Finite Group Databasebeta

← #13074346

Subgroups — #13074346

Order 1408.

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

normal subgroups only

This group has 816 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×#4339C211yesmaximal, characteristic
2C2×C11×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#66999C212yesmaximal, complemented
4#10622825C211yesmaximal, characteristic, complemented
5#10622604C212yesmaximal, complemented
6#10622600C212yesmaximal, complemented
7#10622592C212yesmaximal, complemented
8#10622572C214yesmaximal, complemented
9C22×Q8×C11C2211yescharacteristic
10C23×C4×C11C2211yescharacteristic
11C22×C22⋊D11C2212yes—
12C2×#4339—24no—
13C2×#4339—28no—
14C22×D44C2211yescharacteristic
15C22×C4×D11C2212yes—
16C22×Dic22C2211yescharacteristic
17C11×C22⋊Q8—24no—
18C2×C11×C4⋊C4C2212yes—
19C2×C11×C4⋊C4C2211yescharacteristic
20C2×C11×C22⋊C4C2212yes—
21C2×C22⋊Dic11C2212yes—
22#66999—28no—
23C2×(C2×D11)⋊C4C2214yes—
24C2×(C2×D11)⋊C4C2214yes—
25C2×C4⋊Dic11C2212yes—
26C2×Dic11⋊C4C2214yes—
27C11⋊#95C2212yes—
28C11⋊#95C2214yes—
29C24×C11C2311yescharacteristic
30C23×D11C2312yes—
31C2×Q8×C11—24no—
32C2×Q8×C11—22no—
33C22×C4×C11—24no—
34C22×C4×C11D414yes—
35C22×C4×C11C2312yes—
36C22×C4×C11C2312yes—
37C22×C4×C11C2311yescharacteristic
38C22×C4×C11C2311yescharacteristic
39C2×C22⋊D11—24no—
40C2×C22⋊D11—24no—
41C2×C22⋊D11—24no—
42C22×Dic11C2314yes—
43C22×Dic11C2312yes—
44#4339—232no—
45C2×D44—24no—
46C2×D44—22no—
47C2×C4×D11—24no—
48C2×C4×D11—24no—
49C2×C4×D11D418yes—
50C2×Dic22—24no—
51C2×Dic22—22no—
52C11×C4⋊C4—22no—
53C11×C4⋊C4—24no—
54C11×C22⋊C4—24no—
55C22⋊Dic11—42no—
56(C2×D11)⋊C4—28no—
57(C2×D11)⋊C4—44no—
58C4⋊Dic11—24no—
59Dic11⋊C4—28no—
60#2616—111nomaximal, Sylow
61C23×C11—22no—
62C23×C11—21no—
63C23×C11—22no—
64C23×C11—21no—
65C23×C11C2×D412yes—
66C23×C11C2411yescharacteristic
67C22×D11—22no—
68C22×D11—22no—
69C22×D11—22no—
70C22×D11—22no—
71C22×D11—22no—
72C22×D11—22no—
73C22×D11C2×D414yes—
74Q8×C11—44no—
75C2×C4×C11—22no—
76C2×C4×C11—28no—
77C2×C4×C11—24no—
78C2×C4×C11—22no—
79C2×C4×C11—22no—
80C2×C4×C11—22no—
81C2×C4×C11—22no—
82C2×C4×C11C2×D414yes—
83C2×C4×C11C2×D412yes—
84C2×C4×C11C2×D412yes—
85C22⋊D11—216no—
86C2×Dic11—22no—
87C2×Dic11—22no—
88C2×Dic11—24no—
89C2×Dic11—24no—
90C2×Dic11—24no—
91C2×Dic11C2×D414yes—
92D44—28no—
93C4×D11—216no—
94Dic22—28no—
95C22×#41—111no—
96C2×C22⋊Q8—112no—
97C2×C22⋊Q8D1111yescharacteristic, complemented
98#395—112no—
99#394—112no—
100#394—114no—
101#380—111no—
102#380—112no—
103C22×C11—22no—
104C22×C11—22no—
105C22×C11—22no—
106C22×C11—42no—
107C22×C11—42no—
108C22×C11C22×D412yes—
109C22×C11C22×D411yescharacteristic
110C22×C11H(5,𝔽2)11yescharacteristic
111C22×C112−1+412yes—
112C22×C112−1+411yescharacteristic
113C2×D11—24no—
114C2×D11—24no—
115C2×D11—24no—
116C2×D11—42no—
117C2×D11—42no—
118C2×D11—42no—
119C2×D11—42no—
120C4×C11—22no—
121C4×C11—42no—
122C4×C11—42no—
123C4×C11#12014yes—
124Dic11—24no—
125Dic11—44no—
126C2×#41—228no—
127C2×#41—224no—
128C22×Q8—111no—
129C22×Q8C2×D1111yescharacteristic
130C22×D4—112no—
131C22×D4—111no—
132C23×C4—112no—
133C23×C4C2×D1111yescharacteristic
134C22⋊Q8—24no—
135C22⋊Q8—228no—
136C2×C4⋊C4—112no—
137C2×C4⋊C4—114no—
138C2×C4⋊C4C2×D1111yescharacteristic
139C2×C4⋊C4C2×D1112yes—
140C2×C22⋊C4—114no—
141C2×C22⋊C4—112no—
142C2×C22⋊C4—114no—
143C2×C22⋊C4C2×D1112yes—
144#95—114no—
145#95—112no—
146C2×C11—42no—
147C2×C11C2×#12011yescharacteristic
148C2×C11#53611yescharacteristic
149C2×C11#53612yes—
150C2×C11#53711yescharacteristic
151C2×C11#53712yes—
152D11—44no—
153C24—112no—
154C24C22×D1111yescharacteristic
155#41—2232no—
156C2×Q8—24no—
157C2×Q8—22no—
158C2×Q8—224no—
159C2×Q8—222no—
160C2×D4—224no—
161C2×D4—224no—
162C2×D4—224no—
163C2×D4—224no—
164C2×D4—222no—
165C22×C4—24no—
166C22×C4—112no—
167C22×C4—118no—
168C22×C4—114no—
169C22×C4—224no—
170C22×C4—224no—
171C22×C4C22⋊D1114yes—
172C22×C4C22×D1112yes—
173C22×C4C22×D1112yes—
174C22×C4C22×D1111yescharacteristic
175C22×C4C22×D1111yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—24no—
178C4⋊C4—228no—
179C4⋊C4—224no—
180C22⋊C4—24no—
181C22⋊C4—228no—
182C22⋊C4—444no—
183C22⋊C4—442no—
184C11#261611yescharacteristic, complemented, Sylow
185C23—22no—
186C23—22no—
187C23—21no—
188C23—21no—
189C23—114no—
190C23—222no—
191C23—222no—
192C23—222no—
193C23—222no—
194C23—222no—
195C23—222no—
196C23C2×C22⋊D1112yes—
197C23C23×D1111yescharacteristic
198Q8—44no—
199Q8—228no—
200D4—228no—
201D4—2216no—
202C2×C4—22no—
203C2×C4—28no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—114no—
210C2×C4—222no—
211C2×C4—222no—
212C2×C4—2216no—
213C2×C4—224no—
214C2×C4—224no—
215C2×C4—224no—
216C2×C4D4×D1114yes—
217C2×C4C2×C22⋊D1112yes—
218C2×C4C2×C22⋊D1112yes—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—42no—
223C22—42no—
224C22—224no—
225C22—224no—
226C22—224no—
227C22—442no—
228C22—442no—
229C22—442no—
230C22—442no—
231C22C2×D4×D1112yes—
232C22#6704012yes—
233C22#6704211yescharacteristic
234C22#6704311yescharacteristic
235C22C22×C22⋊D1111yescharacteristic
236C4—22no—
237C4—42no—
238C4—42no—
239C4—224no—
240C4—444no—
241C4#6699014yes—
242C2—42no—
243C2—444no—
244C2#1062315912yes—
245C2#1062316012yes—
246C2C2×#6699011yescharacteristic
247C2#1062334711yescharacteristic
248C2#1062336411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC2×C11×C22⋊Q8C2
prime radicalC22×#4339C2
hypercenterC2×C22⋊Q8D11
hypocenterC11#2616
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C11×C22⋊Q8C2semidirect
3. C2×#66999C2semidirect
4. #10622825C2semidirect
5. #10622604C2semidirect
6. #10622600C2semidirect
7. #10622592C2semidirect
8. #10622572C2semidirect
97. C2×C22⋊Q8D11semidirect
184. C11#2616semidirect

Maximal subgroups

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

subgroupindex
C22×#43392
C2×C11×C22⋊Q82
C2×#669992
#106228252
#106226042
#106226002
#106225922
#106225722
#261611