Finite Group Databasebeta

← #27180924

Subgroups — #27180924

Order 12928.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#20265732C211yesmaximal, characteristic, complemented
2C4×D4×D101C211yesmaximal, characteristic
3#20265480C212yesmaximal, complemented
4#20265479C212yesmaximal, complemented
5#20265468C211yesmaximal, characteristic, complemented
6#20265467C211yesmaximal, characteristic, complemented
7#20265454C211yesmaximal, characteristic, complemented
8#20265218C212yesmaximal, complemented
9#20265199C211yesmaximal, characteristic, complemented
10#20265180C212yesmaximal
11#474×C101C211yesmaximal, characteristic, complemented
12C2×D4×D101C2211yescharacteristic
13C22×C4×D101—21no—
14C2×#13108057C2212yes—
15#15692255C2211yescharacteristic, complemented
16#15692246C2212yescomplemented
17#15692235—21no—
18#15692231—21no—
19Q8×Dic101—21no—
20#15692220—21no—
21C4×D404C2211yescharacteristic
22D4×Dic101C2211yescharacteristic
23#15692217C2211yescharacteristic
24C4×C22⋊D101—21no—
25#15692215—21no—
26#15692213C2212yes—
27C4⋊C4×D101C2211yescharacteristic
28C22⋊C4×D101—21no—
29C42×D101C2211yescharacteristic
30#15692188C2212yescomplemented
31#15692187C2212yescomplemented
32#15692186C2212yescomplemented
33C4⋊#11717579C2211yescharacteristic, complemented
34#15692184C2212yescomplemented
35#15692177C2212yes—
36#15692176C2212yes—
37#15692175C2212yes—
38C4⋊Q8×C101C2211yescharacteristic, complemented
39C4×D4×C101C2211yescharacteristic
40#107×C101C2212yescomplemented
41C4⋊C8×C101C2211yescharacteristic, complemented
42#102×C101C2212yes—
43C23×D101—21no—
44C2×Dic202—21no—
45C2×D404C2311yescharacteristic
46C2×C22⋊D101—21no—
47D4×D101—22no—
48D4×D101Q814yescomplemented
49C22×Dic101—21no—
50C2×C4×D101—21no—
51C2×C4×D101—21no—
52C2×C4×D101—21no—
53C2×C4×D101—44no—
54C2×C4×D101D411yescharacteristic
55C2×C4×D101D411yescharacteristic
56C2×C4×D101C2311yescharacteristic
57#13108057—24no—
58C2×#11717579C2312yes—
59C4⋊Dic101—21no—
60C4⋊Dic101—22no—
61C4⋊Dic101C2312yes—
62C4⋊Dic101C2311yescharacteristic
63#13108053—21no—
64#13108053—22no—
65C22⋊Dic101—21no—
66#13108051—22no—
67#13108051—21no—
68#13108051—21no—
69C4×Dic101—22no—
70C4×Dic101D411yescharacteristic
71C4×Dic101D411yescharacteristic
72C4×Dic101C2311yescharacteristic
73C2×Q8×C101—21no—
74C2×D4×C101C2311yescharacteristic
75C22×C4×C101—21no—
76C2×C8×C101C2312yes—
77C4⋊C4×C101—21no—
78C4⋊C4×C101C2312yes—
79C4⋊C4×C101C2311yescharacteristic
80C22⋊C4×C101—21no—
81C42×C101C2311yescharacteristic
82C23×C101—21no—
83C22×D101—21no—
84C22×D101—24no—
85C22×D101—42no—
86C22×D101—42no—
87C22×D101C2×D411yescharacteristic
88Q8×C101—42no—
89D4×C101—21no—
90D4×C101C2×Q812yes—
91C2×C4×C101—21no—
92C2×C4×C101—22no—
93C2×C4×C101—42no—
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2411yescharacteristic
97C22⋊D101—24no—
98C2×Dic101—22no—
99C2×Dic101—21no—
100C2×Dic101—42no—
101C2×Dic101C2×D411yescharacteristic
102C2×Dic101C2×D411yescharacteristic
103C2×Dic101C2×D411yescharacteristic
104D404—21no—
105D404C2×Q812yes—
106C4×D101—22no—
107C4×D101—41no—
108C4×D101—42no—
109C4×D101—42no—
110C4×D101—42no—
111C4×D101C2×Q812yes—
112C4×D101#4112yes—
113Dic202—42no—
114C8×C101—22no—
115#11717579—22no—
116C22×C101—22no—
117C22×C101—41no—
118C22×C101—41no—
119C22×C101C22×D411yescharacteristic
120C2×D101—21no—
121C2×D101—21no—
122C2×D101—22no—
123C2×D101—48no—
124C2×D101—41no—
125C2×D101—41no—
126C2×D101C22⋊Q812yes—
127C4×C101—21no—
128C4×C101—41no—
129C4×C101—42no—
130C4×C101AGL(1,ℤ/8)12yes—
131C4×C101C22×Q811yescharacteristic
132C4×C101C2×#4111yescharacteristic
133Dic101—21no—
134Dic101—21no—
135Dic101—41no—
136Dic101—42no—
137Dic101C22⋊Q812yes—
138C2×C101—42no—
139C2×C101C2×C22⋊Q811yescharacteristic
140C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
141C2×C101#57511yescharacteristic
142D101—22no—
143D101—42no—
144#3077—1011nomaximal, Sylow
145C101#307711yescharacteristic, complemented, Sylow
146#533—1011no—
147C2×C4×D4—1011no—
148#478—1012no—
149#478—1012no—
150#476—1011no—
151#476—1011no—
152#474—1011no—
153#474D10111yescharacteristic, complemented
154#428—1012no—
155#423—1011no—
156#418—1012no—
157C22×D4—1011no—
158C23×C4—2021no—
159C2×M16—1012no—
160C4⋊Q8—1011no—
161C4⋊Q8C2×D10111yescharacteristic, complemented
162#125—1012no—
163C22⋊Q8—2021no—
164C22⋊Q8—2021no—
165C4×Q8—2021no—
166C4×Q8—2021no—
167C4×D4—1011no—
168C4×D4—1011no—
169C4×D4—1011no—
170C4×D4—2021no—
171C4×D4—2021no—
172C4×D4C2×D10111yescharacteristic
173#117—1012no—
174C2×C4⋊C4—1011no—
175C2×C22⋊C4—2021no—
176C2×C42—1011no—
177#107—1012no—
178#107C2×D10112yescomplemented
179#106—1012no—
180#106—1012no—
181C4⋊C8—1012no—
182C4⋊C8—1011no—
183C4⋊C8C2×D10111yescharacteristic, complemented
184#102—1012no—
185#102—1012no—
186#102—1012no—
187#102C2×D10112yes—
188C24—2021no—
189C2×Q8—21no—
190C2×Q8—2021no—
191C2×D4—1014no—
192C2×D4—1011no—
193C2×D4—2022no—
194C2×D4—2021no—
195C2×D4C22×D10111yescharacteristic
196C22×C4—21no—
197C22×C4—1011no—
198C22×C4—1011no—
199C22×C4—1011no—
200C22×C4—4044no—
201C22×C4—2021no—
202C22×C4—2021no—
203C22×C4—2021no—
204C22×C4—2021no—
205M16—2024no—
206C2×C8—1012no—
207C2×C8C22×D10112yes—
208C4⋊C4—21no—
209C4⋊C4—1012no—
210C4⋊C4—1011no—
211C4⋊C4—2021no—
212C4⋊C4—2022no—
213C4⋊C4—2022no—
214C4⋊C4—2021no—
215C4⋊C4C22×D10112yes—
216C4⋊C4C22×D10111yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—2021no—
219C22⋊C4—2022no—
220C22⋊C4—2021no—
221C22⋊C4—2021no—
222C42—1011no—
223C42—1011no—
224C42—1011no—
225C42—2022no—
226C42C22×D10111yescharacteristic
227C23—21no—
228C23—1011no—
229C23—4042no—
230C23—4042no—
231C23—2024no—
232C23—2021no—
233Q8—42no—
234Q8—4042no—
235D4—21no—
236D4—1012no—
237D4—2024no—
238D4—2021no—
239D4Q8×D10112yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—1011no—
244C2×C4—1012no—
245C2×C4—1012no—
246C2×C4—1011no—
247C2×C4—1011no—
248C2×C4—4041no—
249C2×C4—4042no—
250C2×C4—4042no—
251C2×C4—4042no—
252C2×C4—4042no—
253C2×C4—2022no—
254C2×C4—2021no—
255C2×C4—2022no—
256C2×C4D4×D10111yescharacteristic
257C2×C4D4×D10111yescharacteristic
258C2×C4C23×D10111yescharacteristic
259C8—22no—
260C8—2022no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—1012no—
265C22—4048no—
266C22—4041no—
267C22—4041no—
268C22—2022no—
269C22—2021no—
270C22—2021no—
271C22C2×D4×D10111yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—1012no—
276C4—4041no—
277C4—4042no—
278C4—2021no—
279C4—2021no—
280C4#1569227912yes—
281C4C2×Q8×D10111yescharacteristic
282C4#41×D10111yescharacteristic
283C2—42no—
284C2—4042no—
285C2—2022no—
286C2C22⋊Q8×D10111yescharacteristic
287C2C2×#1569227911yescharacteristic
288C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#474×C101C2
socleC22×C101C22×D4
prime radicalC2×D4×D101C22
hypercenter#474D101
hypocenterC101#3077
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20265732C2semidirect
3. #20265480C2semidirect
4. #20265479C2semidirect
5. #20265468C2semidirect
6. #20265467C2semidirect
7. #20265454C2semidirect
8. #20265218C2semidirect
9. #20265199C2semidirect
11. #474×C101C2semidirect
15. #15692255C22semidirect
16. #15692246C22semidirect
30. #15692188C22semidirect
31. #15692187C22semidirect
32. #15692186C22semidirect
33. C4⋊#11717579C22semidirect
34. #15692184C22semidirect
38. C4⋊Q8×C101C22semidirect
40. #107×C101C22semidirect
41. C4⋊C8×C101C22semidirect
48. D4×D101Q8semidirect
145. C101#3077semidirect
153. #474D101semidirect
161. C4⋊Q8C2×D101semidirect
178. #107C2×D101semidirect
183. C4⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657322
C4×D4×D1012
#202654802
#202654792
#202654682
#202654672
#202654542
#202652182
#202651992
#202651802
#474×C1012
#3077101