Finite Group Databasebeta

← C2×C3×D4×SL(2,𝔽3) #11919716

Subgroups — C2×C3×D4×SL(2,𝔽3) #11919716

Order 1152.

1296 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 270 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
1C23×C3×SL(2,𝔽3)C212yesmaximal, complemented
2C3×D4×SL(2,𝔽3)C214yesmaximal, direct product factor
3C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C3×D4×Q8C311yesmaximal, characteristic, complemented
5C2×D4×SL(2,𝔽3)C313yesmaximal, direct product factor
6C22×C32×D4—41nomaximal
7C22×C3×SL(2,𝔽3)—24no—
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2214yescomplemented
10C3×C4×SL(2,𝔽3)C2212yescomplemented
11C23×C3×Q8C2×C312yescomplemented
12C23×SL(2,𝔽3)C2×C316yescomplemented
13C3×D4×Q8—34no—
14C3×D4×Q8C2×C314yescomplemented
15C2×C3×C4⋊Q8—31no—
16C2×C3×C22⋊Q8—32no—
17C2×C3×C4×Q8C2×C311yescharacteristic, complemented
18C2×C3×C4×D4—31no—
19D4×SL(2,𝔽3)C2×C3112yesdirect product factor
20C2×C4×SL(2,𝔽3)C2×C313yescomplemented
21C24×C32—42no—
22C2×C32×D4—48no—
23C2×C32×D4—44no—
24C22×C32×C4—41no—
25C2×C3×SL(2,𝔽3)—24no—
26C2×C3×SL(2,𝔽3)D412yesdirect product factor
27C2×C3×SL(2,𝔽3)C2311yescharacteristic
28C2×D4×Q8C3211yescharacteristic, complemented, Sylow
29C22×C3×Q8—24no—
30C22×C3×Q8—32no—
31C22×C3×Q8—34no—
32C22×C3×Q8—64no—
33C22×C3×Q8C22×C311yescharacteristic
34C22×C3×Q8C22×C314yescomplemented
35C22×C3×D4—43no—
36C22×C3×D4A411yescharacteristic
37C23×C3×C4—32no—
38C22×SL(2,𝔽3)—212no—
39C22×SL(2,𝔽3)C22×C313yes—
40C22×SL(2,𝔽3)C22×C3112yescomplemented
41C3×C4⋊Q8—34no—
42C3×C4⋊Q8—34no—
43C3×C22⋊Q8—38no—
44C3×C22⋊Q8—38no—
45C3×C4×Q8—32no—
46C3×C4×Q8C22×C312yescomplemented
47C3×C4×D4—34no—
48C3×C4×D4—34no—
49C2×C3×C4⋊C4—31no—
50C2×C3×C4⋊C4—32no—
51C2×C3×C4⋊C4—31no—
52C2×C3×C22⋊C4—32no—
53C2×C3×C42—31no—
54C4×SL(2,𝔽3)C22×C316yescomplemented
55C23×C32—48no—
56C23×C32—44no—
57C23×C32—41no—
58C23×C32—84no—
59C23×C32—84no—
60C32×D4—416no—
61C2×C32×C4—44no—
62C2×C32×C4—42no—
63C3×SL(2,𝔽3)C2×D411yescharacteristic, direct product factor
64C23×Q8C2×C3212yescomplemented
65D4×Q8—34no—
66D4×Q8C2×C3214yescomplemented
67C2×C4⋊Q8—31no—
68C2×C22⋊Q8—32no—
69C2×C4×Q8C2×C3211yescharacteristic, complemented
70C2×C4×D4—31no—
71C24×C3—46no—
72C24×C3C2×A412yes—
73C2×C3×Q8—24no—
74C2×C3×Q8—34no—
75C2×C3×Q8—31no—
76C2×C3×Q8—32no—
77C2×C3×Q8—34no—
78C2×C3×Q8—64no—
79C2×C3×Q8—64no—
80C2×C3×Q8—68no—
81C2×C3×Q8—64no—
82C2×C3×Q8—68no—
83C2×C3×Q8C3×D412yescomplemented
84C2×C3×Q8C23×C311yescharacteristic
85C2×C3×D4—424no—
86C2×C3×D4—412no—
87C2×C3×D4SL(2,𝔽3)18yesdirect product factor
88C2×C3×D4C2×A414yes—
89C22×C3×C4—31no—
90C22×C3×C4—34no—
91C22×C3×C4—34no—
92C22×C3×C4—31no—
93C22×C3×C4—32no—
94C22×C3×C4—43no—
95C22×C3×C4—64no—
96C22×C3×C4—64no—
97C22×C3×C4C2×A411yescharacteristic
98C2×SL(2,𝔽3)—212no—
99C2×SL(2,𝔽3)C3×D416yesdirect product factor
100C2×SL(2,𝔽3)C23×C313yes—
101C3×C4⋊C4—32no—
102C3×C4⋊C4—32no—
103C3×C4⋊C4—38no—
104C3×C4⋊C4—34no—
105C3×C22⋊C4—38no—
106C3×C42—32no—
107C3×C42—32no—
108C22×C32—42no—
109C22×C32—42no—
110C22×C32—48no—
111C22×C32—42no—
112C22×C32—41no—
113C22×C32—84no—
114C22×C32—816no—
115C22×C32—84no—
116C32×C4—44no—
117C22×Q8—24no—
118C22×Q8—32no—
119C22×Q8—34no—
120C22×Q8—64no—
121C22×Q8C22×C3211yescharacteristic
122C22×Q8C22×C3214yescomplemented
123C22×D4C3×A411yescharacteristic
124C23×C4—32no—
125C4⋊Q8—34no—
126C4⋊Q8—34no—
127C22⋊Q8—38no—
128C22⋊Q8—38no—
129C4×Q8—32no—
130C4×Q8C22×C3212yescomplemented
131C4×D4—34no—
132C4×D4—34no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—31no—
136C2×C22⋊C4—32no—
137C2×C42—31no—
138C23×C3—24no—
139C23×C3—24no—
140C23×C3—43no—
141C23×C3—412no—
142C23×C3—424no—
143C23×C3—812no—
144C23×C3—812no—
145C23×C3C2×SL(2,𝔽3)18yescomplemented
146C23×C3C22×A414yes—
147C23×C3C22×A411yescharacteristic
148C3×Q8—32no—
149C3×Q8—31no—
150C3×Q8—32no—
151C3×Q8—64no—
152C3×Q8—64no—
153C3×Q8—68no—
154C3×Q8C2×C3×D411yescharacteristic, complemented
155C3×D4—448no—
156C3×D4C2×SL(2,𝔽3)116yesdirect product factor
157C2×C3×C4—32no—
158C2×C3×C4—32no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—32no—
162C2×C3×C4—34no—
163C2×C3×C4—46no—
164C2×C3×C4—412no—
165C2×C3×C4—64no—
166C2×C3×C4—68no—
167C2×C3×C4—64no—
168C2×C3×C4—64no—
169C2×C3×C4—62no—
170C2×C3×C4C2×SL(2,𝔽3)14yescomplemented
171C2×C3×C4C22×A412yes—
172SL(2,𝔽3)C2×C3×D413yesdirect product factor
173C2×C32—41no—
174C2×C32—41no—
175C2×C32—41no—
176C2×C32—44no—
177C2×C32—88no—
178C24C2×C3×A412yes—
179C2×Q8—24no—
180C2×Q8—34no—
181C2×Q8—31no—
182C2×Q8—32no—
183C2×Q8—34no—
184C2×Q8—64no—
185C2×Q8—64no—
186C2×Q8—68no—
187C2×Q8—64no—
188C2×Q8—68no—
189C2×Q8C32×D412yescomplemented
190C2×Q8C23×C3211yescharacteristic
191C2×D4C3×SL(2,𝔽3)18yesdirect product factor
192C2×D4C2×C3×A414yes—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—32no—
196C22×C4—34no—
197C22×C4—34no—
198C22×C4—64no—
199C22×C4—64no—
200C22×C4C2×C3×A411yescharacteristic
201C4⋊C4—32no—
202C4⋊C4—32no—
203C4⋊C4—38no—
204C4⋊C4—34no—
205C22⋊C4—38no—
206C42—32no—
207C42—32no—
208C22×C3—24no—
209C22×C3—24no—
210C22×C3—216no—
211C22×C3—46no—
212C22×C3—46no—
213C22×C3—424no—
214C22×C3—46no—
215C22×C3—43no—
216C22×C3—812no—
217C22×C3—848no—
218C22×C3—812no—
219C22×C3D4×A412yes—
220C22×C3C22×SL(2,𝔽3)12yes—
221C22×C3C22×SL(2,𝔽3)18yescomplemented
222C22×C32−1+4⋊C312yes—
223C22×C3C23×A411yescharacteristic
224C3×C4—31no—
225C3×C4—32no—
226C3×C4—31no—
227C3×C4—412no—
228C3×C4—64no—
229C3×C4—62no—
230C3×C4C22×SL(2,𝔽3)14yescomplemented
231C32—41noSylow
232C23—24no—
233C23—24no—
234C23C2×C3×SL(2,𝔽3)18yescomplemented
235C23C22×C3×A414yes—
236C23C22×C3×A411yescharacteristic
237Q8—32no—
238Q8—31no—
239Q8—32no—
240Q8—64no—
241Q8—64no—
242Q8—68no—
243Q8C2×C32×D411yescharacteristic, complemented
244D4C2×C3×SL(2,𝔽3)116yesdirect product factor
245C2×C4—32no—
246C2×C4—32no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—32no—
250C2×C4—34no—
251C2×C4—64no—
252C2×C4—64no—
253C2×C4—68no—
254C2×C4—64no—
255C2×C4—62no—
256C2×C4C2×C3×SL(2,𝔽3)14yescomplemented
257C2×C4C22×C3×A412yes—
258C2×C3—28no—
259C2×C3—43no—
260C2×C3—43no—
261C2×C3—43no—
262C2×C3—412no—
263C2×C3—824no—
264C2×C3D4×SL(2,𝔽3)14yesdirect product factor
265C2×C3C2×D4×A411yescharacteristic
266C2×C3C23×SL(2,𝔽3)11yescharacteristic
267C2×C3C2×2−1+4⋊C311yescharacteristic
268C22—24no—
269C22—24no—
270C22—216no—
271C22C3×D4×A412yes—
272C22C22×C3×SL(2,𝔽3)12yes—
273C22C22×C3×SL(2,𝔽3)18yescomplemented
274C22C3×2−1+4⋊C312yes—
275C22C23×C3×A411yescharacteristic
276C4—31no—
277C4—32no—
278C4—31no—
279C4—64no—
280C4—62no—
281C4C22×C3×SL(2,𝔽3)14yescomplemented
282C3—43no—
283C3C2×D4×SL(2,𝔽3)11yescharacteristic, direct product factor
284C2—28no—
285C2C3×D4×SL(2,𝔽3)14yesdirect product factor
286C2C2×C3×D4×A411yescharacteristic
287C2C23×C3×SL(2,𝔽3)11yescharacteristic
288C2C2×C3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC23×C3C22×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC2×C3×D4×Q8C3
hypercenterC22×C3×D4A4
hypocenterQ8C2×C32×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×SL(2,𝔽3)C2semidirect
2. C3×D4×SL(2,𝔽3)C2direct (N × H)
3. C2×C3×C4×SL(2,𝔽3)C2semidirect
4. C2×C3×D4×Q8C3semidirect
5. C2×D4×SL(2,𝔽3)C3direct (N × H)
9. C22×C3×SL(2,𝔽3)C22semidirect
10. C3×C4×SL(2,𝔽3)C22semidirect
11. C23×C3×Q8C2×C3semidirect
12. C23×SL(2,𝔽3)C2×C3semidirect
14. C3×D4×Q8C2×C3semidirect
17. C2×C3×C4×Q8C2×C3semidirect
19. D4×SL(2,𝔽3)C2×C3direct (N × H)
20. C2×C4×SL(2,𝔽3)C2×C3semidirect
26. C2×C3×SL(2,𝔽3)D4direct (N × H)
28. C2×D4×Q8C32semidirect
34. C22×C3×Q8C22×C3semidirect
40. C22×SL(2,𝔽3)C22×C3semidirect
46. C3×C4×Q8C22×C3semidirect
54. C4×SL(2,𝔽3)C22×C3semidirect
63. C3×SL(2,𝔽3)C2×D4direct (N × H)
64. C23×Q8C2×C32semidirect
66. D4×Q8C2×C32semidirect
69. C2×C4×Q8C2×C32semidirect
83. C2×C3×Q8C3×D4semidirect
87. C2×C3×D4SL(2,𝔽3)direct (N × H)
99. C2×SL(2,𝔽3)C3×D4direct (N × H)
122. C22×Q8C22×C32semidirect
130. C4×Q8C22×C32semidirect
145. C23×C3C2×SL(2,𝔽3)semidirect
154. C3×Q8C2×C3×D4semidirect
156. C3×D4C2×SL(2,𝔽3)direct (N × H)
170. C2×C3×C4C2×SL(2,𝔽3)semidirect
172. SL(2,𝔽3)C2×C3×D4direct (N × H)
189. C2×Q8C32×D4semidirect
191. C2×D4C3×SL(2,𝔽3)direct (N × H)
221. C22×C3C22×SL(2,𝔽3)semidirect
230. C3×C4C22×SL(2,𝔽3)semidirect
234. C23C2×C3×SL(2,𝔽3)semidirect
243. Q8C2×C32×D4semidirect
244. D4C2×C3×SL(2,𝔽3)direct (N × H)
256. C2×C4C2×C3×SL(2,𝔽3)semidirect
264. C2×C3D4×SL(2,𝔽3)direct (N × H)
273. C22C22×C3×SL(2,𝔽3)semidirect
281. C4C22×C3×SL(2,𝔽3)semidirect
283. C3C2×D4×SL(2,𝔽3)direct (N × H)
285. C2C3×D4×SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C3×D4×SL(2,𝔽3)2
C2×C3×C4×SL(2,𝔽3)2
C2×C3×D4×Q83
C2×D4×SL(2,𝔽3)3
C22×C32×D44