Finite Group Databasebeta

← #27174446

Subgroups — #27174446

Order 12928.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 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, complemented
2C2×#15692223C212yesmaximal, complemented
3C2×C22⋊C4×D101C211yesmaximal, characteristic, complemented
4#20265090C211yesmaximal, characteristic, complemented
5#20265089C211yesmaximal, characteristic, complemented
6#20265085C211yesmaximal, characteristic, complemented
7#20265057C212yesmaximal, complemented
8#20265054C212yesmaximal, complemented
9#20265053C212yesmaximal, complemented
10#20265039C211yesmaximal, characteristic, complemented
11#392×C101C211yesmaximal, characteristic, complemented
12C24×D101C2211yescharacteristic
13C2×#13108073—28no—
14C2×#13108073—24no—
15C22×Dic202C2211yescharacteristic, complemented
16C22×C22⋊D101C2212yes—
17C22×C22⋊D101C2212yes—
18C23×Dic101C2212yescomplemented
19C22×C4×D101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21#15692223—28no—
22C2×C22⋊Dic101C2211yescharacteristic
23C2×C22⋊Dic101C2212yes—
24C2×C22⋊Dic101C2212yescomplemented
25C2×#13108051C2212yes—
26C2×#13108051C2211yescharacteristic
27C2×#13108051C2212yes—
28C2×#13108051C2211yescharacteristic
29C22⋊C4×D101—24no—
30#15692162C2212yescomplemented
31#15692162C2211yescharacteristic, complemented
32#15692162C2212yescomplemented
33#15692162C2212yescomplemented
34C22×D4×C101C2212yes—
35C22×D4×C101C2211yescharacteristic, complemented
36C2×C22⋊C4×C101C2211yescharacteristic
37C2×C22⋊C4×C101C2212yescomplemented
38#95×C101C2211yescharacteristic, complemented
39C23×D101—22no—
40C23×D101—22no—
41C23×D101—22no—
42C23×D101—24no—
43C23×D101—42no—
44C23×D101C2311yescharacteristic
45C23×D101C2311yescharacteristic
46#13108073—232no—
47C2×Dic202—22no—
48C2×Dic202—24no—
49C2×C22⋊D101—24no—
50C2×C22⋊D101—24no—
51C2×C22⋊D101—24no—
52C2×C22⋊D101—24no—
53C2×C22⋊D101—42no—
54C2×C22⋊D101D418yescomplemented
55C22×Dic101—24no—
56C22×Dic101—24no—
57C22×Dic101—24no—
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2312yes—
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2312yes—
62C2×C4×D101—24no—
63C2×C4×D101—22no—
64C2×C4×D101—42no—
65C2×C4×D101D414yescomplemented
66C22⋊Dic101—22no—
67C22⋊Dic101—24no—
68C22⋊Dic101—42no—
69#13108051—24no—
70#13108051—22no—
71#13108051—22no—
72#13108051—24no—
73C24×C101C2312yes—
74C24×C101C2311yescharacteristic
75C2×D4×C101—24no—
76C2×D4×C101—24no—
77C2×D4×C101—24no—
78C2×D4×C101—22no—
79C2×D4×C101—42no—
80C22×C4×C101C2311yescharacteristic
81C22×C4×C101C2312yes—
82C22×C4×C101C2311yescharacteristic
83C22⋊C4×C101—22no—
84C22⋊C4×C101—42no—
85C23×C101—21no—
86C23×C101—22no—
87C23×C101—22no—
88C23×C101—22no—
89C23×C101—22no—
90C23×C101—22no—
91C23×C101—21no—
92C23×C101—22no—
93C23×C101—22no—
94C23×C101—22no—
95C23×C101C2×D414yes—
96C23×C101#4112yescomplemented
97C23×C101C2411yescharacteristic
98C22×D101—21no—
99C22×D101—21no—
100C22×D101—22no—
101C22×D101—28no—
102C22×D101—21no—
103C22×D101—22no—
104C22×D101—21no—
105C22×D101—22no—
106C22×D101—48no—
107C22×D101—48no—
108C22×D101—44no—
109C22×D101—44no—
110C22×D101C2×D414yes—
111C22×D101C2×D412yes—
112C22×D101#4112yescomplemented
113D4×C101—28no—
114D4×C101—48no—
115C2×C4×C101—22no—
116C2×C4×C101—21no—
117C2×C4×C101—22no—
118C2×C4×C101—22no—
119C2×C4×C101—22no—
120C2×C4×C101—22no—
121C2×C4×C101C2×D412yes—
122C22⋊D101—216no—
123C22⋊D101—48no—
124C2×Dic101—216no—
125C2×Dic101—22no—
126C2×Dic101—22no—
127C2×Dic101—22no—
128C2×Dic101—21no—
129C2×Dic101—22no—
130C2×Dic101—22no—
131C2×Dic101—22no—
132C2×Dic101—22no—
133C2×Dic101C2×D412yes—
134C2×Dic101C2×D414yes—
135C4×D101—28no—
136C4×D101—44no—
137Dic202—28no—
138C22×C101—24no—
139C22×C101—22no—
140C22×C101—24no—
141C22×C101—24no—
142C22×C101—24no—
143C22×C101—41no—
144C22×C101—42no—
145C22×C101—41no—
146C22×C101—42no—
147C22×C101—42no—
148C22×C101—42no—
149C22×C101—42no—
150C22×C101C22×D411yescharacteristic
151C22×C101C22×D412yes—
152C22×C101C2×#4111yescharacteristic
153C22×C101C2×#4111yescharacteristic
154C22×C101H(5,𝔽2)12yes—
155C2×D101—22no—
156C2×D101—22no—
157C2×D101—24no—
158C2×D101—24no—
159C2×D101—22no—
160C2×D101—24no—
161C2×D101—416no—
162C2×D101—41no—
163C2×D101—41no—
164C2×D101—42no—
165C2×D101#12014yes—
166C4×C101—22no—
167C4×C101—41no—
168C4×C101—42no—
169Dic101—24no—
170Dic101—22no—
171Dic101—41no—
172Dic101—42no—
173C2×C101—42no—
174C2×C101—44no—
175C2×C101C2×#12011yescharacteristic
176C2×C101#52511yescharacteristic
177C2×C101#53511yescharacteristic
178C2×C101#54612yes—
179C2×C101#54612yes—
180D101—24no—
181D101—42no—
182#2404—1011nomaximal, Sylow
183C101#240411yescharacteristic, complemented, Sylow
184C22×#41—1011no—
185C2×#120—1012no—
186C22×C22⋊C4—1011no—
187#393—1011no—
188#393—1011no—
189#392—1011no—
190#392D10111yescharacteristic, complemented
191#386—1012no—
192#386—1012no—
193#386—1012no—
194#381—1011no—
195C25—1011no—
196C2×#41—2024no—
197C2×#41—2028no—
198C22×Q8—1011no—
199C22×D4—1012no—
200C22×D4—1012no—
201C22×D4C2×D10112yes—
202C22×D4C2×D10111yescharacteristic, complemented
203C23×C4—1011no—
204C23×C4—1012no—
205C23×C4—1011no—
206#120—2028no—
207C2×C22⋊C4—1012no—
208C2×C22⋊C4—1011no—
209C2×C22⋊C4—1012no—
210C2×C22⋊C4—1011no—
211C2×C22⋊C4—1011no—
212C2×C22⋊C4—1012no—
213C2×C22⋊C4—1012no—
214C2×C22⋊C4—2024no—
215C2×C22⋊C4C2×D10111yescharacteristic
216C2×C22⋊C4C2×D10112yescomplemented
217#95—1012no—
218#95—1011no—
219#95—1012no—
220#95—1012no—
221#95C2×D10111yescharacteristic, complemented
222C24—1011no—
223C24—1011no—
224C24—4042no—
225C24—2024no—
226C24—2022no—
227C24—2022no—
228C24—2022no—
229C24C22×D10112yes—
230C24C22×D10111yescharacteristic
231#41—20232no—
232C2×Q8—2024no—
233C2×Q8—2022no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—1018no—
240C2×D4—4042no—
241C2×D4—2024no—
242C2×D4—2024no—
243C2×D4—2024no—
244C2×D4—2024no—
245C22×C4—1011no—
246C22×C4—1012no—
247C22×C4—1014no—
248C22×C4—1012no—
249C22×C4—1011no—
250C22×C4—4042no—
251C22×C4—2022no—
252C22×C4—2024no—
253C22×C4—2024no—
254C22×C4—2024no—
255C22×C4—2024no—
256C22×C4C22×D10112yes—
257C22×C4C22×D10111yescharacteristic
258C22×C4C22×D10111yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—4042no—
262C22⋊C4—2022no—
263C22⋊C4—2024no—
264C22⋊C4—2024no—
265C22⋊C4—2022no—
266C22⋊C4—2024no—
267C22⋊C4—2022no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—1014no—
279C23—1012no—
280C23—1012no—
281C23—4048no—
282C23—4044no—
283C23—4044no—
284C23—4048no—
285C23—2028no—
286C23—2021no—
287C23—2022no—
288C23—2021no—
289C23—2022no—
290C23—2021no—
291C23—2021no—
292C23—2022no—
293C23D4×D10114yes—
294C23#1310807312yescomplemented
295C23C23×D10111yescharacteristic
296Q8—2028no—
297D4—28no—
298D4—48no—
299D4—4048no—
300D4—20216no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—1012no—
308C2×C4—1014no—
309C2×C4—4044no—
310C2×C4—2028no—
311C2×C4—2021no—
312C2×C4—2022no—
313C2×C4—2022no—
314C2×C4—2022no—
315C2×C4—2022no—
316C2×C4—20216no—
317C2×C4—2022no—
318C2×C4—2022no—
319C2×C4—2022no—
320C2×C4D4×D10112yes—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—22no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—42no—
333C22—1014no—
334C22—40416no—
335C22—4041no—
336C22—4042no—
337C22—4041no—
338C22—2024no—
339C22—2022no—
340C22—2022no—
341C22—2024no—
342C22—2024no—
343C22—2022no—
344C22C2×D4×D10111yescharacteristic
345C22C2×D4×D10112yes—
346C22#41×D10111yescharacteristic
347C22C2×#1310807311yescharacteristic
348C22#1569230012yes—
349C4—22no—
350C4—41no—
351C4—42no—
352C4—4041no—
353C4—4042no—
354C4—2022no—
355C4—2024no—
356C2—42no—
357C2—44no—
358C2—4042no—
359C2—2024no—
360C2#120×D10111yescharacteristic
361C2#2026568711yescharacteristic
362C2#2026574011yescharacteristic
363C2#2026581412yes—
364C2#2026581612yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#392×C101C2
hypercenter#392D101
hypocenterC101#2404
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#13108073C2semidirect
2. C2×#15692223C2semidirect
3. C2×C22⋊C4×D101C2semidirect
4. #20265090C2semidirect
5. #20265089C2semidirect
6. #20265085C2semidirect
7. #20265057C2semidirect
8. #20265054C2semidirect
9. #20265053C2semidirect
10. #20265039C2semidirect
11. #392×C101C2semidirect
15. C22×Dic202C22semidirect
18. C23×Dic101C22semidirect
24. C2×C22⋊Dic101C22semidirect
30. #15692162C22semidirect
31. #15692162C22semidirect
32. #15692162C22semidirect
33. #15692162C22semidirect
35. C22×D4×C101C22semidirect
37. C2×C22⋊C4×C101C22semidirect
38. #95×C101C22semidirect
54. C2×C22⋊D101D4semidirect
65. C2×C4×D101D4semidirect
96. C23×C101#41semidirect
112. C22×D101#41semidirect
183. C101#2404semidirect
190. #392D101semidirect
202. C22×D4C2×D101semidirect
216. C2×C22⋊C4C2×D101semidirect
221. #95C2×D101semidirect
294. C23#13108073semidirect

Maximal subgroups

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

subgroupindex
C22×#131080732
C2×#156922232
C2×C22⋊C4×D1012
#202650902
#202650892
#202650852
#202650572
#202650542
#202650532
#202650392
#392×C1012
#2404101