Finite Group Databasebeta

← #13072023

Subgroups — #13072023

Order 1408.

610 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×C11×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#66982C211yesmaximal, characteristic
3C2×#66982C211yesmaximal, characteristic
4C2×#66979C211yesmaximal, characteristic, complemented
5#10622826C211yesmaximal, characteristic
6#10622824C211yesmaximal, characteristic, complemented
7#10622823C211yesmaximal, characteristic, complemented
8#10622607C212yesmaximal, complemented
9#10622603C212yesmaximal, complemented
10#10622597C212yesmaximal, complemented
11#10622585C212yesmaximal, complemented
12C23×C4×C11C2211yescharacteristic
13C23×C4×C11C2211yescharacteristic
14C23×C4×C11C2211yescharacteristic
15C22×C22⋊D11C2211yescharacteristic
16C2×C11×C4⋊C4—24no—
17C2×C11×C4⋊C4C2212yes—
18C2×C11×C4⋊C4C2212yes—
19C2×C22⋊Dic11C2211yescharacteristic
20C2×C22⋊Dic11C2211yescharacteristic
21C2×C22⋊Dic11C2211yescharacteristic
22#66982—24no—
23#66982—24no—
24C2×(C2×D11)⋊C4C2211yescharacteristic
25C2×(C2×D11)⋊C4C2212yes—
26C2×(C2×D11)⋊C4C2211yescharacteristic
27C2×(C2×D11)⋊C4C2212yes—
28#66979—24no—
29C2×C4⋊Dic11C2212yes—
30C2×Dic11⋊C4C2211yescharacteristic
31C2×Dic11⋊C4C2212yes—
32C2×Dic11⋊C4C2211yescharacteristic
33C2×Dic11⋊C4C2212yes—
34C2×C4×Dic11C2212yes—
35C11⋊#95C2212yes—
36C11⋊#95C2212yes—
37C11⋊#95C2212yes—
38C11⋊#95C2212yes—
39C24×C11C2311yescharacteristic
40C23×D11C2311yescharacteristic
41C22×C4×C11—22no—
42C22×C4×C11—22no—
43C22×C4×C11—22no—
44C22×C4×C11—24no—
45C22×C4×C11—22no—
46C22×C4×C11—22no—
47C22×C4×C11—22no—
48C22×C4×C11D414yes—
49C22×C4×C11C2312yes—
50C22×C4×C11C2311yescharacteristic
51C22×C4×C11C2311yescharacteristic
52C22×C4×C11C2312yes—
53C2×C22⋊D11—22no—
54C2×C22⋊D11—22no—
55C2×C22⋊D11—22no—
56C22×Dic11C2311yescharacteristic
57C22×Dic11C2312yes—
58C22×Dic11C2311yescharacteristic
59C22×Dic11C2311yescharacteristic
60C22×Dic11C2312yes—
61C11×C4⋊C4—24no—
62C11×C4⋊C4—24no—
63C22⋊Dic11—22no—
64C22⋊Dic11—22no—
65C22⋊Dic11D414yes—
66(C2×D11)⋊C4—24no—
67(C2×D11)⋊C4—22no—
68(C2×D11)⋊C4—22no—
69(C2×D11)⋊C4—24no—
70C4⋊Dic11—24no—
71Dic11⋊C4—24no—
72Dic11⋊C4—22no—
73Dic11⋊C4—22no—
74Dic11⋊C4—24no—
75C4×Dic11—24no—
76#2413—111nomaximal, Sylow
77C23×C11—22no—
78C23×C11—22no—
79C23×C11C2×D412yes—
80C23×C11C2×D412yes—
81C23×C11#4112yes—
82C23×C11C2411yescharacteristic
83C22×D11—21no—
84C22×D11—21no—
85C22×D11—22no—
86C22×D11—21no—
87C22×D11—22no—
88C2×C4×C11—28no—
89C2×C4×C11—28no—
90C2×C4×C11—21no—
91C2×C4×C11—21no—
92C2×C4×C11—22no—
93C2×C4×C11—22no—
94C2×C4×C11—21no—
95C2×C4×C11—21no—
96C2×C4×C11—22no—
97C2×C4×C11—22no—
98C2×C4×C11—22no—
99C2×C4×C11—44no—
100C2×C4×C11C2×D412yes—
101C2×C4×C11C2×D412yes—
102C2×C4×C11#4114yes—
103C22⋊D11—44no—
104C2×Dic11—22no—
105C2×Dic11—22no—
106C2×Dic11—22no—
107C2×Dic11—21no—
108C2×Dic11—21no—
109C2×Dic11—21no—
110C2×Dic11—21no—
111C2×Dic11—22no—
112C2×Dic11—22no—
113C2×Dic11—21no—
114C2×Dic11—22no—
115C2×Dic11—22no—
116C2×Dic11C2×D414yes—
117C2×#123—111no—
118C2×#123—111no—
119C2×#117—111no—
120C22×C4⋊C4D1111yescharacteristic, complemented
121#398—112no—
122#396—112no—
123#389—112no—
124#388—112no—
125#386—111no—
126#385—111no—
127#381—111no—
128C22×C11—22no—
129C22×C11—22no—
130C22×C11—24no—
131C22×C11—24no—
132C22×C11#12114yes—
133C22×C11C22×D411yescharacteristic
134C22×C11C22×D411yescharacteristic
135C22×C11C2×#4112yes—
136C22×C11C2×#4111yescharacteristic
137C22×C112−1+412yes—
138C2×D11—41no—
139C2×D11—41no—
140C2×D11—41no—
141C2×D11—42no—
142C2×D11—42no—
143C4×C11—22no—
144C4×C11—22no—
145C4×C11—24no—
146C4×C11—42no—
147Dic11—24no—
148Dic11—41no—
149Dic11—41no—
150Dic11—41no—
151Dic11—42no—
152C22×D4—111no—
153C23×C4C2×D1111yescharacteristic
154C23×C4C2×D1111yescharacteristic
155C23×C4C2×D1111yescharacteristic
156#123—224no—
157#123—224no—
158#117—224no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—112no—
161C2×C4⋊C4—111no—
162C2×C4⋊C4—112no—
163C2×C4⋊C4—111no—
164C2×C4⋊C4—112no—
165C2×C4⋊C4C2×D1112yes—
166C2×C4⋊C4C2×D1112yes—
167C2×C22⋊C4—111no—
168C2×C22⋊C4—111no—
169C2×C22⋊C4—111no—
170C2×C22⋊C4—111no—
171C2×C22⋊C4—112no—
172C2×C22⋊C4—111no—
173C2×C22⋊C4—112no—
174C2×C42—112no—
175#95—112no—
176#95—112no—
177#95—112no—
178#95—112no—
179C2×C11—24no—
180C2×C11C2×#12111yescharacteristic
181C2×C11#52511yescharacteristic
182C2×C11#53611yescharacteristic
183C2×C11#54712yes—
184C2×C11#55212yes—
185D11—81no—
186C24—111no—
187C24C22×D1111yescharacteristic
188C2×D4—222no—
189C2×D4—222no—
190C2×D4—222no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—112no—
199C22×C4—111no—
200C22×C4—112no—
201C22×C4—111no—
202C22×C4—111no—
203C22×C4C22⋊D1114yes—
204C22×C4C22×D1112yes—
205C22×C4C22×D1112yes—
206C22×C4C22×D1111yescharacteristic
207C22×C4C22×D1111yescharacteristic
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—224no—
211C4⋊C4—224no—
212C4⋊C4—224no—
213C4⋊C4—222no—
214C4⋊C4—222no—
215C22⋊C4—114no—
216C22⋊C4—224no—
217C22⋊C4—222no—
218C22⋊C4—222no—
219C22⋊C4—224no—
220C22⋊C4—222no—
221C22⋊C4—222no—
222C42—224no—
223C11#241311yescharacteristic, complemented, Sylow
224C23—22no—
225C23—22no—
226C23—222no—
227C23—221no—
228C23—221no—
229C23—222no—
230C23—221no—
231C23D4×D1112yes—
232C23#434112yes—
233C23C2×C22⋊D1112yes—
234C23C23×D1111yescharacteristic
235D4—444no—
236C2×C4—28no—
237C2×C4—28no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—44no—
248C2×C4—114no—
249C2×C4—222no—
250C2×C4—222no—
251C2×C4—222no—
252C2×C4—222no—
253C2×C4—222no—
254C2×C4—221no—
255C2×C4—221no—
256C2×C4—222no—
257C2×C4—222no—
258C2×C4—221no—
259C2×C4—221no—
260C2×C4—221no—
261C2×C4#433914yes—
262C2×C4C2×C22⋊D1112yes—
263C2×C4C2×C22⋊D1112yes—
264C22—24no—
265C22—24no—
266C22—22no—
267C22—22no—
268C22—442no—
269C22—442no—
270C22—441no—
271C22—441no—
272C22—441no—
273C22#6699214yes—
274C22C2×#433912yes—
275C22C2×D4×D1111yescharacteristic
276C22C2×#434111yescharacteristic
277C22#6704012yes—
278C22C22×C22⋊D1111yescharacteristic
279C4—24no—
280C4—22no—
281C4—22no—
282C4—42no—
283C4—224no—
284C4—442no—
285C4—441no—
286C4—441no—
287C4—441no—
288C2—24no—
289C2—881no—
290C2#1062303812yes—
291C2#1062304412yes—
292C2#1062332211yescharacteristic
293C2C2×#6699211yescharacteristic
294C2#1062334711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC22×C11×C4⋊C4C2
prime radicalC22×C22⋊D11C22
hypercenterC22×C4⋊C4D11
hypocenterC11#2413
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C11×C4⋊C4C2semidirect
4. C2×#66979C2semidirect
6. #10622824C2semidirect
7. #10622823C2semidirect
8. #10622607C2semidirect
9. #10622603C2semidirect
10. #10622597C2semidirect
11. #10622585C2semidirect
120. C22×C4⋊C4D11semidirect
223. C11#2413semidirect

Maximal subgroups

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

subgroupindex
C22×C11×C4⋊C42
C2×#669822
C2×#669792
#106228262
#106228242
#106228232
#106226072
#106226032
#106225972
#106225852
#241311