Finite Group Databasebeta

← C2×C4×D3×SL(2,𝔽3) #11923181

Subgroups — C2×C4×D3×SL(2,𝔽3) #11923181

Order 1152.

1073 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 1073 conjugacy classes of subgroups — too many to draw. Only its 215 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)C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4×D3×SL(2,𝔽3)C214yesmaximal, direct product factor
5C2×C4×D3×Q8C311yesmaximal, characteristic, complemented
6C22×C4×SL(2,𝔽3)—31nomaximal
7C22×C3×C4×D3—41nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×D3×SL(2,𝔽3)C414yesdirect product factor
10C2×D3×SL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2212yescomplemented
12Dic3×SL(2,𝔽3)C2212yescomplemented
13C22×D3×Q8C2×C311yescharacteristic
14C23×SL(2,𝔽3)—31no—
15C2×C3×C4×Q8C2×C311yescharacteristic, complemented
16C2×Q8×Dic3C2×C311yescharacteristic, complemented
17C4×D3×Q8—34no—
18C4×D3×Q8C2×C314yescomplemented
19C2×D3×C4⋊C4—31no—
20C2×#887—31no—
21C2×C42×D3—31no—
22C2×C4×Dic6—31no—
23C2×C4×SL(2,𝔽3)—21no—
24C2×C4×SL(2,𝔽3)—31no—
25C2×C4×SL(2,𝔽3)—34no—
26C2×C4×SL(2,𝔽3)D311yescharacteristic, direct product factor
27C23×C3×D3—41no—
28C22×C32×C4—41no—
29C22×C3×Dic3—41no—
30C2×C3×C4×D3—48no—
31C2×C3×C4×D3—44no—
32C2×C3×SL(2,𝔽3)C2×C412yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34D3×SL(2,𝔽3)C2×C414yesdirect product factor
35C22×C4×Q8—31noSylow
36C22×C3×Q8C22×C311yescharacteristic
37C23×C3×C4—121no—
38C2×D3×Q8—34no—
39C2×D3×Q8—32no—
40C2×D3×Q8C3×C414yescomplemented
41C2×D3×Q8C22×C312yes—
42C22×C4×D3—31no—
43C22×C4×D3—31no—
44C22×C4×D3A411yescharacteristic
45C22×Dic6—31no—
46C22×SL(2,𝔽3)—21no—
47C22×SL(2,𝔽3)—34no—
48C22×SL(2,𝔽3)—32no—
49C22×SL(2,𝔽3)C2×D311yescharacteristic
50C3×C4×Q8—32no—
51C3×C4×Q8C22×C312yescomplemented
52C2×C3×C4⋊C4—31no—
53C2×C3×C42—31no—
54Q8×Dic3—32no—
55Q8×Dic3C22×C312yescomplemented
56C2×C4⋊Dic3—31no—
57C2×Dic3⋊C4—32no—
58C2×C4×Dic3—31no—
59C2×C4×Dic3—31no—
60C2×C4×Dic3—31no—
61D3×C4⋊C4—34no—
62D3×C4⋊C4—34no—
63#887—34no—
64#887—34no—
65C42×D3—34no—
66C42×D3—34no—
67C4×Dic6—34no—
68C4×Dic6—34no—
69C4×SL(2,𝔽3)—22no—
70C4×SL(2,𝔽3)—32no—
71C4×SL(2,𝔽3)C2×D312yesdirect product factor
72C23×C32—41no—
73C22×C3×D3—44no—
74C22×C3×D3—44no—
75C22×C3×D3—44no—
76C22×C3×D3—42no—
77C2×C32×C4—44no—
78C2×C32×C4—42no—
79C2×C3×Dic3—44no—
80C2×C3×Dic3—42no—
81C3×C4×D3—416no—
82C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
83C23×Q8—31no—
84C2×C4×Q8—31no—
85C2×C4×Q8—34no—
86C2×C4×Q8—91no—
87C2×C4×Q8—91no—
88C2×C4×Q8—94no—
89C2×C4×Q8C3×D311yescharacteristic, complemented
90C22×C4⋊C4—91no—
91C22×C42—91no—
92C24×C3—121no—
93C23×D3C2×A411yescharacteristic
94C2×C3×Q8—31no—
95C2×C3×Q8—32no—
96C2×C3×Q8C2×C3×C412yescomplemented
97C2×C3×Q8C23×C311yescharacteristic
98C22×C3×C4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—41no—
101C22×C3×C4—81no—
102C22×C3×C4—124no—
103C22×C3×C4—128no—
104C22×C3×C4—121no—
105C22×C3×C4C2×A411yescharacteristic
106C22×Dic3—31no—
107C22×Dic3—31no—
108C22×Dic3C2×A411yescharacteristic
109D3×Q8—38no—
110D3×Q8—38no—
111D3×Q8—34no—
112D3×Q8C2×C3×C414yescomplemented
113C2×C4×D3—34no—
114C2×C4×D3—34no—
115C2×C4×D3—32no—
116C2×C4×D3—32no—
117C2×C4×D3—34no—
118C2×C4×D3—64no—
119C2×C4×D3SL(2,𝔽3)18yesdirect product factor
120C2×C4×D3C2×A414yes—
121C2×Dic6—32no—
122C2×Dic6—34no—
123C2×Dic6—32no—
124C2×Dic6—34no—
125C2×SL(2,𝔽3)—22no—
126C2×SL(2,𝔽3)—21no—
127C2×SL(2,𝔽3)—34no—
128C2×SL(2,𝔽3)C4×D312yesdirect product factor
129C2×SL(2,𝔽3)C22×D311yescharacteristic
130C3×C4⋊C4—32no—
131C3×C4⋊C4—32no—
132C3×C42—32no—
133C3×C42—32no—
134C4⋊Dic3—32no—
135C4⋊Dic3—32no—
136Dic3⋊C4—38no—
137C4×Dic3—34no—
138C4×Dic3—34no—
139C4×Dic3—32no—
140C4×Dic3—32no—
141C22×C32—42no—
142C22×C32—42no—
143C22×C32—42no—
144C22×C32—41no—
145C2×C3×D3—416no—
146C2×C3×D3—44no—
147C2×C3×D3—44no—
148C2×C3×D3—44no—
149C32×C4—44no—
150C3×Dic3—44no—
151C22×Q8—34no—
152C22×Q8—32no—
153C22×Q8—91no—
154C22×Q8—92no—
155C22×Q8—94no—
156C22×Q8C2×C3×D311yescharacteristic
157C23×C4—31no—
158C23×C4—91no—
159C23×C4—91no—
160C4×Q8—32no—
161C4×Q8—32no—
162C4×Q8—94no—
163C4×Q8—94no—
164C4×Q8—94no—
165C4×Q8—94no—
166C4×Q8—92no—
167C4×Q8C2×C3×D312yescomplemented
168C2×C4⋊C4—31no—
169C2×C4⋊C4—94no—
170C2×C4⋊C4—94no—
171C2×C4⋊C4—91no—
172C2×C4⋊C4—92no—
173C2×C42—31no—
174C2×C42—94no—
175C2×C42—94no—
176C2×C42—91no—
177C2×C42—91no—
178C2×C42—91no—
179C23×C3—41no—
180C23×C3—81no—
181C23×C3—122no—
182C23×C3—124no—
183C23×C3—124no—
184C23×C3—124no—
185C23×C3C22×A411yescharacteristic
186C22×D3C4×A414yes—
187C22×D3C2×SL(2,𝔽3)14yes—
188C22×D3#41⋊C314yes—
189C22×D3C22×A412yes—
190C3×Q8—32no—
191C3×Q8—31no—
192C3×Q8—32no—
193C3×Q8C22×C3×C411yescharacteristic, complemented
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—32no—
199C2×C3×C4—42no—
200C2×C3×C4—44no—
201C2×C3×C4—62no—
202C2×C3×C4—82no—
203C2×C3×C4—84no—
204C2×C3×C4—1216no—
205C2×C3×C4—122no—
206C2×C3×C4—124no—
207C2×C3×C4C2×SL(2,𝔽3)14yescomplemented
208C2×C3×C4C22×A412yes—
209C2×Dic3—34no—
210C2×Dic3—32no—
211C2×Dic3—32no—
212C2×Dic3—62no—
213C2×Dic3C2×SL(2,𝔽3)14yescomplemented
214C2×Dic3C22×A412yes—
215C4×D3—34no—
216C4×D3—38no—
217C4×D3—34no—
218C4×D3—68no—
219C4×D3C2×SL(2,𝔽3)116yesdirect product factor
220Dic6—38no—
221Dic6—34no—
222Dic6—34no—
223SL(2,𝔽3)—21no—
224SL(2,𝔽3)C2×C4×D311yescharacteristic, direct product factor
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—44no—
229C3×D3—48no—
230C24—31no—
231C2×Q8—31no—
232C2×Q8—32no—
233C2×Q8—34no—
234C2×Q8—92no—
235C2×Q8—94no—
236C2×Q8—92no—
237C2×Q8—94no—
238C2×Q8—98no—
239C2×Q8—98no—
240C2×Q8—94no—
241C2×Q8C3×C4×D312yescomplemented
242C2×Q8C22×C3×D311yescharacteristic
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—34no—
246C22×C4—38no—
247C22×C4—31no—
248C22×C4—92no—
249C22×C4—92no—
250C22×C4—94no—
251C22×C4—94no—
252C22×C4—94no—
253C22×C4—91no—
254C22×C4—91no—
255C22×C4—184no—
256C22×C4D3×A411yescharacteristic
257C4⋊C4—32no—
258C4⋊C4—32no—
259C4⋊C4—92no—
260C4⋊C4—98no—
261C4⋊C4—92no—
262C42—32no—
263C42—32no—
264C42—92no—
265C42—92no—
266C42—94no—
267C42—94no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—41no—
272C22×C3—82no—
273C22×C3—82no—
274C22×C3—82no—
275C22×C3—81no—
276C22×C3—124no—
277C22×C3—124no—
278C22×C3—1216no—
279C22×C3—124no—
280C22×C3C2×C4×A412yes—
281C22×C3C22×SL(2,𝔽3)12yes—
282C22×C3C2×#41⋊C312yes—
283C22×C3C23×A411yescharacteristic
284C2×D3C4×SL(2,𝔽3)116yesdirect product factor
285C2×D3C2×C4×A414yes—
286C2×D3C22×SL(2,𝔽3)14yes—
287C2×D3C2×#41⋊C314yes—
288C3×C4—31no—
289C3×C4—32no—
290C3×C4—31no—
291C3×C4—44no—
292C3×C4—62no—
293C3×C4—84no—
294C3×C4—124no—
295C3×C4C22×SL(2,𝔽3)14yescomplemented
296Dic3—34no—
297Dic3—62no—
298Dic3C22×SL(2,𝔽3)14yescomplemented
299C32—41noSylow
300C23—34no—
301C23—34no—
302C23—34no—
303C23—32no—
304C23C2×D3×A411yescharacteristic
305Q8—32no—
306Q8—31no—
307Q8—32no—
308Q8—94no—
309Q8—98no—
310Q8—94no—
311Q8C2×C3×C4×D311yescharacteristic, complemented
312C2×C4—32no—
313C2×C4—32no—
314C2×C4—34no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—32no—
318C2×C4—316no—
319C2×C4—32no—
320C2×C4—62no—
321C2×C4—94no—
322C2×C4—94no—
323C2×C4—98no—
324C2×C4—94no—
325C2×C4—92no—
326C2×C4—92no—
327C2×C4—188no—
328C2×C4—182no—
329C2×C4D3×SL(2,𝔽3)14yesdirect product factor
330C2×C4C2×D3×A412yes—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—44no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3—81no—
338C2×C3—84no—
339C2×C3—128no—
340C2×C3C2×C4×SL(2,𝔽3)14yescomplemented
341C2×C3C22×C4×A411yescharacteristic
342C2×C3C23×SL(2,𝔽3)11yescharacteristic
343C2×C3C22×#41⋊C311yescharacteristic
344D3C2×C4×SL(2,𝔽3)18yesdirect product factor
345C22—34no—
346C22—34no—
347C22—34no—
348C22—316no—
349C22C4×D3×A412yes—
350C22C2×D3×SL(2,𝔽3)12yes—
351C22D3×#41⋊C312yes—
352C22C22×D3×A411yescharacteristic
353C4—31no—
354C4—32no—
355C4—31no—
356C4—34no—
357C4—62no—
358C4—94no—
359C4—182no—
360C4C2×D3×SL(2,𝔽3)14yesdirect product factor
361C3—41no—
362C3—81no—
363C3C22×C4×SL(2,𝔽3)11yescharacteristic, complemented
364C2—38no—
365C2C4×D3×SL(2,𝔽3)14yesdirect product factor
366C2C2×C4×D3×A411yescharacteristic
367C2C22×D3×SL(2,𝔽3)11yescharacteristic
368C2C2×D3×#41⋊C311yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×D3×SL(2,𝔽3)C2
cosocleC2×C3×Q8C23×C3
LCSG ▷ C3×Q8
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. C4×D3×SL(2,𝔽3)C2direct (N × H)
5. C2×C4×D3×Q8C3semidirect
9. C2×D3×SL(2,𝔽3)C4direct (N × H)
11. C3×C4×SL(2,𝔽3)C22semidirect
12. Dic3×SL(2,𝔽3)C22semidirect
15. C2×C3×C4×Q8C2×C3semidirect
16. C2×Q8×Dic3C2×C3semidirect
18. C4×D3×Q8C2×C3semidirect
26. C2×C4×SL(2,𝔽3)D3direct (N × H)
32. C2×C3×SL(2,𝔽3)C2×C4semidirect
34. D3×SL(2,𝔽3)C2×C4direct (N × H)
40. C2×D3×Q8C3×C4semidirect
51. C3×C4×Q8C22×C3semidirect
55. Q8×Dic3C22×C3semidirect
71. C4×SL(2,𝔽3)C2×D3direct (N × H)
82. C3×SL(2,𝔽3)C22×C4semidirect
89. C2×C4×Q8C3×D3semidirect
96. C2×C3×Q8C2×C3×C4semidirect
112. D3×Q8C2×C3×C4semidirect
119. C2×C4×D3SL(2,𝔽3)direct (N × H)
128. C2×SL(2,𝔽3)C4×D3direct (N × H)
167. C4×Q8C2×C3×D3semidirect
193. C3×Q8C22×C3×C4semidirect
207. C2×C3×C4C2×SL(2,𝔽3)semidirect
213. C2×Dic3C2×SL(2,𝔽3)semidirect
219. C4×D3C2×SL(2,𝔽3)direct (N × H)
224. SL(2,𝔽3)C2×C4×D3direct (N × H)
241. C2×Q8C3×C4×D3semidirect
284. C2×D3C4×SL(2,𝔽3)direct (N × H)
295. C3×C4C22×SL(2,𝔽3)semidirect
298. Dic3C22×SL(2,𝔽3)semidirect
311. Q8C2×C3×C4×D3semidirect
329. C2×C4D3×SL(2,𝔽3)direct (N × H)
340. C2×C3C2×C4×SL(2,𝔽3)semidirect
344. D3C2×C4×SL(2,𝔽3)direct (N × H)
360. C4C2×D3×SL(2,𝔽3)direct (N × H)
363. C3C22×C4×SL(2,𝔽3)semidirect
365. C2C4×D3×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
C22×D3×SL(2,𝔽3)2
C2×C3×C4×SL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
C4×D3×SL(2,𝔽3)2
C2×C4×D3×Q83
C22×C4×SL(2,𝔽3)3
C22×C3×C4×D34