Finite Group Databasebeta

← SL(2,𝔽3)×#239 #11923292

Subgroups — SL(2,𝔽3)×#239 #11923292

Order 1152.

721 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 135 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
2SL(2,𝔽3)×D12C213yesmaximal, complemented
3C4×D3×SL(2,𝔽3)C213yesmaximal, complemented
4Q8×#239C311yesmaximal, characteristic, complemented
5#41×SL(2,𝔽3)—31nomaximal
6C2×C3×#239—41nomaximal
7C2×D3×SL(2,𝔽3)C2213yes—
8C3×C4×SL(2,𝔽3)C2213yes—
9Dic3×SL(2,𝔽3)C2211yescharacteristic
10C3×Q82C2×C311yescharacteristic, complemented
11#5730—33no—
12#5728—33no—
13Q8×D12C2×C313yescomplemented
14C4×#239—31no—
15C4×D3×Q8C2×C313yescomplemented
16#5570—31no—
17Q8×SL(2,𝔽3)—21no—
18Q8×SL(2,𝔽3)D311yescharacteristic, complemented
19D4×SL(2,𝔽3)—33no—
20C2×C4×SL(2,𝔽3)—33no—
21C2×C32×Q8—41no—
22C3×#239—48no—
23C2×C3×D12—43no—
24C2×C3×C4×D3—43no—
25C2×C3×SL(2,𝔽3)C2311yescharacteristic
26D3×SL(2,𝔽3)—23no—
27Q8×#41—31noSylow
28C2×C3×#41—121no—
29C2×#239A411yescharacteristic
30C2×D3×Q8C22×C313yes—
31C22×SL(2,𝔽3)—33no—
32C3×C4⋊Q8—33no—
33C3×C4×Q8—31no—
34C3×C4×Q8C22×C313yes—
35#946—36no—
36Q8×Dic3—32no—
37Q8×Dic3—31no—
38Q8×Dic3C22×C311yescharacteristic
39#897—36no—
40#893—33no—
41#892—33no—
42D3×C4⋊C4—33no—
43#890—36no—
44#887—33no—
45C4×D12—33no—
46C42×D3—33no—
47#869—33no—
48C4×Dic6—33no—
49C4×SL(2,𝔽3)—23no—
50C4×SL(2,𝔽3)—31no—
51C4×SL(2,𝔽3)C2×D313yes—
52C22×C3×D3—43no—
53C32×Q8—44no—
54C2×C32×C4—43no—
55C2×C3×Dic3—41no—
56C3×D12—412no—
57C3×C4×D3—412no—
58C3×SL(2,𝔽3)#4111yescharacteristic, complemented
59Q82C3×D311yescharacteristic, complemented
60#557—91no—
61#554—93no—
62D4×Q8—33no—
63#533—93no—
64C4×#41—91no—
65C2×C4×Q8—33no—
66C3×#41—128no—
67C2×C3×Q8—32no—
68C2×C3×Q8—41no—
69C2×C3×Q8—81no—
70C2×C3×Q8C2×A411yescharacteristic
71C2×C3×Q8C23×C311yescharacteristic
72C2×C3×D4—123no—
73C22×C3×C4—123no—
74#239SL(2,𝔽3)18yesdirect product factor
75D3×Q8—23no—
76D3×Q8—63no—
77C2×D12C2×A413yes—
78C2×C4×D3—33no—
79C2×C4×D3—33no—
80C2×C4×D3C2×A413yes—
81C2×Dic6—33no—
82C2×SL(2,𝔽3)—21no—
83C2×SL(2,𝔽3)—63no—
84C2×SL(2,𝔽3)C22×D311yescharacteristic
85C3×C4⋊C4—33no—
86C3×C4⋊C4—33no—
87C3×C42—33no—
88(C2×D3)⋊C4—36no—
89C4⋊Dic3—36no—
90C4⋊Dic3—33no—
91C4⋊Dic3—31no—
92Dic3⋊C4—36no—
93C4×Dic3—33no—
94C4×Dic3—31no—
95C4×Dic3—33no—
96C22×C32—41no—
97C2×C3×D3—46no—
98C2×C3×D3—83no—
99C2×C3×D3—83no—
100C32×C4—46no—
101C3×Dic3—42no—
102C2×#41—31no—
103C22×Q8—33no—
104C4⋊Q8—33no—
105C4⋊Q8—93no—
106#125—96no—
107C22⋊Q8—96no—
108C22⋊Q8—96no—
109C4×Q8—31no—
110C4×Q8—31no—
111C4×Q8—93no—
112C4×Q8—92no—
113C4×Q8—91no—
114C4×Q8—93no—
115C4×Q8C2×C3×D313yes—
116C4×D4—93no—
117C4×D4—93no—
118#117—93no—
119C2×C4⋊C4—93no—
120C2×C42—93no—
121C23×C3—123no—
122C22×D3C22×A413yes—
123C3×Q8—31no—
124C3×Q8—44no—
125C3×Q8—84no—
126C3×Q8—122no—
127C3×Q8C2×SL(2,𝔽3)14yescomplemented
128C3×Q8C3×#4111yescharacteristic, complemented
129C3×D4—1212no—
130C2×C3×C4—31no—
131C2×C3×C4—33no—
132C2×C3×C4—43no—
133C2×C3×C4—83no—
134C2×C3×C4—1212no—
135C2×C3×C4—121no—
136C2×C3×C4C22×A413yes—
137C2×Dic3—31no—
138C2×Dic3—33no—
139C2×Dic3C22×A411yescharacteristic
140D12C2×SL(2,𝔽3)112yescomplemented
141C4×D3—63no—
142C4×D3—63no—
143C4×D3—123no—
144C4×D3C2×SL(2,𝔽3)112yescomplemented
145Dic6—63no—
146Dic6—63no—
147SL(2,𝔽3)—21no—
148SL(2,𝔽3)#23911yescharacteristic, direct product factor
149C2×C32—41no—
150C2×C32—41no—
151C2×C32—41no—
152C3×D3—86no—
153#41—38no—
154C2×Q8—32no—
155C2×Q8—63no—
156C2×Q8—93no—
157C2×Q8—183no—
158C2×Q8D3×A411yescharacteristic
159C2×Q8C22×C3×D311yescharacteristic
160C2×D4—33no—
161C22×C4—33no—
162C22×C4—93no—
163C22×C4—93no—
164C4⋊C4—33no—
165C4⋊C4—33no—
166C4⋊C4—91no—
167C4⋊C4—96no—
168C4⋊C4—96no—
169C4⋊C4—93no—
170C22⋊C4—96no—
171C42—33no—
172C42—91no—
173C42—93no—
174C42—93no—
175C22×C3—41no—
176C22×C3—81no—
177C22×C3—126no—
178C22×C3—243no—
179C22×C3—243no—
180C22×C3C23×A411yescharacteristic
181C2×D3—23no—
182C2×D3—23no—
183C2×D3C22×SL(2,𝔽3)16yes—
184C3×C4—31no—
185C3×C4—31no—
186C3×C4—46no—
187C3×C4—63no—
188C3×C4—86no—
189C3×C4—122no—
190C3×C4C22×SL(2,𝔽3)16yes—
191Dic3—63no—
192Dic3—61no—
193Dic3C22×SL(2,𝔽3)12yes—
194C32—41noSylow
195C23—33no—
196Q8—31no—
197Q8—122no—
198Q8—183no—
199Q8—183no—
200Q8D3×SL(2,𝔽3)14yescomplemented
201Q8C3×#23911yescharacteristic, complemented
202D4—312no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—312no—
206C2×C4—33no—
207C2×C4—91no—
208C2×C4—93no—
209C2×C4—183no—
210C2×C4—183no—
211C2×C4—363no—
212C2×C4C2×D3×A413yes—
213C2×C3—41no—
214C2×C3—41no—
215C2×C3—41no—
216C2×C3—81no—
217C2×C3—81no—
218C2×C3—81no—
219C2×C3—246no—
220C2×C3C23×SL(2,𝔽3)11yescharacteristic
221C2×C3A4×#4111yescharacteristic
222C2×C3#594511yescharacteristic
223D3—26no—
224C22—36no—
225C22—63no—
226C22—63no—
227C22C22×D3×A411yescharacteristic
228C4—32no—
229C4—31no—
230C4—31no—
231C4—63no—
232C4—183no—
233C4—181no—
234C4C2×D3×SL(2,𝔽3)16yes—
235C3—41no—
236C3—81no—
237C3#41×SL(2,𝔽3)11yescharacteristic, complemented
238C2—66no—
239C2C22×D3×SL(2,𝔽3)11yescharacteristic
240C2A4×#23911yescharacteristic
241C2#1059662311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×Q82C2×C3
socleC22×C3C23×A4
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
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. SL(2,𝔽3)×D12C2semidirect
3. C4×D3×SL(2,𝔽3)C2semidirect
4. Q8×#239C3semidirect
10. C3×Q82C2×C3semidirect
13. Q8×D12C2×C3semidirect
15. C4×D3×Q8C2×C3semidirect
18. Q8×SL(2,𝔽3)D3semidirect
58. C3×SL(2,𝔽3)#41semidirect
59. Q82C3×D3semidirect
74. #239SL(2,𝔽3)direct (N × H)
127. C3×Q8C2×SL(2,𝔽3)semidirect
128. C3×Q8C3×#41semidirect
140. D12C2×SL(2,𝔽3)semidirect
144. C4×D3C2×SL(2,𝔽3)semidirect
148. SL(2,𝔽3)#239direct (N × H)
200. Q8D3×SL(2,𝔽3)semidirect
201. Q8C3×#239semidirect
237. C3#41×SL(2,𝔽3)semidirect

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
SL(2,𝔽3)×D122
C4×D3×SL(2,𝔽3)2
Q8×#2393
#41×SL(2,𝔽3)3
C2×C3×#2394