Finite Group Databasebeta

← C2×SL(2,𝔽3)×D12 #11923182

Subgroups — C2×SL(2,𝔽3)×D12 #11923182

Order 1152.

1041 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×D3×SL(2,𝔽3)C212yesmaximal, complemented
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×D12C214yesmaximal, direct product factor
4C2×Q8×D12C311yesmaximal, characteristic, complemented
5C2×D4×SL(2,𝔽3)—31nomaximal
6C22×C3×D12—41nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×D3×SL(2,𝔽3)—24no—
9C2×D3×SL(2,𝔽3)C2214yescomplemented
10C3×C4×SL(2,𝔽3)C2212yescomplemented
11C22×D3×Q8C2×C312yescomplemented
12C23×SL(2,𝔽3)—32no—
13C2×C3×C4×Q8C2×C311yescharacteristic, complemented
14Q8×D12—34no—
15Q8×D12C2×C314yescomplemented
16C2×#897—32no—
17C2×C4×D12—31no—
18C2×#869—31no—
19D4×SL(2,𝔽3)—34no—
20C2×C4×SL(2,𝔽3)—21no—
21C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
22C23×C3×D3—42no—
23C22×C32×C4—41no—
24C2×C3×D12—48no—
25C2×C3×D12—44no—
26C2×C3×SL(2,𝔽3)D412yescomplemented
27C2×C3×SL(2,𝔽3)C2311yescharacteristic
28D3×SL(2,𝔽3)—24no—
29C2×D4×Q8—31noSylow
30C22×C3×Q8C22×C311yescharacteristic
31C22×C3×D4—121no—
32C2×D3×Q8—24no—
33C2×D3×Q8—34no—
34C2×D3×Q8—64no—
35C2×D3×Q8C22×C314yescomplemented
36C22×D12A411yescharacteristic
37C22×C4×D3—32no—
38C22×Dic6—32no—
39C22×SL(2,𝔽3)—21no—
40C22×SL(2,𝔽3)—34no—
41C22×SL(2,𝔽3)—64no—
42C22×SL(2,𝔽3)C2×D311yescharacteristic
43C3×C4×Q8—32no—
44C3×C4×Q8C22×C312yescomplemented
45C2×C3×C4⋊C4—31no—
46C2×C3×C42—31no—
47C2×(C2×D3)⋊C4—32no—
48C2×C4⋊Dic3—32no—
49C2×C4⋊Dic3—31no—
50#897—38no—
51#897—38no—
52C4×D12—34no—
53C4×D12—34no—
54#869—34no—
55#869—34no—
56C4×SL(2,𝔽3)—22no—
57C4×SL(2,𝔽3)C2×D312yescomplemented
58C23×C32—41no—
59C22×C3×D3—48no—
60C22×C3×D3—44no—
61C22×C3×D3—84no—
62C22×C3×D3—84no—
63C2×C32×C4—44no—
64C2×C32×C4—42no—
65C3×D12—416no—
66C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
67C23×Q8—32no—
68D4×Q8—34no—
69D4×Q8—94no—
70C2×C4⋊Q8—91no—
71C2×C22⋊Q8—92no—
72C2×C4×Q8C3×D311yescharacteristic, complemented
73C2×C4×D4—91no—
74C24×C3—122no—
75C23×D3C2×A412yes—
76C2×C3×Q8—31no—
77C2×C3×Q8—32no—
78C2×C3×Q8C3×D412yescomplemented
79C2×C3×Q8C23×C311yescharacteristic
80C2×C3×D4—124no—
81C2×C3×D4—128no—
82C22×C3×C4—31no—
83C22×C3×C4—31no—
84C22×C3×C4—41no—
85C22×C3×C4—81no—
86C22×C3×C4C2×A411yescharacteristic
87C22×Dic3—32no—
88D3×Q8—24no—
89D3×Q8—68no—
90D3×Q8—68no—
91D3×Q8—64no—
92C2×D12SL(2,𝔽3)18yesdirect product factor
93C2×D12C2×A414yes—
94C2×C4×D3—34no—
95C2×C4×D3—34no—
96C2×C4×D3—64no—
97C2×C4×D3—64no—
98C2×Dic6—34no—
99C2×Dic6—34no—
100C2×Dic6—64no—
101C2×Dic6—64no—
102C2×SL(2,𝔽3)—22no—
103C2×SL(2,𝔽3)—21no—
104C2×SL(2,𝔽3)—64no—
105C2×SL(2,𝔽3)D1212yesdirect product factor
106C2×SL(2,𝔽3)C22×D311yescharacteristic
107C3×C4⋊C4—32no—
108C3×C4⋊C4—32no—
109C3×C42—32no—
110C3×C42—32no—
111(C2×D3)⋊C4—38no—
112C4⋊Dic3—38no—
113C4⋊Dic3—34no—
114C22×C32—42no—
115C22×C32—42no—
116C22×C32—42no—
117C22×C32—41no—
118C2×C3×D3—48no—
119C2×C3×D3—84no—
120C2×C3×D3—816no—
121C2×C3×D3—84no—
122C32×C4—44no—
123C22×Q8—34no—
124C22×Q8—64no—
125C22×Q8—92no—
126C22×Q8—94no—
127C22×Q8—184no—
128C22×Q8C2×C3×D311yescharacteristic
129C22×D4—31no—
130C23×C4—92no—
131C4⋊Q8—94no—
132C4⋊Q8—94no—
133C22⋊Q8—98no—
134C22⋊Q8—98no—
135C4×Q8—32no—
136C4×Q8C2×C3×D312yescomplemented
137C4×D4—94no—
138C4×D4—94no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—92no—
141C2×C4⋊C4—91no—
142C2×C22⋊C4—92no—
143C2×C42—31no—
144C23×C3—41no—
145C23×C3—81no—
146C23×C3—124no—
147C23×C3—128no—
148C23×C3—244no—
149C23×C3—244no—
150C23×C3C22×A411yescharacteristic
151C22×D3—24no—
152C22×D3—24no—
153C22×D3C2×SL(2,𝔽3)18yescomplemented
154C22×D3C22×A414yes—
155C3×Q8—32no—
156C3×Q8—31no—
157C3×Q8—32no—
158C3×Q8C2×C3×D411yescharacteristic, complemented
159C3×D4—1216no—
160C2×C3×C4—32no—
161C2×C3×C4—32no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—32no—
165C2×C3×C4—42no—
166C2×C3×C4—44no—
167C2×C3×C4—62no—
168C2×C3×C4—82no—
169C2×C3×C4—84no—
170C2×C3×C4C2×SL(2,𝔽3)14yescomplemented
171C2×C3×C4C22×A412yes—
172C2×Dic3—34no—
173C2×Dic3—64no—
174D12C2×SL(2,𝔽3)116yesdirect product factor
175C4×D3—64no—
176C4×D3—68no—
177C4×D3—64no—
178Dic6—68no—
179Dic6—64no—
180Dic6—64no—
181SL(2,𝔽3)—21no—
182SL(2,𝔽3)C2×D1211yescharacteristic, direct product factor
183C2×C32—41no—
184C2×C32—41no—
185C2×C32—41no—
186C2×C32—44no—
187C3×D3—88no—
188C24—32no—
189C2×Q8—31no—
190C2×Q8—32no—
191C2×Q8—64no—
192C2×Q8—94no—
193C2×Q8—94no—
194C2×Q8—184no—
195C2×Q8—184no—
196C2×Q8—188no—
197C2×Q8—184no—
198C2×Q8—188no—
199C2×Q8C3×D1212yescomplemented
200C2×Q8C22×C3×D311yescharacteristic
201C2×D4—34no—
202C2×D4—38no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—94no—
206C22×C4—94no—
207C22×C4—92no—
208C22×C4—184no—
209C22×C4—184no—
210C22×C4D3×A411yescharacteristic
211C4⋊C4—32no—
212C4⋊C4—32no—
213C4⋊C4—98no—
214C4⋊C4—94no—
215C22⋊C4—98no—
216C42—32no—
217C42—32no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—42no—
221C22×C3—41no—
222C22×C3—82no—
223C22×C3—82no—
224C22×C3—82no—
225C22×C3—81no—
226C22×C3—128no—
227C22×C3—244no—
228C22×C3—2416no—
229C22×C3—244no—
230C22×C3D4×A412yes—
231C22×C3C22×SL(2,𝔽3)12yes—
232C22×C32−1+4⋊C312yes—
233C22×C3C23×A411yescharacteristic
234C2×D3—24no—
235C2×D3—24no—
236C2×D3—216no—
237C2×D3C22×SL(2,𝔽3)18yescomplemented
238C3×C4—32no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—44no—
242C3×C4—62no—
243C3×C4—84no—
244C3×C4C22×SL(2,𝔽3)14yescomplemented
245Dic3—64no—
246C32—41noSylow
247C23—38no—
248C23—34no—
249C23—64no—
250C23—64no—
251C23C2×D3×A411yescharacteristic
252Q8—32no—
253Q8—31no—
254Q8—32no—
255Q8—184no—
256Q8—184no—
257Q8—188no—
258Q8C2×C3×D1211yescharacteristic, complemented
259D4—316no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—32no—
265C2×C4—62no—
266C2×C4—94no—
267C2×C4—184no—
268C2×C4—184no—
269C2×C4—188no—
270C2×C4—184no—
271C2×C4D3×SL(2,𝔽3)14yescomplemented
272C2×C4C2×D3×A412yes—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—44no—
277C2×C3—81no—
278C2×C3—81no—
279C2×C3—81no—
280C2×C3—84no—
281C2×C3—248no—
282C2×C3D4×SL(2,𝔽3)14yescomplemented
283C2×C3C2×D4×A411yescharacteristic
284C2×C3C23×SL(2,𝔽3)11yescharacteristic
285C2×C3C2×2−1+4⋊C311yescharacteristic
286D3—28no—
287C22—38no—
288C22—64no—
289C22—64no—
290C22—616no—
291C22A4×D1212yes—
292C22C2×D3×SL(2,𝔽3)12yes—
293C22#1010⋊C312yes—
294C22C22×D3×A411yescharacteristic
295C4—31no—
296C4—32no—
297C4—31no—
298C4—62no—
299C4—184no—
300C4C2×D3×SL(2,𝔽3)14yescomplemented
301C3—41no—
302C3—81no—
303C3C2×D4×SL(2,𝔽3)11yescharacteristic, complemented
304C2—68no—
305C2SL(2,𝔽3)×D1214yesdirect product factor
306C2C2×A4×D1211yescharacteristic
307C2C22×D3×SL(2,𝔽3)11yescharacteristic
308C2C2×#1010⋊C311yescharacteristic
subgroupG/H
centerC23C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×D3×SL(2,𝔽3)C2semidirect
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)×D12C2direct (N × H)
4. C2×Q8×D12C3semidirect
9. C2×D3×SL(2,𝔽3)C22semidirect
10. C3×C4×SL(2,𝔽3)C22semidirect
11. C22×D3×Q8C2×C3semidirect
13. C2×C3×C4×Q8C2×C3semidirect
15. Q8×D12C2×C3semidirect
21. C2×C4×SL(2,𝔽3)D3semidirect
26. C2×C3×SL(2,𝔽3)D4semidirect
35. C2×D3×Q8C22×C3semidirect
44. C3×C4×Q8C22×C3semidirect
57. C4×SL(2,𝔽3)C2×D3semidirect
66. C3×SL(2,𝔽3)C2×D4semidirect
72. C2×C4×Q8C3×D3semidirect
78. C2×C3×Q8C3×D4semidirect
92. C2×D12SL(2,𝔽3)direct (N × H)
105. C2×SL(2,𝔽3)D12direct (N × H)
136. C4×Q8C2×C3×D3semidirect
153. C22×D3C2×SL(2,𝔽3)semidirect
158. C3×Q8C2×C3×D4semidirect
170. C2×C3×C4C2×SL(2,𝔽3)semidirect
174. D12C2×SL(2,𝔽3)direct (N × H)
182. SL(2,𝔽3)C2×D12direct (N × H)
199. C2×Q8C3×D12semidirect
237. C2×D3C22×SL(2,𝔽3)semidirect
244. C3×C4C22×SL(2,𝔽3)semidirect
258. Q8C2×C3×D12semidirect
271. C2×C4D3×SL(2,𝔽3)semidirect
282. C2×C3D4×SL(2,𝔽3)semidirect
300. C4C2×D3×SL(2,𝔽3)semidirect
303. C3C2×D4×SL(2,𝔽3)semidirect
305. C2SL(2,𝔽3)×D12direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×SL(2,𝔽3)2
C2×C3×C4×SL(2,𝔽3)2
SL(2,𝔽3)×D122
C2×Q8×D123
C2×D4×SL(2,𝔽3)3
C22×C3×D124