Finite Group Databasebeta

← SL(2,𝔽3)×C4⋊Dic3 #11921264

Subgroups — SL(2,𝔽3)×C4⋊Dic3 #11921264

Order 1152.

421 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 103 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
2C2×Dic3×SL(2,𝔽3)C212yesmaximal
3Q8×C4⋊Dic3C311yesmaximal, characteristic, complemented
4C4⋊C4×SL(2,𝔽3)—31nomaximal
5C22×C3×SL(2,𝔽3)C2211yescharacteristic
6C2×C3×C4⋊Dic3—41nomaximal
7C3×C4×SL(2,𝔽3)C412yescomplemented
8Dic3×SL(2,𝔽3)—22no—
9C2×C3×C4×Q8C2×C311yescharacteristic
10C2×Q8×Dic3C2×C312yes—
11C2×C4×SL(2,𝔽3)—21no—
12C2×C4×SL(2,𝔽3)—32no—
13C2×C4×SL(2,𝔽3)D311yescharacteristic
14#4984—32no—
15#4935—31no—
16C4×C4⋊Dic3—31no—
17C22×C32×C4—41no—
18C22×C3×Dic3—42no—
19C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
20C2×C3×SL(2,𝔽3)D411yescharacteristic
21C2×C3×SL(2,𝔽3)Q811yescharacteristic
22C3×C4⋊Dic3—44no—
23Q8×C4⋊C4—31noSylow
24C22×C3×Q8C22×C311yescharacteristic
25C22×SL(2,𝔽3)—21no—
26C22×SL(2,𝔽3)C2×D311yescharacteristic
27C3×C4×Q8—32no—
28C3×C4×Q8C3×C412yescomplemented
29C2×C3×C4⋊C4—31no—
30C2×C3×C4⋊C4—121no—
31C2×C3×C42—31no—
32Q8×Dic3—22no—
33Q8×Dic3—62no—
34C2×C4⋊Dic3—31no—
35C2×C4⋊Dic3—32no—
36C2×C4⋊Dic3—32no—
37C2×C4⋊Dic3A411yescharacteristic
38C2×C4×Dic3—32no—
39C4×SL(2,𝔽3)—22no—
40C4×SL(2,𝔽3)—62no—
41C4×SL(2,𝔽3)Dic312yescomplemented
42C3⋊#95—32no—
43C23×C32—41no—
44C2×C32×C4—42no—
45C2×C32×C4—42no—
46C2×C32×C4—42no—
47C2×C3×Dic3—44no—
48C2×C3×Dic3—82no—
49C2×C3×Dic3—82no—
50C3×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
51C2×C4×Q8—32no—
52C2×C4×Q8C3×D311yescharacteristic
53#389—92no—
54#384—91no—
55C4×C4⋊C4—91no—
56C2×C3×Q8—31no—
57C2×C3×Q8—31no—
58C2×C3×Q8—31no—
59C2×C3×Q8C2×C3×C411yescharacteristic
60C2×C3×Q8C3×D411yescharacteristic
61C2×C3×Q8C3×Q811yescharacteristic
62C22×C3×C4—31no—
63C22×C3×C4—31no—
64C22×C3×C4—41no—
65C22×C3×C4—81no—
66C22×C3×C4—122no—
67C22×C3×C4C2×A411yescharacteristic
68C22×Dic3—32no—
69C22×Dic3C2×A412yes—
70C2×SL(2,𝔽3)—21no—
71C2×SL(2,𝔽3)—21no—
72C2×SL(2,𝔽3)—21no—
73C2×SL(2,𝔽3)Dic611yescharacteristic
74C2×SL(2,𝔽3)D1211yescharacteristic
75C2×SL(2,𝔽3)C2×Dic311yescharacteristic
76C3×C4⋊C4—32no—
77C3×C4⋊C4—32no—
78C3×C4⋊C4—124no—
79C3×C42—32no—
80C3×C42—32no—
81C4⋊Dic3—62no—
82C4⋊Dic3—62no—
83C4⋊Dic3—62no—
84C4⋊Dic3—122no—
85C4⋊Dic3SL(2,𝔽3)14yesdirect product factor
86C4×Dic3—62no—
87C4×Dic3—62no—
88C22×C32—41no—
89C22×C32—41no—
90C22×C32—41no—
91C22×C32—41no—
92C22×C32—41no—
93C22×C32—41no—
94C22×C32—41no—
95C32×C4—44no—
96C3×Dic3—84no—
97C22×Q8C2×C3×D311yescharacteristic
98C4×Q8—32no—
99C4×Q8—62no—
100C4×Q8—182no—
101C4×Q8C3×Dic312yescomplemented
102C2×C4⋊C4—31no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—92no—
105C2×C4⋊C4—91no—
106C2×C4⋊C4—92no—
107C2×C42—31no—
108C2×C42—92no—
109#95—92no—
110C23×C3—41no—
111C23×C3—81no—
112C23×C3C22×A411yescharacteristic
113C3×Q8—31no—
114C3×Q8—31no—
115C3×Q8—31no—
116C3×Q8—32no—
117C3×Q8C3×C4⋊C411yescharacteristic, complemented
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—31no—
122C2×C3×C4—31no—
123C2×C3×C4—31no—
124C2×C3×C4—32no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—61no—
129C2×C3×C4—61no—
130C2×C3×C4—82no—
131C2×C3×C4—82no—
132C2×C3×C4—82no—
133C2×C3×C4—124no—
134C2×C3×C4—242no—
135C2×C3×C4—242no—
136C2×C3×C4C4×A412yes—
137C2×C3×C4C2×SL(2,𝔽3)12yes—
138C2×C3×C4#41⋊C312yes—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—62no—
142C2×Dic3—62no—
143C2×Dic3—62no—
144C2×Dic3C2×SL(2,𝔽3)14yes—
145SL(2,𝔽3)—21no—
146SL(2,𝔽3)C4⋊Dic311yescharacteristic, direct product factor
147C2×C32—41no—
148C2×C32—41no—
149C2×C32—41no—
150C2×C32—41no—
151C2×C32—41no—
152C2×C32—41no—
153C2×C32—41no—
154C2×Q8—31no—
155C2×Q8—31no—
156C2×Q8—31no—
157C2×Q8C3×Dic611yescharacteristic
158C2×Q8C3×D1211yescharacteristic
159C2×Q8C2×C3×Dic311yescharacteristic
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—32no—
163C22×C4—92no—
164C22×C4D3×A411yescharacteristic
165C4⋊C4—32no—
166C4⋊C4—32no—
167C4⋊C4—34no—
168C4⋊C4—182no—
169C4⋊C4—182no—
170C4⋊C4—182no—
171C4⋊C4—362no—
172C42—32no—
173C42—32no—
174C42—182no—
175C42—182no—
176C22×C3—41no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—81no—
184C22×C3—81no—
185C22×C3—81no—
186C22×C3—81no—
187C22×C3—81no—
188C22×C3—81no—
189C22×C3—81no—
190C22×C3C2×C4×A411yescharacteristic
191C22×C3D4×A411yescharacteristic
192C22×C3C22×SL(2,𝔽3)11yescharacteristic
193C22×C3Q8×A411yescharacteristic
194C22×C3C2×#41⋊C311yescharacteristic
195C22×C3#99411yescharacteristic
196C22×C32−1+4⋊C311yescharacteristic
197C3×C4—31no—
198C3×C4—31no—
199C3×C4—31no—
200C3×C4—31no—
201C3×C4—44no—
202C3×C4—62no—
203C3×C4—84no—
204C3×C4—244no—
205C3×C4C4×SL(2,𝔽3)14yescomplemented
206Dic3—24no—
207Dic3—122no—
208C32—41noSylow
209C23C2×D3×A411yescharacteristic
210Q8—31no—
211Q8—31no—
212Q8—31no—
213Q8—32no—
214Q8C3×C4⋊Dic311yescharacteristic, complemented
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—31no—
219C2×C4—31no—
220C2×C4—31no—
221C2×C4—34no—
222C2×C4—32no—
223C2×C4—62no—
224C2×C4—61no—
225C2×C4—62no—
226C2×C4—61no—
227C2×C4—182no—
228C2×C4—182no—
229C2×C4—182no—
230C2×C4C3⋊(#41⋊C3)12yes—
231C2×C4D3×SL(2,𝔽3)12yes—
232C2×C4Dic3×A412yes—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—41no—
239C2×C3—41no—
240C2×C3—81no—
241C2×C3—81no—
242C2×C3—81no—
243C2×C3—81no—
244C2×C3—81no—
245C2×C3—81no—
246C2×C3—81no—
247C2×C3A4×C4⋊C411yescharacteristic
248C2×C3C2×C4×SL(2,𝔽3)11yescharacteristic
249C2×C3#520⋊C311yescharacteristic
250C2×C3D4×SL(2,𝔽3)11yescharacteristic
251C2×C3C4⋊(#41⋊C3)11yescharacteristic
252C2×C3#557⋊C311yescharacteristic
253C2×C3Q8×SL(2,𝔽3)11yescharacteristic
254C22A4×Dic611yescharacteristic
255C22A4×D1211yescharacteristic
256C22C2×C3⋊(#41⋊C3)11yescharacteristic
257C22C2×D3×SL(2,𝔽3)11yescharacteristic
258C22C3⋊#99411yescharacteristic
259C22#1010⋊C311yescharacteristic
260C22C2×Dic3×A411yescharacteristic
261C4—31no—
262C4—31no—
263C4—31no—
264C4—31no—
265C4—64no—
266C4—62no—
267C4—362no—
268C4Dic3×SL(2,𝔽3)14yescomplemented
269C3—41no—
270C3—81no—
271C3C4⋊C4×SL(2,𝔽3)11yescharacteristic, complemented
272C2SL(2,𝔽3)×Dic611yescharacteristic
273C2C4⋊(C3⋊(#41⋊C3))11yescharacteristic
274C2#5570⋊C311yescharacteristic
275C2SL(2,𝔽3)×D1211yescharacteristic
276C2A4×C4⋊Dic311yescharacteristic
277C2C2×Dic3×SL(2,𝔽3)11yescharacteristic
278C2#5827⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×C3×SL(2,𝔽3)C22
cosocleC22×C3×Q8C22×C3
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×C4⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. Q8×C4⋊Dic3C3semidirect
7. C3×C4×SL(2,𝔽3)C4semidirect
28. C3×C4×Q8C3×C4semidirect
41. C4×SL(2,𝔽3)Dic3semidirect
50. C3×SL(2,𝔽3)C4⋊C4semidirect
85. C4⋊Dic3SL(2,𝔽3)direct (N × H)
101. C4×Q8C3×Dic3semidirect
117. C3×Q8C3×C4⋊C4semidirect
146. SL(2,𝔽3)C4⋊Dic3direct (N × H)
205. C3×C4C4×SL(2,𝔽3)semidirect
214. Q8C3×C4⋊Dic3semidirect
268. C4Dic3×SL(2,𝔽3)semidirect
271. C3C4⋊C4×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
C2×Dic3×SL(2,𝔽3)2
Q8×C4⋊Dic33
C4⋊C4×SL(2,𝔽3)3
C2×C3×C4⋊Dic34