Finite Group Databasebeta

← #14669211

Subgroups — #14669211

Order 2816.

1038 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C23×#4313C211yesmaximal, characteristic
2C2×#10622152C214yesmaximal, complemented
3#13070227C211yesmaximal, characteristic, complemented
4C11×#2112C211yesmaximal, characteristic, complemented
5C23×C4×D11C411yescharacteristic
6C23×C8×C11C2211yescharacteristic
7C23×C11⋊C8C411yescharacteristic
8C22×#4313—22no—
9C22×#4313—22no—
10C22×#4313—28no—
11C22×#4313C412yes—
12C22×#4313C2212yes—
13C2×C11×M32C2212yescomplemented
14C2×#66876C2212yescomplemented
15C11×C22⋊C16—22no—
16C22⋊(C11⋊C16)—22no—
17#10622152—28no—
18C24×D11C811yescharacteristic
19C23×C4×C11C2×C411yescharacteristic
20C23×Dic11C811yescharacteristic
21C22×C4×D11—22no—
22C22×C4×D11—22no—
23C22×C4×D11—24no—
24C22×C4×D11—24no—
25C22×C4×D11C812yes—
26C22×C4×D11C2×C412yes—
27C22×C8×C11—24no—
28C22×C8×C11C2×C411yescharacteristic
29C22×C8×C11D412yes—
30C22×C8×C11D412yes—
31C22×C8×C11C2311yescharacteristic
32C22×C11⋊C8—22no—
33C22×C11⋊C8—24no—
34C22×C11⋊C8C2×C412yes—
35C2×#4313—216no—
36C2×#4313—216no—
37C2×#4313—24no—
38C2×#4313—24no—
39C2×#4313—22no—
40C2×#4313—24no—
41C2×#4313—22no—
42C2×#4313—24no—
43C2×#4313D418yes—
44C11×M32—24no—
45C2×C11×C16—22no—
46#66876—24no—
47C2×C11⋊C16—22no—
48#12681—111nomaximal, Sylow
49C24×C11C2×C811yescharacteristic
50C23×D11—22no—
51C23×D11—24no—
52C23×D11—44no—
53C23×D11C2×C812yes—
54C22×C4×C11—22no—
55C22×C4×C11—22no—
56C22×C4×C11C22⋊C412yes—
57C22×C4×C11C2×C811yescharacteristic
58C22×C4×C11M1612yes—
59C22×C4×C11C22×C411yescharacteristic
60C22×Dic11—24no—
61C22×Dic11—22no—
62C22×Dic11C2×C812yes—
63C2×C4×D11—28no—
64C2×C4×D11—28no—
65C2×C4×D11—216no—
66C2×C4×D11—24no—
67C2×C4×D11—24no—
68C2×C4×D11—24no—
69C2×C4×D11—22no—
70C2×C4×D11—22no—
71C2×C4×D11—22no—
72C2×C4×D11—22no—
73C2×C4×D11C22⋊C414yes—
74C2×C4×D11M1614yes—
75C2×C8×C11—24no—
76C2×C8×C11—24no—
77C2×C8×C11—22no—
78C2×C8×C11C22⋊C412yes—
79C2×C8×C11C2×D411yescharacteristic
80C2×C8×C11C2×D414yes—
81C2×C8×C11C2×D411yescharacteristic
82C2×C11⋊C8—28no—
83C2×C11⋊C8—22no—
84C2×C11⋊C8—22no—
85C2×C11⋊C8C22⋊C414yes—
86#4313—216no—
87#4313—28no—
88#4313—28no—
89C11×C16—42no—
90C11⋊C16—42no—
91C23×M16—111no—
92C2×#349—114no—
93#2112—111no—
94#2112D1111yescharacteristic, complemented
95C23×C11—24no—
96C23×C11—22no—
97C23×C11C22⋊C812yes—
98C23×C11C22×C811yescharacteristic
99C22×D11—28no—
100C22×D11—22no—
101C22×D11—22no—
102C22×D11—416no—
103C22×D11—44no—
104C22×D11—44no—
105C22×D11—44no—
106C22×D11C22⋊C814yes—
107C2×C4×C11—22no—
108C2×C4×C11—22no—
109C2×C4×C11—22no—
110C2×C4×C11—24no—
111C2×C4×C11C22⋊C812yes—
112C2×C4×C11C2×C22⋊C412yes—
113C2×C4×C11C2×C22⋊C411yescharacteristic
114C2×C4×C11C2×M1611yescharacteristic
115C2×C4×C11C2×M1612yes—
116C2×Dic11—28no—
117C2×Dic11—22no—
118C2×Dic11—22no—
119C2×Dic11C22⋊C814yes—
120C4×D11—216no—
121C4×D11—24no—
122C4×D11—24no—
123C4×D11—24no—
124C4×D11—24no—
125C8×C11—22no—
126C8×C11#12012yes—
127C8×C11#12012yes—
128C11⋊C8—24no—
129C24×C4—111no—
130C22×M16—112no—
131C22×M16—112no—
132C22×M16—228no—
133C22×M16—222no—
134C22×M16—222no—
135C23×C8—111no—
136C23×C8C2×D1111yescharacteristic
137C2×M32—112no—
138C2×M32C2×D1112yescomplemented
139#349—228no—
140C22⋊C16—22no—
141C22⋊C16—222no—
142C22×C11—28no—
143C22×C11—22no—
144C22×C11—22no—
145C22×C11—22no—
146C22×C11#34914yescomplemented
147C22×C11C2×C22⋊C811yescharacteristic
148C22×C11C2×C22⋊C812yes—
149C2×D11—24no—
150C2×D11—416no—
151C2×D11—44no—
152C2×D11—44no—
153C2×D11—44no—
154C4×C11—22no—
155C4×C11#37911yescharacteristic
156C4×C11#40711yescharacteristic
157C4×C11#40712yes—
158Dic11—24no—
159C25—111no—
160C23×C4—112no—
161C23×C4—111no—
162C23×C4—112no—
163C23×C4—224no—
164C23×C4—224no—
165C23×C4—222no—
166C23×C4—222no—
167C23×C4C4×D1111yescharacteristic
168C2×M16—118no—
169C2×M16—224no—
170C2×M16—224no—
171C2×M16—2216no—
172C2×M16—2216no—
173C2×M16—224no—
174C2×M16—222no—
175C2×M16—224no—
176C2×M16—222no—
177C22×C8—24no—
178C22×C8—112no—
179C22×C8—224no—
180C22×C8—222no—
181C22×C8C4×D1111yescharacteristic
182C22×C8D4412yes—
183C22×C8C22⋊D1112yes—
184C22×C8C22×D1111yescharacteristic
185M32—24no—
186M32—224no—
187C2×C16—22no—
188C2×C16—222no—
189C2×C11—24no—
190C2×C11—22no—
191C2×C11#177911yescharacteristic
192C2×C11C2×#34912yes—
193D11—44no—
194C24—112no—
195C24—224no—
196C24—222no—
197C24—444no—
198C24C8×D1111yescharacteristic
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—112no—
202C22×C4—114no—
203C22×C4—114no—
204C22×C4—224no—
205C22×C4—224no—
206C22×C4—228no—
207C22×C4—228no—
208C22×C4—2216no—
209C22×C4—224no—
210C22×C4—222no—
211C22×C4—224no—
212C22×C4—222no—
213C22×C4—222no—
214C22×C4—222no—
215C22×C4—222no—
216C22×C4C8×D1111yescharacteristic
217C22×C4#431312yes—
218C22×C4(C2×D11)⋊C412yes—
219C22×C4C2×C4×D1111yescharacteristic
220M16—2216no—
221M16—228no—
222M16—228no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8—114no—
227C2×C8—228no—
228C2×C8—222no—
229C2×C8—222no—
230C2×C8(C2×D11)⋊C412yes—
231C2×C8C2×D4411yescharacteristic
232C2×C8D4×D1114yes—
233C2×C8C2×C22⋊D1111yescharacteristic
234C16—42no—
235C16—442no—
236C11#1268111yescharacteristic, complemented, Sylow
237C23—22no—
238C23—24no—
239C23—114no—
240C23—228no—
241C23—222no—
242C23—222no—
243C23—444no—
244C23—4416no—
245C23—444no—
246C23—444no—
247C23(C2×D11)⋊C812yes—
248C23C2×C8×D1111yescharacteristic
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—114no—
254C2×C4—2216no—
255C2×C4—228no—
256C2×C4—222no—
257C2×C4—222no—
258C2×C4—224no—
259C2×C4—224no—
260C2×C4—224no—
261C2×C4—224no—
262C2×C4(C2×D11)⋊C812yes—
263C2×C4C22⋊C4×D1112yes—
264C2×C4C2×#431311yescharacteristic
265C2×C4M16×D1112yes—
266C2×C4C2×(C2×D11)⋊C411yescharacteristic
267C8—22no—
268C8—224no—
269C8#6693512yes—
270C8#6699012yes—
271C22—28no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—224no—
276C22—444no—
277C22—444no—
278C22—4416no—
279C22—444no—
280C22#1062215214yescomplemented
281C22D11×C22⋊C812yes—
282C22C2×(C2×D11)⋊C811yescharacteristic
283C4—22no—
284C4—224no—
285C4#1062233912yes—
286C4#1062256811yescharacteristic
287C4#1062273911yescharacteristic
288C2—22no—
289C2—24no—
290C2—444no—
291C2#1306852511yescharacteristic
292C2C2×#1062215212yes—
subgroupG/H
centerC2×C4C2×(C2×D11)⋊C4
commutatorC23×C11C22×C8
FrattiniC22×C8C22×D11
FittingC11×#2112C2
socleC22×C11C2×C22⋊C8
prime radicalC24×D11C8
cosocleC22×C8×C11C23
hypercenter#2112D11
hypocenterC11#12681
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. C2×#10622152C2semidirect
3. #13070227C2semidirect
4. C11×#2112C2semidirect
13. C2×C11×M32C22semidirect
14. C2×#66876C22semidirect
94. #2112D11semidirect
138. C2×M32C2×D11semidirect
146. C22×C11#349semidirect
236. C11#12681semidirect
280. C22#10622152semidirect

Maximal subgroups

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

subgroupindex
C23×#43132
C2×#106221522
#130702272
C11×#21122
#1268111