Finite Group Databasebeta

← C22×#10622737 #15617465

Subgroups — C22×#10622737 #15617465

Order 2816.

2718 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4318C211yesmaximal, characteristic
2C23×#4318C211yesmaximal, characteristic
3C2×#10622737C2124yesmaximal, direct product factor
4C22×#66886C214yesmaximal, complemented
5C23×C11×M16C211yesmaximal, characteristic
6C24×C4×C11C2211yescharacteristic
7C22×C11×M16—21no—
8C22×C11×M16—26no—
9C22×C11×M16C2212yes—
10C22×C11×M16C22112yes—
11C23×C8×C11—21no—
12C22×#4318—21no—
13C22×#4318—26no—
14C22×#4318C412yes—
15C22×#4318C412yes—
16C22×#4318C4112yes—
17C22×#4318C2212yes—
18C22×#4318C22112yes—
19C22×#4318C22112yes—
20C23×C11⋊C8—21no—
21C23×C11⋊C8C2212yes—
22#10622737C22164yesdirect product factor
23C2×#66886C22148yescomplemented
24C25×C11Q811yescharacteristic
25C23×C4×C11C2×C411yescharacteristic
26C23×C4×C11C2×C411yescharacteristic
27C23×C4×C11C2×C416yes—
28C23×C4×C11D411yescharacteristic
29C23×C4×C11D411yescharacteristic
30C23×C4×C11D416yes—
31C23×C4×C11Q811yescharacteristic
32C23×C4×C11Q816yes—
33C23×C4×C11C2311yescharacteristic
34C23×C4×C11C2316yes—
35C2×C11×M16—212no—
36C2×C11×M16—26no—
37C2×C11×M16—224no—
38C2×C11×M16C23112yes—
39C2×C11×M16C23116yes—
40C22×C8×C11—212no—
41C22×C8×C11—22no—
42C2×#4318—212no—
43C2×#4318—26no—
44C2×#4318—224no—
45C2×#4318C2×C4124yes—
46C2×#4318C2×C4124yes—
47C2×#4318C2×C4148yes—
48C2×#4318C23112yes—
49C2×#4318C23116yes—
50C2×#4318C23116yes—
51C22×C11⋊C8—212no—
52C22×C11⋊C8—22no—
53C22×C11⋊C8C2×C4112yes—
54C22×C11⋊C8C2×C414yes—
55C22×C11⋊C8C23112yes—
56#66886C23164yescomplemented
57C22×#430—111nomaximal, Sylow
58C24×C11—44no—
59C24×C11C4⋊C411yescharacteristic
60C24×C11C4⋊C411yescharacteristic
61C24×C11C4⋊C416yes—
62C24×C11C2×Q811yescharacteristic
63C24×C11C2×Q816yes—
64C22×C4×C11C4⋊C416yes—
65C22×C4×C11C4⋊C416yes—
66C22×C4×C11C4⋊C416yes—
67C22×C4×C11C4⋊C416yes—
68C22×C4×C11C4⋊C416yes—
69C22×C4×C11C4⋊C4124yes—
70C22×C4×C11C4⋊C411yescharacteristic
71C22×C4×C11C4⋊C411yescharacteristic
72C22×C4×C11C22×C413yes—
73C22×C4×C11C22×C411yescharacteristic
74C22×C4×C11C22×C416yes—
75C22×C4×C11C22×C416yes—
76C22×C4×C11C22×C4112yes—
77C22×C4×C11C2×D416yes—
78C22×C4×C11C2×D416yes—
79C22×C4×C11C2×D413yes—
80C22×C4×C11C2×D4112yes—
81C22×C4×C11C2×D411yescharacteristic
82C22×C4×C11C2×Q813yes—
83C22×C4×C11C2×Q816yes—
84C22×C4×C11C2×Q8112yes—
85C22×C4×C11C2413yes—
86C22×C4×C11C2414yes—
87C11×M16—216no—
88C11×M16—212no—
89C11×M16C2418yes—
90C2×C8×C11—212no—
91C2×C8×C11—216no—
92#4318—216no—
93#4318—212no—
94#4318C22×C4132yes—
95#4318C22×C4132yes—
96#4318C2418yes—
97C2×C11⋊C8—212no—
98C2×C11⋊C8—216no—
99C2×C11⋊C8C22×C4124yes—
100C2×C11⋊C8C22×C4124yes—
101C2×C11⋊C8C2418yes—
102C23×M16—111no—
103C23×M16—111no—
104C23×M16D1111yescharacteristic
105C2×#430—1124no—
106C22×#108—114no—
107C23×C11—24no—
108C23×C11—24no—
109C23×C11—24no—
110C23×C11—412no—
111C23×C11—412no—
112C23×C11C2×C4⋊C413yes—
113C23×C11C2×C4⋊C416yes—
114C23×C11C2×C4⋊C416yes—
115C23×C11C2×C4⋊C4112yes—
116C23×C11C2×C4⋊C411yescharacteristic
117C23×C11C22×Q813yes—
118C23×C11C22×Q814yes—
119C2×C4×C11C2×C4⋊C416yes—
120C2×C4×C11C2×C4⋊C4112yes—
121C2×C4×C11C2×C4⋊C4112yes—
122C2×C4×C11C2×C4⋊C4112yes—
123C2×C4×C11C2×C4⋊C4112yes—
124C2×C4×C11C2×C4⋊C4112yes—
125C2×C4×C11C2×C4⋊C413yes—
126C2×C4×C11C2×C4⋊C416yes—
127C2×C4×C11C2×C4⋊C416yes—
128C2×C4×C11C2×C4⋊C413yes—
129C2×C4×C11C23×C413yes—
130C2×C4×C11C23×C413yes—
131C2×C4×C11C23×C414yes—
132C2×C4×C11C23×C414yes—
133C2×C4×C11C22×D413yes—
134C2×C4×C11C22×D414yes—
135C2×C4×C11C22×D414yes—
136C2×C4×C11C22×D413yes—
137C2×C4×C11C22×Q813yes—
138C2×C4×C11C22×Q814yes—
139C2×C4×C11C2511yescharacteristic
140C8×C11—28no—
141C11⋊C8—28no—
142C11⋊C8C23×C4116yes—
143C24×C4C2×D1111yescharacteristic
144C22×M16—26no—
145C22×M16—21no—
146C22×M16—1112no—
147C22×M16—112no—
148C22×M16—112no—
149C22×M16—1112no—
150C22×M16—112no—
151C22×M16—1112no—
152C22×M16—226no—
153C22×M16—221no—
154C22×M16C2×D1112yes—
155C22×M16C2×D11112yes—
156C23×C8—21no—
157C23×C8—112no—
158C23×C8—221no—
159#430—1164no—
160C2×#108—1148no—
161C22×C11—212no—
162C22×C11—212no—
163C22×C11—26no—
164C22×C11—26no—
165C22×C11—412no—
166C22×C11—44no—
167C22×C11#43014yescomplemented
168C22×C11C22×C4⋊C413yes—
169C22×C11C22×C4⋊C414yes—
170C22×C11C22×C4⋊C414yes—
171C22×C11C22×C4⋊C413yes—
172C22×C11C23×Q811yescharacteristic
173C4×C11C22×C4⋊C413yes—
174C4×C11C22×C4⋊C413yes—
175C4×C11C22×C4⋊C414yes—
176C4×C11C22×C4⋊C414yes—
177C4×C11C24×C411yescharacteristic
178C4×C11C23×D411yescharacteristic
179C25Dic2211yescharacteristic
180C23×C4Dic2211yescharacteristic
181C23×C4Dic2216yes—
182C23×C4C4×D1111yescharacteristic
183C23×C4C4×D1111yescharacteristic
184C23×C4C4×D1116yes—
185C23×C4C22⋊D1116yes—
186C23×C4C22⋊D1111yescharacteristic
187C23×C4C22⋊D1111yescharacteristic
188C23×C4C22×D1116yes—
189C23×C4C22×D1111yescharacteristic
190C2×M16—212no—
191C2×M16—26no—
192C2×M16—224no—
193C2×M16—1124no—
194C2×M16—1124no—
195C2×M16—1148no—
196C2×M16—1116no—
197C2×M16—1112no—
198C2×M16—1116no—
199C2×M16—2212no—
200C2×M16—226no—
201C2×M16—2224no—
202C2×M16C22×D11112yes—
203C2×M16C22×D11116yes—
204C22×C8—212no—
205C22×C8—22no—
206C22×C8—114no—
207C22×C8—1112no—
208C22×C8—1112no—
209C22×C8—2212no—
210C22×C8—222no—
211#108—1164no—
212C2×C11—24no—
213C2×C11—24no—
214C2×C11—23no—
215C2×C11—21no—
216C2×C11C2×#43016yescomplemented
217C2×C11C23×C4⋊C411yescharacteristic
218C24—44no—
219C24Dic11⋊C411yescharacteristic
220C24Dic11⋊C411yescharacteristic
221C24Dic11⋊C416yes—
222C24C2×Dic2211yescharacteristic
223C24C2×Dic2216yes—
224C22×C4Dic11⋊C416yes—
225C22×C4Dic11⋊C416yes—
226C22×C4Dic11⋊C416yes—
227C22×C4Dic11⋊C4124yes—
228C22×C4Dic11⋊C411yescharacteristic
229C22×C4Dic11⋊C411yescharacteristic
230C22×C4Dic11⋊C416yes—
231C22×C4Dic11⋊C416yes—
232C22×C4C2×Dic2216yes—
233C22×C4C2×Dic22112yes—
234C22×C4C2×Dic2213yes—
235C22×C4C2×C4×D1111yescharacteristic
236C22×C4C2×C4×D1113yes—
237C22×C4C2×C4×D1116yes—
238C22×C4C2×C4×D1116yes—
239C22×C4C2×C4×D11112yes—
240C22×C4C2×C22⋊D1116yes—
241C22×C4C2×C22⋊D1116yes—
242C22×C4C2×C22⋊D1113yes—
243C22×C4C2×C22⋊D11112yes—
244C22×C4C2×C22⋊D1111yescharacteristic
245C22×C4C23×D1113yes—
246C22×C4C23×D1114yes—
247M16—216no—
248M16—212no—
249M16—1132no—
250M16—1132no—
251M16—118no—
252M16—2216no—
253M16—2212no—
254M16C23×D1118yes—
255C2×C8—212no—
256C2×C8—216no—
257C2×C8—1124no—
258C2×C8—1124no—
259C2×C8—118no—
260C2×C8—2212no—
261C2×C8—2216no—
262C11C22×#43011yescharacteristic, complemented, Sylow
263C23—24no—
264C23—24no—
265C23—24no—
266C23—412no—
267C23—412no—
268C23C2×Dic11⋊C413yes—
269C23C2×Dic11⋊C416yes—
270C23C2×Dic11⋊C416yes—
271C23C2×Dic11⋊C4112yes—
272C23C2×Dic11⋊C411yescharacteristic
273C23C22×Dic2213yes—
274C23C22×Dic2214yes—
275C2×C4C2×Dic11⋊C416yes—
276C2×C4C2×Dic11⋊C4112yes—
277C2×C4C2×Dic11⋊C4112yes—
278C2×C4C2×Dic11⋊C4112yes—
279C2×C4C2×Dic11⋊C4112yes—
280C2×C4C2×Dic11⋊C4112yes—
281C2×C4C2×Dic11⋊C413yes—
282C2×C4C2×Dic11⋊C416yes—
283C2×C4C2×Dic11⋊C416yes—
284C2×C4C2×Dic11⋊C413yes—
285C2×C4C22×Dic2213yes—
286C2×C4C22×Dic2214yes—
287C2×C4C22×C4×D1113yes—
288C2×C4C22×C4×D1113yes—
289C2×C4C22×C4×D1114yes—
290C2×C4C22×C4×D1114yes—
291C2×C4C22×C22⋊D1113yes—
292C2×C4C22×C22⋊D1114yes—
293C2×C4C22×C22⋊D1114yes—
294C2×C4C22×C22⋊D1113yes—
295C2×C4C24×D1111yescharacteristic
296C8—28no—
297C8—1116no—
298C8—228no—
299C22—212no—
300C22—212no—
301C22—26no—
302C22—26no—
303C22—412no—
304C22—44no—
305C22#1062273714yesdirect product factor
306C22C22×Dic11⋊C413yes—
307C22C22×Dic11⋊C414yes—
308C22C22×Dic11⋊C414yes—
309C22C22×Dic11⋊C413yes—
310C22C23×Dic2211yescharacteristic
311C4C22×Dic11⋊C413yes—
312C4C22×Dic11⋊C414yes—
313C4C22×Dic11⋊C414yes—
314C4C22×Dic11⋊C413yes—
315C4C23×C4×D1111yescharacteristic
316C4C23×C22⋊D1111yescharacteristic
317C2—24no—
318C2—24no—
319C2—23no—
320C2—21no—
321C2C2×#1062273716yesdirect product factor
322C2C23×Dic11⋊C411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D11
commutatorC4×C11C24×C4
FrattiniC2×C4C24×D11
FittingC23×C11×M16C2
socleC23×C11C2×C4⋊C4
prime radicalC25×C11Q8
cosocleC2×C4×C11C25
hypercenterC23×M16D11
hypocenterC11C22×#430
LCSG ▷ C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#10622737C2direct (N × H)
4. C22×#66886C2semidirect
22. #10622737C22direct (N × H)
23. C2×#66886C22semidirect
56. #66886C23semidirect
167. C22×C11#430semidirect
216. C2×C11C2×#430semidirect
262. C11C22×#430semidirect
305. C22#10622737direct (N × H)
321. C2C2×#10622737direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#43182
C2×#106227372
C22×#668862
C23×C11×M162
C22×#43011