Finite Group Databasebeta

← SL(2,𝔽3)×#208 #11921259

Subgroups — SL(2,𝔽3)×#208 #11921259

Order 1152.

311 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 79 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊C8×SL(2,𝔽3)C212yesmaximal, complemented
3Q8×#208C311yesmaximal, characteristic, complemented
4M16×SL(2,𝔽3)—31nomaximal
5C22×C3×SL(2,𝔽3)C411yescharacteristic
6C2×C3×#208—41nomaximal
7C3×C4×SL(2,𝔽3)C411yescharacteristic
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×C4×Q8C2×C311yescharacteristic
10C2×C4×SL(2,𝔽3)—21no—
11C2×C4×SL(2,𝔽3)D311yescharacteristic
12#5024—32no—
13Q8×C3⋊C8C2×C312yescomplemented
14#4924—31no—
15C4×#208—31no—
16C8×SL(2,𝔽3)—32no—
17C22×C32×C4—41no—
18C2×C3×SL(2,𝔽3)—21no—
19C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
20C3×#208—44no—
21C2×C3×C3⋊C8—42no—
22Q8×M16—31noSylow
23C22×C3×Q8C3×C411yescharacteristic
24C22×SL(2,𝔽3)—21no—
25C22×SL(2,𝔽3)Dic311yescharacteristic
26C2×C3×M16—121no—
27C3×C4×Q8—31no—
28C3×C4×Q8—31no—
29C3×C4×Q8C3×C411yescharacteristic
30C3×C4×Q8C22×C311yescharacteristic
31C2×C3×C4⋊C4—31no—
32C2×C3×C42—31no—
33C2×#208—31no—
34C2×#208A411yescharacteristic
35C4×SL(2,𝔽3)—21no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)Dic311yescharacteristic
38C4×SL(2,𝔽3)C2×D311yescharacteristic
39C4⋊(C3⋊C8)—32no—
40C4⋊(C3⋊C8)—32no—
41#803—32no—
42C4×C3⋊C8—32no—
43C23×C32—41no—
44C2×C32×C4—41no—
45C2×C32×C4—42no—
46C2×C32×C4—42no—
47C2×C32×C4—41no—
48C3×SL(2,𝔽3)M1611yescharacteristic, complemented
49C3×C3⋊C8—44no—
50C2×C4×Q8C3×D311yescharacteristic
51#446—92no—
52C8×Q8—32no—
53#423—91no—
54C4×M16—91no—
55C2×C3×Q8—21no—
56C2×C3×Q8—31no—
57C2×C3×Q8—61no—
58C2×C3×Q8C2×C3×C411yescharacteristic
59C22×C3×C4—31no—
60C22×C3×C4—31no—
61C22×C3×C4—41no—
62C22×C3×C4—81no—
63C22×C3×C4C2×A411yescharacteristic
64C2×SL(2,𝔽3)—22no—
65C2×SL(2,𝔽3)—21no—
66C2×SL(2,𝔽3)—21no—
67C2×SL(2,𝔽3)C2×Dic311yescharacteristic
68C3×M16—124no—
69C2×C3×C8—122no—
70C3×C4⋊C4—31no—
71C3×C4⋊C4—31no—
72C3×C4⋊C4—32no—
73C3×C42—31no—
74C3×C42—31no—
75C3×C42—31no—
76C3×C42—31no—
77#208—62no—
78#208SL(2,𝔽3)14yesdirect product factor
79C2×C3⋊C8—32no—
80C2×C3⋊C8C2×A412yes—
81C22×C32—42no—
82C22×C32—41no—
83C22×C32—81no—
84C22×C32—81no—
85C32×C4—42no—
86C32×C4—41no—
87C32×C4—41no—
88C22×Q8C3×Dic311yescharacteristic
89C2×M16—31no—
90C2×M16—91no—
91C4×Q8—31no—
92C4×Q8—31no—
93C4×Q8C3×Dic311yescharacteristic
94C4×Q8C2×C3×D311yescharacteristic
95C2×C4⋊C4—31no—
96C2×C42—31no—
97C4⋊C8—92no—
98C4⋊C8—92no—
99#97—92no—
100C4×C8—92no—
101C23×C3—41no—
102C23×C3—81no—
103C23×C3C4×A411yescharacteristic
104C3×Q8—31no—
105C3×Q8—61no—
106C3×Q8—61no—
107C3×Q8C3×M1611yescharacteristic, complemented
108C2×C3×C4—31no—
109C2×C3×C4—31no—
110C2×C3×C4—31no—
111C2×C3×C4—31no—
112C2×C3×C4—41no—
113C2×C3×C4—42no—
114C2×C3×C4—42no—
115C2×C3×C4—41no—
116C2×C3×C4—61no—
117C2×C3×C4—61no—
118C2×C3×C4—81no—
119C2×C3×C4—82no—
120C2×C3×C4—82no—
121C2×C3×C4—81no—
122C2×C3×C4—121no—
123C2×C3×C4C4×A411yescharacteristic
124C2×C3×C4C2×SL(2,𝔽3)12yes—
125C2×C3×C4#41⋊C312yes—
126C2×C3×C4C22×A411yescharacteristic
127SL(2,𝔽3)—21no—
128SL(2,𝔽3)#20811yescharacteristic, direct product factor
129C3×C8—124no—
130C3⋊C8—62no—
131C3⋊C8C2×SL(2,𝔽3)14yescomplemented
132C2×C32—41no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C32—82no—
136C2×Q8—21no—
137C2×Q8—31no—
138C2×Q8—61no—
139C2×Q8C2×C3×Dic311yescharacteristic
140C22×C4—31no—
141C22×C4—31no—
142C22×C4D3×A411yescharacteristic
143M16—34no—
144M16—182no—
145C2×C8—32no—
146C2×C8—92no—
147C4⋊C4—31no—
148C4⋊C4—31no—
149C4⋊C4—32no—
150C42—31no—
151C42—31no—
152C42—31no—
153C42—31no—
154C22×C3—21no—
155C22×C3—21no—
156C22×C3—42no—
157C22×C3—41no—
158C22×C3—81no—
159C22×C3—81no—
160C22×C3—82no—
161C22×C3—82no—
162C22×C3—82no—
163C22×C3—81no—
164C22×C3C4×SL(2,𝔽3)12yes—
165C22×C3C2×C4×A411yescharacteristic
166C3×C4—31no—
167C3×C4—31no—
168C3×C4—42no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—61no—
172C3×C4—61no—
173C3×C4—61no—
174C3×C4—82no—
175C3×C4—81no—
176C3×C4—81no—
177C3×C4C4×SL(2,𝔽3)12yes—
178C3×C4C22×SL(2,𝔽3)11yescharacteristic
179C3×C4C2×#41⋊C311yescharacteristic
180C32—41noSylow
181C23Dic3×A411yescharacteristic
182Q8—31no—
183Q8—61no—
184Q8—61no—
185Q8C3×#20811yescharacteristic, complemented
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—121no—
193C2×C4C3⋊(#41⋊C3)12yes—
194C2×C4D3×SL(2,𝔽3)12yes—
195C2×C4Dic3×A411yescharacteristic
196C2×C4C2×D3×A411yescharacteristic
197C8—34no—
198C8—182no—
199C2×C3—22no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—82no—
204C2×C3—84no—
205C2×C3—81no—
206C2×C3—81no—
207C2×C3—81no—
208C2×C3C2×C4×SL(2,𝔽3)11yescharacteristic
209C2×C3A4×M1611yescharacteristic
210C2×C3#568⋊C311yescharacteristic
211C22—21no—
212C22—21no—
213C22Dic3×SL(2,𝔽3)12yes—
214C22C2×Dic3×A411yescharacteristic
215C4—31no—
216C4—31no—
217C4—61no—
218C4—61no—
219C4—61no—
220C4Dic3×SL(2,𝔽3)12yes—
221C4C2×C3⋊(#41⋊C3)11yescharacteristic
222C4C2×D3×SL(2,𝔽3)11yescharacteristic
223C3—41no—
224C3—81no—
225C3M16×SL(2,𝔽3)11yescharacteristic, complemented
226C2—22no—
227C2A4×#20811yescharacteristic
228C2C2×Dic3×SL(2,𝔽3)11yescharacteristic
229C2#5819⋊C311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC2×C3×C4×Q8C2×C3
socleC22×C3C2×C4×A4
prime radicalC22×C3×SL(2,𝔽3)C4
cosocleC3×C4×Q8C22×C3
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×M16
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊C8×SL(2,𝔽3)C2semidirect
3. Q8×#208C3semidirect
13. Q8×C3⋊C8C2×C3semidirect
48. C3×SL(2,𝔽3)M16semidirect
78. #208SL(2,𝔽3)direct (N × H)
107. C3×Q8C3×M16semidirect
128. SL(2,𝔽3)#208direct (N × H)
131. C3⋊C8C2×SL(2,𝔽3)semidirect
185. Q8C3×#208semidirect
225. C3M16×SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
C3⋊C8×SL(2,𝔽3)2
Q8×#2083
M16×SL(2,𝔽3)3
C2×C3×#2084