Finite Group Databasebeta

← #27174492

Subgroups — #27174492

Order 12928.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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×Q8×D101C211yesmaximal, characteristic, complemented
2C2×#15692233C212yesmaximal, complemented
3C2×#15692217C211yesmaximal, characteristic
4#20265097C211yesmaximal, characteristic, complemented
5#20265090C211yesmaximal, characteristic, complemented
6#20265088C211yesmaximal, characteristic, complemented
7#20265081C212yesmaximal, complemented
8#20265055C212yesmaximal
9#20265048C212yesmaximal, complemented
10#20265036C211yesmaximal, characteristic, complemented
11#395×C101C211yesmaximal, characteristic, complemented
12C22×Dic202C2211yescharacteristic
13C22×Dic202C2212yes—
14C2×Q8×D101—28no—
15C2×Q8×D101—24no—
16C22×D404C2211yescharacteristic
17C22×C4×D101C2212yes—
18C22×C4×D101C2211yescharacteristic
19C22×C4×D101C2211yescharacteristic
20#15692233—28no—
21#15692217—24no—
22C2×#13108053C2212yes—
23C2×#13108053C2212yes—
24C2×#13108051C2212yes—
25C2×#13108051C2211yescharacteristic
26C2×#13108051C2212yes—
27C2×#13108051C2212yes—
28C2×#13108051C2211yescharacteristic
29C2×C4×Dic101C2211yescharacteristic
30#15692162C2211yescharacteristic
31#15692162C2212yes—
32#15692162C2212yes—
33#15692162C2212yes—
34C22×Q8×C101C2211yescharacteristic
35C2×C4⋊C4×C101C2212yes—
36C2×C4⋊C4×C101C2211yescharacteristic
37#95×C101C2211yescharacteristic
38#95×C101C2212yes—
39C23×D101C2311yescharacteristic
40C23×D101C2311yescharacteristic
41C2×Dic202—22no—
42C2×Dic202—24no—
43C2×Dic202—24no—
44C2×Dic202—24no—
45C2×Dic202D418yescomplemented
46Q8×D101—232no—
47C2×D404—22no—
48C2×D404—22no—
49C2×D404Q814yescomplemented
50C22×Dic101C2311yescharacteristic
51C22×Dic101C2312yes—
52C22×Dic101C2312yes—
53C22×Dic101C2311yescharacteristic
54C2×C4×D101—24no—
55C2×C4×D101—24no—
56C2×C4×D101—24no—
57C2×C4×D101—24no—
58C2×C4×D101—22no—
59C2×C4×D101—24no—
60C2×C4×D101D414yes—
61#13108053—24no—
62#13108053—24no—
63#13108051—22no—
64#13108051—24no—
65#13108051—22no—
66#13108051—24no—
67#13108051—24no—
68C4×Dic101—22no—
69C2×Q8×C101—24no—
70C2×Q8×C101—22no—
71C22×C4×C101C2311yescharacteristic
72C22×C4×C101C2312yes—
73C22×C4×C101C2312yes—
74C22×C4×C101C2311yescharacteristic
75C22×C4×C101C2311yescharacteristic
76C4⋊C4×C101—22no—
77C4⋊C4×C101—24no—
78C23×C101C2411yescharacteristic
79C22×D101—22no—
80C22×D101—22no—
81C22×D101—22no—
82C22×D101—21no—
83C22×D101—21no—
84C22×D101—22no—
85C22×D101C2×D412yes—
86C22×D101C2×Q812yes—
87C22×D101C2×Q812yescomplemented
88C22×D101#4112yes—
89Q8×C101—28no—
90C2×C4×C101—21no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101—22no—
94C2×C4×C101—21no—
95C2×C4×C101—22no—
96C2×C4×C101—21no—
97C2×C4×C101—22no—
98C2×C4×C101—22no—
99C2×C4×C101—22no—
100C2×C4×C101C2×D414yes—
101C2×C4×C101C2×D412yes—
102C2×C4×C101C2×Q812yescomplemented
103C2×Dic101—22no—
104C2×Dic101—22no—
105C2×Dic101—21no—
106C2×Dic101—22no—
107C2×Dic101—22no—
108C2×Dic101—22no—
109C2×Dic101—22no—
110C2×Dic101C2×D414yes—
111C2×Dic101C2×D412yes—
112C2×Dic101C2×D414yes—
113D404—28no—
114C4×D101—216no—
115C4×D101—28no—
116C4×D101—44no—
117Dic202—28no—
118Dic202—216no—
119C22×C101C22×D411yescharacteristic
120C22×C101C22×D412yes—
121C22×C101C22×Q811yescharacteristic
122C22×C101C2×#4111yescharacteristic
123C22×C1012−1+412yes—
124C2×D101—22no—
125C2×D101—22no—
126C2×D101—24no—
127C2×D101—24no—
128C2×D101—22no—
129C2×D101—24no—
130C2×D101—41no—
131C2×D101—41no—
132C2×D101—42no—
133C2×D101C22⋊Q814yescomplemented
134C4×C101—22no—
135C4×C101—24no—
136C4×C101—22no—
137C4×C101—41no—
138C4×C101—42no—
139Dic101—24no—
140Dic101—41no—
141Dic101—42no—
142Dic101#12014yes—
143C2×C101C2×#12011yescharacteristic
144C2×C101C2×C22⋊Q811yescharacteristic
145C2×C101#53611yescharacteristic
146C2×C101#54812yes—
147C2×C101D4×Q812yes—
148D101—24no—
149D101—42no—
150#2409—1011nomaximal, Sylow
151C101#240911yescharacteristic, complemented, Sylow
152C23×Q8—1011no—
153C2×C22⋊Q8—1012no—
154C2×C4×D4—1011no—
155#395—1011no—
156#395D10111yescharacteristic, complemented
157#393—1011no—
158#393—1011no—
159#391—1012no—
160#386—1012no—
161#385—1012no—
162#380—1011no—
163C22×Q8—1012no—
164C22×Q8—1011no—
165C22×Q8—2024no—
166C22×Q8—2028no—
167C22×Q8C2×D10111yescharacteristic
168C22×D4—1011no—
169C23×C4—1011no—
170C23×C4—1011no—
171C23×C4—1012no—
172C22⋊Q8—2028no—
173C4×D4—2024no—
174C2×C4⋊C4—1012no—
175C2×C4⋊C4—1012no—
176C2×C4⋊C4C2×D10111yescharacteristic
177C2×C4⋊C4C2×D10112yes—
178C2×C22⋊C4—1011no—
179C2×C22⋊C4—1011no—
180C2×C22⋊C4—1012no—
181C2×C22⋊C4—1012no—
182C2×C22⋊C4—1012no—
183C2×C42—1011no—
184#95—1011no—
185#95—1012no—
186#95—1012no—
187#95—1012no—
188#95C2×D10111yescharacteristic
189#95C2×D10112yes—
190C24—1011no—
191C24—1011no—
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—1018no—
195C2×Q8—2024no—
196C2×Q8—2024no—
197C2×Q8—2024no—
198C2×Q8—20232no—
199C2×Q8—2022no—
200C2×D4—1014no—
201C2×D4—2022no—
202C2×D4—2022no—
203C22×C4—1011no—
204C22×C4—1012no—
205C22×C4—1014no—
206C22×C4—1011no—
207C22×C4—1012no—
208C22×C4—2022no—
209C22×C4—2024no—
210C22×C4—2024no—
211C22×C4—2024no—
212C22×C4—2024no—
213C22×C4—2024no—
214C22×C4C22×D10112yes—
215C22×C4C22×D10111yescharacteristic
216C22×C4C22×D10111yescharacteristic
217C22×C4C22×D10112yes—
218C22×C4C22×D10111yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—2024no—
222C4⋊C4—2024no—
223C22⋊C4—2024no—
224C22⋊C4—2024no—
225C22⋊C4—2022no—
226C22⋊C4—2022no—
227C22⋊C4—2024no—
228C42—2022no—
229C23—1012no—
230C23—1012no—
231C23—1012no—
232C23—1012no—
233C23—2022no—
234C23—2021no—
235C23—2021no—
236C23—2022no—
237C23—2022no—
238C23—2022no—
239C23C23×D10111yescharacteristic
240Q8—28no—
241Q8—20216no—
242Q8—2028no—
243D4—2028no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—1012no—
255C2×C4—1014no—
256C2×C4—1014no—
257C2×C4—4044no—
258C2×C4—2028no—
259C2×C4—20216no—
260C2×C4—2021no—
261C2×C4—2022no—
262C2×C4—2022no—
263C2×C4—2022no—
264C2×C4—2022no—
265C2×C4—2022no—
266C2×C4—2022no—
267C2×C4D4×D10112yes—
268C2×C4D4×D10114yes—
269C2×C4Q8×D10112yescomplemented
270C22—1014no—
271C22—4041no—
272C22—4042no—
273C22—4041no—
274C22—2024no—
275C22—2022no—
276C22—2022no—
277C22—2024no—
278C22—2024no—
279C22—2022no—
280C22C2×D4×D10111yescharacteristic
281C22C2×D4×D10112yes—
282C22C2×Q8×D10111yescharacteristic
283C22#41×D10111yescharacteristic
284C22#1569230312yes—
285C4—22no—
286C4—24no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—1014no—
291C4—4041no—
292C4—4042no—
293C4—2024no—
294C2—4042no—
295C2—2024no—
296C2#120×D10111yescharacteristic
297C2C22⋊Q8×D10111yescharacteristic
298C2#2026575211yescharacteristic
299C2#2026583712yes—
300C2#2026584212yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#395×C101C2
prime radicalC22×D404C22
hypercenter#395D101
hypocenterC101#2409
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D101C2semidirect
2. C2×#15692233C2semidirect
4. #20265097C2semidirect
5. #20265090C2semidirect
6. #20265088C2semidirect
7. #20265081C2semidirect
9. #20265048C2semidirect
10. #20265036C2semidirect
11. #395×C101C2semidirect
45. C2×Dic202D4semidirect
49. C2×D404Q8semidirect
87. C22×D101C2×Q8semidirect
102. C2×C4×C101C2×Q8semidirect
133. C2×D101C22⋊Q8semidirect
151. C101#2409semidirect
156. #395D101semidirect
269. C2×C4Q8×D101semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1012
C2×#156922332
C2×#156922172
#202650972
#202650902
#202650882
#202650812
#202650552
#202650482
#202650362
#395×C1012
#2409101