—
82C2×C3⋊D3—42no—
83C2×C3⋊D3—81no—
84C2×C3⋊D3—81no—
85C32×C4—41no—
86C32×C4—41no—
87C3⋊Dic3—42no—
88C2×SD16—91no—
89C2×M16—181no—
90C4×Q8C2×C3⋊D311yescharacteristic
91C4×D4—91no—
92#106—91no—
93C4⋊C8—31no—
94C4⋊C8—91no—
95#103—91no—
96#102—91no—
97C22⋊C8—181no—
98C4×C8—31no—
99C4×C8—91no—
100C4×C8—91no—
101C22×D3—61no—
102C22×D3—123no—
103C3×Q8—31no—
104C3×Q8ΓL(1,𝔽25)11yescharacteristic, complemented
105C2×C3×C4—31no—
106C2×C3×C4—31no—
107C2×C3×C4—43no—
108C2×C3×C4C2×S411yescharacteristic
109C2×Dic3—61no—
110C2×Dic3—123no—
111D12—61no—
112D12—61no—
113C4×D3—126no—
114C4×D3—126no—
115C4×D3—121no—
116C4×D3—121no—
117SL(2,𝔽3)ΓL(1,𝔽25)13yescomplemented
118C3×C8—46no—
119C3×C8—61no—
120C3×C8GL(2,𝔽3)12yescomplemented
121C3⋊C8—61no—
122C3⋊C8—61no—
123C3⋊C8—126no—
124C3⋊C8—121no—
125C3⋊C8—121no—
126C2×C32—41no—
127C2×C32—41no—
128C2×C32—41no—
129C3⋊D3—82no—
130C2×Q8C4×C3⋊D311yescharacteristic
131C2×D4—91no—
132C22×C4—181no—
133SD16—181no—
134SD16—181no—
135M16—362no—
136C2×C8—31no—
137C2×C8—91no—
138C2×C8—91no—
139C2×C8—181no—
140C2×C8—181no—
141C2×C8C3⋊S411yescharacteristic
142C4⋊C4—31no—
143C4⋊C4—91no—
144C22⋊C4—181no—
145C42—31no—
146C22×C3—43no—
147C22×C3C4×S411yescharacteristic
148C2×D3—126no—
149C2×D3—121no—
150C2×D3—121no—
151C2×D3—121no—
152C2×D3—243no—
153C2×D3—243no—
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—43no—
157C3×C4—43no—
158C3×C4—61no—
159C3×C4C2×GL(2,𝔽3)11yescharacteristic
160C3×C4#98511yescharacteristic
161Dic3—126no—
162Dic3—121no—
163C32—41noSylow
164C23—181no—
165Q8—31no—
166Q8#374311yescharacteristic, complemented
167D4—181no—
168D4—181no—
169C2×C4—31no—
170C2×C4—31no—
171C2×C4—181no—
172C2×C4—361no—
173C2×C4—361no—
174C2×C4C2×C3⋊S411yescharacteristic
175C8—61no—
176C8—181no—
177C8—181no—
178C8—361no—
179C8—361no—
180C8C3⋊GL(2,𝔽3)12yescomplemented
181C2×C3—43no—
182C2×C3—43no—
183C2×C3—43no—
184C2×C3C4×GL(2,𝔽3)11yescharacteristic
185C2×C3#540011yescharacteristic
186C2×C3#540511yescharacteristic
187D3—246no—
188D3—241no—
189C22—361no—
190C22—361no—
191C22—361no—
192C22C4×C3⋊S411yescharacteristic
193C4—31no—
194C4—31no—
195C4—61no—
196C4—361no—
197C4C2×C3⋊GL(2,𝔽3)11yescharacteristic
198C4#6426611yescharacteristic
199C3—43no—
200C3#7301811yescharacteristic, complemented
201C2—721no—
202C2C4×C3⋊GL(2,𝔽3)11yescharacteristic
203C2#1059363411yescharacteristic
204C2#1059363911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×C3⋊S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC3×C8×Q8D3
socleC22×C3C4×S4
prime radicalC2×C3⋊GL(2,𝔽3)C4
cosocleC3×C4×SL(2,𝔽3)C22
hypercenterC2×C8C3⋊S4
hypocenterC3×SL(2,𝔽3)M16
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C3×C8×SL(2,𝔽3)C2semidirect
3. C3⋊(SL(2,𝔽3)⋊C8)C2semidirect
11. C3×C8×Q8D3semidirect
14. C8×SL(2,𝔽3)D3semidirect
49. C3×SL(2,𝔽3)M16semidirect
52. C8×Q8C3⋊D3semidirect
104. C3×Q8ΓL(1,𝔽25)semidirect
117. SL(2,𝔽3)ΓL(1,𝔽25)semidirect
120. C3×C8GL(2,𝔽3)semidirect
166. Q8#3743semidirect
180. C8C3⋊GL(2,𝔽3)semidirect
200. C3#73018semidirect

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
C3⋊(SL(2,𝔽3)⋊C8)2
#730183
#709333
C2×#37434