Finite Group Databasebeta

← #27174462

Subgroups — #27174462

Order 12928.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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×#13108073C211yesmaximal, characteristic
2C2×#15692237C212yesmaximal, complemented
3C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
4#20265098C211yesmaximal, characteristic, complemented
5#20265097C211yesmaximal, characteristic, complemented
6#20265091C211yesmaximal, characteristic, complemented
7#20265053C212yesmaximal, complemented
8#20265052C212yesmaximal, complemented
9#20265049C212yesmaximal, complemented
10#20265039C211yesmaximal, characteristic, complemented
11#394×C101C211yesmaximal, characteristic, complemented
12C2×#13108073—28no—
13C2×#13108073—24no—
14C22×Dic202C2211yescharacteristic
15C22×C22⋊D101C2212yes—
16C23×Dic101C2212yes—
17C22×C4×D101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19C22×C4×D101C2211yescharacteristic
20#15692237—28no—
21C2×C4⋊Dic101C2211yescharacteristic
22C2×C4⋊Dic101C2212yes—
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2212yes—
25C2×#13108053C2211yescharacteristic
26C4⋊C4×D101—24no—
27C2×C22⋊Dic101C2212yes—
28C2×C22⋊Dic101C2212yes—
29C2×#13108051C2212yes—
30#15692162C2212yes—
31#15692162C2211yescharacteristic
32#15692162C2212yes—
33#15692162C2212yes—
34C22×D4×C101C2211yescharacteristic
35C2×C4⋊C4×C101C2211yescharacteristic
36C2×C22⋊C4×C101C2212yes—
37C2×C22⋊C4×C101C2212yes—
38#95×C101C2211yescharacteristic
39C23×D101C2311yescharacteristic
40#13108073—232no—
41C2×Dic202—22no—
42C2×Dic202—24no—
43C2×C22⋊D101—24no—
44C2×C22⋊D101—24no—
45C2×C22⋊D101D418yes—
46C22×Dic101—24no—
47C22×Dic101—24no—
48C22×Dic101—24no—
49C22×Dic101C2311yescharacteristic
50C22×Dic101C2312yes—
51C22×Dic101C2311yescharacteristic
52C22×Dic101C2312yes—
53C22×Dic101C2311yescharacteristic
54C2×C4×D101—24no—
55C2×C4×D101—22no—
56C2×C4×D101—22no—
57C2×C4×D101—22no—
58C2×C4×D101—22no—
59C2×C4×D101—24no—
60C2×C4×D101D414yes—
61C4⋊Dic101—22no—
62C4⋊Dic101—24no—
63#13108053—22no—
64#13108053—22no—
65#13108053—24no—
66C22⋊Dic101—24no—
67C22⋊Dic101—24no—
68#13108051—24no—
69C24×C101C2312yes—
70C2×D4×C101—24no—
71C2×D4×C101—22no—
72C22×C4×C101C2311yescharacteristic
73C22×C4×C101C2312yes—
74C22×C4×C101C2311yescharacteristic
75C22×C4×C101C2311yescharacteristic
76C4⋊C4×C101—22no—
77C22⋊C4×C101—24no—
78C22⋊C4×C101—24no—
79C23×C101—22no—
80C23×C101—22no—
81C23×C101—22no—
82C23×C101—22no—
83C23×C101—22no—
84C23×C101—22no—
85C23×C101C2×D414yes—
86C23×C101C2411yescharacteristic
87C22×D101—21no—
88C22×D101—21no—
89C22×D101—22no—
90C22×D101C2×D414yes—
91C22×D101C2×D412yes—
92D4×C101—28no—
93C2×C4×C101—21no—
94C2×C4×C101—22no—
95C2×C4×C101—21no—
96C2×C4×C101—22no—
97C2×C4×C101—22no—
98C2×C4×C101—21no—
99C2×C4×C101—22no—
100C2×C4×C101—22no—
101C2×C4×C101C2×D412yes—
102C2×C4×C101#4112yescomplemented
103C22⋊D101—216no—
104C2×Dic101—216no—
105C2×Dic101—21no—
106C2×Dic101—22no—
107C2×Dic101—22no—
108C2×Dic101—22no—
109C2×Dic101—21no—
110C2×Dic101—22no—
111C2×Dic101—22no—
112C2×Dic101—21no—
113C2×Dic101—22no—
114C2×Dic101—22no—
115C2×Dic101C2×D412yes—
116C2×Dic101C2×D414yes—
117C2×Dic101#4112yescomplemented
118C4×D101—28no—
119C4×D101—28no—
120C4×D101—44no—
121Dic202—28no—
122C22×C101—24no—
123C22×C101—24no—
124C22×C101—24no—
125C22×C101—42no—
126C22×C101—42no—
127C22×C101—42no—
128C22×C101—42no—
129C22×C101C22×D411yescharacteristic
130C22×C101C22×D412yes—
131C22×C101C2×#4111yescharacteristic
132C22×C101C2×#4111yescharacteristic
133C22×C1012−1+412yes—
134C2×D101—22no—
135C2×D101—22no—
136C2×D101—24no—
137C2×D101—24no—
138C2×D101#12014yes—
139C4×C101—22no—
140C4×C101—22no—
141C4×C101—41no—
142C4×C101—42no—
143Dic101—24no—
144Dic101—22no—
145Dic101—22no—
146Dic101—41no—
147Dic101—42no—
148C2×C101—44no—
149C2×C101C2×#12011yescharacteristic
150C2×C101#52511yescharacteristic
151C2×C101#53611yescharacteristic
152C2×C101#54712yes—
153C2×C101#54712yes—
154D101—24no—
155#2406—1011nomaximal, Sylow
156C101#240611yescharacteristic, complemented, Sylow
157C22×#41—1011no—
158C2×#123—1012no—
159C22×C4⋊C4—1011no—
160#395—1011no—
161#395—1011no—
162#394—1011no—
163#394D10111yescharacteristic, complemented
164#386—1012no—
165#385—1012no—
166#385—1012no—
167#381—1011no—
168C2×#41—2024no—
169C2×#41—2028no—
170C22×Q8—1011no—
171C22×D4—1012no—
172C22×D4C2×D10111yescharacteristic
173C23×C4—1011no—
174C23×C4—1012no—
175C23×C4—1011no—
176C23×C4—1011no—
177#123—2028no—
178C2×C4⋊C4—1011no—
179C2×C4⋊C4—1011no—
180C2×C4⋊C4—1011no—
181C2×C4⋊C4—1012no—
182C2×C4⋊C4—1012no—
183C2×C4⋊C4—2024no—
184C2×C4⋊C4C2×D10111yescharacteristic
185C2×C22⋊C4—1012no—
186C2×C22⋊C4—1012no—
187C2×C22⋊C4—1012no—
188C2×C22⋊C4C2×D10112yes—
189C2×C22⋊C4C2×D10112yes—
190#95—1012no—
191#95—1011no—
192#95—1012no—
193#95—1012no—
194#95C2×D10111yescharacteristic
195C24—1011no—
196C24C22×D10112yes—
197#41—20232no—
198C2×Q8—2024no—
199C2×Q8—2022no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—1018no—
203C2×D4—2024no—
204C2×D4—2024no—
205C22×C4—1011no—
206C22×C4—1011no—
207C22×C4—1012no—
208C22×C4—1014no—
209C22×C4—1012no—
210C22×C4—1011no—
211C22×C4—2022no—
212C22×C4—2022no—
213C22×C4—2024no—
214C22×C4—2022no—
215C22×C4—2024no—
216C22×C4—2024no—
217C22×C4—2024no—
218C22×C4—2024no—
219C22×C4—2022no—
220C22×C4C22×D10112yes—
221C22×C4C22×D10111yescharacteristic
222C22×C4C22×D10111yescharacteristic
223C22×C4C22×D10111yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—2022no—
226C4⋊C4—2024no—
227C4⋊C4—2022no—
228C4⋊C4—2024no—
229C4⋊C4—2022no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—2024no—
233C22⋊C4—2024no—
234C22⋊C4—2024no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—1014no—
242C23—1012no—
243C23—2021no—
244C23—2021no—
245C23—2022no—
246C23D4×D10114yes—
247C23C23×D10111yescharacteristic
248Q8—2028no—
249D4—28no—
250D4—20216no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—1012no—
260C2×C4—1014no—
261C2×C4—1012no—
262C2×C4—4044no—
263C2×C4—2028no—
264C2×C4—2028no—
265C2×C4—2022no—
266C2×C4—2022no—
267C2×C4—2021no—
268C2×C4—2022no—
269C2×C4—2022no—
270C2×C4—20216no—
271C2×C4—2021no—
272C2×C4—2022no—
273C2×C4—2022no—
274C2×C4—2022no—
275C2×C4—2021no—
276C2×C4D4×D10112yes—
277C2×C4#1310807312yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—1014no—
286C22—2022no—
287C22—2024no—
288C22—2024no—
289C22—2022no—
290C22C2×D4×D10111yescharacteristic
291C22C2×D4×D10112yes—
292C22#41×D10111yescharacteristic
293C22C2×#1310807311yescharacteristic
294C22#1569230212yes—
295C4—22no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—4042no—
300C4—4041no—
301C4—2022no—
302C4—2024no—
303C4—2022no—
304C2—44no—
305C2—2024no—
306C2#120×D10111yescharacteristic
307C2#2026568711yescharacteristic
308C2#2026574711yescharacteristic
309C2#2026582512yes—
310C2#2026582712yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#394×C101C2
prime radicalC22×#13108073C2
hypercenter#394D101
hypocenterC101#2406
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#15692237C2semidirect
3. C2×C4⋊C4×D101C2semidirect
4. #20265098C2semidirect
5. #20265097C2semidirect
6. #20265091C2semidirect
7. #20265053C2semidirect
8. #20265052C2semidirect
9. #20265049C2semidirect
10. #20265039C2semidirect
11. #394×C101C2semidirect
102. C2×C4×C101#41semidirect
117. C2×Dic101#41semidirect
156. C101#2406semidirect
163. #394D101semidirect
277. C2×C4#13108073semidirect

Maximal subgroups

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

subgroupindex
C22×#131080732
C2×#156922372
C2×C4⋊C4×D1012
#202650982
#202650972
#202650912
#202650532
#202650522
#202650492
#202650392
#394×C1012
#2406101