Finite Group Databasebeta

← #27174442

Subgroups — #27174442

Order 12928.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C23×C22⋊D101C211yesmaximal, characteristic, complemented
2C2×#15692224C212yesmaximal, complemented
3C2×#15692223C212yesmaximal, complemented
4#20265085C214yesmaximal, complemented
5#20265057C212yesmaximal, complemented
6#20265056C212yesmaximal, complemented
7#20265033C211yesmaximal, characteristic, complemented
8C23×D4×C101C211yesmaximal, characteristic, complemented
9C24×D101C2211yescharacteristic, complemented
10C22×C22⋊D101—28no—
11C22×C22⋊D101—24no—
12C22×C22⋊D101C2212yes—
13C22×C22⋊D101C2212yescomplemented
14C22×C22⋊D101C2214yescomplemented
15C23×Dic101C2211yescharacteristic, complemented
16#15692224—28no—
17#15692223—28no—
18C2×C22⋊Dic101C2212yescomplemented
19C2×C22⋊Dic101C2212yescomplemented
20C2×C22⋊Dic101C2214yescomplemented
21C2×C22⋊Dic101C2212yescomplemented
22C2×#13108051C2214yescomplemented
23#15692162C2214yescomplemented
24C25×C101C2211yescharacteristic
25C25×C101C2211yescharacteristic, complemented
26C22×D4×C101—24no—
27C22×D4×C101—28no—
28C22×D4×C101C2212yes—
29C22×D4×C101C2212yescomplemented
30C23×C4×C101C2211yescharacteristic, complemented
31C23×D101—22no—
32C23×D101—24no—
33C23×D101—24no—
34C23×D101—42no—
35C23×D101C2312yes—
36C2×C22⋊D101—232no—
37C2×C22⋊D101—24no—
38C2×C22⋊D101—24no—
39C2×C22⋊D101—24no—
40C2×C22⋊D101—24no—
41C2×C22⋊D101—28no—
42C2×C22⋊D101—28no—
43C2×C22⋊D101—24no—
44C2×C22⋊D101—44no—
45C2×C22⋊D101D418yescomplemented
46C22×Dic101—22no—
47C22×Dic101—24no—
48C22×Dic101C2314yes—
49C22×Dic101C2312yes—
50C22⋊Dic101—24no—
51C22⋊Dic101—24no—
52C22⋊Dic101—28no—
53C22⋊Dic101—42no—
54#13108051—28no—
55C24×C101—24no—
56C24×C101—24no—
57C24×C101—22no—
58C24×C101—24no—
59C24×C101—24no—
60C24×C101—42no—
61C24×C101—42no—
62C24×C101D414yescomplemented
63C24×C101C2311yescharacteristic
64C24×C101C2312yes—
65C24×C101C2311yescharacteristic
66C24×C101C2311yescharacteristic
67C2×D4×C101—232no—
68C2×D4×C101—24no—
69C2×D4×C101—24no—
70C2×D4×C101—24no—
71C2×D4×C101—24no—
72C2×D4×C101—24no—
73C2×D4×C101D418yescomplemented
74C22×C4×C101—22no—
75C22×C4×C101—24no—
76C22×C4×C101C2312yes—
77C23×C101—22no—
78C23×C101—22no—
79C23×C101—22no—
80C23×C101—22no—
81C23×C101—28no—
82C23×C101—28no—
83C23×C101—28no—
84C23×C101—22no—
85C23×C101—22no—
86C23×C101—22no—
87C23×C101—24no—
88C23×C101—22no—
89C23×C101—22no—
90C23×C101—24no—
91C23×C101—44no—
92C23×C101—416no—
93C23×C101—416no—
94C23×C101—44no—
95C23×C101—44no—
96C23×C101C2×D414yes—
97C23×C101C2×D412yes—
98C23×C101C2×D412yes—
99C23×C101C2×D412yes—
100C23×C101C2×D414yes—
101C23×C101C2×D414yescomplemented
102C23×C101C2411yescharacteristic
103C22×D101—28no—
104C22×D101—24no—
105C22×D101—22no—
106C22×D101—22no—
107C22×D101—24no—
108C22×D101—416no—
109C22×D101—44no—
110C22×D101—44no—
111C22×D101C2×D414yescomplemented
112D4×C101—216no—
113D4×C101—216no—
114C2×C4×C101—28no—
115C2×C4×C101—22no—
116C2×C4×C101—22no—
117C2×C4×C101C2×D414yescomplemented
118C22⋊D101—216no—
119C22⋊D101—216no—
120C22⋊D101—416no—
121C2×Dic101—28no—
122C2×Dic101—22no—
123C2×Dic101—22no—
124C2×Dic101—24no—
125C2×Dic101—24no—
126C2×Dic101—24no—
127C2×Dic101C2×D414yescomplemented
128C22×C101—22no—
129C22×C101—22no—
130C22×C101—22no—
131C22×C101—22no—
132C22×C101—28no—
133C22×C101—28no—
134C22×C101—22no—
135C22×C101—22no—
136C22×C101—24no—
137C22×C101—22no—
138C22×C101—28no—
139C22×C101—416no—
140C22×C101—416no—
141C22×C101—42no—
142C22×C101—42no—
143C22×C101—44no—
144C22×C101—44no—
145C22×C101#12014yescomplemented
146C22×C101#12014yescomplemented
147C22×C101C22×D412yes—
148C22×C101C22×D411yescharacteristic
149C22×C101C22×D412yes—
150C22×C101H(5,𝔽2)12yes—
151C2×D101—24no—
152C2×D101—28no—
153C2×D101—416no—
154C2×D101—42no—
155C2×D101—42no—
156C2×D101—44no—
157C4×C101—24no—
158Dic101—24no—
159Dic101—44no—
160C2×C101—24no—
161C2×C101—24no—
162C2×C101—42no—
163C2×C101—44no—
164C2×C101C2×#12011yescharacteristic
165C2×C101C2×#12011yescharacteristic
166C2×C101#53411yescharacteristic
167C2×C101D4214yes—
168D101—44no—
169#2403—1011nomaximal, Sylow
170C101#240311yescharacteristic, complemented, Sylow
171C23×D4—1011no—
172C23×D4D10111yescharacteristic, complemented
173C2×#120—1012no—
174C2×#120—1012no—
175#392—1014no—
176#386—1012no—
177#386—1012no—
178#379—1011no—
179C25—1011no—
180C25C2×D10111yescharacteristic
181C25C2×D10111yescharacteristic, complemented
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—1012no—
185C22×D4—1014no—
186C22×D4—1012no—
187C22×D4—2028no—
188C22×D4—2024no—
189C22×D4C2×D10112yes—
190C22×D4C2×D10112yescomplemented
191C23×C4—1011no—
192C23×C4C2×D10111yescharacteristic, complemented
193#120—2028no—
194#120—2028no—
195C2×C22⋊C4—1012no—
196C2×C22⋊C4—1012no—
197C2×C22⋊C4—1012no—
198C2×C22⋊C4—1014no—
199C2×C22⋊C4—1014no—
200#95—1014no—
201C24—24no—
202C24—24no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—42no—
207C24—42no—
208C24—1012no—
209C24—4042no—
210C24—2024no—
211C24—2024no—
212C24—2022no—
213C24C22⋊D10114yescomplemented
214C24C22×D10111yescharacteristic
215C24C22×D10112yes—
216C24C22×D10111yescharacteristic
217C24C22×D10111yescharacteristic
218C2×D4—232no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—1018no—
225C2×D4—4044no—
226C2×D4—2024no—
227C2×D4—2028no—
228C2×D4—2028no—
229C2×D4—2024no—
230C2×D4—20232no—
231C2×D4—2024no—
232C2×D4—2024no—
233C2×D4—2024no—
234C2×D4C22⋊D10118yescomplemented
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—1012no—
238C22×C4—1014no—
239C22×C4—2024no—
240C22×C4—2022no—
241C22×C4C22×D10112yes—
242C22⋊C4—4042no—
243C22⋊C4—2028no—
244C22⋊C4—2024no—
245C22⋊C4—2024no—
246C22⋊C4—2028no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—24no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—24no—
259C23—28no—
260C23—28no—
261C23—44no—
262C23—416no—
263C23—416no—
264C23—44no—
265C23—44no—
266C23—1014no—
267C23—4044no—
268C23—40416no—
269C23—4044no—
270C23—2028no—
271C23—2022no—
272C23—2024no—
273C23—2022no—
274C23—2024no—
275C23D4×D10114yes—
276C23C2×C22⋊D10112yes—
277C23C2×C22⋊D10112yes—
278C23C2×C22⋊D10114yes—
279C23C2×C22⋊D10112yes—
280C23C2×C22⋊D10114yescomplemented
281C23C23×D10111yescharacteristic
282D4—216no—
283D4—216no—
284D4—40416no—
285D4—20216no—
286D4—20216no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—1014no—
291C2×C4—2024no—
292C2×C4—2024no—
293C2×C4—2024no—
294C2×C4—2028no—
295C2×C4—2022no—
296C2×C4—2022no—
297C2×C4C2×C22⋊D10114yescomplemented
298C22—22no—
299C22—22no—
300C22—28no—
301C22—28no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—28no—
306C22—22no—
307C22—22no—
308C22—24no—
309C22—416no—
310C22—416no—
311C22—42no—
312C22—44no—
313C22—44no—
314C22—42no—
315C22—40416no—
316C22—4042no—
317C22—4044no—
318C22—4042no—
319C22—2028no—
320C22—2024no—
321C22#1569222314yescomplemented
322C22#1569222414yescomplemented
323C22C2×D4×D10112yes—
324C22C22×C22⋊D10111yescharacteristic
325C22C22×C22⋊D10112yes—
326C22#1569230012yes—
327C4—24no—
328C4—4044no—
329C4—2024no—
330C2—24no—
331C2—24no—
332C2—42no—
333C2—44no—
334C2—4044no—
335C2C2×#1569222311yescharacteristic
336C2C2×#1569222411yescharacteristic
337C2#2026573611yescharacteristic
338C2D4×C22⋊D10114yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
FittingC23×D4×C101C2
hypercenterC23×D4D101
hypocenterC101#2403
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D101C2semidirect
2. C2×#15692224C2semidirect
3. C2×#15692223C2semidirect
4. #20265085C2semidirect
5. #20265057C2semidirect
6. #20265056C2semidirect
7. #20265033C2semidirect
8. C23×D4×C101C2semidirect
9. C24×D101C22semidirect
13. C22×C22⋊D101C22semidirect
14. C22×C22⋊D101C22semidirect
15. C23×Dic101C22semidirect
18. C2×C22⋊Dic101C22semidirect
19. C2×C22⋊Dic101C22semidirect
20. C2×C22⋊Dic101C22semidirect
21. C2×C22⋊Dic101C22semidirect
22. C2×#13108051C22semidirect
23. #15692162C22semidirect
25. C25×C101C22semidirect
29. C22×D4×C101C22semidirect
30. C23×C4×C101C22semidirect
45. C2×C22⋊D101D4semidirect
62. C24×C101D4semidirect
73. C2×D4×C101D4semidirect
101. C23×C101C2×D4semidirect
111. C22×D101C2×D4semidirect
117. C2×C4×C101C2×D4semidirect
127. C2×Dic101C2×D4semidirect
145. C22×C101#120semidirect
146. C22×C101#120semidirect
170. C101#2403semidirect
172. C23×D4D101semidirect
181. C25C2×D101semidirect
190. C22×D4C2×D101semidirect
192. C23×C4C2×D101semidirect
213. C24C22⋊D101semidirect
234. C2×D4C22⋊D101semidirect
280. C23C2×C22⋊D101semidirect
297. C2×C4C2×C22⋊D101semidirect
321. C22#15692223semidirect
322. C22#15692224semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1012
C2×#156922242
C2×#156922232
#202650852
#202650572
#202650562
#202650332
C23×D4×C1012
#2403101