Finite Group Databasebeta

← #38786697

Subgroups — #38786697

Order 25664.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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#27165732C211yesmaximal, characteristic, complemented
2#27165731C211yesmaximal, characteristic, complemented
3#27165725C211yesmaximal, characteristic, complemented
4#27165723C211yesmaximal, characteristic, complemented
5#27165722C211yesmaximal, characteristic, complemented
6#27165720C211yesmaximal, characteristic, complemented
7#27165714C211yesmaximal, characteristic, complemented
8#27165711C211yesmaximal, characteristic, complemented
9#27165710C211yesmaximal, characteristic, complemented
10#27165709C211yesmaximal, characteristic, complemented
11#27165707C211yesmaximal, characteristic, complemented
12#27165705C211yesmaximal, characteristic, complemented
13C4×C22⋊D401C211yesmaximal, characteristic, complemented
14C2×#20264343C211yesmaximal, characteristic, complemented
15C4×D4×C401C211yesmaximal, characteristic, complemented
16C23×D401C2211yescharacteristic
17C2×Dic802C2211yescharacteristic, complemented
18C2×D1604C2211yescharacteristic
19C2×D1604C2211yescharacteristic
20C2×C22⋊D401C2211yescharacteristic
21C2×C22⋊D401C2211yescharacteristic
22C2×C22⋊D401C2211yescharacteristic
23C2×C22⋊D401C2211yescharacteristic, complemented
24C22×Dic401C2211yescharacteristic
25C2×C4×D401C2211yescharacteristic, complemented
26C4⋊Dic401C2211yescharacteristic, complemented
27#20264345C2211yescharacteristic, complemented
28#20264345C2211yescharacteristic, complemented
29#20264345C2211yescharacteristic, complemented
30#20264345C2211yescharacteristic, complemented
31C22⋊Dic401C2211yescharacteristic
32C22⋊Dic401C2211yescharacteristic, complemented
33#20264343C2211yescharacteristic
34#20264343C2211yescharacteristic
35#20264343C2211yescharacteristic
36#20264343C2211yescharacteristic
37#20264343C2211yescharacteristic
38#20264343C2211yescharacteristic
39#20264343C2211yescharacteristic
40#20264343C2211yescharacteristic
41#20264343C2211yescharacteristic, complemented
42#20264343C2211yescharacteristic, complemented
43C4×Dic401C2211yescharacteristic, complemented
44C2×D4×C401C2211yescharacteristic
45C22×C4×C401C2211yescharacteristic
46C22×C4×C401C2211yescharacteristic, complemented
47C4⋊C4×C401C2211yescharacteristic, complemented
48C22⋊C4×C401C2211yescharacteristic
49C22⋊C4×C401C2211yescharacteristic, complemented
50C42×C401C2211yescharacteristic, complemented
51C22×D401—22no—
52C22×D401—41no—
53C22×D401—41no—
54C22×D401C2311yescharacteristic
55C22×D401C2311yescharacteristic
56C22×D401C2311yescharacteristic
57Dic802—41no—
58D1604—41no—
59D1604—41no—
60C22⋊D401—22no—
61C22⋊D401—41no—
62C22⋊D401—41no—
63C22⋊D401—41no—
64C2×Dic401—41no—
65C2×Dic401C2311yescharacteristic
66C2×Dic401C2311yescharacteristic
67C2×Dic401C2311yescharacteristic
68C2×Dic401C2311yescharacteristic
69C2×Dic401C2311yescharacteristic
70C4×D401—41no—
71C23×C401C2311yescharacteristic
72C23×C401C2311yescharacteristic
73D4×C401—22no—
74C2×C4×C401—41no—
75C2×C4×C401—41no—
76C2×C4×C401C2311yescharacteristic
77C2×C4×C401C2311yescharacteristic
78C2×C4×C401C2311yescharacteristic
79C2×C4×C401C2311yescharacteristic
80C2×C4×C401C2311yescharacteristic
81C22×C401—21no—
82C22×C401—21no—
83C22×C401—21no—
84C22×C401—21no—
85C22×C401—21no—
86C22×C401#4112yes—
87C22×C401C2411yescharacteristic
88C2×D401—21no—
89C2×D401—21no—
90C2×D401—21no—
91C2×D401—21no—
92C2×D401—21no—
93C2×D401—21no—
94C2×D401—21no—
95C2×D401—21no—
96C2×D401—21no—
97C2×D401—44no—
98C4×C401—21no—
99C4×C401—21no—
100C4×C401—21no—
101C4×C401—21no—
102C4×C401—21no—
103Dic401—21no—
104Dic401—21no—
105Dic401—21no—
106Dic401—21no—
107Dic401—21no—
108C2×C401—22no—
109C2×C401—41no—
110C2×C401C2×#4111yescharacteristic
111C2×C401H(5,𝔽2)11yescharacteristic
112C2×C401H(5,𝔽2)11yescharacteristic
113D401—41no—
114D401—41no—
115D401—41no—
116C401#53811yescharacteristic, complemented, Sylow
117#538—4011nomaximal, Sylow
118#126—4011no—
119#126—4011no—
120#124—4011no—
121#124—4011no—
122#123—4011no—
123#123—4011no—
124C22⋊Q8—4011no—
125#121—4011no—
126#121—4011no—
127#121—4011no—
128#120—4011no—
129#120—4011no—
130C4×D4—4011no—
131C4×D4D40111yescharacteristic, complemented
132C2×C22⋊C4—4011no—
133C24—4011no—
134C2×Q8—4011no—
135C2×D4—4011no—
136C2×D4—4011no—
137C2×D4—4011no—
138C2×D4—4011no—
139C2×D4—4011no—
140C2×D4—4011no—
141C2×D4C2×D40111yescharacteristic
142C22×C4—4011no—
143C22×C4—4011no—
144C22×C4C2×D40111yescharacteristic
145C22×C4C2×D40111yescharacteristic, complemented
146C4⋊C4—4011no—
147C4⋊C4—4011no—
148C4⋊C4—4011no—
149C4⋊C4—4011no—
150C4⋊C4—4011no—
151C4⋊C4C2×D40111yescharacteristic, complemented
152C22⋊C4—4011no—
153C22⋊C4—4011no—
154C22⋊C4—4011no—
155C22⋊C4—4011no—
156C22⋊C4—4011no—
157C22⋊C4—4011no—
158C22⋊C4—4011no—
159C22⋊C4—4011no—
160C22⋊C4—4011no—
161C22⋊C4—4011no—
162C22⋊C4—4011no—
163C22⋊C4—4011no—
164C22⋊C4C2×D40111yescharacteristic
165C22⋊C4C2×D40111yescharacteristic, complemented
166C42—4011no—
167C42C2×D40111yescharacteristic, complemented
168C23—8022no—
169C23—16041no—
170C23—16041no—
171C23—4011no—
172C23—4011no—
173C23—4011no—
174C23C22×D40111yescharacteristic
175C23C22×D40111yescharacteristic
176Q8—16041no—
177D4—22no—
178D4—8022no—
179D4—16041no—
180D4—16041no—
181D4—16041no—
182D4—16041no—
183D4—16041no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—16041no—
187C2×C4—16041no—
188C2×C4—4011no—
189C2×C4—4011no—
190C2×C4—4011no—
191C2×C4—4011no—
192C2×C4—4011no—
193C2×C4C22×D40111yescharacteristic
194C2×C4C22×D40111yescharacteristic
195C2×C4C22×D40111yescharacteristic
196C2×C4C22×D40111yescharacteristic
197C2×C4C22×D40111yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—8021no—
204C22—8021no—
205C22—8021no—
206C22—8021no—
207C22—8021no—
208C22—8021no—
209C22—8021no—
210C22—8021no—
211C22—8021no—
212C22—16044no—
213C22#2026436612yes—
214C22C23×D40111yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—8021no—
221C4—8021no—
222C4—8021no—
223C4—8021no—
224C4—8021no—
225C2—22no—
226C2—41no—
227C2—16041no—
228C2—16041no—
229C2—16041no—
230C2C2×#2026436611yescharacteristic
231C2#2716578211yescharacteristic
232C2#2716578311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D401
commutator, socle, cosocleC22×C401C24
FittingC4×D4×C401C2
hypercenterC4×D4D401
hypocenterC401#538
LCSG ▷ C22×C401 ▷ C401
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #27165732C2semidirect
2. #27165731C2semidirect
3. #27165725C2semidirect
4. #27165723C2semidirect
5. #27165722C2semidirect
6. #27165720C2semidirect
7. #27165714C2semidirect
8. #27165711C2semidirect
9. #27165710C2semidirect
10. #27165709C2semidirect
11. #27165707C2semidirect
12. #27165705C2semidirect
13. C4×C22⋊D401C2semidirect
14. C2×#20264343C2semidirect
15. C4×D4×C401C2semidirect
17. C2×Dic802C22semidirect
23. C2×C22⋊D401C22semidirect
25. C2×C4×D401C22semidirect
26. C4⋊Dic401C22semidirect
27. #20264345C22semidirect
28. #20264345C22semidirect
29. #20264345C22semidirect
30. #20264345C22semidirect
32. C22⋊Dic401C22semidirect
41. #20264343C22semidirect
42. #20264343C22semidirect
43. C4×Dic401C22semidirect
46. C22×C4×C401C22semidirect
47. C4⋊C4×C401C22semidirect
49. C22⋊C4×C401C22semidirect
50. C42×C401C22semidirect
116. C401#538semidirect
131. C4×D4D401semidirect
145. C22×C4C2×D401semidirect
151. C4⋊C4C2×D401semidirect
165. C22⋊C4C2×D401semidirect
167. C42C2×D401semidirect

Maximal subgroups

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

subgroupindex
#271657322
#271657312
#271657252
#271657232
#271657222
#271657202
#271657142
#271657112
#271657102
#271657092
#271657072
#271657052
C4×C22⋊D4012
C2×#202643432
C4×D4×C4012
#538401