Finite Group Databasebeta

← #27174481

Subgroups — #27174481

Order 12928.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 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×D4×D101C211yesmaximal, characteristic, complemented
2C2×#15692227C212yesmaximal, complemented
3C2×#15692223C212yesmaximal, complemented
4C2×#15692215C211yesmaximal, characteristic, complemented
5#20265095C211yesmaximal, characteristic, complemented
6#20265091C211yesmaximal, characteristic, complemented
7#20265086C211yesmaximal, characteristic, complemented
8#20265077C212yesmaximal, complemented
9#20265036C211yesmaximal, characteristic, complemented
10#20265032C212yesmaximal, complemented
11#392×C101C211yesmaximal, characteristic, complemented
12C24×D101C2212yes—
13C22×D404C2211yescharacteristic
14C22×C22⋊D101C2212yes—
15C22×C22⋊D101C2211yescharacteristic
16C22×C22⋊D101C2212yescomplemented
17C2×D4×D101—28no—
18C2×D4×D101—24no—
19C23×Dic101C2211yescharacteristic, complemented
20C22×C4×D101C2211yescharacteristic
21#15692227—28no—
22#15692223—28no—
23#15692215—24no—
24C2×#13108053C2212yescomplemented
25C2×#13108053C2211yescharacteristic, complemented
26C2×C22⋊Dic101C2212yescomplemented
27C2×C22⋊Dic101C2212yescomplemented
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2212yes—
30C2×#13108051C2212yescomplemented
31C2×#13108051C2212yescomplemented
32C2×#13108051C2212yescomplemented
33C2×C4×Dic101C2211yescharacteristic, complemented
34#15692162C2211yescharacteristic, complemented
35C22×D4×C101C2211yescharacteristic
36C22×D4×C101C2212yescomplemented
37C2×C22⋊C4×C101C2211yescharacteristic, complemented
38C2×C22⋊C4×C101C2212yescomplemented
39#95×C101C2211yescharacteristic, complemented
40C23×D101—24no—
41C23×D101—24no—
42C23×D101—24no—
43C23×D101—24no—
44C23×D101—24no—
45C23×D101—42no—
46C23×D101—42no—
47C23×D101C2312yes—
48C23×D101C2311yescharacteristic
49C2×D404—24no—
50C2×D404—22no—
51C2×C22⋊D101—22no—
52C2×C22⋊D101—22no—
53C2×C22⋊D101—24no—
54C2×C22⋊D101—24no—
55C2×C22⋊D101—24no—
56C2×C22⋊D101—24no—
57C2×C22⋊D101—42no—
58C2×C22⋊D101D418yescomplemented
59C2×C22⋊D101D414yescomplemented
60D4×D101—232no—
61C22×Dic101—22no—
62C22×Dic101—24no—
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2312yes—
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C2×C4×D101—24no—
68C2×C4×D101D414yescomplemented
69#13108053—22no—
70#13108053—24no—
71C22⋊Dic101—24no—
72C22⋊Dic101—24no—
73#13108051—22no—
74#13108051—24no—
75#13108051—24no—
76#13108051—24no—
77#13108051—42no—
78C4×Dic101—22no—
79C24×C101C2312yes—
80C24×C101C2311yescharacteristic
81C2×D4×C101—24no—
82C2×D4×C101—24no—
83C2×D4×C101—24no—
84C2×D4×C101—22no—
85C2×D4×C101—42no—
86C22×C4×C101C2312yes—
87C22×C4×C101C2311yescharacteristic
88C22×C4×C101C2311yescharacteristic
89C22⋊C4×C101—22no—
90C22⋊C4×C101—42no—
91C23×C101—21no—
92C23×C101—22no—
93C23×C101—22no—
94C23×C101—22no—
95C23×C101—22no—
96C23×C101—21no—
97C23×C101—22no—
98C23×C101—22no—
99C23×C101—22no—
100C23×C101—22no—
101C23×C101C2×D414yes—
102C23×C101C2×D412yescomplemented
103C23×C101C2411yescharacteristic
104C22×D101—22no—
105C22×D101—22no—
106C22×D101—216no—
107C22×D101—22no—
108C22×D101—22no—
109C22×D101—22no—
110C22×D101—22no—
111C22×D101—22no—
112C22×D101—22no—
113C22×D101—48no—
114C22×D101—48no—
115C22×D101—48no—
116C22×D101—48no—
117C22×D101—48no—
118C22×D101—48no—
119C22×D101C2×D414yes—
120C22×D101C2×D412yes—
121C22×D101C2×D412yes—
122C22×D101#4112yes—
123D4×C101—28no—
124D4×C101—48no—
125C2×C4×C101—22no—
126C2×C4×C101—21no—
127C2×C4×C101—22no—
128C2×C4×C101—22no—
129C2×C4×C101—22no—
130C2×C4×C101—22no—
131C2×C4×C101C2×D412yes—
132C22⋊D101—28no—
133C22⋊D101—216no—
134C22⋊D101—48no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—22no—
138C2×Dic101—22no—
139C2×Dic101—21no—
140C2×Dic101—22no—
141C2×Dic101—22no—
142C2×Dic101—44no—
143C2×Dic101C2×D412yes—
144C2×Dic101C2×D414yes—
145C2×Dic101C2×D412yescomplemented
146D404—28no—
147C4×D101—28no—
148C22×C101—24no—
149C22×C101—22no—
150C22×C101—24no—
151C22×C101—24no—
152C22×C101—24no—
153C22×C101—41no—
154C22×C101—42no—
155C22×C101—42no—
156C22×C101—41no—
157C22×C101—42no—
158C22×C101—42no—
159C22×C101—42no—
160C22×C101C22×D411yescharacteristic
161C22×C101C22×D412yes—
162C22×C101C22×D411yescharacteristic
163C22×C101C2×#4111yescharacteristic
164C22×C101H(5,𝔽2)12yes—
165C2×D101—22no—
166C2×D101—22no—
167C2×D101—24no—
168C2×D101—24no—
169C2×D101—24no—
170C2×D101—24no—
171C2×D101—24no—
172C2×D101—432no—
173C2×D101—42no—
174C2×D101—42no—
175C2×D101—42no—
176C2×D101—42no—
177C2×D101#12114yescomplemented
178C4×C101—22no—
179C4×C101—41no—
180C4×C101—42no—
181Dic101—22no—
182Dic101—41no—
183Dic101—42no—
184Dic101#12014yescomplemented
185C2×C101—44no—
186C2×C101—42no—
187C2×C101C2×#12011yescharacteristic
188C2×C101C2×#12111yescharacteristic
189C2×C101#53511yescharacteristic
190C2×C101D4212yes—
191C2×C101#54612yes—
192D101—24no—
193D101—44no—
194#2408—1011nomaximal, Sylow
195C101#240811yescharacteristic, complemented, Sylow
196C23×D4—1011no—
197C2×#121—1012no—
198C2×#120—1012no—
199C2×C4×D4—1011no—
200#394—1011no—
201#394—1011no—
202#392—1011no—
203#392D10111yescharacteristic, complemented
204#390—1012no—
205#380—1011no—
206#379—1012no—
207C25—1012no—
208C22×D4—1011no—
209C22×D4—1012no—
210C22×D4—1012no—
211C22×D4—1011no—
212C22×D4—2024no—
213C22×D4—2028no—
214C22×D4C2×D10111yescharacteristic
215C22×D4C2×D10112yescomplemented
216C23×C4—1011no—
217C23×C4—1011no—
218#121—2028no—
219#120—2028no—
220C4×D4—2024no—
221C2×C4⋊C4—1011no—
222C2×C4⋊C4—1012no—
223C2×C22⋊C4—1012no—
224C2×C22⋊C4—1012no—
225C2×C22⋊C4—1012no—
226C2×C22⋊C4—1011no—
227C2×C22⋊C4—1012no—
228C2×C22⋊C4—1012no—
229C2×C22⋊C4—1012no—
230C2×C22⋊C4C2×D10111yescharacteristic, complemented
231C2×C22⋊C4C2×D10112yescomplemented
232C2×C42—1011no—
233#95—1011no—
234#95C2×D10111yescharacteristic, complemented
235C24—1011no—
236C24—1012no—
237C24—4042no—
238C24—4042no—
239C24—2024no—
240C24—2024no—
241C24—2024no—
242C24—2024no—
243C24—2024no—
244C24C22×D10112yes—
245C24C22×D10111yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—1018no—
252C2×D4—1014no—
253C2×D4—4042no—
254C2×D4—2024no—
255C2×D4—2024no—
256C2×D4—2024no—
257C2×D4—20232no—
258C2×D4—2024no—
259C2×D4—2024no—
260C2×D4—2022no—
261C2×D4—2022no—
262C2×D4—2022no—
263C22×C4—1011no—
264C22×C4—1011no—
265C22×C4—1014no—
266C22×C4—1012no—
267C22×C4—1011no—
268C22×C4—2024no—
269C22×C4—2022no—
270C22×C4—2024no—
271C22×C4C22×D10112yes—
272C22×C4C22×D10111yescharacteristic
273C22×C4C22×D10111yescharacteristic
274C4⋊C4—2024no—
275C4⋊C4—2022no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—4042no—
279C22⋊C4—2024no—
280C22⋊C4—2022no—
281C22⋊C4—2024no—
282C22⋊C4—2024no—
283C22⋊C4—2024no—
284C22⋊C4—2024no—
285C42—2022no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—21no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—1012no—
297C23—1012no—
298C23—1012no—
299C23—1014no—
300C23—4048no—
301C23—4048no—
302C23—4048no—
303C23—4048no—
304C23—4048no—
305C23—4048no—
306C23—20216no—
307C23—2022no—
308C23—2022no—
309C23—2022no—
310C23—2022no—
311C23—2022no—
312C23—2022no—
313C23—2022no—
314C23—2022no—
315C23D4×D10114yes—
316C23D4×D10112yescomplemented
317C23C23×D10111yescharacteristic
318D4—28no—
319D4—48no—
320D4—4048no—
321D4—2028no—
322D4—20216no—
323D4—2028no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—1012no—
331C2×C4—1014no—
332C2×C4—1012no—
333C2×C4—4044no—
334C2×C4—2028no—
335C2×C4—2022no—
336C2×C4—2022no—
337C2×C4—2021no—
338C2×C4—2022no—
339C2×C4—2022no—
340C2×C4—2021no—
341C2×C4—2021no—
342C2×C4D4×D10112yes—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—42no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—1014no—
356C22—40432no—
357C22—4042no—
358C22—4042no—
359C22—4042no—
360C22—4042no—
361C22—2024no—
362C22—2024no—
363C22—2024no—
364C22—2022no—
365C22—2024no—
366C22—2024no—
367C22—2022no—
368C22C2×D4×D10111yescharacteristic
369C22C2×D4×D10111yescharacteristic
370C22C2×D4×D10112yes—
371C22#41×D10111yescharacteristic
372C22#1569230012yes—
373C4—22no—
374C4—41no—
375C4—42no—
376C4—1014no—
377C4—4042no—
378C4—4041no—
379C4—2022no—
380C2—44no—
381C2—42no—
382C2—4044no—
383C2—2024no—
384C2#120×D10111yescharacteristic
385C2#121×D10111yescharacteristic
386C2#2026574011yescharacteristic
387C2#2026580912yes—
388C2#2026581612yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#392×C101C2
hypercenter#392D101
hypocenterC101#2408
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D101C2semidirect
2. C2×#15692227C2semidirect
3. C2×#15692223C2semidirect
4. C2×#15692215C2semidirect
5. #20265095C2semidirect
6. #20265091C2semidirect
7. #20265086C2semidirect
8. #20265077C2semidirect
9. #20265036C2semidirect
10. #20265032C2semidirect
11. #392×C101C2semidirect
16. C22×C22⋊D101C22semidirect
19. C23×Dic101C22semidirect
24. C2×#13108053C22semidirect
25. C2×#13108053C22semidirect
26. C2×C22⋊Dic101C22semidirect
27. C2×C22⋊Dic101C22semidirect
30. C2×#13108051C22semidirect
31. C2×#13108051C22semidirect
32. C2×#13108051C22semidirect
33. C2×C4×Dic101C22semidirect
34. #15692162C22semidirect
36. C22×D4×C101C22semidirect
37. C2×C22⋊C4×C101C22semidirect
38. C2×C22⋊C4×C101C22semidirect
39. #95×C101C22semidirect
58. C2×C22⋊D101D4semidirect
59. C2×C22⋊D101D4semidirect
68. C2×C4×D101D4semidirect
102. C23×C101C2×D4semidirect
145. C2×Dic101C2×D4semidirect
177. C2×D101#121semidirect
184. Dic101#120semidirect
195. C101#2408semidirect
203. #392D101semidirect
215. C22×D4C2×D101semidirect
230. C2×C22⋊C4C2×D101semidirect
231. C2×C22⋊C4C2×D101semidirect
234. #95C2×D101semidirect
316. C23D4×D101semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1012
C2×#156922272
C2×#156922232
C2×#156922152
#202650952
#202650912
#202650862
#202650772
#202650362
#202650322
#392×C1012
#2408101