Finite Group Databasebeta

← SL(2,𝔽3)×#216 #11921314

Subgroups — SL(2,𝔽3)×#216 #11921314

Order 1152.

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

normal subgroups only

This group has 267 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
2SL(2,𝔽3)×Dic6C211yesmaximal, characteristic
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic
4Q8×#216C311yesmaximal, characteristic, complemented
5Q16×SL(2,𝔽3)—31nomaximal
6C2×C3×#216—41nomaximal
7C3×C4×SL(2,𝔽3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9Dic3×SL(2,𝔽3)—21no—
10C3×Q82C2×C311yescharacteristic
11Q8×Dic6C2×C311yescharacteristic
12Q8×SL(2,𝔽3)—21no—
13Q8×SL(2,𝔽3)—31no—
14Q8×SL(2,𝔽3)D311yescharacteristic
15#5091—31no—
16#5078—31no—
17C4×#216—31no—
18Q8×C3⋊C8C2×C311yescharacteristic
19#5021—31no—
20C8×SL(2,𝔽3)—31no—
21C2×C32×Q8—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3×#216—44no—
25C2×C3×C3⋊C8—41no—
26Q8×Q16—31noSylow
27C2×C3×Q16—121no—
28C3×C4⋊Q8—31no—
29C3×C4⋊Q8—61no—
30C3×C4×Q8—21no—
31C3×C4×Q8—31no—
32C3×C4×Q8C22×C311yescharacteristic
33Q8×Dic3—21no—
34C2×#216A411yescharacteristic
35#888—61no—
36#869—31no—
37C4×Dic6—31no—
38C4×SL(2,𝔽3)—22no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)—21no—
41C4×SL(2,𝔽3)—61no—
42C4×SL(2,𝔽3)C2×D311yescharacteristic
43#835—31no—
44#810—31no—
45#807—32no—
46#807—31no—
47C4⋊(C3⋊C8)—31no—
48C4×C3⋊C8—31no—
49C32×Q8—42no—
50C32×Q8—81no—
51C2×C32×C4—41no—
52C2×C32×C4—81no—
53C2×C3×Dic3—81no—
54C3×Dic6—42no—
55C3×Dic6—81no—
56C3×SL(2,𝔽3)Q1611yescharacteristic, complemented
57C3×C3⋊C8—42no—
58Q82—31no—
59Q82C3×D311yescharacteristic
60#500—91no—
61#477—91no—
62#477—91no—
63C8×Q8—31no—
64C4×Q16—91no—
65C2×C3×Q8—41no—
66C2×C3×Q8—61no—
67C2×C3×Q8—81no—
68C2×C3×Q8—121no—
69C2×C3×Q8C3×D411yescharacteristic
70C2×C3×Q8C2×A411yescharacteristic
71C2×Dic6—61no—
72C2×Dic6C2×A411yescharacteristic
73C2×SL(2,𝔽3)—21no—
74C2×SL(2,𝔽3)#6711yescharacteristic
75C3×Q16—124no—
76C2×C3×C8—121no—
77C3×C4⋊C4—31no—
78C3×C4⋊C4—31no—
79C3×C4⋊C4—61no—
80C3×C4⋊C4—61no—
81C3×C42—31no—
82C3×C42—61no—
83#216SL(2,𝔽3)14yesdirect product factor
84C4⋊Dic3—31no—
85C4⋊Dic3—61no—
86Dic3⋊C4—61no—
87C4×Dic3—61no—
88C2×C3⋊C8—31no—
89C2×C3⋊C8C2×A411yescharacteristic
90C22×C32—41no—
91C32×C4—41no—
92C32×C4—41no—
93C32×C4—82no—
94C3×Dic3—82no—
95C2×Q16—31no—
96C4⋊Q8—31no—
97C4⋊Q8—61no—
98C4⋊Q8—91no—
99C4⋊Q8—181no—
100C4×Q8—21no—
101C4×Q8—31no—
102C4×Q8—61no—
103C4×Q8—91no—
104C4×Q8C2×C3×D311yescharacteristic
105#107—91no—
106#107—92no—
107C4⋊C8—91no—
108#103—91no—
109#103—91no—
110C4×C8—91no—
111C3×Q8—21no—
112C3×Q8—31no—
113C3×Q8—42no—
114C3×Q8—81no—
115C3×Q8—82no—
116C3×Q8—82no—
117C3×Q8—122no—
118C3×Q8—241no—
119C3×Q8—241no—
120C3×Q8C3×Q1611yescharacteristic, complemented
121C3×Q8C2×SL(2,𝔽3)12yes—
122C2×C3×C4—21no—
123C2×C3×C4—31no—
124C2×C3×C4—31no—
125C2×C3×C4—41no—
126C2×C3×C4—61no—
127C2×C3×C4—81no—
128C2×C3×C4—82no—
129C2×C3×C4—81no—
130C2×C3×C4—241no—
131C2×C3×C4C22×A411yescharacteristic
132C2×Dic3—21no—
133C2×Dic3—61no—
134Dic6—21no—
135Dic6—241no—
136Dic6C2×SL(2,𝔽3)12yes—
137SL(2,𝔽3)—21no—
138SL(2,𝔽3)#21611yescharacteristic, direct product factor
139C3×C8—122no—
140C3⋊C8—61no—
141C3⋊C8C2×SL(2,𝔽3)12yes—
142C2×C32—41no—
143C2×C32—41no—
144C2×C32—41no—
145C2×Q8—31no—
146C2×Q8—61no—
147C2×Q8—181no—
148C2×Q8C3×#6711yescharacteristic
149C2×Q8D3×A411yescharacteristic
150Q16—34no—
151C2×C8—31no—
152C2×C8—91no—
153C4⋊C4—31no—
154C4⋊C4—31no—
155C4⋊C4—61no—
156C4⋊C4—61no—
157C4⋊C4—91no—
158C4⋊C4—181no—
159C4⋊C4—181no—
160C42—31no—
161C42—61no—
162C42—181no—
163C22×C3—41no—
164C22×C3—81no—
165C22×C3D4×A411yescharacteristic
166C3×C4—22no—
167C3×C4—31no—
168C3×C4—31no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—61no—
172C3×C4—82no—
173C3×C4—84no—
174C3×C4—81no—
175C3×C4—81no—
176C3×C4—121no—
177C3×C4—242no—
178C3×C4C22×SL(2,𝔽3)11yescharacteristic
179C3×C42−1+4⋊C311yescharacteristic
180Dic3—22no—
181Dic3—121no—
182C32—41noSylow
183Q8—21no—
184Q8—31no—
185Q8—32no—
186Q8—61no—
187Q8—241no—
188Q8—721no—
189Q8C3×#21611yescharacteristic, complemented
190Q8D3×SL(2,𝔽3)12yes—
191C2×C4—21no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—181no—
197C2×C4C2×D3×A411yescharacteristic
198C8—32no—
199C8—181no—
200C2×C3—41no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—81no—
204C2×C3—81no—
205C2×C3—81no—
206C2×C3D4×SL(2,𝔽3)11yescharacteristic
207C2×C3A4×Q1611yescharacteristic
208C2×C3#576⋊C311yescharacteristic
209C22A4×#6711yescharacteristic
210C4—22no—
211C4—31no—
212C4—31no—
213C4—62no—
214C4—61no—
215C4—121no—
216C4—361no—
217C4C2×D3×SL(2,𝔽3)11yescharacteristic
218C4#1007⋊C311yescharacteristic
219C3—41no—
220C3—81no—
221C3Q16×SL(2,𝔽3)11yescharacteristic, complemented
222C2A4×#21611yescharacteristic
223C2SL(2,𝔽3)×#6711yescharacteristic
224C2#5835⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×Q82C2×C3
socleC22×C3D4×A4
prime radicalC2×C3×SL(2,𝔽3)D4
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C3×Q16
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
4. Q8×#216C3semidirect
56. C3×SL(2,𝔽3)Q16semidirect
83. #216SL(2,𝔽3)direct (N × H)
120. C3×Q8C3×Q16semidirect
138. SL(2,𝔽3)#216direct (N × H)
189. Q8C3×#216semidirect
221. C3Q16×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)×Dic62
C3⋊C8×SL(2,𝔽3)2
Q8×#2163
Q16×SL(2,𝔽3)3
C2×C3×#2164