Finite Group Databasebeta

← SL(2,𝔽3)×#215 #11921313

Subgroups — SL(2,𝔽3)×#215 #11921313

Order 1152.

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

normal subgroups only

This group has 295 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×D12C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4Q8×#215C311yesmaximal, characteristic, complemented
5SD16×SL(2,𝔽3)—31nomaximal
6C2×D3×SL(2,𝔽3)—21no—
7C2×C3×#215—41nomaximal
8C3×C4×SL(2,𝔽3)—21no—
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10C3×Q82C2×C311yescharacteristic, complemented
11Q8×D12C2×C311yescharacteristic
12Q8×SL(2,𝔽3)—21no—
13Q8×SL(2,𝔽3)D311yescharacteristic, complemented
14D4×SL(2,𝔽3)—31no—
15#5084—31no—
16#5080—31no—
17C4×#215—31no—
18Q8×C3⋊C8C2×C311yescharacteristic, complemented
19#5020—31no—
20C8×SL(2,𝔽3)—31no—
21C2×C32×Q8—41no—
22C2×C3×D12—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24D3×SL(2,𝔽3)—41no—
25C3×#215—44no—
26C2×C3×C3⋊C8—41no—
27Q8×SD16—31noSylow
28C2×D3×Q8—21no—
29C22×SL(2,𝔽3)—61no—
30C2×C3×SD16—121no—
31C3×C4⋊Q8—31no—
32C3×C4⋊Q8—61no—
33C3×C4×Q8—21no—
34C3×C4×Q8—31no—
35C3×C4×Q8C22×C311yescharacteristic
36C2×#215A411yescharacteristic
37#897—61no—
38C4×D12—31no—
39#869—31no—
40C4×SL(2,𝔽3)—22no—
41C4×SL(2,𝔽3)—21no—
42C4×SL(2,𝔽3)—21no—
43C4×SL(2,𝔽3)C2×D311yescharacteristic
44#835—31no—
45#809—31no—
46#808—32no—
47#808—31no—
48C4⋊(C3⋊C8)—31no—
49C4×C3⋊C8—31no—
50C22×C3×D3—81no—
51C32×Q8—42no—
52C32×Q8—81no—
53C2×C32×C4—41no—
54C2×C32×C4—81no—
55C3×D12—42no—
56C3×D12—81no—
57C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
58C3×C3⋊C8—42no—
59Q82C3×D311yescharacteristic, complemented
60D4×Q8—31no—
61#498—91no—
62#476—91no—
63#475—91no—
64C8×Q8—31no—
65C4×SD16—91no—
66C2×C3×Q8—41no—
67C2×C3×Q8—61no—
68C2×C3×Q8—81no—
69C2×C3×Q8C3×D411yescharacteristic
70C2×C3×Q8C2×A411yescharacteristic
71C2×C3×D4—121no—
72D3×Q8—41no—
73D3×Q8—121no—
74C2×D12C2×A411yescharacteristic
75C2×C4×D3—61no—
76C2×Dic6—61no—
77C2×SL(2,𝔽3)—21no—
78C2×SL(2,𝔽3)—121no—
79C2×SL(2,𝔽3)#6711yescharacteristic
80C3×SD16—124no—
81C2×C3×C8—121no—
82C3×C4⋊C4—31no—
83C3×C4⋊C4—31no—
84C3×C4⋊C4—61no—
85C3×C4⋊C4—61no—
86C3×C42—31no—
87C3×C42—61no—
88#215SL(2,𝔽3)14yesdirect product factor
89(C2×D3)⋊C4—61no—
90C4⋊Dic3—31no—
91C4⋊Dic3—61no—
92C2×C3⋊C8—31no—
93C2×C3⋊C8C2×A411yescharacteristic
94C22×C32—41no—
95C2×C3×D3—82no—
96C2×C3×D3—161no—
97C2×C3×D3—161no—
98C32×C4—41no—
99C32×C4—41no—
100C32×C4—82no—
101C22×Q8—61no—
102C2×SD16—31no—
103C4⋊Q8—31no—
104C4⋊Q8—61no—
105C4⋊Q8—91no—
106C22⋊Q8—181no—
107C4×Q8—21no—
108C4×Q8—31no—
109C4×Q8C2×C3×D311yescharacteristic
110C4×D4—91no—
111#106—91no—
112#106—92no—
113C4⋊C8—91no—
114#103—91no—
115#102—91no—
116C4×C8—91no—
117C23×C3—241no—
118C22×D3—21no—
119C3×Q8—21no—
120C3×Q8—31no—
121C3×Q8—42no—
122C3×Q8—81no—
123C3×Q8—82no—
124C3×Q8—82no—
125C3×Q8—241no—
126C3×Q8C3×SD1611yescharacteristic, complemented
127C3×Q8C2×SL(2,𝔽3)12yescomplemented
128C3×D4—122no—
129C3×D4—241no—
130C2×C3×C4—21no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—41no—
134C2×C3×C4—61no—
135C2×C3×C4—81no—
136C2×C3×C4—82no—
137C2×C3×C4—81no—
138C2×C3×C4C22×A411yescharacteristic
139C2×Dic3—61no—
140D12—21no—
141D12C2×SL(2,𝔽3)12yes—
142C4×D3—121no—
143C4×D3—121no—
144Dic6—121no—
145Dic6—121no—
146SL(2,𝔽3)—21no—
147SL(2,𝔽3)#21511yescharacteristic, direct product factor
148C3×C8—122no—
149C3⋊C8—61no—
150C3⋊C8C2×SL(2,𝔽3)12yescomplemented
151C2×C32—41no—
152C2×C32—41no—
153C2×C32—41no—
154C3×D3—162no—
155C2×Q8—61no—
156C2×Q8—121no—
157C2×Q8—181no—
158C2×Q8—361no—
159C2×Q8C3×#6711yescharacteristic
160C2×Q8D3×A411yescharacteristic
161C2×D4—31no—
162C22×C4—181no—
163SD16—34no—
164C2×C8—31no—
165C2×C8—91no—
166C4⋊C4—31no—
167C4⋊C4—31no—
168C4⋊C4—61no—
169C4⋊C4—61no—
170C4⋊C4—91no—
171C4⋊C4—181no—
172C22⋊C4—181no—
173C42—31no—
174C42—61no—
175C22×C3—41no—
176C22×C3—81no—
177C22×C3—242no—
178C22×C3—481no—
179C22×C3—481no—
180C22×C3D4×A411yescharacteristic
181C2×D3—22no—
182C2×D3—41no—
183C2×D3—41no—
184C3×C4—22no—
185C3×C4—31no—
186C3×C4—31no—
187C3×C4—41no—
188C3×C4—41no—
189C3×C4—61no—
190C3×C4—82no—
191C3×C4—84no—
192C3×C4—81no—
193C3×C4—81no—
194C3×C4—121no—
195C3×C4C22×SL(2,𝔽3)11yescharacteristic
196C3×C42−1+4⋊C311yescharacteristic
197Dic3—121no—
198C32—41noSylow
199C23—61no—
200Q8—21no—
201Q8—31no—
202Q8—241no—
203Q8—361no—
204Q8—361no—
205Q8C3×#21511yescharacteristic, complemented
206Q8D3×SL(2,𝔽3)12yescomplemented
207D4—32no—
208D4—61no—
209C2×C4—21no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—61no—
213C2×C4—181no—
214C2×C4—361no—
215C2×C4—361no—
216C2×C4C2×D3×A411yescharacteristic
217C8—32no—
218C8—181no—
219C2×C3—41no—
220C2×C3—41no—
221C2×C3—41no—
222C2×C3—81no—
223C2×C3—81no—
224C2×C3—81no—
225C2×C3—482no—
226C2×C3D4×SL(2,𝔽3)11yescharacteristic
227C2×C3A4×SD1611yescharacteristic
228C2×C3#577⋊C311yescharacteristic
229D3—42no—
230C22—62no—
231C22—121no—
232C22—121no—
233C22A4×#6711yescharacteristic
234C4—22no—
235C4—31no—
236C4—31no—
237C4—61no—
238C4—121no—
239C4—361no—
240C4C2×D3×SL(2,𝔽3)11yescharacteristic
241C4#1007⋊C311yescharacteristic
242C3—41no—
243C3—81no—
244C3SD16×SL(2,𝔽3)11yescharacteristic, complemented
245C2—122no—
246C2A4×#21511yescharacteristic
247C2SL(2,𝔽3)×#6711yescharacteristic
248C2#5836⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×Q82C2×C3
socleC22×C3D4×A4
prime radicalSL(2,𝔽3)×D12C2
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C3×SD16
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. Q8×#215C3semidirect
10. C3×Q82C2×C3semidirect
13. Q8×SL(2,𝔽3)D3semidirect
18. Q8×C3⋊C8C2×C3semidirect
57. C3×SL(2,𝔽3)SD16semidirect
59. Q82C3×D3semidirect
88. #215SL(2,𝔽3)direct (N × H)
126. C3×Q8C3×SD16semidirect
127. C3×Q8C2×SL(2,𝔽3)semidirect
147. SL(2,𝔽3)#215direct (N × H)
150. C3⋊C8C2×SL(2,𝔽3)semidirect
205. Q8C3×#215semidirect
206. Q8D3×SL(2,𝔽3)semidirect
244. C3SD16×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
C3⋊C8×SL(2,𝔽3)2
Q8×#2153
SD16×SL(2,𝔽3)3
C2×C3×#2154