Finite Group Databasebeta

← #14913902

Subgroups — #14913902

Order 2816.

2858 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13071589C218yesmaximal, complemented
2#13071565C218yesmaximal, complemented
3#13071214C214yesmaximal, complemented
4#13071198C212yesmaximal
5#13071023C212yesmaximal, complemented
6#13070999C214yesmaximal, complemented
7#13070973C212yesmaximal, complemented
8C22×C4×D4×C11C211yesmaximal, characteristic
9C23×D4×C11C2211yescharacteristic
10C24×C4×C11C2212yes—
11C2×C4×D4×C11C4116yescomplemented
12C2×C4×D4×C11C2214yes—
13C2×C4×D4×C11C2214yes—
14C22×C11×C4⋊C4C2211yescharacteristic
15C22×C11×C22⋊C4C2212yes—
16C22×C42×C11C2211yescharacteristic
17C2×#66979C2214yes—
18C2×#66979C2218yescomplemented
19#10622860C2218yes—
20#10622834C22116yes—
21#10622825C2214yes—
22#10622825C2214yes—
23#10622825C2218yescomplemented
24#10622824C2218yescomplemented
25C4×C22⋊Dic11C2218yes—
26C4×C22⋊Dic11C2218yescomplemented
27C4⋊C4×Dic11C22116yescomplemented
28#10622564C22132yescomplemented
29#10622549C2214yescomplemented
30#10622548C2214yescomplemented
31C4×C4⋊Dic11C2214yescomplemented
32C4×C4⋊Dic11C2214yescomplemented
33C25×C11C2312yes—
34C22×D4×C11—28no—
35C22×D4×C11C2×C418yes—
36C22×D4×C11C2314yes—
37C23×C4×C11—48no—
38C23×C4×C11C2×C4116yes—
39C23×C4×C11C2314yes—
40C23×C4×C11C2311yescharacteristic
41C23×C4×C11C2312yes—
42C23×C4×C11C2314yes—
43C23×C4×C11C2311yescharacteristic
44C23×C4×C11C2311yescharacteristic
45C4×D4×C11C2×C4132yes—
46C4×D4×C11C2×C4132yes—
47C2×C11×C4⋊C4C2×C418yes—
48C2×C11×C4⋊C4C2314yes—
49C2×C11×C22⋊C4C2×C4116yes—
50C2×C11×C22⋊C4C2318yes—
51C2×C42×C11C2×C418yes—
52C2×C42×C11C2312yes—
53C2×C42×C11C2312yes—
54C2×C22⋊Dic11C2318yes—
55C2×C22⋊Dic11C23116yes—
56C2×C22⋊Dic11C23116yes—
57#66979—216no—
58#66979—416no—
59C2×C4⋊Dic11C2318yes—
60C2×C4⋊Dic11C2318yes—
61C2×C4⋊Dic11C2318yes—
62C2×C4×Dic11C2318yes—
63C2×C4×Dic11C23116yes—
64C11⋊#95C2318yes—
65C11⋊#95C2318yes—
66C11⋊#95C23116yes—
67#34186—111nomaximal, Sylow
68C24×C11—28no—
69C24×C11—44no—
70C24×C11—44no—
71C24×C11C22×C418yes—
72C24×C11C2414yes—
73C24×C11C2411yescharacteristic
74C2×D4×C11—232no—
75C2×D4×C11—28no—
76C2×D4×C11—28no—
77C2×D4×C11C22×C4116yes—
78C22×C4×C11—216no—
79C22×C4×C11—24no—
80C22×C4×C11—432no—
81C22×C4×C11—44no—
82C22×C4×C11—48no—
83C22×C4×C11—42no—
84C22×C4×C11C22×C414yes—
85C22×C4×C11C22×C4116yes—
86C22×C4×C11C22×C414yes—
87C22×C4×C11C22×C418yes—
88C22×C4×C11C22×C4116yes—
89C22×C4×C11C22×C414yes—
90C22×C4×C11#4118yes—
91C22×C4×C11C2411yescharacteristic
92C22×C4×C11C2414yes—
93C22×C4×C11C2412yes—
94C22×C4×C11C2412yes—
95C22×C4×C11C2411yescharacteristic
96C22×Dic11C24116yes—
97C11×C4⋊C4C22×C4116yes—
98C11×C22⋊C4C22×C4132yes—
99C42×C11C22×C418yes—
100C42×C11C22×C418yes—
101C22⋊Dic11—216no—
102C22⋊Dic11—416no—
103C22⋊Dic11—416no—
104C4⋊Dic11—216no—
105C4⋊Dic11—216no—
106C4⋊Dic11—48no—
107C4×Dic11—216no—
108C4×Dic11—416no—
109C22×C4×D4D1111yescharacteristic
110#2357—118no—
111#2354—118no—
112#2312—114no—
113#2309—112no—
114#2285—112no—
115#2280—114no—
116#2274—112no—
117C23×C11—22no—
118C23×C11—22no—
119C23×C11—216no—
120C23×C11—28no—
121C23×C11—28no—
122C23×C11—28no—
123C23×C11—48no—
124C23×C11—48no—
125C23×C11—432no—
126C23×C11C23×C418yes—
127C23×C11C23×C412yes—
128C23×C11C2×#4114yes—
129C23×C11C2511yescharacteristic
130D4×C11—232no—
131C2×C4×C11—28no—
132C2×C4×C11—22no—
133C2×C4×C11—24no—
134C2×C4×C11—28no—
135C2×C4×C11—232no—
136C2×C4×C11—416no—
137C2×C4×C11—48no—
138C2×C4×C11—44no—
139C2×C4×C11#117116yescomplemented
140C2×C4×C11C23×C414yes—
141C2×C4×C11C23×C412yes—
142C2×C4×C11C23×C418yes—
143C2×C4×C11C23×C414yes—
144C2×C4×C11C23×C412yes—
145C2×C4×C11C2×#4114yes—
146C2×C4×C11C2×#4114yes—
147C2×C4×C11H(5,𝔽2)14yes—
148C2×Dic11—216no—
149C2×Dic11—216no—
150C2×Dic11—216no—
151C23×D4C2×D1111yescharacteristic
152C24×C4C2×D1112yes—
153C2×C4×D4Dic11116yescomplemented
154C2×C4×D4C2×D1114yes—
155C2×C4×D4C2×D1114yes—
156C2×#117—114no—
157C2×#117—118no—
158C22×C4⋊C4C2×D1111yescharacteristic
159C22×C22⋊C4C2×D1112yes—
160C22×C42C2×D1111yescharacteristic
161#390—1116no—
162#388—1132no—
163#384—114no—
164#382—114no—
165#381—118no—
166#380—114no—
167#380—118no—
168#380—114no—
169#379—118no—
170C4×C4⋊C4—1116no—
171C4×C4⋊C4—114no—
172C4×C4⋊C4—114no—
173C4×C22⋊C4—118no—
174C4×C22⋊C4—118no—
175C22×C11—22no—
176C22×C11—22no—
177C22×C11—28no—
178C22×C11—28no—
179C22×C11—28no—
180C22×C11—28no—
181C22×C11—432no—
182C22×C11—416no—
183C22×C11C2×#11714yes—
184C22×C11C24×C411yescharacteristic
185C22×C11C22×#4112yes—
186C22×C11C2×H(5,𝔽2)12yes—
187C22×C11#58512yes—
188C4×C11—22no—
189C4×C11—28no—
190C4×C11—24no—
191C4×C11C2×#11718yes—
192C4×C11#51814yes—
193Dic11—416no—
194C25C22×D1112yes—
195C22×D4—28no—
196C22×D4C2×Dic1118yes—
197C22×D4C22×D1114yes—
198C23×C4—48no—
199C23×C4C2×Dic11116yes—
200C23×C4C22×D1114yes—
201C23×C4C22×D1111yescharacteristic
202C23×C4C22×D1111yescharacteristic
203C23×C4C22×D1112yes—
204C23×C4C22×D1114yes—
205C23×C4C22×D1111yescharacteristic
206C4×D4C2×Dic11132yes—
207C4×D4C2×Dic11132yes—
208#117—2216no—
209#117—4416no—
210C2×C4⋊C4—118no—
211C2×C4⋊C4—118no—
212C2×C4⋊C4—118no—
213C2×C4⋊C4C2×Dic1118yes—
214C2×C4⋊C4C22×D1114yes—
215C2×C22⋊C4—118no—
216C2×C22⋊C4—1116no—
217C2×C22⋊C4—1116no—
218C2×C22⋊C4C2×Dic11116yes—
219C2×C22⋊C4C22×D1118yes—
220C2×C42—118no—
221C2×C42—1116no—
222C2×C42C2×Dic1118yes—
223C2×C42C22×D1112yes—
224C2×C42C22×D1112yes—
225#95—118no—
226#95—1116no—
227#95—118no—
228C2×C11—24no—
229C2×C11—48no—
230C2×C11C22×#11711yescharacteristic
231C2×C11C2×#51811yescharacteristic
232C2×C11#342811yescharacteristic
233C2×C11#350614yes—
234C24—28no—
235C24—44no—
236C24—44no—
237C24C22×Dic1118yes—
238C24C23×D1114yes—
239C24C23×D1111yescharacteristic
240C2×D4—232no—
241C2×D4—28no—
242C2×D4—28no—
243C2×D4C22×Dic11116yes—
244C22×C4—216no—
245C22×C4—24no—
246C22×C4—432no—
247C22×C4—44no—
248C22×C4—48no—
249C22×C4—42no—
250C22×C4—1116no—
251C22×C4#433918yes—
252C22×C4C22×Dic1114yes—
253C22×C4C22×Dic11116yes—
254C22×C4C22×Dic1114yes—
255C22×C4C22×Dic1118yes—
256C22×C4C22×Dic11116yes—
257C22×C4C22×Dic1114yes—
258C22×C4C23×D1111yescharacteristic
259C22×C4C23×D1114yes—
260C22×C4C23×D1112yes—
261C22×C4C23×D1112yes—
262C22×C4C23×D1111yescharacteristic
263C4⋊C4—2216no—
264C4⋊C4—2216no—
265C4⋊C4—448no—
266C4⋊C4C22×Dic11116yes—
267C22⋊C4—2216no—
268C22⋊C4—4416no—
269C22⋊C4—4416no—
270C22⋊C4C22×Dic11132yes—
271C42—2216no—
272C42—4416no—
273C42C22×Dic1118yes—
274C42C22×Dic1118yes—
275C11#3418611yescharacteristic, complemented, Sylow
276C23—22no—
277C23—22no—
278C23—216no—
279C23—28no—
280C23—28no—
281C23—28no—
282C23—48no—
283C23—48no—
284C23—432no—
285C23C2×#433914yes—
286C23C23×Dic1118yes—
287C23C23×Dic1112yes—
288C23C24×D1111yescharacteristic
289D4—232no—
290C2×C4—28no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—28no—
294C2×C4—232no—
295C2×C4—416no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—2216no—
299C2×C4—2216no—
300C2×C4—2216no—
301C2×C4#66979116yescomplemented
302C2×C4C2×#433914yes—
303C2×C4C2×#433914yes—
304C2×C4#6703714yes—
305C2×C4C23×Dic1114yes—
306C2×C4C23×Dic1112yes—
307C2×C4C23×Dic1118yes—
308C2×C4C23×Dic1114yes—
309C2×C4C23×Dic1112yes—
310C22—28no—
311C22—22no—
312C22—22no—
313C22—28no—
314C22—28no—
315C22—28no—
316C22—432no—
317C22—416no—
318C22C2×#6697914yes—
319C22C22×#433912yes—
320C22C2×#6703712yes—
321C22#1062345312yes—
322C22C24×Dic1111yescharacteristic
323C4—22no—
324C4—28no—
325C4—24no—
326C4—4416no—
327C4C2×#6697918yes—
328C4#1062332914yes—
329C2—24no—
330C2—48no—
331C2C22×#6697911yescharacteristic
332C2C2×#1062332911yescharacteristic
333C2#1308181911yescharacteristic
334C2#1308246814yes—
subgroupG/H
center, FrattiniC23C24×D11
commutatorC22×C11C24×C4
FittingC22×C4×D4×C11C2
socle, cosocleC23×C11C25
prime radicalC23×D4×C11C22
hypercenterC22×C4×D4D11
hypocenterC11#34186
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. #13071589C2semidirect
2. #13071565C2semidirect
3. #13071214C2semidirect
5. #13071023C2semidirect
6. #13070999C2semidirect
7. #13070973C2semidirect
11. C2×C4×D4×C11C4semidirect
18. C2×#66979C22semidirect
23. #10622825C22semidirect
24. #10622824C22semidirect
26. C4×C22⋊Dic11C22semidirect
27. C4⋊C4×Dic11C22semidirect
28. #10622564C22semidirect
29. #10622549C22semidirect
30. #10622548C22semidirect
31. C4×C4⋊Dic11C22semidirect
32. C4×C4⋊Dic11C22semidirect
139. C2×C4×C11#117semidirect
153. C2×C4×D4Dic11semidirect
275. C11#34186semidirect
301. C2×C4#66979semidirect

Maximal subgroups

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

subgroupindex
#130715892
#130715652
#130712142
#130711982
#130710232
#130709992
#130709732
C22×C4×D4×C112
#3418611