Finite Group Databasebeta

← C2×C3×Q8×S4 #11923692

Subgroups — C2×C3×Q8×S4 #11923692

Order 1152.

1660 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1660 conjugacy classes of subgroups — too many to draw. Only its 232 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×Q8×A4C211yesmaximal, characteristic, complemented
2C3×Q8×S4C218yesmaximal, direct product factor
3C2×C3×C4×S4C213yesmaximal
4C2×C3×A4⋊Q8C213yesmaximal, complemented
5C2×Q8×S4C311yesmaximal, characteristic, direct product factor
6C2×C3×D4×Q8—31nomaximal
7C22×C3×S4C2211yescharacteristic
8C2×C3×D3×Q8—41nomaximal
9C3×Q8×A4C2214yescomplemented
10C2×C3×C4×A4C2213yes—
11C2×C3×SL(2,ℤ/4)C2213yes—
12C3×C4×S4C22112yes—
13C3×A4⋊Q8C22112yescomplemented
14C23×C3×Q8—31no—
15C23×C3×Q8D311yescharacteristic, complemented
16C2×Q8×A4—21no—
17C2×Q8×A4C2×C311yescharacteristic, complemented
18Q8×S4C2×C318yesdirect product factor
19C2×C4×S4C2×C313yes—
20C2×A4⋊Q8C2×C313yescomplemented
21C3×D4×Q8—38no—
22C3×D4×Q8—38no—
23C2×C3×C4⋊Q8—33no—
24C2×C3×C22⋊Q8—33no—
25C2×C3×C22⋊Q8—33no—
26C2×C3×C4×Q8—31no—
27C2×C3×C4×D4—33no—
28C22×C3×A4C2311yescharacteristic
29C2×C3×S4Q814yesdirect product factor
30C2×C3×S4C2312yes—
31C2×C32×Q8—41no—
32C3×D3×Q8—48no—
33C2×C3×C4×D3—43no—
34C2×C3×Dic6—43no—
35C3×C4×A4C2316yes—
36C3×SL(2,ℤ/4)C2316yes—
37C2×D4×Q8—31noSylow
38C22×S4C22×C311yescharacteristic
39C22×C3×Q8—31no—
40C22×C3×Q8—33no—
41C22×C3×Q8—34no—
42C22×C3×Q8—33no—
43C22×C3×Q8—38no—
44C22×C3×Q8—64no—
45C22×C3×Q8—63no—
46C22×C3×Q8—61no—
47C22×C3×Q8C2×D314yescomplemented
48C22×C3×D4—31no—
49C23×C3×C4—33no—
50C23×C3×C4C2×D313yes—
51C2×D3×Q8—41no—
52Q8×A4—24no—
53Q8×A4C22×C314yescomplemented
54C2×C4×A4—23no—
55C2×C4×A4C22×C313yes—
56C2×SL(2,ℤ/4)C22×C313yes—
57C4×S4C22×C3112yes—
58A4⋊Q8C22×C3112yescomplemented
59C3×C4⋊Q8—324no—
60C3×C22⋊Q8—312no—
61C3×C22⋊Q8—324no—
62C3×C22⋊Q8—312no—
63C3×C4×Q8—38no—
64C3×C4×D4—312no—
65C3×C4×D4—312no—
66C2×C3×C4⋊C4—33no—
67C2×C3×C4⋊C4—33no—
68C2×C3×C4⋊C4—36no—
69C2×C3×C22⋊C4—33no—
70C2×C3×C22⋊C4—33no—
71C2×C3×C42—33no—
72C22×C3×D3—41no—
73C2×C3×A4C2×Q812yescomplemented
74C2×C3×A4C2411yescharacteristic
75C3×S4C2×Q814yesdirect product factor
76C32×Q8—44no—
77C2×C32×C4—43no—
78C2×C3×Dic3—43no—
79C3×C4×D3—412no—
80C3×Dic6—412no—
81C23×Q8—31no—
82C23×Q8C3×D311yescharacteristic, complemented
83D4×Q8—38no—
84D4×Q8—38no—
85C2×C4⋊Q8—33no—
86C2×C22⋊Q8—33no—
87C2×C22⋊Q8—33no—
88C2×C4×Q8—31no—
89C2×C4×D4—33no—
90C24×C3—31no—
91C24×C3C22×D311yescharacteristic
92C22×A4—21no—
93C22×A4C23×C311yescharacteristic
94C2×S4C3×Q814yesdirect product factor
95C2×S4C23×C312yes—
96C2×C3×Q8—34no—
97C2×C3×Q8—34no—
98C2×C3×Q8—312no—
99C2×C3×Q8—312no—
100C2×C3×Q8—33no—
101C2×C3×Q8—41no—
102C2×C3×Q8—61no—
103C2×C3×Q8—66no—
104C2×C3×Q8—63no—
105C2×C3×Q8—66no—
106C2×C3×Q8—624no—
107C2×C3×Q8—68no—
108C2×C3×Q8—63no—
109C2×C3×Q8—81no—
110C2×C3×Q8S411yescharacteristic, direct product factor
111C2×C3×D4—32no—
112C2×C3×D4—34no—
113C2×C3×D4—34no—
114C2×C3×D4—32no—
115C22×C3×C4—31no—
116C22×C3×C4—33no—
117C22×C3×C4—33no—
118C22×C3×C4—36no—
119C22×C3×C4—33no—
120C22×C3×C4—33no—
121C22×C3×C4—312no—
122C22×C3×C4—63no—
123C22×C3×C4—66no—
124C22×C3×C4—63no—
125C22×C3×C4C22×D316yes—
126D3×Q8—48no—
127C2×C4×D3—43no—
128C2×Dic6—43no—
129C4×A4—26no—
130C4×A4C23×C316yes—
131SL(2,ℤ/4)C23×C316yes—
132C3×C4⋊C4—312no—
133C3×C4⋊C4—312no—
134C3×C4⋊C4—324no—
135C3×C22⋊C4—312no—
136C3×C22⋊C4—36no—
137C3×C22⋊C4—36no—
138C3×C42—312no—
139C22×C32—41no—
140C2×C3×D3—44no—
141C2×C3×D3—42no—
142C3×A4C22×Q811yescharacteristic, complemented
143C32×C4—46no—
144C3×Dic3—46no—
145C22×Q8—31no—
146C22×Q8—33no—
147C22×Q8—34no—
148C22×Q8—33no—
149C22×Q8—38no—
150C22×Q8—64no—
151C22×Q8—63no—
152C22×Q8—61no—
153C22×Q8C2×C3×D314yescomplemented
154C22×D4—31no—
155C23×C4—33no—
156C23×C4C2×C3×D313yes—
157C4⋊Q8—324no—
158C22⋊Q8—312no—
159C22⋊Q8—324no—
160C22⋊Q8—312no—
161C4×Q8—38no—
162C4×D4—312no—
163C4×D4—312no—
164C2×C4⋊C4—33no—
165C2×C4⋊C4—33no—
166C2×C4⋊C4—36no—
167C2×C22⋊C4—33no—
168C2×C22⋊C4—33no—
169C2×C42—33no—
170C23×C3—31no—
171C23×C3—31no—
172C23×C3—32no—
173C23×C3—32no—
174C23×C3—34no—
175C23×C3—61no—
176C23×C3—62no—
177C23×C3—61no—
178C23×C3D3×Q812yescomplemented
179C23×C3C23×D311yescharacteristic
180C22×D3—41no—
181C2×A4—21no—
182C2×A4—22no—
183C2×A4C2×C3×Q812yescomplemented
184C2×A4C24×C311yescharacteristic
185S4C2×C3×Q814yesdirect product factor
186C3×Q8—312no—
187C3×Q8—44no—
188C3×Q8—64no—
189C3×Q8—612no—
190C3×Q8—612no—
191C3×Q8—84no—
192C3×Q8C2×S414yesdirect product factor
193C3×D4—38no—
194C3×D4—34no—
195C3×D4—34no—
196C2×C3×C4—32no—
197C2×C3×C4—34no—
198C2×C3×C4—33no—
199C2×C3×C4—36no—
200C2×C3×C4—36no—
201C2×C3×C4—36no—
202C2×C3×C4—36no—
203C2×C3×C4—36no—
204C2×C3×C4—43no—
205C2×C3×C4—63no—
206C2×C3×C4—612no—
207C2×C3×C4—66no—
208C2×C3×C4—63no—
209C2×C3×C4—63no—
210C2×C3×C4—612no—
211C2×C3×C4—83no—
212C2×C3×C4C2×S413yes—
213C2×Dic3—43no—
214C4×D3—412no—
215Dic6—412no—
216C2×C32—42no—
217C2×C32—41no—
218C3×D3—44no—
219C24—31no—
220C24C22×C3×D311yescharacteristic
221C2×Q8—34no—
222C2×Q8—34no—
223C2×Q8—312no—
224C2×Q8—312no—
225C2×Q8—33no—
226C2×Q8—61no—
227C2×Q8—66no—
228C2×Q8—63no—
229C2×Q8—66no—
230C2×Q8—624no—
231C2×Q8—68no—
232C2×Q8—63no—
233C2×Q8C3×S411yescharacteristic, direct product factor
234C2×D4—32no—
235C2×D4—34no—
236C2×D4—32no—
237C2×D4—34no—
238C22×C4—31no—
239C22×C4—33no—
240C22×C4—33no—
241C22×C4—36no—
242C22×C4—33no—
243C22×C4—33no—
244C22×C4—312no—
245C22×C4—63no—
246C22×C4—66no—
247C22×C4—63no—
248C22×C4C22×C3×D316yes—
249C4⋊C4—312no—
250C4⋊C4—312no—
251C4⋊C4—324no—
252C22⋊C4—312no—
253C22⋊C4—36no—
254C22⋊C4—36no—
255C42—312no—
256C22×C3—32no—
257C22×C3—31no—
258C22×C3—32no—
259C22×C3—32no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—34no—
263C22×C3—41no—
264C22×C3—61no—
265C22×C3—62no—
266C22×C3—64no—
267C22×C3—64no—
268C22×C3—62no—
269C22×C3—81no—
270C22×C3C2×D3×Q811yescharacteristic, complemented
271C22×C3C22×S411yescharacteristic
272C2×D3—42no—
273C2×D3—44no—
274A4—21no—
275A4C22×C3×Q811yescharacteristic, complemented
276C3×C4—34no—
277C3×C4—36no—
278C3×C4—46no—
279C3×C4—66no—
280C3×C4—66no—
281C3×C4—86no—
282C3×C4C22×S416yes—
283Dic3—46no—
284C32—41noSylow
285C23—31no—
286C23—31no—
287C23—32no—
288C23—32no—
289C23—34no—
290C23—61no—
291C23—62no—
292C23—61no—
293C23C3×D3×Q812yescomplemented
294C23C23×C3×D311yescharacteristic
295Q8—312no—
296Q8—64no—
297Q8—612no—
298Q8—612no—
299Q8C2×C3×S414yesdirect product factor
300D4—34no—
301D4—38no—
302D4—34no—
303C2×C4—32no—
304C2×C4—34no—
305C2×C4—33no—
306C2×C4—36no—
307C2×C4—36no—
308C2×C4—36no—
309C2×C4—36no—
310C2×C4—36no—
311C2×C4—63no—
312C2×C4—612no—
313C2×C4—63no—
314C2×C4—66no—
315C2×C4—63no—
316C2×C4—612no—
317C2×C4C2×C3×S413yes—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—32no—
321C2×C3—41no—
322C2×C3—42no—
323C2×C3—64no—
324C2×C3—81no—
325C2×C3—82no—
326C2×C3Q8×S412yesdirect product factor
327C2×C3C23×S411yescharacteristic
328D3—44no—
329C22—31no—
330C22—32no—
331C22—31no—
332C22—31no—
333C22—32no—
334C22—32no—
335C22—34no—
336C22—61no—
337C22—62no—
338C22—64no—
339C22—62no—
340C22—64no—
341C22C2×C3×D3×Q811yescharacteristic, complemented
342C22C22×C3×S411yescharacteristic
343C4—34no—
344C4—36no—
345C4—66no—
346C4—66no—
347C4C22×C3×S416yes—
348C3—41no—
349C3—81no—
350C3C2×Q8×S411yescharacteristic, direct product factor
351C2—31no—
352C2—31no—
353C2—32no—
354C2—64no—
355C2C3×Q8×S412yesdirect product factor
356C2C23×C3×S411yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutator, cosocleC2×A4C24×C3
FrattiniC2C23×C3×S4
FittingC23×C3×Q8D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
hypercenterC2×C3×Q8S4
hypocenterA4C22×C3×Q8
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
2. C3×Q8×S4C2direct (N × H)
4. C2×C3×A4⋊Q8C2semidirect
5. C2×Q8×S4C3direct (N × H)
9. C3×Q8×A4C22semidirect
13. C3×A4⋊Q8C22semidirect
15. C23×C3×Q8D3semidirect
17. C2×Q8×A4C2×C3semidirect
18. Q8×S4C2×C3direct (N × H)
20. C2×A4⋊Q8C2×C3semidirect
29. C2×C3×S4Q8direct (N × H)
47. C22×C3×Q8C2×D3semidirect
53. Q8×A4C22×C3semidirect
58. A4⋊Q8C22×C3semidirect
73. C2×C3×A4C2×Q8semidirect
75. C3×S4C2×Q8direct (N × H)
82. C23×Q8C3×D3semidirect
94. C2×S4C3×Q8direct (N × H)
110. C2×C3×Q8S4direct (N × H)
142. C3×A4C22×Q8semidirect
153. C22×Q8C2×C3×D3semidirect
178. C23×C3D3×Q8semidirect
183. C2×A4C2×C3×Q8semidirect
185. S4C2×C3×Q8direct (N × H)
192. C3×Q8C2×S4direct (N × H)
233. C2×Q8C3×S4direct (N × H)
270. C22×C3C2×D3×Q8semidirect
275. A4C22×C3×Q8semidirect
293. C23C3×D3×Q8semidirect
299. Q8C2×C3×S4direct (N × H)
326. C2×C3Q8×S4direct (N × H)
341. C22C2×C3×D3×Q8semidirect
350. C3C2×Q8×S4direct (N × H)
355. C2C3×Q8×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
C3×Q8×S42
C2×C3×C4×S42
C2×C3×A4⋊Q82
C2×Q8×S43
C2×C3×D4×Q83
C2×C3×D3×Q84