Finite Group Databasebeta

← C22×C3×C22⋊SL(2,𝔽3) #11919892

Subgroups — C22×C3×C22⋊SL(2,𝔽3) #11919892

Order 1152.

2392 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 2392 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C2×C3×C22⋊SL(2,𝔽3)C213yesmaximal, direct product factor
2C24×C3×Q8C311yesmaximal, characteristic, complemented
3C22×C22⋊SL(2,𝔽3)C313yesmaximal, direct product factor
4C23×C3×A4—41nomaximal
5C3×C22⋊SL(2,𝔽3)C2211yescharacteristic, direct product factor
6C22×C3×SL(2,𝔽3)—44nomaximal
7C23×C3×Q8—312no—
8C23×C3×Q8—34no—
9C23×C3×Q8—33no—
10C23×C3×Q8C2×C313yescomplemented
11C24×C3×C4—31no—
12C2×C22⋊SL(2,𝔽3)C2×C319yesdirect product factor
13C22×C3×A4—44no—
14C22×C3×A4—43no—
15C2×C3×SL(2,𝔽3)—412no—
16C24×Q8C3211yescharacteristic, complemented, Sylow
17C25×C3A411yescharacteristic
18C23×A4—43no—
19C22×C3×Q8—312no—
20C22×C3×Q8—324no—
21C22×C3×Q8—336no—
22C22×C3×Q8—336no—
23C22×C3×Q8—336no—
24C22×C3×Q8—38no—
25C22×C3×Q8—32no—
26C22×C3×Q8—34no—
27C22×C3×Q8—312no—
28C22×C3×Q8—312no—
29C22×C3×Q8—33no—
30C22×C3×Q8A414yescomplemented
31C22×C3×Q8C22×C311yescharacteristic, complemented
32C23×C3×C4—318no—
33C23×C3×C4—36no—
34C23×C3×C4—33no—
35C23×C3×C4—33no—
36C22⋊SL(2,𝔽3)C22×C313yesdirect product factor
37C22×SL(2,𝔽3)—412no—
38C23×C32—161no—
39C2×C3×A4—46no—
40C2×C3×A4—41no—
41C3×SL(2,𝔽3)—44no—
42C23×Q8—312no—
43C23×Q8—34no—
44C23×Q8—33no—
45C23×Q8C2×C3213yescomplemented
46C24×C4—31no—
47C24×C3—34no—
48C24×C3—33no—
49C24×C3—31no—
50C24×C3SL(2,𝔽3)14yescomplemented
51C24×C3C2×A413yes—
52C22×A4—412no—
53C22×A4—49no—
54C2×C3×Q8—336no—
55C2×C3×Q8—336no—
56C2×C3×Q8—372no—
57C2×C3×Q8—372no—
58C2×C3×Q8—324no—
59C2×C3×Q8—336no—
60C2×C3×Q8—312no—
61C2×C3×Q8—312no—
62C2×C3×Q8—34no—
63C2×C3×Q8—312no—
64C2×C3×Q8C2×A4112yescomplemented
65C22×C3×C4—318no—
66C22×C3×C4—318no—
67C22×C3×C4—324no—
68C22×C3×C4—318no—
69C22×C3×C4—336no—
70C22×C3×C4—318no—
71C22×C3×C4—33no—
72C22×C3×C4—31no—
73C22×C3×C4—34no—
74C2×SL(2,𝔽3)—436no—
75C22×C32—164no—
76C22×C32—163no—
77C3×A4—41no—
78C25C3×A411yescharacteristic
79C22×Q8—312no—
80C22×Q8—324no—
81C22×Q8—336no—
82C22×Q8—336no—
83C22×Q8—336no—
84C22×Q8—38no—
85C22×Q8—32no—
86C22×Q8—34no—
87C22×Q8—312no—
88C22×Q8—312no—
89C22×Q8—33no—
90C22×Q8C3×A414yescomplemented
91C22×Q8C22×C3211yescharacteristic, complemented
92C23×C4—318no—
93C23×C4—36no—
94C23×C4—33no—
95C23×C4—33no—
96C23×C3—33no—
97C23×C3—312no—
98C23×C3—36no—
99C23×C3—32no—
100C23×C3—312no—
101C23×C3—34no—
102C23×C3—34no—
103C23×C3—33no—
104C23×C3—33no—
105C23×C3—163no—
106C23×C3C2×SL(2,𝔽3)16yescomplemented
107C23×C3C22×A411yescharacteristic
108C23×C3C22⋊A411yescharacteristic
109C2×A4—43no—
110C2×A4—418no—
111C3×Q8—312no—
112C3×Q8—324no—
113C3×Q8—38no—
114C3×Q8—336no—
115C3×Q8—34no—
116C3×Q8C22×A414yescomplemented
117C2×C3×C4—336no—
118C2×C3×C4—318no—
119C2×C3×C4—336no—
120C2×C3×C4—36no—
121C2×C3×C4—312no—
122C2×C3×C4—312no—
123SL(2,𝔽3)—412no—
124C2×C32—166no—
125C2×C32—161no—
126C24—33no—
127C24—34no—
128C24—31no—
129C24C3×SL(2,𝔽3)14yescomplemented
130C24C2×C3×A413yes—
131C2×Q8—336no—
132C2×Q8—336no—
133C2×Q8—372no—
134C2×Q8—372no—
135C2×Q8—324no—
136C2×Q8—312no—
137C2×Q8—336no—
138C2×Q8—34no—
139C2×Q8—312no—
140C2×Q8—312no—
141C2×Q8C2×C3×A4112yescomplemented
142C22×C4—318no—
143C22×C4—318no—
144C22×C4—324no—
145C22×C4—318no—
146C22×C4—336no—
147C22×C4—318no—
148C22×C4—31no—
149C22×C4—33no—
150C22×C4—34no—
151C22×C3—36no—
152C22×C3—31no—
153C22×C3—36no—
154C22×C3—38no—
155C22×C3—312no—
156C22×C3—36no—
157C22×C3—36no—
158C22×C3—33no—
159C22×C3—31no—
160C22×C3—1612no—
161C22×C3—169no—
162C22×C3C22×SL(2,𝔽3)11yescharacteristic, complemented
163C22×C3C22⋊SL(2,𝔽3)14yesdirect product factor
164C22×C3C2×C22⋊A413yes—
165A4—43no—
166C3×C4—312no—
167C3×C4—34no—
168C32—161noSylow
169C23—33no—
170C23—312no—
171C23—36no—
172C23—32no—
173C23—312no—
174C23—34no—
175C23—34no—
176C23—33no—
177C23—33no—
178C23C2×C3×SL(2,𝔽3)16yescomplemented
179C23C22×C3×A411yescharacteristic
180C23C3×C22⋊A411yescharacteristic
181Q8—312no—
182Q8—34no—
183Q8—336no—
184Q8—324no—
185Q8—38no—
186Q8C22×C3×A414yescomplemented
187C2×C4—36no—
188C2×C4—336no—
189C2×C4—318no—
190C2×C4—336no—
191C2×C4—312no—
192C2×C4—312no—
193C2×C3—36no—
194C2×C3—31no—
195C2×C3—31no—
196C2×C3—1618no—
197C2×C3—163no—
198C2×C3C2×C22⋊SL(2,𝔽3)16yesdirect product factor
199C2×C3C22×C22⋊A411yescharacteristic
200C22—31no—
201C22—36no—
202C22—36no—
203C22—38no—
204C22—36no—
205C22—312no—
206C22—36no—
207C22—31no—
208C22—33no—
209C22C22×C3×SL(2,𝔽3)11yescharacteristic, complemented
210C22C3×C22⋊SL(2,𝔽3)14yesdirect product factor
211C22C2×C3×C22⋊A413yes—
212C4—34no—
213C4—312no—
214C3—163no—
215C3C22×C22⋊SL(2,𝔽3)11yescharacteristic, direct product factor
216C2—31no—
217C2—31no—
218C2—36no—
219C2C2×C3×C22⋊SL(2,𝔽3)16yesdirect product factor
220C2C22×C3×C22⋊A411yescharacteristic
subgroupG/H
center, hypercenterC23×C3C22⋊A4
commutator, cosocle, hypocenterC22×Q8C22×C32
FrattiniC2C22×C3×C22⋊A4
FittingC24×C3×Q8C3
socleC25×C3A4
LCSG ▷ C22×Q8
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊SL(2,𝔽3)C2direct (N × H)
2. C24×C3×Q8C3semidirect
3. C22×C22⋊SL(2,𝔽3)C3direct (N × H)
5. C3×C22⋊SL(2,𝔽3)C22direct (N × H)
10. C23×C3×Q8C2×C3semidirect
12. C2×C22⋊SL(2,𝔽3)C2×C3direct (N × H)
16. C24×Q8C32semidirect
30. C22×C3×Q8A4semidirect
31. C22×C3×Q8C22×C3semidirect
36. C22⋊SL(2,𝔽3)C22×C3direct (N × H)
45. C23×Q8C2×C32semidirect
50. C24×C3SL(2,𝔽3)semidirect
64. C2×C3×Q8C2×A4semidirect
90. C22×Q8C3×A4semidirect
91. C22×Q8C22×C32semidirect
106. C23×C3C2×SL(2,𝔽3)semidirect
116. C3×Q8C22×A4semidirect
129. C24C3×SL(2,𝔽3)semidirect
141. C2×Q8C2×C3×A4semidirect
162. C22×C3C22×SL(2,𝔽3)semidirect
163. C22×C3C22⋊SL(2,𝔽3)direct (N × H)
178. C23C2×C3×SL(2,𝔽3)semidirect
186. Q8C22×C3×A4semidirect
198. C2×C3C2×C22⋊SL(2,𝔽3)direct (N × H)
209. C22C22×C3×SL(2,𝔽3)semidirect
210. C22C3×C22⋊SL(2,𝔽3)direct (N × H)
215. C3C22×C22⋊SL(2,𝔽3)direct (N × H)
219. C2C2×C3×C22⋊SL(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
C2×C3×C22⋊SL(2,𝔽3)2
C24×C3×Q83
C22×C22⋊SL(2,𝔽3)3
C23×C3×A44
C22×C3×SL(2,𝔽3)4