Finite Group Databasebeta

← #14926814

Subgroups — #14926814

Order 2816.

2650 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#66977C212yesmaximal
2#13076968C214yesmaximal, complemented
3#13074730C214yesmaximal, complemented
4#13073991C218yesmaximal, complemented
5#13073094C214yesmaximal, complemented
6#13072260C214yesmaximal, complemented
7#13071230C214yesmaximal
8C2×C4×C11×C22⋊C4C211yesmaximal, characteristic
9C24×C4×C11C2211yescharacteristic
10C23×Dic22C2212yes—
11C22×C11×C22⋊C4C2212yes—
12C22×C42×C11C2212yes—
13C22×C22⋊Dic11C2214yes—
14C22×C4⋊Dic11C2212yes—
15C2×#66977—216no—
16C2×#66977C2214yes—
17C2×#66977C2214yes—
18C2×#66977C2214yes—
19C2×#66977C2214yes—
20C22×Dic11⋊C4C2214yes—
21C2×#66922C2214yescomplemented
22C4×C11×C22⋊C4C2214yes—
23C4×C11×C22⋊C4C2214yes—
24C2×C11×#95C2212yes—
25#10622825C22116yes—
26#10622563C22116yes—
27#10622563C22116yes—
28#10622548C2214yescomplemented
29#10622542C2218yescomplemented
30#10622541C22116yescomplemented
31#10622274C22116yescomplemented
32#10622269C22116yescomplemented
33C25×C11C2311yescharacteristic
34C23×C4×C11—44no—
35C23×C4×C11D418yes—
36C23×C4×C11C2311yescharacteristic
37C23×C4×C11C2312yes—
38C23×C4×C11C2312yes—
39C23×C4×C11C2311yescharacteristic
40C23×C4×C11C2314yes—
41C23×Dic11C2314yes—
42C23×Dic11C2314yes—
43C22×Dic22—28no—
44C22×Dic22—48no—
45C22×Dic22C2314yes—
46C22×Dic22C2314yes—
47C2×C11×C22⋊C4Q8116yescomplemented
48C2×C11×C22⋊C4C2318yes—
49C2×C42×C11—44no—
50C2×C42×C11C2314yes—
51C2×C42×C11C2314yes—
52C2×C22⋊Dic11—216no—
53C2×C22⋊Dic11C2318yes—
54C2×C22⋊Dic11C2318yes—
55C2×C4⋊Dic11—28no—
56C2×C4⋊Dic11C2314yes—
57C2×C4⋊Dic11C2314yes—
58C2×C4⋊Dic11C23116yes—
59C2×C4⋊Dic11C23116yes—
60#66977—216no—
61#66977—216no—
62#66977—216no—
63#66977—216no—
64C2×Dic11⋊C4—216no—
65C2×Dic11⋊C4C2318yes—
66C2×Dic11⋊C4C2318yes—
67#66922—48no—
68C11×#95C2318yes—
69C11⋊#95C23132yes—
70#35011—111nomaximal, Sylow
71C24×C11—42no—
72C24×C11—42no—
73C24×C11C2×D414yes—
74C24×C11C2×Q818yes—
75C24×C11C2411yescharacteristic
76C24×C11C2412yes—
77C22×C4×C11—24no—
78C22×C4×C11—28no—
79C22×C4×C11—416no—
80C22×C4×C11—44no—
81C22×C4×C11—44no—
82C22×C4×C11—42no—
83C22×C4×C11—44no—
84C22×C4×C11C2×D414yes—
85C22×C4×C11C2×D418yes—
86C22×C4×C11C2×D418yes—
87C22×C4×C11C2×D414yes—
88C22×C4×C11C2×Q8116yescomplemented
89C22×C4×C11C2412yes—
90C22×C4×C11C2411yescharacteristic
91C22×C4×C11C2418yes—
92C22×C4×C11C2411yescharacteristic
93C22×Dic11—28no—
94C22×Dic11—28no—
95C22×Dic11—48no—
96C22×Dic11—48no—
97C22×Dic11C2418yes—
98C22×Dic11C2418yes—
99C2×Dic22—28no—
100C2×Dic22—28no—
101C2×Dic22—216no—
102C2×Dic22—48no—
103C2×Dic22—432no—
104C11×C22⋊C4C2×Q8132yescomplemented
105C42×C11—28no—
106C42×C11—44no—
107C22⋊Dic11—216no—
108C22⋊Dic11—216no—
109C4⋊Dic11—28no—
110C4⋊Dic11—28no—
111C4⋊Dic11—416no—
112C4⋊Dic11—416no—
113Dic11⋊C4—216no—
114Dic11⋊C4—216no—
115C22×C22⋊Q8—112no—
116#2857—114no—
117#2659—114no—
118#2583—118no—
119#2502—114no—
120#2434—114no—
121#2315—114no—
122C2×C4×C22⋊C4D1111yescharacteristic
123C23×C11—24no—
124C23×C11—22no—
125C23×C11—24no—
126C23×C11—22no—
127C23×C11—44no—
128C23×C11—44no—
129C23×C11—416no—
130C23×C11C4⋊Q8116yescomplemented
131C23×C11C22×D412yes—
132C23×C11C22×D414yes—
133C23×C11C22×Q818yes—
134C23×C11C22×Q814yes—
135C23×C11C2511yescharacteristic
136C2×C4×C11—28no—
137C2×C4×C11—22no—
138C2×C4×C11—24no—
139C2×C4×C11—28no—
140C2×C4×C11—216no—
141C2×C4×C11—28no—
142C2×C4×C11—48no—
143C2×C4×C11—44no—
144C2×C4×C11—416no—
145C2×C4×C11C22×D412yes—
146C2×C4×C11C22×D414yes—
147C2×C4×C11C22×D412yes—
148C2×C4×C11C22×Q8116yes—
149C2×C4×C11H(5,𝔽2)14yes—
150C2×Dic11—28no—
151C2×Dic11—28no—
152C2×Dic11—216no—
153C2×Dic11—216no—
154C2×Dic11—416no—
155C2×Dic11—416no—
156Dic22—416no—
157Dic22—416no—
158C23×Q8—112no—
159C24×C4C2×D1111yescharacteristic
160C2×C4⋊Q8—114no—
161C2×C22⋊Q8—114no—
162C2×C22⋊Q8—114no—
163C2×C22⋊Q8—114no—
164C2×C22⋊Q8—114no—
165C2×C22⋊Q8—2216no—
166C22×C4⋊C4—114no—
167C22×C4⋊C4—112no—
168C22×C22⋊C4—114no—
169C22×C22⋊C4C2×D1112yes—
170C22×C42C2×D1112yes—
171#398—1116no—
172#395—1116no—
173#391—118no—
174#389—1116no—
175#385—1116no—
176#385—1116no—
177#384—114no—
178#380—1116no—
179C4×C22⋊C4C2×D1114yes—
180C4×C22⋊C4C2×D1114yes—
181C2×#95C2×D1112yes—
182C22×C11—22no—
183C22×C11—24no—
184C22×C11—22no—
185C22×C11—24no—
186C22×C11—28no—
187C22×C11—216no—
188C22×C11—416no—
189C22×C11C2×C4⋊Q814yes—
190C22×C11C2×C4⋊Q818yes—
191C22×C11C23×D411yescharacteristic
192C22×C11C23×Q812yes—
193C22×C11C2×H(5,𝔽2)12yes—
194C22×C11#58512yes—
195C4×C11—22no—
196C4×C11—24no—
197C4×C11—216no—
198C4×C11#53414yes—
199Dic11—48no—
200Dic11—48no—
201C25C22×D1111yescharacteristic
202C22×Q8—114no—
203C22×Q8—114no—
204C22×Q8—228no—
205C22×Q8—448no—
206C23×C4—44no—
207C23×C4—114no—
208C23×C4—114no—
209C23×C4D4418yes—
210C23×C4C22×D1112yes—
211C23×C4C22×D1111yescharacteristic
212C23×C4C22×D1114yes—
213C23×C4C22×D1112yes—
214C23×C4C22×D1111yescharacteristic
215C4⋊Q8—448no—
216C22⋊Q8—2216no—
217C22⋊Q8—2216no—
218C22⋊Q8—2216no—
219C22⋊Q8—2216no—
220C2×C4⋊C4—114no—
221C2×C4⋊C4—118no—
222C2×C4⋊C4—118no—
223C2×C4⋊C4—114no—
224C2×C4⋊C4—1116no—
225C2×C4⋊C4—1116no—
226C2×C4⋊C4—2216no—
227C2×C4⋊C4—228no—
228C2×C22⋊C4—118no—
229C2×C22⋊C4—118no—
230C2×C22⋊C4—2216no—
231C2×C22⋊C4Dic22116yescomplemented
232C2×C22⋊C4C22×D1118yes—
233C2×C42—44no—
234C2×C42C22×D1114yes—
235C2×C42C22×D1114yes—
236#95—1132no—
237#95C22×D1118yes—
238C2×C11—24no—
239C2×C11—28no—
240C2×C11C22×C4⋊Q811yescharacteristic
241C2×C11C2×#53411yescharacteristic
242C2×C11#344911yescharacteristic
243C2×C11#350114yes—
244C24—42no—
245C24—42no—
246C24C2×Dic2218yes—
247C24C2×D4414yes—
248C24C23×D1111yescharacteristic
249C24C23×D1112yes—
250C2×Q8—228no—
251C2×Q8—228no—
252C2×Q8—2216no—
253C2×Q8—448no—
254C2×Q8—4432no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—416no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—118no—
263C22×C4—118no—
264C22×C4—228no—
265C22×C4—228no—
266C22×C4—448no—
267C22×C4—448no—
268C22×C4C2×Dic22116yescomplemented
269C22×C4C2×D4418yes—
270C22×C4C2×D4414yes—
271C22×C4C2×D4418yes—
272C22×C4C2×D4414yes—
273C22×C4C23×D1118yes—
274C22×C4C23×D1112yes—
275C22×C4C23×D1111yescharacteristic
276C22×C4C23×D1111yescharacteristic
277C4⋊C4—2216no—
278C4⋊C4—2216no—
279C4⋊C4—228no—
280C4⋊C4—228no—
281C4⋊C4—4416no—
282C4⋊C4—4416no—
283C22⋊C4—2216no—
284C22⋊C4—2216no—
285C22⋊C4C2×Dic22132yescomplemented
286C42—28no—
287C42—44no—
288C11#3501111yescharacteristic, complemented, Sylow
289C23—22no—
290C23—24no—
291C23—22no—
292C23—24no—
293C23—44no—
294C23—416no—
295C23—44no—
296C23#66922116yescomplemented
297C23C22×Dic2218yes—
298C23C22×Dic2214yes—
299C23C22×D4412yes—
300C23C22×D4414yes—
301C23C24×D1111yescharacteristic
302Q8—4416no—
303Q8—4416no—
304C2×C4—28no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—216no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—416no—
313C2×C4—228no—
314C2×C4—228no—
315C2×C4—2216no—
316C2×C4—2216no—
317C2×C4—4416no—
318C2×C4—4416no—
319C2×C4C22×Dic22116yes—
320C2×C4C22×D4412yes—
321C2×C4C22×D4414yes—
322C2×C4C22×D4412yes—
323C2×C4#6703714yes—
324C22—28no—
325C22—216no—
326C22—24no—
327C22—22no—
328C22—24no—
329C22—22no—
330C22—416no—
331C22C2×#6692218yes—
332C22C2×#6692214yes—
333C22C23×Dic2212yes—
334C22C23×D4411yescharacteristic
335C22C2×#6703712yes—
336C22#1062345312yes—
337C4—216no—
338C4—24no—
339C4—22no—
340C4—448no—
341C4—448no—
342C4#1062302414yes—
343C2—24no—
344C2—28no—
345C2C22×#6692211yescharacteristic
346C2C2×#1062302411yescharacteristic
347C2#1308194911yescharacteristic
348C2#1308241914yes—
subgroupG/H
center, FrattiniC23C24×D11
commutator, socle, cosocleC23×C11C25
FittingC2×C4×C11×C22⋊C4C2
prime radicalC25×C11C23
hypercenterC2×C4×C22⋊C4D11
hypocenterC11#35011
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #13076968C2semidirect
3. #13074730C2semidirect
4. #13073991C2semidirect
5. #13073094C2semidirect
6. #13072260C2semidirect
21. C2×#66922C22semidirect
28. #10622548C22semidirect
29. #10622542C22semidirect
30. #10622541C22semidirect
31. #10622274C22semidirect
32. #10622269C22semidirect
47. C2×C11×C22⋊C4Q8semidirect
88. C22×C4×C11C2×Q8semidirect
104. C11×C22⋊C4C2×Q8semidirect
130. C23×C11C4⋊Q8semidirect
231. C2×C22⋊C4Dic22semidirect
268. C22×C4C2×Dic22semidirect
285. C22⋊C4C2×Dic22semidirect
288. C11#35011semidirect
296. C23#66922semidirect

Maximal subgroups

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

subgroupindex
C22×#669772
#130769682
#130747302
#130739912
#130730942
#130722602
#130712302
C2×C4×C11×C22⋊C42
#3501111