Finite Group Databasebeta

← #27173977

Subgroups — #27173977

Order 12928.

792 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692211C211yesmaximal, characteristic, complemented
2C22×C22⋊Dic101C211yesmaximal, characteristic
3#20265071C212yesmaximal, complemented
4#20265065C212yesmaximal, complemented
5#20265050C212yesmaximal, complemented
6#20265038C211yesmaximal, characteristic, complemented
7#20265035C211yesmaximal, characteristic, complemented
8#20265033C211yesmaximal, characteristic
9C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
10C4×C22⋊Dic101C212yesmaximal, complemented
11C22×C22⋊C4×C101C211yesmaximal, characteristic
12C23×Dic101C2211yescharacteristic
13C23×Dic101C2211yescharacteristic
14#15692211—24no—
15C2×#13108053C2212yescomplemented
16C2×#13108053C2212yescomplemented
17C2×C22⋊Dic101C418yescomplemented
18C2×C22⋊Dic101C2212yes—
19C2×C22⋊Dic101C2211yescharacteristic
20C2×C22⋊Dic101C2211yescharacteristic
21C2×C22⋊Dic101C2212yes—
22C2×C22⋊Dic101C2211yescharacteristic
23C2×C22⋊Dic101C2211yescharacteristic
24C2×C22⋊Dic101C2211yescharacteristic
25C2×C22⋊Dic101C2211yescharacteristic
26C2×C4×Dic101C2212yescomplemented
27C2×C4×Dic101C2212yescomplemented
28#15692162C2212yescomplemented
29#15692162C2212yescomplemented
30#15692162C2212yescomplemented
31#15692162C2212yescomplemented
32C25×C101C2211yescharacteristic
33C23×C4×C101C2212yes—
34C2×C22⋊C4×C101—24no—
35C2×C22⋊C4×C101C2211yescharacteristic
36C2×C22⋊C4×C101C2212yes—
37C2×C22⋊C4×C101C2211yescharacteristic
38C22×Dic101—22no—
39C22×Dic101—22no—
40C22×Dic101—42no—
41C22×Dic101C2×C414yescomplemented
42C22×Dic101C2×C414yescomplemented
43C22×Dic101C2312yes—
44C22×Dic101C2312yes—
45C22×Dic101C2311yescharacteristic
46C22×Dic101C2311yescharacteristic
47C22×Dic101C2311yescharacteristic
48C22×Dic101C2311yescharacteristic
49#13108053—24no—
50#13108053—42no—
51C22⋊Dic101—22no—
52C22⋊Dic101—24no—
53C22⋊Dic101—22no—
54C22⋊Dic101—42no—
55C22⋊Dic101C2×C414yes—
56C22⋊Dic101C2×C414yes—
57C22⋊Dic101C2×C414yescomplemented
58C22⋊Dic101C2×C414yescomplemented
59C4×Dic101—24no—
60C4×Dic101—42no—
61C24×C101—22no—
62C24×C101—24no—
63C24×C101—22no—
64C24×C101—42no—
65C24×C101C2×C414yes—
66C24×C101C2311yescharacteristic
67C24×C101C2311yescharacteristic
68C24×C101C2311yescharacteristic
69C22×C4×C101—24no—
70C22×C4×C101—24no—
71C22×C4×C101—42no—
72C22×C4×C101C2312yes—
73C22×C4×C101C2312yes—
74C22⋊C4×C101—22no—
75C22⋊C4×C101—24no—
76C22⋊C4×C101D414yes—
77C23×C101—21no—
78C23×C101—22no—
79C23×C101—21no—
80C23×C101—22no—
81C23×C101—22no—
82C23×C101—28no—
83C23×C101—28no—
84C23×C101—21no—
85C23×C101—22no—
86C23×C101—21no—
87C23×C101—22no—
88C23×C101—44no—
89C23×C101—48no—
90C23×C101—48no—
91C23×C101C22×C412yes—
92C23×C101C22×C412yes—
93C23×C101C22×C412yes—
94C23×C101C2×D412yes—
95C23×C101#4114yes—
96C23×C101#4112yes—
97C23×C101C2411yescharacteristic
98C2×C4×C101—22no—
99C2×C4×C101—22no—
100C2×C4×C101—22no—
101C2×C4×C101—22no—
102C2×C4×C101—22no—
103C2×C4×C101—48no—
104C2×C4×C101C2×D414yes—
105C2×Dic101—28no—
106C2×Dic101—21no—
107C2×Dic101—21no—
108C2×Dic101—21no—
109C2×Dic101—21no—
110C2×Dic101—22no—
111C2×Dic101—22no—
112C2×Dic101—22no—
113C2×Dic101—22no—
114C2×Dic101—22no—
115C2×Dic101—22no—
116C2×Dic101—22no—
117C2×Dic101—22no—
118C2×Dic101—44no—
119C2×Dic101C22×C412yes—
120C2×Dic101C22×C412yes—
121C2×Dic101C22×C412yes—
122C2×Dic101C22×C412yes—
123C22×C101—22no—
124C22×C101—22no—
125C22×C101—22no—
126C22×C101—22no—
127C22×C101—24no—
128C22×C101—24no—
129C22×C101—24no—
130C22×C101—24no—
131C22×C101—22no—
132C22×C101—22no—
133C22×C101—22no—
134C22×C101—416no—
135C22×C101—42no—
136C22×C101—42no—
137C22×C101#11714yescomplemented
138C22×C101C4×D414yes—
139C22×C101C23×C411yescharacteristic
140C22×C101C22×D411yescharacteristic
141C22×C101C2×#4112yes—
142C22×C101C2×#4111yescharacteristic
143C22×C101H(5,𝔽2)12yes—
144C4×C101—24no—
145C4×C101—42no—
146Dic101—22no—
147Dic101—22no—
148Dic101—22no—
149Dic101—22no—
150Dic101—42no—
151Dic101—42no—
152C2×C101—24no—
153C2×C101—24no—
154C2×C101—42no—
155C2×C101C2×#11711yescharacteristic
156C2×C101C2×C4×D411yescharacteristic
157C2×C101#51811yescharacteristic
158C2×C101#54612yes—
159C2×C101#55112yes—
160#2342—1011nomaximal, Sylow
161C101#234211yescharacteristic, complemented, Sylow
162C2×#117—1011no—
163C22×C22⋊C4—1011no—
164C22×C22⋊C4D10111yescharacteristic
165#388—1012no—
166#388—1012no—
167#385—1012no—
168#381—1011no—
169#380—1011no—
170#379—1011no—
171C4×C22⋊C4—1012no—
172C4×C22⋊C4—1011no—
173C25C2×D10111yescharacteristic
174C23×C4—1011no—
175C23×C4—1011no—
176C23×C4C2×D10112yes—
177#117—2024no—
178C2×C4⋊C4—1012no—
179C2×C4⋊C4—1012no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—1018no—
182C2×C22⋊C4—1011no—
183C2×C22⋊C4—1011no—
184C2×C22⋊C4—1011no—
185C2×C22⋊C4—1011no—
186C2×C22⋊C4—1012no—
187C2×C22⋊C4—1011no—
188C2×C22⋊C4—1011no—
189C2×C22⋊C4—1012no—
190C2×C22⋊C4C2×D10111yescharacteristic
191C2×C22⋊C4C2×D10112yes—
192C2×C22⋊C4C2×D10111yescharacteristic
193C2×C42—1012no—
194C2×C42—1012no—
195#95—1012no—
196#95—1012no—
197#95—1012no—
198#95—1012no—
199C24—22no—
200C24—24no—
201C24—22no—
202C24—42no—
203C24C4×D10114yes—
204C24C22×D10111yescharacteristic
205C24C22×D10111yescharacteristic
206C24C22×D10111yescharacteristic
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—1014no—
211C22×C4—1014no—
212C22×C4—1011no—
213C22×C4—1011no—
214C22×C4—1011no—
215C22×C4—1011no—
216C22×C4—1012no—
217C22×C4—1012no—
218C22×C4—4042no—
219C22×C4—2022no—
220C22×C4—2022no—
221C22×C4C22×D10112yes—
222C22×C4C22×D10112yes—
223C4⋊C4—4042no—
224C4⋊C4—2024no—
225C22⋊C4—22no—
226C22⋊C4—24no—
227C22⋊C4—1014no—
228C22⋊C4—1014no—
229C22⋊C4—1014no—
230C22⋊C4—1014no—
231C22⋊C4—4042no—
232C22⋊C4—2022no—
233C22⋊C4—2024no—
234C22⋊C4—2022no—
235C22⋊C4C22⋊D10114yes—
236C42—4042no—
237C42—2024no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—22no—
245C23—21no—
246C23—22no—
247C23—28no—
248C23—28no—
249C23—44no—
250C23—48no—
251C23—48no—
252C23C2×C4×D10112yes—
253C23C2×C4×D10112yes—
254C23C2×C4×D10112yes—
255C23C2×C22⋊D10112yes—
256C23#1310807314yes—
257C23#1310807412yes—
258C23C23×D10111yescharacteristic
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—48no—
265C2×C4—1012no—
266C2×C4—1012no—
267C2×C4—1012no—
268C2×C4—1012no—
269C2×C4—4044no—
270C2×C4—2022no—
271C2×C4—2022no—
272C2×C4—2022no—
273C2×C4—2022no—
274C2×C4—2022no—
275C2×C4—2022no—
276C2×C4—2021no—
277C2×C4—2021no—
278C2×C4—2021no—
279C2×C4—2021no—
280C2×C4—2022no—
281C2×C4—2022no—
282C2×C4—2028no—
283C2×C4C2×C22⋊D10114yes—
284C22—24no—
285C22—24no—
286C22—24no—
287C22—24no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—416no—
296C22—42no—
297C22—42no—
298C22#1569221114yescomplemented
299C22C4×C22⋊D10114yes—
300C22C22×C4×D10111yescharacteristic
301C22C22×C22⋊D10111yescharacteristic
302C22C2×#1310807312yes—
303C22C2×#1310807411yescharacteristic
304C22#1569230012yes—
305C4—24no—
306C4—42no—
307C4—4042no—
308C4—4042no—
309C4—2022no—
310C4—2022no—
311C4—2022no—
312C4—2022no—
313C2—24no—
314C2—24no—
315C2—42no—
316C2C2×#1569221111yescharacteristic
317C2C2×C4×C22⋊D10111yescharacteristic
318C2#2026565011yescharacteristic
319C2#2026582112yes—
320C2#2026585312yes—
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
FittingC22×C22⋊C4×C101C2
socle, cosocleC23×C101C24
prime radicalC25×C101C22
hypercenterC22×C22⋊C4D101
hypocenterC101#2342
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#15692211C2semidirect
3. #20265071C2semidirect
4. #20265065C2semidirect
5. #20265050C2semidirect
6. #20265038C2semidirect
7. #20265035C2semidirect
9. C22⋊C4×Dic101C2semidirect
10. C4×C22⋊Dic101C2semidirect
15. C2×#13108053C22semidirect
16. C2×#13108053C22semidirect
17. C2×C22⋊Dic101C4semidirect
26. C2×C4×Dic101C22semidirect
27. C2×C4×Dic101C22semidirect
28. #15692162C22semidirect
29. #15692162C22semidirect
30. #15692162C22semidirect
31. #15692162C22semidirect
41. C22×Dic101C2×C4semidirect
42. C22×Dic101C2×C4semidirect
57. C22⋊Dic101C2×C4semidirect
58. C22⋊Dic101C2×C4semidirect
137. C22×C101#117semidirect
161. C101#2342semidirect
298. C22#15692211semidirect

Maximal subgroups

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

subgroupindex
C2×#156922112
C22×C22⋊Dic1012
#202650712
#202650652
#202650502
#202650382
#202650352
#202650332
C22⋊C4×Dic1012
C4×C22⋊Dic1012
C22×C22⋊C4×C1012
#2342101