Finite Group Databasebeta

← C2×C4⋊(C3⋊GL(2,𝔽3)) #11922873

Subgroups — C2×C4⋊(C3⋊GL(2,𝔽3)) #11922873

Order 1152.

1088 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 113 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×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊GL(2,𝔽3))C214yesmaximal, direct product factor
4C2×C4⋊GL(2,𝔽3)—33nomaximal
5C2×#5027—31nomaximal
6C22×C4⋊(C3⋊D3)—41nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3⋊GL(2,𝔽3)—24no—
9C2×C3⋊GL(2,𝔽3)C2214yescomplemented
10C3×C4×SL(2,𝔽3)C2212yescomplemented
11C22×GL(2,𝔽3)—36no—
12C2×C3×C4×Q8D311yescharacteristic, complemented
13C22×#215—32no—
14C2×#874—31no—
15C2×C4×SL(2,𝔽3)D313yescomplemented
16C4⋊GL(2,𝔽3)—312no—
17#5027—34no—
18#5027—34no—
19C2×#809—32no—
20C2×C4⋊(C3⋊C8)—31no—
21C23×C3⋊D3—42no—
22C22×C32×C4—41no—
23C2×C4⋊(C3⋊D3)—44no—
24C2×C4⋊(C3⋊D3)—48no—
25C2×C3×SL(2,𝔽3)D412yescomplemented
26C2×C3×SL(2,𝔽3)C2311yescharacteristic
27C3⋊GL(2,𝔽3)—24no—
28C2×#460—91noSylow
29C22×C3×Q8C2×D311yescharacteristic
30C22×D12—32no—
31C22×D12—61no—
32C22×D12—61no—
33C22×D12—123no—
34C22×SL(2,𝔽3)C2×D313yes—
35C2×GL(2,𝔽3)—312no—
36C2×GL(2,𝔽3)—612no—
37C3×C4×Q8—32no—
38C3×C4×Q8C2×D312yescomplemented
39C2×C3×C4⋊C4—31no—
40C2×C3×C42—31no—
41C2×#215—34no—
42C2×#215—34no—
43C2×#215—64no—
44C2×#215—64no—
45C22×C3⋊C8—32no—
46#874—34no—
47#874—62no—
48C4×SL(2,𝔽3)C2×D316yescomplemented
49#809—38no—
50C4⋊(C3⋊C8)—34no—
51C23×C32—41no—
52C22×C3⋊D3—44no—
53C22×C3⋊D3—48no—
54C22×C3⋊D3—84no—
55C22×C3⋊D3—84no—
56C2×C32×C4—44no—
57C2×C32×C4—42no—
58C4⋊(C3⋊D3)—416no—
59C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
60C22×SD16—92no—
61C2×#127—91no—
62C2×C4×Q8C3⋊D311yescharacteristic, complemented
63#460—94no—
64#460—94no—
65C2×C4⋊C8—91no—
66C2×#102—92no—
67C23×D3—62no—
68C23×D3—126no—
69C2×C3×Q8—31no—
70C2×C3×Q8—32no—
71C2×C3×Q8D1212yescomplemented
72C2×C3×Q8C22×D311yescharacteristic
73C22×C3×C4—31no—
74C22×C3×C4—31no—
75C22×C3×C4—43no—
76C22×C3×C4S411yescharacteristic
77C2×D12—34no—
78C2×D12—64no—
79C2×D12—64no—
80C2×D12—64no—
81C2×D12—62no—
82C2×D12—64no—
83C2×D12—1212no—
84C2×D12—1224no—
85C2×D12—124no—
86C2×D12—124no—
87C2×SL(2,𝔽3)D1216yescomplemented
88C2×SL(2,𝔽3)C22×D313yes—
89GL(2,𝔽3)—612no—
90C3×C4⋊C4—32no—
91C3×C4⋊C4—61no—
92C3×C42—32no—
93C3×C42—32no—
94#215—68no—
95#215—64no—
96#215—64no—
97C2×C3⋊C8—34no—
98C2×C3⋊C8—64no—
99C22×C32—42no—
100C22×C32—42no—
101C22×C32—42no—
102C22×C32—41no—
103C2×C3⋊D3—48no—
104C2×C3⋊D3—84no—
105C2×C3⋊D3—84no—
106C2×C3⋊D3—816no—
107C32×C4—44no—
108C22×Q8C2×C3⋊D311yescharacteristic
109C22×D4—92no—
110C22×D4—181no—
111C22×D4—181no—
112C2×SD16—94no—
113C2×SD16—94no—
114C2×SD16—184no—
115C2×SD16—184no—
116C22×C8—92no—
117#127—94no—
118#127—182no—
119C4×Q8—32no—
120C4×Q8C2×C3⋊D312yescomplemented
121C2×C4⋊C4—31no—
122C2×C42—31no—
123C4⋊C8—94no—
124#102—98no—
125C23×C3—43no—
126C23×C3C2×S411yescharacteristic
127C22×D3—64no—
128C22×D3—1212no—
129C22×D3—1224no—
130C22×D3—124no—
131C22×D3—124no—
132C22×D3—124no—
133C22×D3—2412no—
134C22×D3—2412no—
135C3×Q8—32no—
136C3×Q8—31no—
137C3×Q8—61no—
138C3×Q8C2×D1211yescharacteristic, complemented
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—32no—
144C2×C3×C4—46no—
145C2×C3×C4—412no—
146C2×C3×C4—62no—
147C2×C3×C4GL(2,𝔽3)14yescomplemented
148C2×C3×C4C2×S412yes—
149D12—64no—
150D12—64no—
151D12—124no—
152D12—1248no—
153D12—128no—
154D12—128no—
155SL(2,𝔽3)C2×D1213yescomplemented
156C3⋊C8—64no—
157C2×C32—44no—
158C2×C32—41no—
159C2×C32—41no—
160C2×C32—41no—
161C3⋊D3—88no—
162C24—182no—
163C2×Q8—31no—
164C2×Q8—32no—
165C2×Q8C4⋊(C3⋊D3)12yescomplemented
166C2×Q8C22×C3⋊D311yescharacteristic
167C2×D4—94no—
168C2×D4—184no—
169C2×D4—184no—
170C2×D4—184no—
171C2×D4—182no—
172C2×D4—184no—
173C2×D4—364no—
174C2×D4—364no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4C3⋊S411yescharacteristic
178SD16—188no—
179SD16—184no—
180SD16—184no—
181C2×C8—94no—
182C2×C8—184no—
183C4⋊C4—32no—
184C4⋊C4—61no—
185C42—32no—
186C42—32no—
187C22×C3—46no—
188C22×C3—46no—
189C22×C3—46no—
190C22×C3—43no—
191C22×C3C4⋊S412yes—
192C22×C3C2×GL(2,𝔽3)12yes—
193C22×C3#98612yes—
194C22×C3C22×S411yescharacteristic
195C2×D3—1224no—
196C2×D3—124no—
197C2×D3—124no—
198C2×D3—124no—
199C2×D3—2412no—
200C2×D3—2412no—
201C2×D3—2448no—
202C2×D3—248no—
203C3×C4—31no—
204C3×C4—31no—
205C3×C4—32no—
206C3×C4—412no—
207C3×C4—62no—
208C3×C4C2×GL(2,𝔽3)14yescomplemented
209C32—41noSylow
210C23—184no—
211C23—364no—
212C23—364no—
213C23—364no—
214C23C2×C3⋊S411yescharacteristic
215Q8—32no—
216Q8—31no—
217Q8—61no—
218Q8C2×C4⋊(C3⋊D3)11yescharacteristic, complemented
219D4—184no—
220D4—184no—
221D4—364no—
222D4—368no—
223D4—368no—
224C2×C4—32no—
225C2×C4—32no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—32no—
229C2×C4—62no—
230C2×C4C3⋊GL(2,𝔽3)14yescomplemented
231C2×C4C2×C3⋊S412yes—
232C8—184no—
233C2×C3—412no—
234C2×C3—43no—
235C2×C3—43no—
236C2×C3—43no—
237C2×C3C4⋊GL(2,𝔽3)14yescomplemented
238C2×C3C2×C4⋊S411yescharacteristic
239C2×C3C22×GL(2,𝔽3)11yescharacteristic
240C2×C3C2×#98611yescharacteristic
241D3—2424no—
242D3—244no—
243C22—364no—
244C22—364no—
245C22—364no—
246C22—728no—
247C22C4⋊(C3⋊S4)12yes—
248C22C2×C3⋊GL(2,𝔽3)12yes—
249C22#6426712yes—
250C22C22×C3⋊S411yescharacteristic
251C4—31no—
252C4—31no—
253C4—32no—
254C4—62no—
255C4C2×C3⋊GL(2,𝔽3)14yescomplemented
256C3—43no—
257C3C2×C4⋊GL(2,𝔽3)11yescharacteristic, complemented
258C2—724no—
259C2C4⋊(C3⋊GL(2,𝔽3))14yesdirect product factor
260C2C2×C4⋊(C3⋊S4)11yescharacteristic
261C2C22×C3⋊GL(2,𝔽3)11yescharacteristic
262C2C2×#6426711yescharacteristic
subgroupG/H
centerC23C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊GL(2,𝔽3)C2semidirect
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. C4⋊(C3⋊GL(2,𝔽3))C2direct (N × H)
9. C2×C3⋊GL(2,𝔽3)C22semidirect
10. C3×C4×SL(2,𝔽3)C22semidirect
12. C2×C3×C4×Q8D3semidirect
15. C2×C4×SL(2,𝔽3)D3semidirect
25. C2×C3×SL(2,𝔽3)D4semidirect
38. C3×C4×Q8C2×D3semidirect
48. C4×SL(2,𝔽3)C2×D3semidirect
59. C3×SL(2,𝔽3)C2×D4semidirect
62. C2×C4×Q8C3⋊D3semidirect
71. C2×C3×Q8D12semidirect
87. C2×SL(2,𝔽3)D12semidirect
120. C4×Q8C2×C3⋊D3semidirect
138. C3×Q8C2×D12semidirect
147. C2×C3×C4GL(2,𝔽3)semidirect
155. SL(2,𝔽3)C2×D12semidirect
165. C2×Q8C4⋊(C3⋊D3)semidirect
208. C3×C4C2×GL(2,𝔽3)semidirect
218. Q8C2×C4⋊(C3⋊D3)semidirect
230. C2×C4C3⋊GL(2,𝔽3)semidirect
237. C2×C3C4⋊GL(2,𝔽3)semidirect
255. C4C2×C3⋊GL(2,𝔽3)semidirect
257. C3C2×C4⋊GL(2,𝔽3)semidirect
259. C2C4⋊(C3⋊GL(2,𝔽3))direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊GL(2,𝔽3)2
C2×C3×C4×SL(2,𝔽3)2
C4⋊(C3⋊GL(2,𝔽3))2
C2×C4⋊GL(2,𝔽3)3
C2×#50273
C22×C4⋊(C3⋊D3)4