Finite Group Databasebeta

← SL(2,𝔽3)×#213 #11921301

Subgroups — SL(2,𝔽3)×#213 #11921301

Order 1152.

333 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 55 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4Q8×#213C311yesmaximal, characteristic, complemented
5D8×SL(2,𝔽3)—31nomaximal
6C22×C3×SL(2,𝔽3)—21no—
7C2×D3×SL(2,𝔽3)—21no—
8C2×C3×#213—41nomaximal
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10C3×D4×Q8C2×C311yescharacteristic, complemented
11Q8×D12C2×C311yescharacteristic, complemented
12D4×SL(2,𝔽3)—21no—
13D4×SL(2,𝔽3)—31no—
14D4×SL(2,𝔽3)D311yescharacteristic, complemented
15#5087—31no—
16#5078—31no—
17Q8×C3⋊C8C2×C311yescharacteristic, complemented
18C4×#213—31no—
19#5007—31no—
20C8×SL(2,𝔽3)—31no—
21C2×C32×D4—41no—
22C2×C3×D12—41no—
23C2×C3×SL(2,𝔽3)—41no—
24C2×C3×SL(2,𝔽3)D411yescharacteristic
25D3×SL(2,𝔽3)—41no—
26C3×#213—44no—
27C2×C3×C3⋊C8—41no—
28Q8×D8—31noSylow
29C22×C3×Q8—21no—
30C2×D3×Q8—21no—
31C22×SL(2,𝔽3)—22no—
32C22×SL(2,𝔽3)—21no—
33C22×SL(2,𝔽3)—61no—
34C2×C3×D8—121no—
35C3×C4⋊Q8—31no—
36C3×C22⋊Q8—61no—
37C3×C4×Q8C22×C311yescharacteristic
38C3×C4×D4—31no—
39C2×#213A411yescharacteristic
40#897—61no—
41C4×D12—31no—
42#869—31no—
43C4×SL(2,𝔽3)—21no—
44C4×SL(2,𝔽3)C2×D311yescharacteristic
45#832—31no—
46#809—31no—
47#807—32no—
48#807—31no—
49C4⋊(C3⋊C8)—31no—
50C4×C3⋊C8—31no—
51C23×C32—81no—
52C22×C3×D3—81no—
53C32×D4—42no—
54C32×D4—81no—
55C2×C32×C4—41no—
56C3×D12—42no—
57C3×D12—81no—
58C3×SL(2,𝔽3)D811yescharacteristic, complemented
59C3×C3⋊C8—42no—
60D4×Q8—31no—
61D4×Q8C3×D311yescharacteristic, complemented
62#500—91no—
63#474—91no—
64#474—91no—
65C8×Q8—31no—
66C4×D8—91no—
67C2×C3×Q8—41no—
68C2×C3×Q8—61no—
69C2×C3×Q8—121no—
70C2×C3×Q8C3×D411yescharacteristic
71C2×C3×D4—41no—
72C2×C3×D4—81no—
73C2×C3×D4—121no—
74C2×C3×D4C2×A411yescharacteristic
75C22×C3×C4—61no—
76D3×Q8—41no—
77D3×Q8—121no—
78C2×D12C2×A411yescharacteristic
79C2×C4×D3—61no—
80C2×Dic6—61no—
81C2×SL(2,𝔽3)—21no—
82C2×SL(2,𝔽3)—42no—
83C2×SL(2,𝔽3)—41no—
84C2×SL(2,𝔽3)—121no—
85C2×SL(2,𝔽3)#6711yescharacteristic
86C3×D8—124no—
87C2×C3×C8—121no—
88C3×C4⋊C4—31no—
89C3×C4⋊C4—31no—
90C3×C4⋊C4—61no—
91C3×C22⋊C4—61no—
92C3×C42—31no—
93#213SL(2,𝔽3)14yesdirect product factor
94(C2×D3)⋊C4—61no—
95C4⋊Dic3—31no—
96C4⋊Dic3—61no—
97C2×C3⋊C8—31no—
98C2×C3⋊C8C2×A411yescharacteristic
99C22×C32—41no—
100C22×C32—82no—
101C22×C32—161no—
102C22×C32—161no—
103C2×C3×D3—82no—
104C2×C3×D3—161no—
105C2×C3×D3—161no—
106C32×C4—41no—
107C32×C4—41no—
108C22×Q8—21no—
109C22×Q8—61no—
110C2×D8—31no—
111C4⋊Q8—31no—
112C4⋊Q8—91no—
113C22⋊Q8—61no—
114C22⋊Q8—181no—
115C4×Q8C2×C3×D311yescharacteristic
116C4×D4—31no—
117C4×D4—91no—
118#107—91no—
119#107—92no—
120C4⋊C8—91no—
121#102—91no—
122#102—91no—
123C4×C8—91no—
124C23×C3—21no—
125C23×C3—81no—
126C23×C3—82no—
127C23×C3—241no—
128C22×D3—21no—
129C3×Q8—31no—
130C3×Q8—121no—
131C3×Q8—121no—
132C3×Q8C3×D811yescharacteristic, complemented
133C3×D4—21no—
134C3×D4—42no—
135C3×D4—81no—
136C3×D4—82no—
137C3×D4—82no—
138C3×D4—122no—
139C3×D4—241no—
140C3×D4C2×SL(2,𝔽3)12yescomplemented
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—41no—
144C2×C3×C4—61no—
145C2×C3×C4—81no—
146C2×C3×C4—121no—
147C2×C3×C4—121no—
148C2×C3×C4C22×A411yescharacteristic
149C2×Dic3—61no—
150D12—21no—
151D12C2×SL(2,𝔽3)12yescomplemented
152C4×D3—121no—
153C4×D3—121no—
154Dic6—121no—
155Dic6—121no—
156SL(2,𝔽3)—21no—
157SL(2,𝔽3)#21311yescharacteristic, direct product factor
158C3×C8—122no—
159C3⋊C8—61no—
160C3⋊C8C2×SL(2,𝔽3)12yescomplemented
161C2×C32—41no—
162C2×C32—41no—
163C2×C32—41no—
164C2×C32—162no—
165C3×D3—162no—
166C2×Q8—41no—
167C2×Q8—61no—
168C2×Q8—121no—
169C2×Q8—121no—
170C2×Q8—181no—
171C2×Q8—361no—
172C2×Q8C3×#6711yescharacteristic
173C2×D4—31no—
174C2×D4D3×A411yescharacteristic
175C22×C4—61no—
176C22×C4—181no—
177D8—34no—
178C2×C8—31no—
179C2×C8—91no—
180C4⋊C4—31no—
181C4⋊C4—31no—
182C4⋊C4—61no—
183C4⋊C4—91no—
184C4⋊C4—181no—
185C22⋊C4—61no—
186C22⋊C4—181no—
187C42—31no—
188C22×C3—22no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—82no—
193C22×C3—84no—
194C22×C3—81no—
195C22×C3—161no—
196C22×C3—161no—
197C22×C3—162no—
198C22×C3—162no—
199C22×C3—242no—
200C22×C3—481no—
201C22×C3—481no—
202C22×C3D4×A411yescharacteristic
203C2×D3—22no—
204C2×D3—41no—
205C2×D3—41no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—41no—
209C3×C4—41no—
210C3×C4—61no—
211C3×C4—81no—
212C3×C4—81no—
213C3×C4—121no—
214C3×C4C22×SL(2,𝔽3)11yescharacteristic
215C3×C42−1+4⋊C311yescharacteristic
216Dic3—121no—
217C32—41noSylow
218C23—21no—
219C23—61no—
220Q8—31no—
221Q8—121no—
222Q8—121no—
223Q8—361no—
224Q8—361no—
225Q8C3×#21311yescharacteristic, complemented
226D4—21no—
227D4—32no—
228D4—61no—
229D4D3×SL(2,𝔽3)12yescomplemented
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—61no—
233C2×C4—121no—
234C2×C4—121no—
235C2×C4—181no—
236C2×C4—361no—
237C2×C4—361no—
238C2×C4C2×D3×A411yescharacteristic
239C8—32no—
240C8—181no—
241C2×C3—42no—
242C2×C3—41no—
243C2×C3—41no—
244C2×C3—41no—
245C2×C3—81no—
246C2×C3—81no—
247C2×C3—81no—
248C2×C3—162no—
249C2×C3—164no—
250C2×C3—482no—
251C2×C3D4×SL(2,𝔽3)11yescharacteristic
252C2×C3A4×D811yescharacteristic
253C2×C3#578⋊C311yescharacteristic
254D3—42no—
255C22—22no—
256C22—41no—
257C22—41no—
258C22—62no—
259C22—121no—
260C22—121no—
261C22A4×#6711yescharacteristic
262C4—31no—
263C4—31no—
264C4—61no—
265C4—121no—
266C4—361no—
267C4C2×D3×SL(2,𝔽3)11yescharacteristic
268C4#1007⋊C311yescharacteristic
269C3—41no—
270C3—81no—
271C3D8×SL(2,𝔽3)11yescharacteristic, complemented
272C2—42no—
273C2—122no—
274C2A4×#21311yescharacteristic
275C2SL(2,𝔽3)×#6711yescharacteristic
276C2#5838⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×D4×Q8C2×C3
socleC22×C3D4×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×D8
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×SL(2,𝔽3)C2semidirect
2. SL(2,𝔽3)×D12C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. Q8×#213C3semidirect
10. C3×D4×Q8C2×C3semidirect
11. Q8×D12C2×C3semidirect
14. D4×SL(2,𝔽3)D3semidirect
17. Q8×C3⋊C8C2×C3semidirect
58. C3×SL(2,𝔽3)D8semidirect
61. D4×Q8C3×D3semidirect
93. #213SL(2,𝔽3)direct (N × H)
132. C3×Q8C3×D8semidirect
140. C3×D4C2×SL(2,𝔽3)semidirect
151. D12C2×SL(2,𝔽3)semidirect
157. SL(2,𝔽3)#213direct (N × H)
160. C3⋊C8C2×SL(2,𝔽3)semidirect
225. Q8C3×#213semidirect
229. D4D3×SL(2,𝔽3)semidirect
271. C3D8×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×D4×SL(2,𝔽3)2
SL(2,𝔽3)×D122
C3⋊C8×SL(2,𝔽3)2
Q8×#2133
D8×SL(2,𝔽3)3
C2×C3×#2134