Finite Group Databasebeta

← #13072004

Subgroups — #13072004

Order 1408.

786 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66990C211yesmaximal, characteristic
2C2×#66988C211yesmaximal, characteristic, complemented
3C22×(C2×D11)⋊C4C211yesmaximal, characteristic, complemented
4C2×#66930C211yesmaximal, characteristic, complemented
5C11×#386C211yesmaximal, characteristic, complemented
6#10622823C211yesmaximal, characteristic, complemented
7#10622572C212yesmaximal, complemented
8#10622570C212yesmaximal, complemented
9#10622568C211yesmaximal, characteristic, complemented
10#10622289C212yesmaximal, complemented
11#10622282C212yesmaximal, complemented
12C24×D11C2211yescharacteristic
13C22×D4×C11C2211yescharacteristic
14C23×C4×C11C2211yescharacteristic
15C22×C22⋊D11C2212yes—
16C23×Dic11C2211yescharacteristic
17C2×C11×C22⋊C4C2211yescharacteristic
18C2×C11×C22⋊C4C2212yescomplemented
19C2×C22⋊Dic11C2211yescharacteristic
20C2×C22⋊Dic11C2211yescharacteristic
21C2×C22⋊Dic11C2211yescharacteristic, complemented
22C2×C22⋊Dic11C2211yescharacteristic, complemented
23#66990—24no—
24#66988—24no—
25C2×(C2×D11)⋊C4—24no—
26C2×(C2×D11)⋊C4C2212yes—
27C2×(C2×D11)⋊C4C2212yes—
28C2×(C2×D11)⋊C4C2212yes—
29C2×(C2×D11)⋊C4C2212yes—
30C2×(C2×D11)⋊C4C2212yes—
31C2×Dic11⋊C4C2212yescomplemented
32C2×Dic11⋊C4C2211yescharacteristic, complemented
33C2×Dic11⋊C4C2211yescharacteristic, complemented
34C2×Dic11⋊C4C2212yescomplemented
35C2×C4×Dic11C2212yescomplemented
36#66930—24no—
37C11×#95C2212yescomplemented
38C11⋊#95C2212yescomplemented
39C24×C11C2311yescharacteristic
40C24×C11C2311yescharacteristic
41C23×D11—22no—
42C23×D11—24no—
43C23×D11—22no—
44C23×D11—22no—
45C23×D11—42no—
46C23×D11C2312yes—
47C2×D4×C11—22no—
48C2×D4×C11—22no—
49C2×D4×C11—22no—
50C22×C4×C11—22no—
51C22×C4×C11—42no—
52C22×C4×C11C2312yes—
53C22×C4×C11C2311yescharacteristic
54C22×C4×C11C2312yes—
55C2×C22⋊D11—24no—
56C2×C22⋊D11—24no—
57C2×C22⋊D11—42no—
58C22×Dic11—22no—
59C22×Dic11—22no—
60C22×Dic11D414yescomplemented
61C22×Dic11C2311yescharacteristic
62C22×Dic11C2311yescharacteristic
63C22×Dic11C2312yes—
64C22×Dic11C2311yescharacteristic
65C22×Dic11C2311yescharacteristic
66C11×C22⋊C4—22no—
67C11×C22⋊C4—42no—
68C22⋊Dic11—22no—
69C22⋊Dic11—22no—
70C22⋊Dic11—22no—
71C22⋊Dic11D414yes—
72(C2×D11)⋊C4—24no—
73(C2×D11)⋊C4—24no—
74(C2×D11)⋊C4—24no—
75(C2×D11)⋊C4—24no—
76(C2×D11)⋊C4—42no—
77Dic11⋊C4—24no—
78Dic11⋊C4—22no—
79Dic11⋊C4—22no—
80Dic11⋊C4—42no—
81C4×Dic11—24no—
82#2411—111nomaximal, Sylow
83C23×C11—22no—
84C23×C11—22no—
85C23×C11—21no—
86C23×C11—21no—
87C23×C11—22no—
88C23×C11—21no—
89C23×C11—22no—
90C23×C11C2×D412yes—
91C23×C11C2×D412yes—
92C23×C11#4112yes—
93C23×C11C2411yescharacteristic
94C22×D11—28no—
95C22×D11—22no—
96C22×D11—22no—
97C22×D11—22no—
98C22×D11—22no—
99C22×D11—22no—
100C22×D11—22no—
101C22×D11—48no—
102C22×D11—48no—
103C22×D11—44no—
104C22×D11—44no—
105C22×D11#4114yes—
106D4×C11—44no—
107C2×C4×C11—22no—
108C2×C4×C11—21no—
109C2×C4×C11—22no—
110C2×C4×C11—21no—
111C2×C4×C11—22no—
112C2×C4×C11—21no—
113C2×C4×C11—22no—
114C2×C4×C11—22no—
115C2×C4×C11—22no—
116C2×C4×C11—44no—
117C22⋊D11—48no—
118C2×Dic11—28no—
119C2×Dic11—21no—
120C2×Dic11—21no—
121C2×Dic11—21no—
122C2×Dic11—21no—
123C2×Dic11—22no—
124C2×Dic11—22no—
125C2×Dic11—21no—
126C2×Dic11—21no—
127C2×Dic11—22no—
128C2×Dic11—22no—
129C2×Dic11C2×D414yes—
130C2×Dic11C2×D412yes—
131C2×Dic11C2×D412yes—
132C2×#123—111no—
133C2×#120—111no—
134C2×#117—111no—
135C22×C22⋊C4—111no—
136#396—112no—
137#394—112no—
138#390—112no—
139#388—112no—
140#386D1111yescharacteristic, complemented
141#385—111no—
142#379—111no—
143C22×C11—22no—
144C22×C11—22no—
145C22×C11—24no—
146C22×C11—24no—
147C22×C11—41no—
148C22×C11—41no—
149C22×C11—41no—
150C22×C11—42no—
151C22×C11—42no—
152C22×C11#12114yes—
153C22×C11C22×D411yescharacteristic
154C22×C11C22×D411yescharacteristic
155C22×C11C2×#4111yescharacteristic
156C22×C11C2×#4112yes—
157C22×C11H(5,𝔽2)12yes—
158C2×D11—24no—
159C2×D11—24no—
160C2×D11—24no—
161C2×D11—416no—
162C2×D11—42no—
163C2×D11—42no—
164C2×D11—42no—
165C2×D11—42no—
166C4×C11—41no—
167C4×C11—42no—
168C4×C11—42no—
169Dic11—22no—
170Dic11—22no—
171Dic11—24no—
172Dic11—41no—
173Dic11—41no—
174C25—111no—
175C22×D4—112no—
176C22×D4C2×D1111yescharacteristic
177C23×C4—111no—
178C23×C4C2×D1111yescharacteristic
179#123—224no—
180#120—224no—
181#117—224no—
182C2×C4⋊C4—112no—
183C2×C4⋊C4—111no—
184C2×C4⋊C4—111no—
185C2×C4⋊C4—112no—
186C2×C22⋊C4—111no—
187C2×C22⋊C4—111no—
188C2×C22⋊C4—111no—
189C2×C22⋊C4—111no—
190C2×C22⋊C4—112no—
191C2×C22⋊C4—112no—
192C2×C22⋊C4—112no—
193C2×C22⋊C4—112no—
194C2×C22⋊C4—112no—
195C2×C22⋊C4—224no—
196C2×C22⋊C4C2×D1111yescharacteristic
197C2×C22⋊C4C2×D1112yescomplemented
198C2×C42—112no—
199#95—112no—
200#95C2×D1112yescomplemented
201C2×C11—24no—
202C2×C11—81no—
203C2×C11C2×#12111yescharacteristic
204C2×C11#52511yescharacteristic
205C2×C11#53511yescharacteristic
206C2×C11#54612yes—
207C2×C11#55112yes—
208D11—44no—
209C24—112no—
210C24—224no—
211C24—222no—
212C24—222no—
213C24—222no—
214C24—442no—
215C24C22×D1111yescharacteristic
216C24C22×D1111yescharacteristic
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—224no—
221C2×D4—224no—
222C2×D4—442no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—114no—
226C22×C4—112no—
227C22×C4—111no—
228C22×C4—111no—
229C22×C4—111no—
230C22×C4—111no—
231C22×C4—222no—
232C22×C4—222no—
233C22×C4C22×D1112yes—
234C22×C4C22×D1111yescharacteristic
235C22×C4C22×D1112yes—
236C4⋊C4—224no—
237C4⋊C4—222no—
238C4⋊C4—222no—
239C4⋊C4—442no—
240C22⋊C4—22no—
241C22⋊C4—42no—
242C22⋊C4—114no—
243C22⋊C4—224no—
244C22⋊C4—224no—
245C22⋊C4—224no—
246C22⋊C4—224no—
247C22⋊C4—222no—
248C22⋊C4—222no—
249C22⋊C4—222no—
250C22⋊C4—442no—
251C42—224no—
252C11#241111yescharacteristic, complemented, Sylow
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—114no—
261C23—228no—
262C23—222no—
263C23—222no—
264C23—222no—
265C23—222no—
266C23—222no—
267C23—222no—
268C23—448no—
269C23—448no—
270C23—444no—
271C23—444no—
272C23#433912yes—
273C23D4×D1112yes—
274C23D4×D1112yes—
275C23C23×D1111yescharacteristic
276D4—44no—
277D4—448no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—114no—
289C2×C4—112no—
290C2×C4—112no—
291C2×C4—228no—
292C2×C4—221no—
293C2×C4—221no—
294C2×C4—221no—
295C2×C4—221no—
296C2×C4—222no—
297C2×C4—222no—
298C2×C4—221no—
299C2×C4—221no—
300C2×C4—222no—
301C2×C4—222no—
302C22—24no—
303C22—22no—
304C22—24no—
305C22—22no—
306C22—41no—
307C22—42no—
308C22—41no—
309C22—42no—
310C22—41no—
311C22—224no—
312C22—224no—
313C22—224no—
314C22—4416no—
315C22—442no—
316C22—442no—
317C22—442no—
318C22—442no—
319C22#6693714yes—
320C22C2×#433911yescharacteristic
321C22C2×D4×D1111yescharacteristic
322C22C2×D4×D1111yescharacteristic
323C22#6703712yes—
324C22#41×D1112yes—
325C4—41no—
326C4—42no—
327C4—42no—
328C4—222no—
329C4—222no—
330C4—224no—
331C4—441no—
332C4—441no—
333C2—24no—
334C2—81no—
335C2—444no—
336C2C2×#6693711yescharacteristic
337C2#1062308712yes—
338C2#1062312311yescharacteristic
339C2#1062314112yes—
340C2#1062318711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#386C2
prime radicalC2×#66990C2
hypercenter#386D11
hypocenterC11#2411
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#66988C2semidirect
3. C22×(C2×D11)⋊C4C2semidirect
4. C2×#66930C2semidirect
5. C11×#386C2semidirect
6. #10622823C2semidirect
7. #10622572C2semidirect
8. #10622570C2semidirect
9. #10622568C2semidirect
10. #10622289C2semidirect
11. #10622282C2semidirect
18. C2×C11×C22⋊C4C22semidirect
21. C2×C22⋊Dic11C22semidirect
22. C2×C22⋊Dic11C22semidirect
31. C2×Dic11⋊C4C22semidirect
32. C2×Dic11⋊C4C22semidirect
33. C2×Dic11⋊C4C22semidirect
34. C2×Dic11⋊C4C22semidirect
35. C2×C4×Dic11C22semidirect
37. C11×#95C22semidirect
38. C11⋊#95C22semidirect
60. C22×Dic11D4semidirect
140. #386D11semidirect
197. C2×C22⋊C4C2×D11semidirect
200. #95C2×D11semidirect
252. C11#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#669902
C2×#669882
C22×(C2×D11)⋊C42
C2×#669302
C11×#3862
#106228232
#106225722
#106225702
#106225682
#106222892
#106222822
#241111