Finite Group Databasebeta

← C3×C22⋊C4×C3⋊C8 #11783637

Subgroups — C3×C22⋊C4×C3⋊C8 #11783637

Order 1152.

897 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊C8C211yesmaximal, characteristic
2C32×C4×C22⋊C4C211yesmaximal, characteristic
3C2×C3×#830C211yesmaximal, characteristic
4C2×C3×C4×C3⋊C8C212yesmaximal, complemented
5C3×#4524C212yesmaximal, complemented
6C3×C8×C22⋊C4—31nomaximal
7C22⋊C4×C3⋊C8C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2211yescharacteristic
9C2×C32×C22⋊C4C412yes—
10C2×C32×C42C2212yes—
11C22×C3×C3⋊C8—24no—
12C22×C3×C3⋊C8C414yescomplemented
13C22×C3×C3⋊C8C2212yes—
14C22×C3×C3⋊C8C2211yescharacteristic
15C22×C3×C3⋊C8C2211yescharacteristic
16C32×#95C412yes—
17C3×#830C414yescomplemented
18C3×C4×C3⋊C8—24no—
19C23×C3×C8—31no—
20C23×C3⋊C8C2×C311yescharacteristic
21C2×C3×C22⋊C8—31no—
22C2×C3×C4×C8—32no—
23C3×C4×C22⋊C4—21no—
24C3×C4×C22⋊C4D311yescharacteristic
25C3×C4×C22⋊C4C2×C311yescharacteristic
26C2×#830C2×C311yescharacteristic
27C2×C4×C3⋊C8C2×C312yescomplemented
28C3×#336—32no—
29#4524C2×C312yescomplemented
30C24×C32C2×C411yescharacteristic
31C22×C32×C4—22no—
32C22×C32×C4—22no—
33C22×C32×C4C2×C411yescharacteristic
34C22×C32×C4C2×C412yes—
35C22×C32×C4C2×C412yes—
36C22×C32×C4C2×C414yes—
37C22×C32×C4C2311yescharacteristic
38C32×C22⋊C4C818yescomplemented
39C32×C42—22no—
40C32×C42—22no—
41C2×C3×C3⋊C8—22no—
42C2×C3×C3⋊C8—22no—
43C2×C3×C3⋊C8—22no—
44C2×C3×C3⋊C8—28no—
45C2×C3×C3⋊C8C2×C412yes—
46C2×C3×C3⋊C8C2×C414yes—
47C2×C3×C3⋊C8C2×C412yes—
48C2×C3×C3⋊C8D414yes—
49C8×C22⋊C4—31noSylow
50C23×C3×C4—21no—
51C23×C3×C4C2×D311yescharacteristic
52C23×C3×C4C22×C311yescharacteristic
53C22×C3×C8—31no—
54C22×C3×C8—34no—
55C22×C3×C8—31no—
56C22×C3×C8—32no—
57C22×C3×C8—64no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4Dic312yes—
60C2×C3×C22⋊C4C3×C412yes—
61C2×C3×C42—22no—
62C2×C3×C42C2×D312yes—
63C2×C3×C42C22×C312yes—
64C22×C3⋊C8—24no—
65C22×C3⋊C8C3×C414yescomplemented
66C22×C3⋊C8C22×C312yes—
67C22×C3⋊C8C22×C311yescharacteristic
68C22×C3⋊C8C22×C311yescharacteristic
69C3×C22⋊C8—34no—
70C3×C4×C8—64no—
71C3×#95—22no—
72C3×#95Dic312yes—
73C3×#95C3×C412yes—
74#830C3×C414yescomplemented
75C4×C3⋊C8—24no—
76C23×C32—22no—
77C23×C32—22no—
78C23×C32C4212yes—
79C23×C32C2×C814yes—
80C23×C32C22×C411yescharacteristic
81C2×C32×C4—24no—
82C2×C32×C4—22no—
83C2×C32×C4—24no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—24no—
87C2×C32×C4C4212yes—
88C2×C32×C4C2×C818yescomplemented
89C2×C32×C4C22×C411yescharacteristic
90C2×C32×C4C22×C411yescharacteristic
91C2×C32×C4C2×D412yes—
92C2×C32×C4#4112yes—
93C3×C3⋊C8—22no—
94C3×C3⋊C8—24no—
95C3×C3⋊C8C22⋊C414yesdirect product factor
96C23×C8—31no—
97C2×C22⋊C8—31no—
98C2×C4×C8—32no—
99C4×C22⋊C4C3×D311yescharacteristic
100#336—32no—
101C24×C3—21no—
102C24×C3C2×Dic311yescharacteristic
103C24×C3C2×C3×C411yescharacteristic
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—24no—
112C22×C3×C4—21no—
113C22×C3×C4—42no—
114C22×C3×C4—42no—
115C22×C3×C4C4×D312yes—
116C22×C3×C4C4×D312yes—
117C22×C3×C4C2×Dic314yes—
118C22×C3×C4C2×Dic311yescharacteristic
119C22×C3×C4C2×C3×C412yes—
120C22×C3×C4C2×C3×C412yes—
121C22×C3×C4C2×C3×C411yescharacteristic
122C22×C3×C4C2×C3×C414yes—
123C22×C3×C4C22×D311yescharacteristic
124C22×C3×C4C23×C311yescharacteristic
125C2×C3×C8—32no—
126C2×C3×C8—34no—
127C2×C3×C8—34no—
128C2×C3×C8—32no—
129C2×C3×C8—68no—
130C2×C3×C8—62no—
131C2×C3×C8—62no—
132C2×C3×C8—62no—
133C3×C22⋊C4—28no—
134C3×C22⋊C4C3⋊C818yesdirect product factor
135C3×C22⋊C4C3×C818yescomplemented
136C3×C42—22no—
137C3×C42—22no—
138C3×C42—22no—
139C3×C42—22no—
140C3×C42—42no—
141C3×C42—42no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—28no—
146C2×C3⋊C8C2×C3×C414yes—
147C2×C3⋊C8C2×C3×C412yes—
148C2×C3⋊C8C2×C3×C412yes—
149C2×C3⋊C8C3×D414yes—
150C22×C32—22no—
151C22×C32—22no—
152C22×C32—28no—
153C22×C32C4×C814yescomplemented
154C22×C32C2×C4211yescharacteristic
155C22×C32C4×D412yes—
156C22×C32C22×C812yes—
157C22×C32#13112yes—
158C32×C4—22no—
159C32×C4—28no—
160C32×C4C2×C22⋊C411yescharacteristic
161C32×C4#11711yescharacteristic
162C32×C4C4×D412yes—
163C23×C4C2×C3×D311yescharacteristic
164C22×C8—31no—
165C22×C8—34no—
166C22×C8—31no—
167C22×C8—32no—
168C22×C8—64no—
169C2×C22⋊C4C3×Dic312yes—
170C2×C42C2×C3×D312yes—
171C22⋊C8—34no—
172C4×C8—64no—
173#95C3×Dic312yes—
174C23×C3—22no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—24no—
179C23×C3—22no—
180C23×C3—21no—
181C23×C3—42no—
182C23×C3—42no—
183C23×C3C2×C3⋊C814yes—
184C23×C3C4×Dic312yes—
185C23×C3C3×C4212yes—
186C23×C3C2×C3×C814yes—
187C23×C3C22×Dic311yescharacteristic
188C23×C3C22×C3×C411yescharacteristic
189C2×C3×C4—24no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—22no—
194C2×C3×C4—24no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—24no—
201C2×C3×C4—28no—
202C2×C3×C4—22no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4—44no—
212C2×C3×C4—42no—
213C2×C3×C4C2×C3⋊C818yescomplemented
214C2×C3×C4C4×Dic312yes—
215C2×C3×C4C3×C4212yes—
216C2×C3×C4C2×C3×C818yescomplemented
217C2×C3×C4C2×C4×D311yescharacteristic
218C2×C3×C4C2×C4×D311yescharacteristic
219C2×C3×C4D3×D412yes—
220C2×C3×C4#23712yes—
221C2×C3×C4C22×C3×C411yescharacteristic
222C2×C3×C4C22×C3×C411yescharacteristic
223C2×C3×C4C2×C3×D412yes—
224C2×C3×C4C3×#4112yes—
225C3×C8—34no—
226C3×C8—62no—
227C3×C8—64no—
228C3⋊C8—22no—
229C3⋊C8—24no—
230C3⋊C8C3×C22⋊C414yesdirect product factor
231C2×C32—24no—
232C2×C32C4×C22⋊C411yescharacteristic
233C2×C32C2×C4×C811yescharacteristic
234C2×C32#40511yescharacteristic
235C2×C32C8×D414yes—
236C24C2×C3×Dic311yescharacteristic
237C22×C4—22no—
238C22×C4—22no—
239C22×C4C3×C4×D312yes—
240C22×C4C3×C4×D312yes—
241C22×C4C2×C3×Dic314yes—
242C22×C4C2×C3×Dic311yescharacteristic
243C22×C4C22×C3×D311yescharacteristic
244C2×C8—32no—
245C2×C8—34no—
246C2×C8—34no—
247C2×C8—32no—
248C2×C8—68no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—62no—
252C22⋊C4C3×C3⋊C818yesdirect product factor
253C42—22no—
254C42—22no—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—28no—
259C22×C3—22no—
260C22×C3—28no—
261C22×C3—22no—
262C22×C3—21no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—24no—
266C22×C3—42no—
267C22×C3—42no—
268C22×C3—48no—
269C22×C3C4×C3⋊C814yescomplemented
270C22×C3C3×C4×C814yescomplemented
271C22×C3C22×C3⋊C812yes—
272C22×C3C2×C4×Dic311yescharacteristic
273C22×C3D4×Dic312yes—
274C22×C3#94812yes—
275C22×C3C2×C3×C4211yescharacteristic
276C22×C3C3×C4×D412yes—
277C22×C3C22×C3×C812yes—
278C22×C3C3×#13112yes—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—28no—
282C3×C4—28no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—42no—
287C3×C4—48no—
288C3×C4#87711yescharacteristic
289C3×C4D3×C22⋊C411yescharacteristic
290C3×C4#67⋊C412yes—
291C3×C4C2×C3×C22⋊C411yescharacteristic
292C3×C4C3×#11711yescharacteristic
293C3×C4C3×C4×D412yes—
294C32C8×C22⋊C411yescharacteristic, complemented, Sylow
295C23—22no—
296C23—22no—
297C23C2×C3×C3⋊C814yes—
298C23C3×C4×Dic312yes—
299C23C22×C3×Dic311yescharacteristic
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4C2×C3×C3⋊C818yescomplemented
307C2×C4C3×C4×Dic312yes—
308C2×C4C2×C3×C4×D311yescharacteristic
309C2×C4C2×C3×C4×D311yescharacteristic
310C2×C4C3×D3×D412yes—
311C2×C4C3×#23712yes—
312C8—34no—
313C8—62no—
314C8—64no—
315C2×C3—24no—
316C2×C3—24no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—24no—
321C2×C3—44no—
322C2×C3C2×C4×C3⋊C811yescharacteristic
323C2×C3Dic3×C22⋊C411yescharacteristic
324C2×C3#499711yescharacteristic
325C2×C3D4×C3⋊C814yes—
326C2×C3C3×C4×C22⋊C411yescharacteristic
327C2×C3C2×C3×C4×C811yescharacteristic
328C2×C3C3×#40511yescharacteristic
329C2×C3C3×C8×D414yes—
330C22—22no—
331C22—22no—
332C22—28no—
333C22C3×C4×C3⋊C814yescomplemented
334C22C22×C3×C3⋊C812yes—
335C22C2×C3×C4×Dic311yescharacteristic
336C22C3×D4×Dic312yes—
337C22C3×#94812yes—
338C4—22no—
339C4—28no—
340C4C3×#87711yescharacteristic
341C4C3×D3×C22⋊C411yescharacteristic
342C4C3×#67⋊C412yes—
343C3—21no—
344C3C22⋊C4×C3⋊C811yescharacteristic, direct product factor
345C3C3×C8×C22⋊C411yescharacteristic, complemented
346C2—24no—
347C2C2×C3×C4×C3⋊C811yescharacteristic
348C2C3×Dic3×C22⋊C411yescharacteristic
349C2C3×#499711yescharacteristic
350C2C3×D4×C3⋊C814yes—
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC2×C3C2×C3×C4×C8
FrattiniC22×C4C22×C3×D3
FittingC32×C4×C22⋊C4C2
socleC23×C32C22×C4
prime radicalC24×C32C2×C4
cosocleC22×C3×C4C23×C3
hypercenterC3×C4×C22⋊C4D3
hypocenterC3C3×C8×C22⋊C4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C3×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C3×C4×C3⋊C8C2semidirect
5. C3×#4524C2semidirect
7. C22⋊C4×C3⋊C8C3direct (N × H)
12. C22×C3×C3⋊C8C4semidirect
17. C3×#830C4semidirect
27. C2×C4×C3⋊C8C2×C3semidirect
29. #4524C2×C3semidirect
38. C32×C22⋊C4C8semidirect
65. C22×C3⋊C8C3×C4semidirect
74. #830C3×C4semidirect
88. C2×C32×C4C2×C8semidirect
95. C3×C3⋊C8C22⋊C4direct (N × H)
134. C3×C22⋊C4C3⋊C8direct (N × H)
135. C3×C22⋊C4C3×C8semidirect
153. C22×C32C4×C8semidirect
213. C2×C3×C4C2×C3⋊C8semidirect
216. C2×C3×C4C2×C3×C8semidirect
230. C3⋊C8C3×C22⋊C4direct (N × H)
252. C22⋊C4C3×C3⋊C8direct (N × H)
269. C22×C3C4×C3⋊C8semidirect
270. C22×C3C3×C4×C8semidirect
294. C32C8×C22⋊C4semidirect
306. C2×C4C2×C3×C3⋊C8semidirect
333. C22C3×C4×C3⋊C8semidirect
344. C3C22⋊C4×C3⋊C8direct (N × H)
345. C3C3×C8×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊C82
C32×C4×C22⋊C42
C2×C3×#8302
C2×C3×C4×C3⋊C82
C3×#45242
C3×C8×C22⋊C43
C22⋊C4×C3⋊C83