Finite Group Databasebeta

← #15617225

Subgroups — #15617225

Order 2816.

5134 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623350C212yesmaximal, complemented
2#13077065C2116yesmaximal, complemented
3#13077061C216yesmaximal, complemented
4C2×#10622820C216yesmaximal, complemented
5C23×C11×#41C211yesmaximal, characteristic
6C22×C11×#41—216no—
7C22×C11×#41C418yes—
8C22×C11×#41C411yescharacteristic
9C22×C11×#41C413yes—
10C22×C11×#41C2213yes—
11C22×C11×#41C2218yes—
12C22×C11×#41C2211yescharacteristic
13C23×Q8×C11C411yescharacteristic
14C23×D4×C11C413yes—
15C24×C4×C11C2213yes—
16#10623350—216no—
17C2×#67001C22116yescomplemented
18C22×#4318C22112yescomplemented
19#10622847C22148yescomplemented
20#10622820C2214yes—
21#10622820C22124yescomplemented
22#10622820C22112yescomplemented
23C2×C22⋊(C11⋊C8)C22124yescomplemented
24C25×C11C2×C413yes—
25C2×C11×#41—264no—
26C2×C11×#41—264no—
27C2×C11×#41—28no—
28C2×C11×#41—224no—
29C2×C11×#41—224no—
30C2×C11×#41C2×C418yes—
31C2×C11×#41C2×C4124yes—
32C2×C11×#41C2×C4112yes—
33C2×C11×#41C2×C414yes—
34C2×C11×#41D4116yes—
35C2×C11×#41C23112yes—
36C2×C11×#41C2314yes—
37C22×Q8×C11—28no—
38C22×Q8×C11C2×C418yes—
39C22×Q8×C11C2×C413yes—
40C22×Q8×C11C2×C411yescharacteristic
41C22×D4×C11—224no—
42C22×D4×C11C2×C416yes—
43C22×D4×C11C2×C4124yes—
44C22×D4×C11C2×C413yes—
45C22×D4×C11C2×C413yes—
46C23×C4×C11—224no—
47C23×C4×C11C2×C413yes—
48C23×C4×C11C2×C413yes—
49C23×C4×C11C2×C4112yes—
50C23×C4×C11C2×C413yes—
51C23×C4×C11C23112yes—
52C23×C4×C11C2311yescharacteristic
53C23×C4×C11C2313yes—
54C23×C4×C11C2313yes—
55#67001—264no—
56C2×#4318—248no—
57C2×#4318C2318yes—
58C2×#4318C23124yescomplemented
59C22×C11⋊C8C23124yescomplemented
60C22⋊(C11⋊C8)C23116yes—
61C22⋊(C11⋊C8)C23148yescomplemented
62#60064—111nomaximal, Sylow
63C24×C11—212no—
64C24×C11—412no—
65C24×C11C22×C411yescharacteristic
66C24×C11C22×C4112yes—
67C24×C11C22×C413yes—
68C24×C11C22×C413yes—
69C11×#41—232no—
70C11×#41—296no—
71C11×#41—248no—
72C11×#41—248no—
73C11×#41C22⋊C4132yes—
74C11×#41C2×D4132yes—
75C2×Q8×C11—232no—
76C2×Q8×C11—212no—
77C2×Q8×C11C22⋊C418yes—
78C2×Q8×C11C22×C4112yes—
79C2×Q8×C11C22×C414yes—
80C2×D4×C11—224no—
81C2×D4×C11—296no—
82C2×D4×C11—212no—
83C2×D4×C11C22⋊C4124yes—
84C2×D4×C11C22×C4124yes—
85C2×D4×C11C22×C4112yes—
86C2×D4×C11C22×C4112yes—
87C22×C4×C11—212no—
88C22×C4×C11—212no—
89C22×C4×C11—212no—
90C22×C4×C11—248no—
91C22×C4×C11—24no—
92C22×C4×C11—212no—
93C22×C4×C11—248no—
94C22×C4×C11C22×C411yescharacteristic
95C22×C4×C11C22×C412yes—
96C22×C4×C11C22×C413yes—
97C22×C4×C11C22×C4112yes—
98C22×C4×C11C22×C4112yes—
99C22×C4×C11C22×C4112yes—
100C22×C4×C11C22×C416yes—
101C22×C4×C11C22×C413yes—
102C22×C4×C11C22×C416yes—
103C22×C4×C11C2×D4124yes—
104C22×C4×C11C2412yes—
105C22×C4×C11C2416yes—
106C22×C4×C11C2416yes—
107C22×C4×C11C2411yescharacteristic
108#4318—216no—
109#4318—248no—
110C2×C11⋊C8—248no—
111C2×C11⋊C8C24116yes—
112C23×#41D1111yescharacteristic
113C2×#568—112no—
114#2880—1116no—
115#2879—116no—
116C2×#407—116no—
117C23×C11—22no—
118C23×C11—24no—
119C23×C11—224no—
120C23×C11—26no—
121C23×C11—224no—
122C23×C11—46no—
123C23×C11—412no—
124C23×C11—46no—
125C23×C11—448no—
126C23×C11C2×C22⋊C4112yes—
127C23×C11C23×C416yes—
128C23×C11C23×C411yescharacteristic
129C23×C11C23×C416yes—
130C23×C11C23×C412yes—
131Q8×C11—224no—
132Q8×C11C2×C22⋊C4116yes—
133D4×C11—248no—
134D4×C11—224no—
135D4×C11C2×C22⋊C4148yes—
136C2×C4×C11—22no—
137C2×C4×C11—26no—
138C2×C4×C11—224no—
139C2×C4×C11—24no—
140C2×C4×C11—224no—
141C2×C4×C11—224no—
142C2×C4×C11—212no—
143C2×C4×C11—24no—
144C2×C4×C11—212no—
145C2×C4×C11C2×C22⋊C4112yes—
146C2×C4×C11C2×C22⋊C4124yes—
147C2×C4×C11C23×C412yes—
148C2×C4×C11C23×C416yes—
149C2×C4×C11C23×C411yescharacteristic
150C2×C4×C11C23×C416yes—
151C2×C4×C11C22×D4124yes—
152C2×C4×C11C22×D414yes—
153C2×C4×C11C2511yescharacteristic
154C11⋊C8—216no—
155C22×#41—216no—
156C22×#41Dic1118yes—
157C22×#41Dic1111yescharacteristic
158C22×#41Dic1113yes—
159C22×#41C2×D1113yes—
160C22×#41C2×D1118yes—
161C22×#41C2×D1111yescharacteristic
162C23×Q8Dic1111yescharacteristic
163C23×D4Dic1113yes—
164C24×C4C2×D1113yes—
165#568—2216no—
166C2×#131—1116no—
167C22×M16—1112no—
168#408—1148no—
169#407—114no—
170#407—1112no—
171#407—1124no—
172C2×C22⋊C8—1124no—
173C22×C11—28no—
174C22×C11—22no—
175C22×C11—22no—
176C22×C11—22no—
177C22×C11—212no—
178C22×C11—26no—
179C22×C11—212no—
180C22×C11—412no—
181C22×C11—448no—
182C22×C11—412no—
183C22×C11C22×C22⋊C412yes—
184C22×C11C22×C22⋊C4112yes—
185C22×C11#56814yescomplemented
186C22×C11C24×C411yescharacteristic
187C4×C11—22no—
188C4×C11—26no—
189C4×C11C22×C22⋊C412yes—
190C4×C11C22×C22⋊C4112yes—
191C4×C11C23×D412yes—
192C25C2×Dic1113yes—
193C2×#41—264no—
194C2×#41—264no—
195C2×#41—28no—
196C2×#41—224no—
197C2×#41—224no—
198C2×#41C2×Dic1118yes—
199C2×#41C2×Dic11124yes—
200C2×#41C2×Dic11112yes—
201C2×#41C2×Dic1114yes—
202C2×#41C22⋊D11116yes—
203C2×#41C22×D11112yes—
204C2×#41C22×D1114yes—
205C22×Q8—28no—
206C22×Q8C2×Dic1118yes—
207C22×Q8C2×Dic1113yes—
208C22×Q8C2×Dic1111yescharacteristic
209C22×D4—224no—
210C22×D4C2×Dic1116yes—
211C22×D4C2×Dic11124yes—
212C22×D4C2×Dic1113yes—
213C22×D4C2×Dic1113yes—
214C23×C4—224no—
215C23×C4C2×Dic1113yes—
216C23×C4C2×Dic1113yes—
217C23×C4C2×Dic11112yes—
218C23×C4C2×Dic1113yes—
219C23×C4C22×D1111yescharacteristic
220C23×C4C22×D11112yes—
221C23×C4C22×D1113yes—
222C23×C4C22×D1113yes—
223#131—2264no—
224C2×M16—118no—
225C2×M16—1124no—
226C2×M16—2248no—
227C22×C8—1124no—
228C22⋊C8—1116no—
229C22⋊C8—1148no—
230C2×C11—22no—
231C2×C11—24no—
232C2×C11—212no—
233C2×C11—46no—
234C2×C11C23×C22⋊C411yescharacteristic
235C2×C11C2×#56812yes—
236C24—212no—
237C24—412no—
238C24C22×Dic1111yescharacteristic
239C24C22×Dic11112yes—
240C24C22×Dic1113yes—
241C24C22×Dic1113yes—
242#41—232no—
243#41—296no—
244#41—248no—
245#41—248no—
246#41C22⋊Dic11132yes—
247#41C2×C22⋊D11132yes—
248C2×Q8—232no—
249C2×Q8—212no—
250C2×Q8C22⋊Dic1118yes—
251C2×Q8C22×Dic11112yes—
252C2×Q8C22×Dic1114yes—
253C2×D4—224no—
254C2×D4—296no—
255C2×D4—212no—
256C2×D4C22⋊Dic11124yes—
257C2×D4C22×Dic11124yes—
258C2×D4C22×Dic11112yes—
259C2×D4C22×Dic11112yes—
260C22×C4—212no—
261C22×C4—212no—
262C22×C4—212no—
263C22×C4—248no—
264C22×C4—24no—
265C22×C4—212no—
266C22×C4—248no—
267C22×C4C22×Dic1111yescharacteristic
268C22×C4C22×Dic1112yes—
269C22×C4C22×Dic1113yes—
270C22×C4C22×Dic11112yes—
271C22×C4C22×Dic11112yes—
272C22×C4C22×Dic11112yes—
273C22×C4C22×Dic1116yes—
274C22×C4C22×Dic1113yes—
275C22×C4C22×Dic1116yes—
276C22×C4C2×C22⋊D11124yes—
277C22×C4C23×D1112yes—
278C22×C4C23×D1116yes—
279C22×C4C23×D1116yes—
280C22×C4C23×D1111yescharacteristic
281M16—2216no—
282M16—2248no—
283C2×C8—1116no—
284C2×C8—2248no—
285C11#6006411yescharacteristic, complemented, Sylow
286C23—24no—
287C23—22no—
288C23—224no—
289C23—224no—
290C23—26no—
291C23—46no—
292C23—412no—
293C23—46no—
294C23—448no—
295C23C2×C22⋊Dic11112yes—
296C23C23×Dic1116yes—
297C23C23×Dic1112yes—
298C23C23×Dic1111yescharacteristic
299C23C23×Dic1116yes—
300Q8—224no—
301Q8C2×C22⋊Dic11116yes—
302D4—248no—
303D4—224no—
304D4C2×C22⋊Dic11148yes—
305C2×C4—22no—
306C2×C4—26no—
307C2×C4—224no—
308C2×C4—24no—
309C2×C4—224no—
310C2×C4—224no—
311C2×C4—212no—
312C2×C4—24no—
313C2×C4—212no—
314C2×C4C2×C22⋊Dic11112yes—
315C2×C4C2×C22⋊Dic11124yes—
316C2×C4C23×Dic1112yes—
317C2×C4C23×Dic1116yes—
318C2×C4C23×Dic1111yescharacteristic
319C2×C4C23×Dic1116yes—
320C2×C4C22×C22⋊D11124yes—
321C2×C4C22×C22⋊D1114yes—
322C2×C4C24×D1111yescharacteristic
323C8—2216no—
324C22—28no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—212no—
329C22—26no—
330C22—212no—
331C22—412no—
332C22—448no—
333C22—412no—
334C22#1062335014yescomplemented
335C22C22×C22⋊Dic1112yes—
336C22C22×C22⋊Dic11112yes—
337C22C24×Dic1111yescharacteristic
338C4—22no—
339C4—26no—
340C4C22×C22⋊Dic1112yes—
341C4C22×C22⋊Dic11112yes—
342C4C23×C22⋊D1112yes—
343C2—24no—
344C2—22no—
345C2—212no—
346C2—46no—
347C2C23×C22⋊Dic1111yescharacteristic
348C2C2×#1062335012yes—
subgroupG/H
center, FrattiniC2×C4C24×D11
commutator, socleC22×C11C24×C4
Fitting, prime radicalC23×C11×#41C2
cosocleC2×C4×C11C25
hypercenterC23×#41D11
hypocenterC11#60064
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C2×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C2×#10623350C2semidirect
2. #13077065C2semidirect
3. #13077061C2semidirect
4. C2×#10622820C2semidirect
17. C2×#67001C22semidirect
18. C22×#4318C22semidirect
19. #10622847C22semidirect
21. #10622820C22semidirect
22. #10622820C22semidirect
23. C2×C22⋊(C11⋊C8)C22semidirect
58. C2×#4318C23semidirect
59. C22×C11⋊C8C23semidirect
61. C22⋊(C11⋊C8)C23semidirect
185. C22×C11#568semidirect
285. C11#60064semidirect
334. C22#10623350semidirect

Maximal subgroups

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

subgroupindex
C2×#106233502
#130770652
#130770612
C2×#106228202
C23×C11×#412
#6006411