Finite Group Databasebeta

← #14646408

Subgroups — #14646408

Order 2816.

1154 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 123 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×#67000C211yesmaximal, characteristic
2#13068540C211yesmaximal, characteristic, complemented
3#13066942C214yesmaximal, complemented
4C11×#1784C211yesmaximal, characteristic, complemented
5C23×Q8×C11C2211yescharacteristic
6C23×D44C411yescharacteristic
7C2×#67000—22no—
8C2×#67000—22no—
9C2×#67000—28no—
10C22×(C2×D11)⋊C4C412yes—
11C22×(C2×D11)⋊C4C2212yes—
12C22×C4×Dic11C411yescharacteristic
13C2×C11×#101—22no—
14C11×#407C2212yes—
15C2×#66900—22no—
16#10622820C2212yes—
17#10622106—28no—
18C24×D11C2×C412yes—
19C22×Q8×C11—24no—
20C22×Q8×C11—44no—
21C22×Q8×C11D412yes—
22C22×Q8×C11D412yes—
23C23×C4×C11C2×C411yescharacteristic
24C23×C4×C11C2×C411yescharacteristic
25C23×C4×C11C2311yescharacteristic
26C23×Dic11C2×C412yes—
27C22×D44—24no—
28C22×D44—22no—
29C22×D44—44no—
30C2×C11×M16—22no—
31#67000—216no—
32#67000—216no—
33C2×(C2×D11)⋊C4—24no—
34C2×(C2×D11)⋊C4—24no—
35C2×(C2×D11)⋊C4—22no—
36C2×(C2×D11)⋊C4—24no—
37C2×(C2×D11)⋊C4—22no—
38C2×(C2×D11)⋊C4—24no—
39C2×(C2×D11)⋊C4D418yes—
40C2×C4×Dic11—22no—
41C2×C4×Dic11—24no—
42C2×#4318—22no—
43C11×#101—44no—
44C11×C22⋊C8—24no—
45#66900—44no—
46C22⋊(C11⋊C8)—24no—
47#8552—111nomaximal, Sylow
48C24×C11C22×C411yescharacteristic
49C23×D11—22no—
50C23×D11—22no—
51C23×D11—44no—
52C23×D11—44no—
53C23×D11—44no—
54C23×D11C22⋊C414yes—
55C2×Q8×C11—24no—
56C2×Q8×C11—24no—
57C2×Q8×C11—24no—
58C2×Q8×C11—24no—
59C2×Q8×C11—42no—
60C2×Q8×C11—42no—
61C2×Q8×C11—416no—
62C22×C4×C11—22no—
63C22×C4×C11—24no—
64C22×C4×C11—22no—
65C22×C4×C11—42no—
66C22×C4×C11—42no—
67C22×C4×C11C22⋊C412yes—
68C22×C4×C11C22⋊C412yes—
69C22×C4×C11C2×D411yescharacteristic
70C22×C4×C11C2×D411yescharacteristic
71C22×C4×C11C2×D414yes—
72C22×Dic11—22no—
73C22×Dic11—22no—
74C22×Dic11—44no—
75C22×Dic11C22⋊C414yes—
76C2×D44—28no—
77C2×D44—416no—
78C2×D44—44no—
79C2×D44—44no—
80C11×M16—44no—
81C2×C8×C11—42no—
82(C2×D11)⋊C4—216no—
83(C2×D11)⋊C4—28no—
84(C2×D11)⋊C4—28no—
85C4×Dic11—28no—
86#4318—44no—
87C2×C11⋊C8—42no—
88C22×#124—111no—
89#1784—111no—
90#1784D1111yescharacteristic, complemented
91#1462—114no—
92C23×C11—22no—
93C23×C11—24no—
94C23×C11C2×C22⋊C412yes—
95C23×C11C2×C22⋊C411yescharacteristic
96C22×D11—24no—
97C22×D11—28no—
98C22×D11—44no—
99C22×D11—44no—
100C22×D11—44no—
101C22×D11—44no—
102C22×D11—44no—
103C22×D11—44no—
104C22×D11—44no—
105C22×D11—44no—
106C22×D11—816no—
107Q8×C11—48no—
108Q8×C11—48no—
109C2×C4×C11—28no—
110C2×C4×C11—22no—
111C2×C4×C11—22no—
112C2×C4×C11—22no—
113C2×C4×C11—22no—
114C2×C4×C11—22no—
115C2×C4×C11—22no—
116C2×C4×C11—22no—
117C2×C4×C11—44no—
118C2×C4×C11—44no—
119C2×C4×C11#9914yes—
120C2×C4×C11#12012yes—
121C2×C4×C11#12012yes—
122C2×Dic11—24no—
123C2×Dic11—44no—
124C2×Dic11—44no—
125C2×Dic11—44no—
126D44—416no—
127C8×C11—82no—
128C11⋊C8—82no—
129C23×Q8C2×D1111yescharacteristic
130C23×D4—111no—
131C2×#124—228no—
132C2×#124—222no—
133C2×#124—222no—
134C22×C22⋊C4—112no—
135C22×C22⋊C4—112no—
136C22×C42—111no—
137C2×#101—22no—
138C2×#101—222no—
139#407—112no—
140#407C2×D1112yes—
141#323—228no—
142C22×C11—28no—
143C22×C11—22no—
144C22×C11—22no—
145C22×C11—22no—
146C22×C11#37911yescharacteristic
147C22×C11C2×#9912yes—
148C22×C11#41314yes—
149C2×D11—44no—
150C2×D11—44no—
151C2×D11—44no—
152C2×D11—88no—
153C2×D11—88no—
154C2×D11—88no—
155C4×C11—24no—
156C4×C11—22no—
157C4×C11—22no—
158C4×C11—42no—
159Dic11—44no—
160C25—112no—
161C22×Q8—24no—
162C22×Q8—44no—
163C22×Q8D4412yes—
164C22×Q8C22⋊D1112yes—
165C22×D4—224no—
166C22×D4—222no—
167C22×D4—444no—
168C23×C4—112no—
169C23×C4C4×D1111yescharacteristic
170C23×C4C4×D1111yescharacteristic
171C23×C4C22×D1111yescharacteristic
172C2×M16—22no—
173C2×M16—222no—
174#124—2216no—
175#124—2216no—
176C2×C22⋊C4—118no—
177C2×C22⋊C4—224no—
178C2×C22⋊C4—224no—
179C2×C22⋊C4—224no—
180C2×C22⋊C4—222no—
181C2×C22⋊C4—224no—
182C2×C22⋊C4—222no—
183C2×C42—224no—
184C2×C42—222no—
185#101—44no—
186#101—444no—
187C22⋊C8—24no—
188C22⋊C8—224no—
189C2×C11—22no—
190C2×C11—24no—
191C2×C11#178111yescharacteristic
192C2×C11#178512yes—
193D11—84no—
194C24—114no—
195C24—222no—
196C24—222no—
197C24—444no—
198C24—444no—
199C24—444no—
200C24C2×C4×D1111yescharacteristic
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—416no—
208C2×D4—228no—
209C2×D4—4416no—
210C2×D4—444no—
211C2×D4—444no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—114no—
218C22×C4—222no—
219C22×C4—222no—
220C22×C4—444no—
221C22×C4(C2×D11)⋊C412yes—
222C22×C4(C2×D11)⋊C412yes—
223C22×C4C2×D4411yescharacteristic
224C22×C4D4×D1114yes—
225C22×C4C2×C22⋊D1111yescharacteristic
226M16—44no—
227M16—444no—
228C2×C8—42no—
229C2×C8—442no—
230C22⋊C4—2216no—
231C22⋊C4—228no—
232C22⋊C4—228no—
233C42—228no—
234C11#855211yescharacteristic, complemented, Sylow
235C23—22no—
236C23—24no—
237C23—224no—
238C23—228no—
239C23—444no—
240C23—444no—
241C23—444no—
242C23—444no—
243C23—444no—
244C23—444no—
245C23—444no—
246C23—444no—
247C23—8816no—
248C23C22⋊C4×D1112yes—
249C23C2×(C2×D11)⋊C411yescharacteristic
250Q8—48no—
251Q8—48no—
252D4—4416no—
253C2×C4—28no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—224no—
264C2×C4—444no—
265C2×C4—444no—
266C2×C4—444no—
267C2×C4#6687014yes—
268C2×C4#6693512yes—
269C2×C4#6699012yes—
270C8—82no—
271C8—882no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—444no—
277C22—444no—
278C22—444no—
279C22—888no—
280C22—888no—
281C22—888no—
282C22#1062236414yes—
283C22C2×#6687012yes—
284C22#1062256811yescharacteristic
285C4—22no—
286C4—22no—
287C4—24no—
288C4—42no—
289C4—444no—
290C2—22no—
291C2—24no—
292C2—884no—
293C2#1306853011yescharacteristic
294C2#1306854312yes—
subgroupG/H
centerC22#10622568
commutatorC24×C11C22×C4
FrattiniC23×C4C22×D11
FittingC11×#1784C2
socleC22×C11#379
prime radicalC23×D44C4
cosocleC23×C4×C11C23
hypercenter#1784D11
hypocenterC11#8552
LCSG ▷ C24×C11 ▷ C22×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. #13068540C2semidirect
3. #13066942C2semidirect
4. C11×#1784C2semidirect
90. #1784D11semidirect
234. C11#8552semidirect

Maximal subgroups

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

subgroupindex
C22×#670002
#130685402
#130669422
C11×#17842
#855211