Finite Group Databasebeta

← C3×D3×C8×Q8 #11861245

Subgroups — C3×D3×C8×Q8 #11861245

Order 1152.

931 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×D3×Q8C211yesmaximal, characteristic
2C32×C8×Q8C211yesmaximal, characteristic, complemented
3C3×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×D3×C4⋊C8C213yesmaximal
5C3×#4830C213yesmaximal, complemented
6C3×C4×D3×C8C213yesmaximal
7C3×C8×Dic6C213yesmaximal, complemented
8C2×C3×C8×Q8—31nomaximal
9D3×C8×Q8C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×Q8C411yescharacteristic
11C32×C4×Q8C2211yescharacteristic
12C3×Q8×Dic3C411yescharacteristic
13C2×C3×D3×C8C2213yes—
14C2×C3×D3×C8C2211yescharacteristic
15C3×D3×C4⋊C4C413yes—
16C3×#887C413yes—
17C3×C42×D3C2213yes—
18C3×C4×Dic6C2213yes—
19C32×C4⋊C8C2213yes—
20C32×C4×C8C2213yes—
21C3×Dic3⋊C8C2216yes—
22C3×C8×Dic3C2213yes—
23C3×C8×Dic3C2213yes—
24C3×C4⋊(C3⋊C8)C2213yes—
25C3×C4×C3⋊C8C2213yes—
26C2×C3×C4×Q8—31no—
27C4×D3×Q8C2×C311yescharacteristic
28C3×C8×Q8—21no—
29C3×C8×Q8—31no—
30C3×C8×Q8—33no—
31C3×C8×Q8—33no—
32C3×C8×Q8D311yescharacteristic, direct product factor
33C3×C8×Q8C2×C311yescharacteristic, complemented
34C2×C3×C4⋊C8—33no—
35C2×C3×C4×C8—33no—
36Q8×C3⋊C8C2×C311yescharacteristic, complemented
37D3×C4⋊C8C2×C313yes—
38#4830C2×C313yescomplemented
39C4×D3×C8C2×C313yes—
40C8×Dic6C2×C313yescomplemented
41C2×C32×Q8C2×C411yescharacteristic
42C3×D3×Q8C818yesdirect product factor
43C2×C3×C4×D3C2×C413yes—
44C2×C3×C4×D3C2×C413yes—
45C2×C3×C4×D3C2311yescharacteristic
46C2×C3×Dic6C2×C413yes—
47C2×C32×C8C2313yes—
48C2×C32×C8C2311yescharacteristic
49C32×C4⋊C4C2×C413yes—
50C32×C42C2313yes—
51C3×C4⋊Dic3C2×C413yes—
52C3×Dic3⋊C4C2×C416yes—
53C3×C4×Dic3C2×C413yes—
54C3×C4×Dic3C2×C413yes—
55C3×C4×Dic3C2313yes—
56C2×C3×C3⋊C8C2313yes—
57C2×C3×C3⋊C8C2311yescharacteristic
58C3×D3×C8—26no—
59C3×D3×C8Q814yesdirect product factor
60C2×C8×Q8—31noSylow
61C22×C3×Q8—31no—
62C2×D3×Q8C3×C411yescharacteristic
63C22×C3×C8—31no—
64C22×C3×C8—33no—
65C3×C4×Q8—21no—
66C3×C4×Q8—31no—
67C3×C4×Q8—33no—
68C3×C4×Q8—33no—
69C3×C4×Q8C2×D311yescharacteristic
70C3×C4×Q8C22×C311yescharacteristic
71C2×C3×C4⋊C4—33no—
72C2×C3×C42—33no—
73Q8×Dic3C3×C411yescharacteristic
74C2×D3×C8C22×C313yes—
75C2×D3×C8C22×C311yescharacteristic
76D3×C4⋊C4C3×C413yes—
77#887C3×C413yes—
78C42×D3C22×C313yes—
79C4×Dic6C22×C313yes—
80C3×C4⋊C8—23no—
81C3×C4⋊C8—36no—
82C3×C4⋊C8—33no—
83C3×C4⋊C8C2×D313yes—
84C3×C4⋊C8C22×C313yes—
85C3×C4×C8—23no—
86C3×C4×C8—33no—
87C3×C4×C8—33no—
88C3×C4×C8—33no—
89C3×C4×C8C2×D313yes—
90C3×C4×C8C22×C313yes—
91Dic3⋊C8C22×C316yes—
92C8×Dic3C22×C313yes—
93C8×Dic3C22×C313yes—
94C4⋊(C3⋊C8)C22×C313yes—
95C4×C3⋊C8C22×C313yes—
96C22×C3×D3C22×C411yescharacteristic
97C32×Q8C2×C814yescomplemented
98C2×C32×C4C22×C413yes—
99C2×C32×C4C22×C413yes—
100C2×C32×C4C2411yescharacteristic
101C2×C3×Dic3C22×C413yes—
102C2×C3×Dic3C22×C411yescharacteristic
103C2×C3×Dic3C22×C413yes—
104C3×C4×D3—26no—
105C3×C4×D3C2×C8112yes—
106C3×C4×D3C2×Q812yes—
107C3×C4×D3#4112yes—
108C3×Dic6C2×C8112yescomplemented
109C32×C8—23no—
110C32×C8C2×Q812yescomplemented
111C3×C3⋊C8—23no—
112C3×C3⋊C8C2×Q812yescomplemented
113C2×C4×Q8—31no—
114C8×Q8—31no—
115C8×Q8—33no—
116C8×Q8—33no—
117C8×Q8C3×D311yescharacteristic, direct product factor
118C2×C4⋊C8—33no—
119C2×C4×C8—33no—
120C2×C3×Q8—21no—
121C2×C3×Q8—38no—
122C2×C3×Q8—33no—
123C2×C3×Q8C4×D311yescharacteristic
124C2×C3×Q8C2×C3×C411yescharacteristic
125C22×C3×C4—33no—
126C22×C3×C4—31no—
127C22×C3×C4—33no—
128D3×Q8C3×C818yesdirect product factor
129C2×C4×D3C2×C3×C413yes—
130C2×C4×D3C2×C3×C413yes—
131C2×C4×D3C23×C311yescharacteristic
132C2×Dic6C2×C3×C413yes—
133C2×C3×C8—21no—
134C2×C3×C8—23no—
135C2×C3×C8—31no—
136C2×C3×C8—34no—
137C2×C3×C8—33no—
138C2×C3×C8—66no—
139C2×C3×C8C22×D313yes—
140C2×C3×C8C22×D311yescharacteristic
141C2×C3×C8C23×C313yes—
142C2×C3×C8C23×C311yescharacteristic
143C3×C4⋊C4—23no—
144C3×C4⋊C4—36no—
145C3×C4⋊C4—33no—
146C3×C4⋊C4C4×D313yes—
147C3×C4⋊C4C2×C3×C413yes—
148C3×C42—23no—
149C3×C42—33no—
150C3×C42—33no—
151C3×C42—33no—
152C3×C42C22×D313yes—
153C3×C42C23×C313yes—
154C4⋊Dic3C2×C3×C413yes—
155Dic3⋊C4C2×C3×C416yes—
156C4×Dic3C2×C3×C413yes—
157C4×Dic3C2×C3×C413yes—
158C4×Dic3C23×C313yes—
159C2×C3⋊C8C23×C313yes—
160C2×C3⋊C8C23×C311yescharacteristic
161D3×C8—26no—
162D3×C8C3×Q814yesdirect product factor
163C22×C32C23×C411yescharacteristic
164C2×C3×D3C4×Q812yes—
165C2×C3×D3C22×C812yes—
166C2×C3×D3#13112yes—
167C32×C4—23no—
168C32×C4C22×C816yes—
169C32×C4C22×Q811yescharacteristic
170C32×C4C2×#4111yescharacteristic
171C3×Dic3—23no—
172C3×Dic3C4×Q812yes—
173C3×Dic3C22×C816yes—
174C22×Q8—31no—
175C22×C8—31no—
176C22×C8—33no—
177C4×Q8—31no—
178C4×Q8—33no—
179C4×Q8—33no—
180C4×Q8C2×C3×D311yescharacteristic
181C2×C4⋊C4—33no—
182C2×C42—33no—
183C4⋊C8—36no—
184C4⋊C8—33no—
185C4⋊C8C2×C3×D313yes—
186C4×C8—33no—
187C4×C8—33no—
188C4×C8—33no—
189C4×C8C2×C3×D313yes—
190C23×C3—31no—
191C22×D3C22×C3×C411yescharacteristic
192C3×Q8—24no—
193C3×Q8—312no—
194C3×Q8D3×C814yesdirect product factor
195C3×Q8C2×C3×C814yescomplemented
196C2×C3×C4—21no—
197C2×C3×C4—23no—
198C2×C3×C4—23no—
199C2×C3×C4—31no—
200C2×C3×C4—33no—
201C2×C3×C4—312no—
202C2×C3×C4—32no—
203C2×C3×C4—32no—
204C2×C3×C4—33no—
205C2×C3×C4—66no—
206C2×C3×C4C2×C4×D313yes—
207C2×C3×C4C2×C4×D313yes—
208C2×C3×C4C22×C3×C413yes—
209C2×C3×C4C22×C3×C413yes—
210C2×C3×C4C23×D311yescharacteristic
211C2×C3×C4C24×C311yescharacteristic
212C2×Dic3C22×C3×C413yes—
213C2×Dic3C22×C3×C411yescharacteristic
214C2×Dic3C22×C3×C413yes—
215C4×D3—26no—
216C4×D3C2×C3×C8112yes—
217C4×D3C2×C3×Q812yes—
218C4×D3C3×#4112yes—
219Dic6C2×C3×C8112yescomplemented
220C3×C8—23no—
221C3×C8—23no—
222C3×C8—22no—
223C3×C8—32no—
224C3×C8—43no—
225C3×C8—63no—
226C3×C8D3×Q812yesdirect product factor
227C3×C8C2×C3×Q812yescomplemented
228C3⋊C8—23no—
229C3⋊C8C2×C3×Q812yescomplemented
230C2×C32C2×C4×Q811yescharacteristic
231C2×C32C23×C811yescharacteristic
232C2×C32C2×#13111yescharacteristic
233C3×D3C8×Q814yesdirect product factor
234C2×Q8—38no—
235C2×Q8—33no—
236C2×Q8C3×C4×D311yescharacteristic
237C22×C4—31no—
238C22×C4—33no—
239C22×C4—33no—
240C2×C8—31no—
241C2×C8—34no—
242C2×C8—33no—
243C2×C8—66no—
244C2×C8C22×C3×D313yes—
245C2×C8C22×C3×D311yescharacteristic
246C4⋊C4—36no—
247C4⋊C4—33no—
248C4⋊C4C3×C4×D313yes—
249C42—33no—
250C42—33no—
251C42—33no—
252C42C22×C3×D313yes—
253C22×C3—21no—
254C22×C3—32no—
255C22×C3—32no—
256C22×C3—32no—
257C22×C3C22×C4×D311yescharacteristic
258C22×C3C23×C3×C411yescharacteristic
259C2×D3C3×C4×Q812yes—
260C2×D3C22×C3×C812yes—
261C2×D3C3×#13112yes—
262C3×C4—23no—
263C3×C4—23no—
264C3×C4—26no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—36no—
268C3×C4—32no—
269C3×C4—43no—
270C3×C4—63no—
271C3×C4C2×D3×C816yes—
272C3×C4C22×C3×C816yes—
273C3×C4C2×D3×Q811yescharacteristic
274C3×C4D3×#4111yescharacteristic
275C3×C4C22×C3×Q811yescharacteristic
276C3×C4C2×C3×#4111yescharacteristic
277Dic3—23no—
278Dic3C3×C4×Q812yes—
279Dic3C22×C3×C816yes—
280C32C2×C8×Q811yescharacteristic, complemented, Sylow
281C23—31no—
282Q8—312no—
283Q8C3×D3×C814yesdirect product factor
284C2×C4—31no—
285C2×C4—33no—
286C2×C4—312no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—33no—
290C2×C4—66no—
291C2×C4C2×C3×C4×D313yes—
292C2×C4C2×C3×C4×D313yes—
293C2×C4C23×C3×D311yescharacteristic
294C8—23no—
295C8—32no—
296C8—63no—
297C8C3×D3×Q812yesdirect product factor
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—34no—
302C2×C3C4×D3×Q811yescharacteristic
303C2×C3C22×D3×C811yescharacteristic
304C2×C3D3×#13111yescharacteristic
305C2×C3C2×C3×C4×Q811yescharacteristic
306C2×C3C23×C3×C811yescharacteristic
307C2×C3C2×C3×#13111yescharacteristic
308D3C3×C8×Q814yesdirect product factor
309C22—32no—
310C22—32no—
311C22—32no—
312C22C22×C3×C4×D311yescharacteristic
313C4—23no—
314C4—32no—
315C4—36no—
316C4—63no—
317C4C2×C3×D3×C816yes—
318C4C2×C3×D3×Q811yescharacteristic
319C4C3×D3×#4111yescharacteristic
320C3—21no—
321C3D3×C8×Q811yescharacteristic, direct product factor
322C3C2×C3×C8×Q811yescharacteristic, complemented
323C2—34no—
324C2C3×C4×D3×Q811yescharacteristic
325C2C22×C3×D3×C811yescharacteristic
326C2C3×D3×#13111yescharacteristic
subgroupG/H
centerC2×C3×C8C22×D3
commutatorC2×C3C23×C3×C8
FrattiniC2×C4C23×C3×D3
FittingC32×C8×Q8C2
socleC22×C32C23×C4
prime radicalC22×C3×D3C22×C4
cosocleC2×C3×C4C24×C3
hypercenterC3×C8×Q8D3
hypocenterC3C2×C3×C8×Q8
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C8 ◁ C3×C8×Q8

Complements

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

NH ≅ G/Ntype
2. C32×C8×Q8C2semidirect
3. C3×Q8×C3⋊C8C2semidirect
5. C3×#4830C2semidirect
7. C3×C8×Dic6C2semidirect
9. D3×C8×Q8C3direct (N × H)
32. C3×C8×Q8D3direct (N × H)
33. C3×C8×Q8C2×C3semidirect
36. Q8×C3⋊C8C2×C3semidirect
38. #4830C2×C3semidirect
40. C8×Dic6C2×C3semidirect
42. C3×D3×Q8C8direct (N × H)
59. C3×D3×C8Q8direct (N × H)
97. C32×Q8C2×C8semidirect
108. C3×Dic6C2×C8semidirect
110. C32×C8C2×Q8semidirect
112. C3×C3⋊C8C2×Q8semidirect
117. C8×Q8C3×D3direct (N × H)
128. D3×Q8C3×C8direct (N × H)
162. D3×C8C3×Q8direct (N × H)
194. C3×Q8D3×C8direct (N × H)
195. C3×Q8C2×C3×C8semidirect
219. Dic6C2×C3×C8semidirect
226. C3×C8D3×Q8direct (N × H)
227. C3×C8C2×C3×Q8semidirect
229. C3⋊C8C2×C3×Q8semidirect
233. C3×D3C8×Q8direct (N × H)
280. C32C2×C8×Q8semidirect
283. Q8C3×D3×C8direct (N × H)
297. C8C3×D3×Q8direct (N × H)
308. D3C3×C8×Q8direct (N × H)
321. C3D3×C8×Q8direct (N × H)
322. C3C2×C3×C8×Q8semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D3×Q82
C32×C8×Q82
C3×Q8×C3⋊C82
C3×D3×C4⋊C82
C3×#48302
C3×C4×D3×C82
C3×C8×Dic62
C2×C3×C8×Q83
D3×C8×Q83