Finite Group Databasebeta

← #13072236

Subgroups — #13072236

Order 1408.

952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 169 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×#4339C211yesmaximal, characteristic, complemented
2C2×C11×#121C211yesmaximal, characteristic, complemented
3C2×#66993C211yesmaximal, characteristic, complemented
4C2×#66989C211yesmaximal, characteristic, complemented
5#10622836C214yesmaximal, complemented
6C4×C22⋊Dic11C211yesmaximal, characteristic, complemented
7#10622601C211yesmaximal, characteristic, complemented
8#10622587C211yesmaximal, characteristic, complemented
9#10622569C214yesmaximal, complemented
10C22×D4×C11C2211yescharacteristic
11C22×D4×C11C2212yes—
12C23×C4×C11C2211yescharacteristic
13C22×C22⋊D11C2212yes—
14C22×C22⋊D11C2214yes—
15C2×#4339—24no—
16C2×#4339—24no—
17C2×#4339—24no—
18C22×D44C2211yescharacteristic
19C22×C4×D11C2212yes—
20C22×Dic22C2211yescharacteristic
21C11×#121—24no—
22C2×C11×C4⋊C4C2211yescharacteristic
23C2×C11×C22⋊C4C2212yes—
24C2×C22⋊Dic11C2214yes—
25C2×C22⋊Dic11C2212yes—
26#66993—24no—
27#66989—24no—
28C2×(C2×D11)⋊C4C2214yes—
29C2×C4×Dic11C2211yescharacteristic
30C2×C4×Dic11C2211yescharacteristic
31C11⋊#95C2212yes—
32C11⋊#95C2214yes—
33C24×C11C2312yes—
34C24×C11C2311yescharacteristic
35C23×D11C2312yes—
36C2×D4×C11—24no—
37C2×D4×C11—22no—
38C2×D4×C11—22no—
39C2×D4×C11—22no—
40C2×D4×C11—44no—
41C22×C4×C11—22no—
42C22×C4×C11—22no—
43C22×C4×C11D414yescomplemented
44C22×C4×C11C2312yes—
45C22×C4×C11C2311yescharacteristic
46C22×C4×C11C2311yescharacteristic
47C2×C22⋊D11—24no—
48C2×C22⋊D11—28no—
49C2×C22⋊D11—24no—
50C2×C22⋊D11—24no—
51C2×C22⋊D11—28no—
52C2×C22⋊D11—44no—
53C22×Dic11C2312yes—
54C22×Dic11C2312yes—
55C22×Dic11C2312yes—
56#4339—232no—
57C2×D44—22no—
58C2×D44—22no—
59C2×D44D414yescomplemented
60C2×C4×D11—24no—
61C2×C4×D11—24no—
62C2×C4×D11—24no—
63C2×Dic22—22no—
64C2×Dic22—22no—
65C2×Dic22D414yescomplemented
66C11×C4⋊C4—22no—
67C11×C22⋊C4—24no—
68C22⋊Dic11—28no—
69C22⋊Dic11—24no—
70(C2×D11)⋊C4—44no—
71C4×Dic11—22no—
72C4×Dic11—22no—
73#2431—111nomaximal, Sylow
74C23×C11—21no—
75C23×C11—21no—
76C23×C11—22no—
77C23×C11—22no—
78C23×C11—22no—
79C23×C11—22no—
80C23×C11—22no—
81C23×C11—24no—
82C23×C11—24no—
83C23×C11C2×D412yes—
84C23×C11C2411yescharacteristic
85C22×D11—22no—
86C22×D11—22no—
87C22×D11—24no—
88C22×D11—24no—
89C22×D11C2×D414yes—
90D4×C11—48no—
91D4×C11—44no—
92C2×C4×C11—21no—
93C2×C4×C11—21no—
94C2×C4×C11—28no—
95C2×C4×C11—22no—
96C2×C4×C11—24no—
97C2×C4×C11C2×D412yes—
98C2×C4×C11C2×D412yes—
99C2×C4×C11C2×D412yes—
100C2×C4×C11C2×D412yes—
101C22⋊D11—216no—
102C22⋊D11—416no—
103C2×Dic11—22no—
104C2×Dic11—22no—
105C2×Dic11—22no—
106C2×Dic11—22no—
107C2×Dic11—22no—
108C2×Dic11—22no—
109C2×Dic11—22no—
110C2×Dic11C2×D414yes—
111C2×Dic11#4114yescomplemented
112D44—28no—
113C4×D11—216no—
114Dic22—28no—
115C22×#41—111no—
116C2×#127—111no—
117C2×#124—111no—
118C2×#121D1111yescharacteristic, complemented
119#392—114no—
120#391—111no—
121#390—111no—
122#386—114no—
123C4×C22⋊C4—111no—
124C22×C11—22no—
125C22×C11—22no—
126C22×C11—22no—
127C22×C11—42no—
128C22×C11—42no—
129C22×C11—42no—
130C22×C11—42no—
131C22×C11—42no—
132C22×C11—44no—
133C22×C11—44no—
134C22×C11C22×D411yescharacteristic
135C22×C11C22×D411yescharacteristic
136C22×C11C22×D411yescharacteristic
137C22×C11C2×#4112yes—
138C22×C11H(5,𝔽2)12yes—
139C2×D11—24no—
140C2×D11—28no—
141C2×D11—42no—
142C2×D11—42no—
143C2×D11—44no—
144C4×C11—22no—
145C4×C11—42no—
146C4×C11#12014yes—
147Dic11—24no—
148Dic11—24no—
149Dic11—42no—
150C2×#41—224no—
151C2×#41—224no—
152C2×#41—224no—
153C22×Q8—111no—
154C22×D4—112no—
155C22×D4—114no—
156C22×D4—111no—
157C22×D4C2×D1111yescharacteristic
158C22×D4C2×D1112yes—
159C23×C4—112no—
160C23×C4C2×D1111yescharacteristic
161#127—224no—
162#124—224no—
163#121—24no—
164C2×C4⋊C4C2×D1111yescharacteristic
165C2×C22⋊C4—114no—
166C2×C22⋊C4—112no—
167C2×C22⋊C4—114no—
168C2×C22⋊C4C2×D1112yes—
169C2×C42—111no—
170C2×C42—111no—
171#95—112no—
172#95—114no—
173C2×C11—42no—
174C2×C11—82no—
175C2×C11C2×#12011yescharacteristic
176C2×C11#53211yescharacteristic
177C2×C11#53511yescharacteristic
178C2×C11#54612yes—
179C2×C11#55012yes—
180D11—44no—
181C24—112no—
182C24C22×D1112yes—
183C24C22×D1111yescharacteristic
184#41—2232no—
185C2×Q8—114no—
186C2×Q8—222no—
187C2×Q8—222no—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—44no—
193C2×D4—114no—
194C2×D4—224no—
195C2×D4—228no—
196C2×D4—224no—
197C2×D4—224no—
198C2×D4—228no—
199C2×D4—222no—
200C2×D4—222no—
201C2×D4—444no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—112no—
205C22×C4—112no—
206C22×C4—112no—
207C22×C4—224no—
208C22×C4—224no—
209C22×C4—224no—
210C22×C4C22⋊D1114yescomplemented
211C22×C4C22×D1112yes—
212C22×C4C22×D1111yescharacteristic
213C22×C4C22×D1111yescharacteristic
214C4⋊C4—22no—
215C22⋊C4—24no—
216C22⋊C4—228no—
217C22⋊C4—224no—
218C22⋊C4—444no—
219C42—222no—
220C42—222no—
221C11#243111yescharacteristic, complemented, Sylow
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—24no—
229C23—24no—
230C23—22no—
231C23—114no—
232C23—222no—
233C23—224no—
234C23—224no—
235C23—222no—
236C23C2×C22⋊D1112yes—
237C23C23×D1111yescharacteristic
238Q8—228no—
239D4—48no—
240D4—44no—
241D4—2216no—
242D4—228no—
243D4—4416no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—28no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—114no—
250C2×C4—114no—
251C2×C4—2216no—
252C2×C4—222no—
253C2×C4—222no—
254C2×C4—222no—
255C2×C4—222no—
256C2×C4—222no—
257C2×C4—222no—
258C2×C4—222no—
259C2×C4D4×D1112yes—
260C2×C4D4×D1112yes—
261C2×C4C2×C22⋊D1112yes—
262C2×C4C2×C22⋊D1112yes—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—44no—
272C22—44no—
273C22—224no—
274C22—228no—
275C22—442no—
276C22—442no—
277C22—444no—
278C22C2×D4×D1111yescharacteristic
279C22C2×D4×D1111yescharacteristic
280C22#6703712yes—
281C22#41×D1112yes—
282C22C22×C22⋊D1111yescharacteristic
283C4—22no—
284C4—42no—
285C4—224no—
286C4—224no—
287C4—442no—
288C4#6699014yes—
289C2—42no—
290C2—82no—
291C2—444no—
292C2#1062313012yes—
293C2#1062314312yes—
294C2C2×#6699011yescharacteristic
295C2#1062333611yescharacteristic
296C2#1062336311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC2×C11×#121C2
hypercenterC2×#121D11
hypocenterC11#2431
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#4339C2semidirect
2. C2×C11×#121C2semidirect
3. C2×#66993C2semidirect
4. C2×#66989C2semidirect
5. #10622836C2semidirect
6. C4×C22⋊Dic11C2semidirect
7. #10622601C2semidirect
8. #10622587C2semidirect
9. #10622569C2semidirect
43. C22×C4×C11D4semidirect
59. C2×D44D4semidirect
65. C2×Dic22D4semidirect
111. C2×Dic11#41semidirect
118. C2×#121D11semidirect
210. C22×C4C22⋊D11semidirect
221. C11#2431semidirect

Maximal subgroups

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

subgroupindex
C22×#43392
C2×C11×#1212
C2×#669932
C2×#669892
#106228362
C4×C22⋊Dic112
#106226012
#106225872
#106225692
#243111