Finite Group Databasebeta

← #13074384

Subgroups — #13074384

Order 1408.

528 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 143 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×#66949C212yesmaximal
2C2×#66945C211yesmaximal, characteristic
3C2×#66941C211yesmaximal, characteristic, complemented
4C11×#398C211yesmaximal, characteristic, complemented
5#10622606C211yesmaximal, characteristic
6#10622597C212yesmaximal, complemented
7#10622596C211yesmaximal, characteristic, complemented
8#10622288C212yesmaximal
9#10622287C211yesmaximal, characteristic
10#10622263C211yesmaximal, characteristic, complemented
11#10622259C212yesmaximal, complemented
12C22×C4×D11C2211yescharacteristic
13C22×Dic22C2212yes—
14C2×C11×C4⋊C4C2211yescharacteristic
15C2×C11×C4⋊C4C2211yescharacteristic
16C2×C11×C4⋊C4C2212yes—
17C2×(C2×D11)⋊C4C2211yescharacteristic
18C2×(C2×D11)⋊C4C2211yescharacteristic
19C2×(C2×D11)⋊C4C2212yes—
20C2×(C2×D11)⋊C4C2212yes—
21C2×C4⋊Dic11C2211yescharacteristic
22C2×C4⋊Dic11C2211yescharacteristic
23C2×Dic11⋊C4C2212yes—
24C2×Dic11⋊C4C2212yes—
25C2×Dic11⋊C4C2212yes—
26C2×C4×Dic11C2211yescharacteristic
27C2×C4×Dic11C2211yescharacteristic
28#66949—28no—
29#66945—24no—
30#66941—24no—
31C11×#95C2212yes—
32C11×#95C2211yescharacteristic
33C11⋊#95C2212yes—
34C11⋊#95C2212yes—
35C11⋊#95C2212yes—
36C11⋊#95C2212yes—
37C11⋊#95C2211yescharacteristic
38C23×D11C2311yescharacteristic
39C22×C4×C11C2311yescharacteristic
40C22×C4×C11C2312yes—
41C22×C4×C11C2312yes—
42C22×C4×C11C2311yescharacteristic
43C22×C4×C11C2311yescharacteristic
44C22×Dic11C2311yescharacteristic
45C22×Dic11C2312yes—
46C22×Dic11C2312yes—
47C22×Dic11C2311yescharacteristic
48C22×Dic11C2311yescharacteristic
49C2×C4×D11—22no—
50C2×C4×D11—22no—
51C2×C4×D11D414yes—
52C2×Dic22—24no—
53C2×Dic22—24no—
54C2×Dic22—24no—
55C11×C4⋊C4—22no—
56C11×C4⋊C4—22no—
57C11×C4⋊C4—24no—
58(C2×D11)⋊C4—22no—
59(C2×D11)⋊C4—24no—
60(C2×D11)⋊C4—24no—
61(C2×D11)⋊C4—22no—
62C4⋊Dic11—22no—
63C4⋊Dic11—22no—
64Dic11⋊C4—24no—
65Dic11⋊C4—24no—
66Dic11⋊C4—24no—
67C4×Dic11—22no—
68C4×Dic11D414yes—
69#2621—111nomaximal, Sylow
70C23×C11C2411yescharacteristic
71C22×D11—21no—
72C22×D11—21no—
73C22×D11—22no—
74C22×D11—22no—
75C22×D11C2×D412yes—
76C2×C4×C11—21no—
77C2×C4×C11—21no—
78C2×C4×C11—22no—
79C2×C4×C11—21no—
80C2×C4×C11—22no—
81C2×C4×C11—22no—
82C2×C4×C11—22no—
83C2×C4×C11—22no—
84C2×C4×C11—22no—
85C2×C4×C11—22no—
86C2×C4×C11—22no—
87C2×C4×C11C2×D412yes—
88C2×C4×C11C2×D412yes—
89C2×Dic11—22no—
90C2×Dic11—21no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11—22no—
94C2×Dic11—22no—
95C2×Dic11—22no—
96C2×Dic11—22no—
97C2×Dic11—22no—
98C2×Dic11C2×D412yes—
99C2×Dic11C2×D412yes—
100C2×Dic11C2×D412yes—
101C2×Dic11#4112yes—
102C4×D11—28no—
103Dic22—48no—
104C2×C4⋊Q8—111no—
105C2×C22⋊Q8—112no—
106C2×#117—111no—
107#398—111no—
108#398—112no—
109#398D1111yescharacteristic, complemented
110#397—111no—
111#397—111no—
112#397—112no—
113#391—112no—
114#382—111no—
115C22×C11C22×D411yescharacteristic
116C22×C11C22×D411yescharacteristic
117C22×C11C2×#4111yescharacteristic
118C22×C112−1+412yes—
119C22×C112−1+412yes—
120C2×D11—22no—
121C2×D11—24no—
122C2×D11—41no—
123C2×D11—41no—
124C2×D11—42no—
125C4×C11—22no—
126C4×C11—22no—
127C4×C11—42no—
128C4×C11—42no—
129C4×C11—41no—
130Dic11—22no—
131Dic11—42no—
132Dic11—41no—
133Dic11—42no—
134Dic11#12114yes—
135C22×Q8—112no—
136C23×C4—111no—
137C4⋊Q8—224no—
138C22⋊Q8—228no—
139#117—224no—
140C2×C4⋊C4—111no—
141C2×C4⋊C4—112no—
142C2×C4⋊C4—112no—
143C2×C4⋊C4—112no—
144C2×C4⋊C4—111no—
145C2×C4⋊C4C2×D1111yescharacteristic
146C2×C4⋊C4C2×D1112yes—
147C2×C4⋊C4C2×D1111yescharacteristic
148C2×C22⋊C4—111no—
149C2×C22⋊C4—111no—
150C2×C22⋊C4—112no—
151C2×C22⋊C4—112no—
152C2×C42—111no—
153C2×C42—111no—
154#95—112no—
155#95—111no—
156#95—112no—
157#95—112no—
158#95—112no—
159#95C2×D1111yescharacteristic
160#95C2×D1112yes—
161C2×C11C2×#12111yescharacteristic
162C2×C11#53611yescharacteristic
163C2×C11#53612yes—
164C2×C11#53611yescharacteristic
165C2×C11#54112yes—
166D11—42no—
167C24—111no—
168C2×Q8—224no—
169C2×Q8—224no—
170C2×Q8—224no—
171C22×C4—111no—
172C22×C4—111no—
173C22×C4—114no—
174C22×C4—111no—
175C22×C4—112no—
176C22×C4—112no—
177C22×C4—222no—
178C22×C4—222no—
179C22×C4C22×D1111yescharacteristic
180C22×C4C22×D1112yes—
181C22×C4C22×D1112yes—
182C22×C4C22×D1111yescharacteristic
183C22×C4C22×D1111yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—222no—
188C4⋊C4—224no—
189C4⋊C4—224no—
190C4⋊C4—224no—
191C4⋊C4—222no—
192C22⋊C4—222no—
193C22⋊C4—224no—
194C22⋊C4—224no—
195C22⋊C4—222no—
196C42—114no—
197C42—222no—
198C11#262111yescharacteristic, complemented, Sylow
199C23—112no—
200C23—221no—
201C23—222no—
202C23—221no—
203C23—222no—
204C23C23×D1111yescharacteristic
205Q8—448no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—112no—
218C2×C4—112no—
219C2×C4—112no—
220C2×C4—112no—
221C2×C4—228no—
222C2×C4—222no—
223C2×C4—221no—
224C2×C4—222no—
225C2×C4—222no—
226C2×C4—222no—
227C2×C4—222no—
228C2×C4—222no—
229C2×C4—222no—
230C2×C4—222no—
231C2×C4D4×D1112yes—
232C2×C4D4×D1112yes—
233C22—222no—
234C22—224no—
235C22—441no—
236C22—441no—
237C22—442no—
238C22C2×D4×D1111yescharacteristic
239C22C2×D4×D1111yescharacteristic
240C22#6704012yes—
241C22#41×D1111yescharacteristic
242C22#6704312yes—
243C4—22no—
244C4—22no—
245C4—42no—
246C4—42no—
247C4—41no—
248C4—114no—
249C4—222no—
250C4—441no—
251C4—442no—
252C4—442no—
253C2—442no—
254C2D11×#12111yescharacteristic
255C2#1062315912yes—
256C2#1062317911yescharacteristic
257C2#1062322712yes—
258C2#1062325511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#398C2
prime radicalC23×D11C23
hypercenter#398D11
hypocenterC11#2621
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C2×#66941C2semidirect
4. C11×#398C2semidirect
6. #10622597C2semidirect
7. #10622596C2semidirect
10. #10622263C2semidirect
11. #10622259C2semidirect
109. #398D11semidirect
198. C11#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#669492
C2×#669452
C2×#669412
C11×#3982
#106226062
#106225972
#106225962
#106222882
#106222872
#106222632
#106222592
#262111