Finite Group Databasebeta

← C3×Q8×#239 #11903752

Subgroups — C3×Q8×#239 #11903752

Order 1152.

2242 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×Q82C211yesmaximal, characteristic, complemented
2C3×#5730C219yesmaximal, complemented
3C3×#5728C219yesmaximal, complemented
4C3×Q8×D12C213yesmaximal, complemented
5C3×C4×#239C213yesmaximal
6C3×C4×D3×Q8C213yesmaximal, complemented
7C3×#5570C213yesmaximal, complemented
8C3×Q8×#41—31nomaximal
9Q8×#239C311yesmaximal, characteristic, direct product factor
10C2×C3×#239C2211yescharacteristic
11C2×C3×D3×Q8C2213yes—
12C32×C4⋊Q8C2219yes—
13C32×C4×Q8C2213yes—
14C32×C4×Q8C2213yes—
15C3×#946C22118yes—
16C3×Q8×Dic3C2213yes—
17C3×Q8×Dic3C2211yescharacteristic
18C3×Q8×Dic3C2216yes—
19C3×#897C22118yes—
20C3×#893C2219yes—
21C3×#892C2219yes—
22C3×D3×C4⋊C4C2219yes—
23C3×#890C22118yes—
24C3×#887C2219yes—
25C3×C4×D12C2219yes—
26C3×C42×D3C2219yes—
27C3×#869C2219yes—
28C3×C4×Dic6C2219yes—
29C3×Q82—21no—
30C3×Q82D311yescharacteristic, complemented
31C3×Q82C2×C311yescharacteristic, complemented
32C3×#557—33no—
33C3×#554—39no—
34C3×D4×Q8—33no—
35C3×#533—39no—
36C3×C4×#41—33no—
37C2×C3×C4×Q8—33no—
38#5730C2×C319yescomplemented
39#5728C2×C319yescomplemented
40Q8×D12C2×C313yescomplemented
41C4×#239C2×C313yes—
42C4×D3×Q8C2×C313yescomplemented
43#5570C2×C313yescomplemented
44C2×C32×Q8C2316yes—
45C2×C32×Q8C2311yescharacteristic
46C2×C32×Q8C2311yescharacteristic
47C3×#239Q818yesdirect product factor
48C3×D3×Q8—212no—
49C2×C3×D12C2313yes—
50C2×C3×C4×D3C2319yes—
51C2×C3×C4×D3C2319yes—
52C2×C3×C4×D3C2313yes—
53C2×C3×Dic6C2319yes—
54C32×C4⋊C4C2319yes—
55C32×C4⋊C4C2319yes—
56C32×C42C2319yes—
57C3×(C2×D3)⋊C4C23118yes—
58C3×C4⋊Dic3C2319yes—
59C3×C4⋊Dic3C23118yes—
60C3×C4⋊Dic3C2313yes—
61C3×Dic3⋊C4C23118yes—
62C3×C4×Dic3C2313yes—
63C3×C4×Dic3C2319yes—
64C3×C4×Dic3C2319yes—
65Q8×#41—31noSylow
66C2×C3×#41—31no—
67C22×C3×Q8—33no—
68C2×#239C22×C311yescharacteristic
69C2×D3×Q8C22×C313yes—
70C3×C4⋊Q8—29no—
71C3×C4⋊Q8—39no—
72C3×C4⋊Q8C2×D319yes—
73C3×C4⋊Q8C22×C319yes—
74C3×#125—318no—
75C3×C22⋊Q8—318no—
76C3×C22⋊Q8—318no—
77C3×C4×Q8—23no—
78C3×C4×Q8—23no—
79C3×C4×Q8—31no—
80C3×C4×Q8—33no—
81C3×C4×Q8—36no—
82C3×C4×Q8—39no—
83C3×C4×Q8—39no—
84C3×C4×Q8C2×D313yes—
85C3×C4×Q8C2×D313yes—
86C3×C4×Q8C22×C313yes—
87C3×C4×Q8C22×C313yes—
88C3×C4×D4—39no—
89C3×C4×D4—39no—
90C3×#117—39no—
91C2×C3×C4⋊C4—39no—
92C2×C3×C42—39no—
93#946C22×C3118yes—
94Q8×Dic3C22×C316yes—
95Q8×Dic3C22×C313yes—
96Q8×Dic3C22×C311yescharacteristic
97#897C22×C3118yes—
98#893C22×C319yes—
99#892C22×C319yes—
100D3×C4⋊C4C22×C319yes—
101#890C22×C3118yes—
102#887C22×C319yes—
103C4×D12C22×C319yes—
104C42×D3C22×C319yes—
105#869C22×C319yes—
106C4×Dic6C22×C319yes—
107C22×C3×D3C2413yes—
108C32×Q8—46no—
109C32×Q8C2×Q814yescomplemented
110C32×Q8#4114yescomplemented
111C2×C32×C4C2419yes—
112C2×C32×C4C2413yes—
113C2×C32×C4C2413yes—
114C2×C3×Dic3C2413yes—
115C2×C3×Dic3C2419yes—
116C2×C3×Dic3C2411yescharacteristic
117C3×D12C2×Q8112yescomplemented
118C3×C4×D3—218no—
119C3×C4×D3—49no—
120C3×C4×D3C2×Q8112yescomplemented
121C3×Dic6—218no—
122Q82C3×D311yescharacteristic, complemented
123#557—33no—
124#554—39no—
125D4×Q8—33no—
126#533—39no—
127C4×#41—33no—
128C2×C4×Q8—33no—
129C3×#41—38no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—26no—
133C2×C3×Q8—39no—
134C2×C3×Q8—612no—
135C2×C3×Q8C22×D316yes—
136C2×C3×Q8C22×D311yescharacteristic
137C2×C3×Q8C22×D311yescharacteristic
138C2×C3×Q8C23×C316yes—
139C2×C3×Q8C23×C311yescharacteristic
140C2×C3×Q8C23×C311yescharacteristic
141C2×C3×D4—33no—
142C22×C3×C4—33no—
143C22×C3×C4—39no—
144C22×C3×C4—39no—
145#239C3×Q818yesdirect product factor
146D3×Q8—212no—
147C2×D12C23×C313yes—
148C2×C4×D3C23×C319yes—
149C2×C4×D3C23×C319yes—
150C2×C4×D3C23×C313yes—
151C2×Dic6C23×C319yes—
152C3×C4⋊C4—29no—
153C3×C4⋊C4—29no—
154C3×C4⋊C4—318no—
155C3×C4⋊C4—33no—
156C3×C4⋊C4—39no—
157C3×C4⋊C4—318no—
158C3×C4⋊C4C22×D319yes—
159C3×C4⋊C4C22×D319yes—
160C3×C4⋊C4C23×C319yes—
161C3×C4⋊C4C23×C319yes—
162C3×C22⋊C4—318no—
163C3×C42—29no—
164C3×C42—33no—
165C3×C42—39no—
166C3×C42—39no—
167C3×C42C22×D319yes—
168C3×C42C23×C319yes—
169(C2×D3)⋊C4C23×C3118yes—
170C4⋊Dic3C23×C313yes—
171C4⋊Dic3C23×C319yes—
172C4⋊Dic3C23×C3118yes—
173Dic3⋊C4C23×C3118yes—
174C4×Dic3C23×C319yes—
175C4×Dic3C23×C313yes—
176C4×Dic3C23×C319yes—
177C22×C32C2511yescharacteristic
178C2×C3×D3—23no—
179C2×C3×D3—23no—
180C2×C3×D3C22×Q816yes—
181C32×C4—29no—
182C32×C4C22×Q816yes—
183C32×C4C2×#4116yes—
184C3×Dic3—29no—
185C3×Dic3—23no—
186C3×Dic3C22×Q812yes—
187C2×#41—31no—
188C22×Q8—33no—
189C4⋊Q8—39no—
190C4⋊Q8C2×C3×D319yes—
191#125—318no—
192C22⋊Q8—318no—
193C22⋊Q8—318no—
194C4×Q8—31no—
195C4×Q8—33no—
196C4×Q8—36no—
197C4×Q8—39no—
198C4×Q8—39no—
199C4×Q8C2×C3×D313yes—
200C4×Q8C2×C3×D313yes—
201C4×D4—39no—
202C4×D4—39no—
203#117—39no—
204C2×C4⋊C4—39no—
205C2×C42—39no—
206C23×C3—33no—
207C22×D3C24×C313yes—
208C3×Q8—24no—
209C3×Q8—24no—
210C3×Q8—46no—
211C3×Q8—46no—
212C3×Q8—618no—
213C3×Q8—86no—
214C3×Q8D3×Q814yescomplemented
215C3×Q8#23914yesdirect product factor
216C3×Q8C2×C3×Q814yescomplemented
217C3×Q8C3×#4114yescomplemented
218C3×D4—312no—
219C2×C3×C4—23no—
220C2×C3×C4—23no—
221C2×C3×C4—29no—
222C2×C3×C4—31no—
223C2×C3×C4—312no—
224C2×C3×C4—33no—
225C2×C3×C4—39no—
226C2×C3×C4—618no—
227C2×C3×C4—129no—
228C2×C3×C4C23×D319yes—
229C2×C3×C4C23×D313yes—
230C2×C3×C4C23×D313yes—
231C2×C3×C4C24×C319yes—
232C2×C3×C4C24×C313yes—
233C2×C3×C4C24×C313yes—
234C2×Dic3C24×C313yes—
235C2×Dic3C24×C319yes—
236C2×Dic3C24×C311yescharacteristic
237D12C2×C3×Q8112yescomplemented
238C4×D3—218no—
239C4×D3—49no—
240C4×D3C2×C3×Q8112yescomplemented
241Dic6—218no—
242C2×C32C23×Q811yescharacteristic
243C2×C32C22×#4111yescharacteristic
244C2×C32#58511yescharacteristic
245C3×D3—26no—
246#41—38no—
247C2×Q8—39no—
248C2×Q8—612no—
249C2×Q8C22×C3×D316yes—
250C2×Q8C22×C3×D311yescharacteristic
251C2×Q8C22×C3×D311yescharacteristic
252C2×D4—33no—
253C22×C4—33no—
254C22×C4—39no—
255C22×C4—39no—
256C4⋊C4—33no—
257C4⋊C4—318no—
258C4⋊C4—39no—
259C4⋊C4—318no—
260C4⋊C4C22×C3×D319yes—
261C4⋊C4C22×C3×D319yes—
262C22⋊C4—318no—
263C42—33no—
264C42—39no—
265C42—39no—
266C42C22×C3×D319yes—
267C22×C3—21no—
268C22×C3—36no—
269C22×C3—63no—
270C22×C3—63no—
271C22×C3C24×D311yescharacteristic
272C22×C3C25×C311yescharacteristic
273C2×D3—23no—
274C2×D3—23no—
275C2×D3C22×C3×Q816yes—
276C3×C4—29no—
277C3×C4—29no—
278C3×C4—26no—
279C3×C4—26no—
280C3×C4—32no—
281C3×C4—49no—
282C3×C4—69no—
283C3×C4—63no—
284C3×C4C2×D3×Q816yes—
285C3×C4C2×#23916yes—
286C3×C4C22×C3×Q816yes—
287C3×C4C2×C3×#4116yes—
288Dic3—29no—
289Dic3—23no—
290Dic3C22×C3×Q812yes—
291C32Q8×#4111yescharacteristic, complemented, Sylow
292C23—33no—
293Q8—46no—
294Q8—618no—
295Q8C3×D3×Q814yescomplemented
296Q8C3×#23914yesdirect product factor
297D4—312no—
298C2×C4—31no—
299C2×C4—312no—
300C2×C4—33no—
301C2×C4—39no—
302C2×C4—618no—
303C2×C4—129no—
304C2×C4C23×C3×D319yes—
305C2×C4C23×C3×D313yes—
306C2×C4C23×C3×D313yes—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—66no—
311C2×C3C22×D3×Q811yescharacteristic
312C2×C3C22×#23911yescharacteristic
313C2×C3#596611yescharacteristic
314C2×C3C23×C3×Q811yescharacteristic
315C2×C3C22×C3×#4111yescharacteristic
316C2×C3C3×#58511yescharacteristic
317D3—26no—
318C22—36no—
319C22—63no—
320C22—63no—
321C22C24×C3×D311yescharacteristic
322C4—29no—
323C4—32no—
324C4—69no—
325C4—63no—
326C4C2×C3×D3×Q816yes—
327C4C2×C3×#23916yes—
328C3—21no—
329C3Q8×#23911yescharacteristic, direct product factor
330C3C3×Q8×#4111yescharacteristic, complemented
331C2—66no—
332C2C22×C3×D3×Q811yescharacteristic
333C2C22×C3×#23911yescharacteristic
334C2C3×#596611yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×Q82C2
socleC22×C32C25
prime radicalC2×C3×#239C22
hypercenterC3×Q82D3
hypocenterC3C3×Q8×#41
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×Q82

Complements

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

NH ≅ G/Ntype
1. C32×Q82C2semidirect
2. C3×#5730C2semidirect
3. C3×#5728C2semidirect
4. C3×Q8×D12C2semidirect
6. C3×C4×D3×Q8C2semidirect
7. C3×#5570C2semidirect
9. Q8×#239C3direct (N × H)
30. C3×Q82D3semidirect
31. C3×Q82C2×C3semidirect
38. #5730C2×C3semidirect
39. #5728C2×C3semidirect
40. Q8×D12C2×C3semidirect
42. C4×D3×Q8C2×C3semidirect
43. #5570C2×C3semidirect
47. C3×#239Q8direct (N × H)
109. C32×Q8C2×Q8semidirect
110. C32×Q8#41semidirect
117. C3×D12C2×Q8semidirect
120. C3×C4×D3C2×Q8semidirect
122. Q82C3×D3semidirect
145. #239C3×Q8direct (N × H)
214. C3×Q8D3×Q8semidirect
215. C3×Q8#239direct (N × H)
216. C3×Q8C2×C3×Q8semidirect
217. C3×Q8C3×#41semidirect
237. D12C2×C3×Q8semidirect
240. C4×D3C2×C3×Q8semidirect
291. C32Q8×#41semidirect
295. Q8C3×D3×Q8semidirect
296. Q8C3×#239direct (N × H)
329. C3Q8×#239direct (N × H)
330. C3C3×Q8×#41semidirect

Maximal subgroups

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

subgroupindex
C32×Q822
C3×#57302
C3×#57282
C3×Q8×D122
C3×C4×#2392
C3×C4×D3×Q82
C3×#55702
C3×Q8×#413
Q8×#2393