Finite Group Databasebeta

← #13072975

Subgroups — #13072975

Order 1408.

632 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66983C211yesmaximal, characteristic
2C2×#66941C211yesmaximal, characteristic, complemented
3C2×#66938C212yesmaximal, complemented
4C11×#380C211yesmaximal, characteristic, complemented
5C4×C22⋊Dic11C211yesmaximal, characteristic, complemented
6#10622601C211yesmaximal, characteristic, complemented
7#10622601C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic11C211yesmaximal, characteristic, complemented
9#10622572C212yesmaximal, complemented
10#10622286C212yesmaximal, complemented
11#10622282C212yesmaximal, complemented
12C23×C4×C11C2211yescharacteristic
13C22×C22⋊D11C2212yes—
14C22×D44C2211yescharacteristic
15C22×Dic22C2212yes—
16C2×C11×C4⋊C4C2211yescharacteristic
17C2×C11×C4⋊C4C2211yescharacteristic
18C2×C11×C22⋊C4C2212yes—
19C2×C22⋊Dic11C2211yescharacteristic
20C2×C22⋊Dic11C2211yescharacteristic
21#66983—24no—
22C2×(C2×D11)⋊C4C2212yes—
23C2×(C2×D11)⋊C4C2212yes—
24C2×(C2×D11)⋊C4C2212yes—
25C2×(C2×D11)⋊C4C2212yes—
26C2×(C2×D11)⋊C4C2212yes—
27C2×C4⋊Dic11C2211yescharacteristic
28C2×Dic11⋊C4C2212yes—
29C2×C4×Dic11C2212yes—
30C2×C4×Dic11C2211yescharacteristic
31C2×C4×Dic11C2211yescharacteristic
32#66941—24no—
33#66938—28no—
34C11×#95C2212yes—
35C11⋊#95C2211yescharacteristic
36C11⋊#95C2211yescharacteristic
37C11⋊#95C2212yes—
38C24×C11C2311yescharacteristic
39C23×D11C2312yes—
40C22×C4×C11—22no—
41C22×C4×C11—22no—
42C22×C4×C11—22no—
43C22×C4×C11C2311yescharacteristic
44C22×C4×C11C2312yes—
45C22×C4×C11C2312yes—
46C22×C4×C11C2311yescharacteristic
47C2×C22⋊D11—24no—
48C2×C22⋊D11—24no—
49C2×C22⋊D11—42no—
50C22×Dic11C2311yescharacteristic
51C22×Dic11C2311yescharacteristic
52C22×Dic11C2312yes—
53C22×Dic11C2311yescharacteristic
54C22×Dic11C2311yescharacteristic
55C2×D44—22no—
56C2×D44—22no—
57C2×D44—22no—
58C2×Dic22—24no—
59C2×Dic22—24no—
60C2×Dic22—42no—
61C11×C4⋊C4—22no—
62C11×C4⋊C4—22no—
63C11×C22⋊C4—24no—
64C22⋊Dic11—22no—
65C22⋊Dic11D414yescomplemented
66(C2×D11)⋊C4—24no—
67(C2×D11)⋊C4—24no—
68(C2×D11)⋊C4—24no—
69(C2×D11)⋊C4—24no—
70(C2×D11)⋊C4—42no—
71C4⋊Dic11D414yes—
72Dic11⋊C4—24no—
73C4×Dic11—24no—
74C4×Dic11—22no—
75C4×Dic11—22no—
76#2491—111nomaximal, Sylow
77C23×C11—21no—
78C23×C11—21no—
79C23×C11—21no—
80C23×C11—21no—
81C23×C11—21no—
82C23×C11—21no—
83C23×C11C2×D412yes—
84C23×C11C2411yescharacteristic
85C22×D11—22no—
86C22×D11—22no—
87C22×D11—22no—
88C22×D11—22no—
89C22×D11—22no—
90C22×D11—22no—
91C22×D11—22no—
92C2×C4×C11—22no—
93C2×C4×C11—22no—
94C2×C4×C11—21no—
95C2×C4×C11—22no—
96C2×C4×C11—22no—
97C2×C4×C11—22no—
98C2×C4×C11—21no—
99C2×C4×C11—21no—
100C2×C4×C11—22no—
101C2×C4×C11—44no—
102C2×C4×C11C2×D412yes—
103C2×C4×C11#4112yes—
104C2×C4×C11#4112yes—
105C22⋊D11—48no—
106C2×Dic11—21no—
107C2×Dic11—21no—
108C2×Dic11—22no—
109C2×Dic11—21no—
110C2×Dic11—21no—
111C2×Dic11—22no—
112C2×Dic11—21no—
113C2×Dic11—21no—
114C2×Dic11—21no—
115C2×Dic11—21no—
116C2×Dic11—21no—
117C2×Dic11C2×D412yes—
118C2×Dic11C2×D414yes—
119C2×Dic11C2×D412yes—
120C2×Dic11#4112yes—
121D44—44no—
122Dic22—48no—
123C2×C4⋊Q8—111no—
124C2×#124—112no—
125C2×#121—111no—
126#394—112no—
127#393—112no—
128#390—111no—
129#390—111no—
130#388—112no—
131#380D1111yescharacteristic, complemented
132C4×C4⋊C4—111no—
133C4×C22⋊C4—111no—
134C22×C11—22no—
135C22×C11—22no—
136C22×C11—22no—
137C22×C11—41no—
138C22×C11—41no—
139C22×C11—41no—
140C22×C11—41no—
141C22×C11C22×D411yescharacteristic
142C22×C11C22×D411yescharacteristic
143C22×C11C2×#4111yescharacteristic
144C22×C11C2×#4111yescharacteristic
145C22×C11C2×#4111yescharacteristic
146C22×C11H(5,𝔽2)11yescharacteristic
147C22×C112−1+411yescharacteristic
148C2×D11—42no—
149C2×D11—42no—
150C2×D11—42no—
151C2×D11—42no—
152C2×D11—42no—
153C2×D11—42no—
154C2×D11—42no—
155C4×C11—41no—
156C4×C11—42no—
157C4×C11—42no—
158C4×C11#12414yescomplemented
159Dic11—22no—
160Dic11—24no—
161Dic11—22no—
162Dic11—22no—
163Dic11—41no—
164C22×Q8—112no—
165C22×D4—112no—
166C22×D4—111no—
167C23×C4C2×D1111yescharacteristic
168C4⋊Q8—224no—
169#124—228no—
170#121—224no—
171C2×C4⋊C4—111no—
172C2×C4⋊C4—112no—
173C2×C4⋊C4C2×D1111yescharacteristic
174C2×C4⋊C4C2×D1111yescharacteristic
175C2×C22⋊C4—111no—
176C2×C22⋊C4—111no—
177C2×C22⋊C4—112no—
178C2×C22⋊C4—112no—
179C2×C22⋊C4—112no—
180C2×C22⋊C4—112no—
181C2×C22⋊C4—112no—
182C2×C22⋊C4C2×D1112yes—
183C2×C42—112no—
184C2×C42—111no—
185C2×C42—111no—
186#95—111no—
187#95—111no—
188#95—112no—
189#95C2×D1112yes—
190C2×C11—42no—
191C2×C11C2×#12411yescharacteristic
192C2×C11#53211yescharacteristic
193C2×C11#54211yescharacteristic
194C2×C11D4211yescharacteristic
195C2×C11#54711yescharacteristic
196C2×C11#54811yescharacteristic
197C2×C11#55511yescharacteristic
198D11—82no—
199C24—112no—
200C24C22×D1111yescharacteristic
201C2×Q8—224no—
202C2×Q8—224no—
203C2×Q8—442no—
204C2×D4—224no—
205C2×D4—224no—
206C2×D4—222no—
207C2×D4—222no—
208C2×D4—222no—
209C2×D4—442no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—111no—
214C22×C4—111no—
215C22×C4—111no—
216C22×C4—112no—
217C22×C4—111no—
218C22×C4C22×D1111yescharacteristic
219C22×C4C22×D1112yes—
220C22×C4C22×D1112yes—
221C22×C4C22×D1111yescharacteristic
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—114no—
225C4⋊C4—224no—
226C22⋊C4—24no—
227C22⋊C4—114no—
228C22⋊C4—224no—
229C22⋊C4—224no—
230C22⋊C4—224no—
231C22⋊C4—224no—
232C22⋊C4—222no—
233C22⋊C4—442no—
234C42—224no—
235C42—222no—
236C42—222no—
237C11#249111yescharacteristic, complemented, Sylow
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—222no—
245C23—222no—
246C23—222no—
247C23—222no—
248C23—222no—
249C23—222no—
250C23—222no—
251C23D4×D1112yes—
252C23C23×D1111yescharacteristic
253Q8—448no—
254D4—448no—
255D4—444no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—44no—
266C2×C4—112no—
267C2×C4—112no—
268C2×C4—114no—
269C2×C4—112no—
270C2×C4—221no—
271C2×C4—221no—
272C2×C4—222no—
273C2×C4—221no—
274C2×C4—221no—
275C2×C4—221no—
276C2×C4—222no—
277C2×C4—221no—
278C2×C4—221no—
279C2×C4—221no—
280C2×C4—221no—
281C2×C4#433912yes—
282C2×C4D4×D1112yes—
283C2×C4#434112yes—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—442no—
292C22—442no—
293C22—442no—
294C22—442no—
295C22—442no—
296C22—442no—
297C22—442no—
298C22C2×#433911yescharacteristic
299C22C2×D4×D1111yescharacteristic
300C22C2×D4×D1111yescharacteristic
301C22C2×#434111yescharacteristic
302C22#6704011yescharacteristic
303C22#41×D1111yescharacteristic
304C22#6704211yescharacteristic
305C4—41no—
306C4—42no—
307C4—42no—
308C4—222no—
309C4—224no—
310C4—222no—
311C4—222no—
312C4—441no—
313C4#6693814yescomplemented
314C2—42no—
315C2—882no—
316C2C2×#6693811yescharacteristic
317C2#1062303811yescharacteristic
318C2#1062306311yescharacteristic
319C2#1062312711yescharacteristic
320C2#1062314011yescharacteristic
321C2#1062316411yescharacteristic
322C2#1062317411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#380C2
prime radicalC2×#66983C2
hypercenter#380D11
hypocenterC11#2491
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#66941C2semidirect
3. C2×#66938C2semidirect
4. C11×#380C2semidirect
5. C4×C22⋊Dic11C2semidirect
6. #10622601C2semidirect
7. #10622601C2semidirect
8. C4⋊C4×Dic11C2semidirect
9. #10622572C2semidirect
10. #10622286C2semidirect
11. #10622282C2semidirect
65. C22⋊Dic11D4semidirect
131. #380D11semidirect
158. C4×C11#124semidirect
237. C11#2491semidirect
313. C4#66938semidirect

Maximal subgroups

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

subgroupindex
C2×#669832
C2×#669412
C2×#669382
C11×#3802
C4×C22⋊Dic112
#106226012
C4⋊C4×Dic112
#106225722
#106222862
#106222822
#249111