Finite Group Databasebeta

← #30657322

Subgroups — #30657322

Order 16448.

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#22551585C211yesmaximal, characteristic, complemented
2#22551585C211yesmaximal, characteristic, complemented
3#22551579C211yesmaximal, characteristic, complemented
4#22551578C211yesmaximal, characteristic, complemented
5#22551574C211yesmaximal, characteristic, complemented
6#22551573C211yesmaximal, characteristic, complemented
7#22551569C211yesmaximal, characteristic, complemented
8#22551564C211yesmaximal, characteristic, complemented
9#22551564C211yesmaximal, characteristic, complemented
10#22551563C211yesmaximal, characteristic, complemented
11#22551561C211yesmaximal, characteristic, complemented
12D4×Dic257C211yesmaximal, characteristic, complemented
13#22551552C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic257C211yesmaximal, characteristic, complemented
15#120×C257C211yesmaximal, characteristic, complemented
16C2×Dic514C2211yescharacteristic, complemented
17C2×C22⋊D257C2211yescharacteristic
18C2×C22⋊D257C2211yescharacteristic
19C2×C22⋊D257C2211yescharacteristic
20C2×C22⋊D257C2211yescharacteristic, complemented
21C22×Dic257C2211yescharacteristic
22C22×Dic257C2211yescharacteristic
23C22×Dic257C2211yescharacteristic, complemented
24C2×C4×D257C2211yescharacteristic, complemented
25C4⋊Dic257C2211yescharacteristic, complemented
26C4⋊Dic257C2211yescharacteristic, complemented
27#16861597C2211yescharacteristic, complemented
28#16861597C2211yescharacteristic, complemented
29#16861597C2211yescharacteristic, complemented
30#16861597C2211yescharacteristic, complemented
31C22⋊Dic257C2211yescharacteristic
32C22⋊Dic257C2211yescharacteristic
33C22⋊Dic257C2211yescharacteristic
34C22⋊Dic257C2211yescharacteristic
35C22⋊Dic257C2211yescharacteristic
36C22⋊Dic257C2211yescharacteristic
37C22⋊Dic257C2211yescharacteristic
38C22⋊Dic257C2211yescharacteristic, complemented
39C22⋊Dic257C2211yescharacteristic, complemented
40#16861595C2211yescharacteristic, complemented
41#16861595C2211yescharacteristic, complemented
42C4×Dic257C2211yescharacteristic, complemented
43C4×Dic257C2211yescharacteristic, complemented
44C24×C257C2211yescharacteristic
45C2×D4×C257C2211yescharacteristic
46C2×D4×C257C2211yescharacteristic
47C2×D4×C257C2211yescharacteristic
48C22⋊C4×C257C2211yescharacteristic
49C22⋊C4×C257C2211yescharacteristic
50C22⋊C4×C257C2211yescharacteristic
51C22×D257C2311yescharacteristic
52Dic514—41no—
53C22⋊D257—22no—
54C22⋊D257—41no—
55C22⋊D257—41no—
56C22⋊D257—41no—
57C2×Dic257—41no—
58C2×Dic257—41no—
59C2×Dic257—41no—
60C2×Dic257C2311yescharacteristic
61C2×Dic257C2311yescharacteristic
62C2×Dic257C2311yescharacteristic
63C2×Dic257C2311yescharacteristic
64C2×Dic257C2311yescharacteristic
65C2×Dic257C2311yescharacteristic
66C2×Dic257C2311yescharacteristic
67C4×D257—41no—
68C23×C257—22no—
69C23×C257—41no—
70C23×C257—41no—
71C23×C257C2311yescharacteristic
72C23×C257C2311yescharacteristic
73C23×C257C2311yescharacteristic
74C23×C257C2311yescharacteristic
75D4×C257—22no—
76D4×C257—41no—
77D4×C257—41no—
78C2×C4×C257C2311yescharacteristic
79C2×C4×C257C2311yescharacteristic
80C2×C4×C257C2311yescharacteristic
81C22×C257—21no—
82C22×C257—21no—
83C22×C257—21no—
84C22×C257—21no—
85C22×C257—21no—
86C22×C257—21no—
87C22×C257—21no—
88C22×C257—21no—
89C22×C257—21no—
90C22×C257—21no—
91C22×C257—21no—
92C22×C257—44no—
93C22×C257#4112yes—
94C22×C257C2411yescharacteristic
95C2×D257—21no—
96C2×D257—21no—
97C2×D257—21no—
98C4×C257—21no—
99C4×C257—21no—
100C4×C257—21no—
101Dic257—21no—
102Dic257—21no—
103Dic257—21no—
104Dic257—21no—
105Dic257—21no—
106Dic257—21no—
107Dic257—21no—
108C2×C257—22no—
109C2×C257—41no—
110C2×C257—41no—
111C2×C257—41no—
112C2×C257C2×#4111yescharacteristic
113C2×C257H(5,𝔽2)11yescharacteristic
114C2×C257H(5,𝔽2)11yescharacteristic
115D257—41no—
116C257#53811yescharacteristic, complemented, Sylow
117#538—2571nomaximal, Sylow
118#126—2571no—
119#126—2571no—
120#124—2571no—
121#124—2571no—
122#123—2571no—
123#123—2571no—
124C22⋊Q8—2571no—
125#121—2571no—
126#121—2571no—
127#121—2571no—
128#120—2571no—
129#120D25711yescharacteristic, complemented
130C4×D4—2571no—
131C4×D4—2571no—
132C2×C22⋊C4—2571no—
133C24C2×D25711yescharacteristic
134C2×Q8—2571no—
135C2×D4—2571no—
136C2×D4—2571no—
137C2×D4—2571no—
138C2×D4—2571no—
139C2×D4C2×D25711yescharacteristic
140C2×D4C2×D25711yescharacteristic
141C2×D4C2×D25711yescharacteristic
142C22×C4—2571no—
143C22×C4—2571no—
144C22×C4—2571no—
145C22×C4—2571no—
146C4⋊C4—2571no—
147C4⋊C4—2571no—
148C4⋊C4—2571no—
149C4⋊C4—2571no—
150C4⋊C4—2571no—
151C4⋊C4—2571no—
152C22⋊C4—2571no—
153C22⋊C4—2571no—
154C22⋊C4—2571no—
155C22⋊C4—2571no—
156C22⋊C4—2571no—
157C22⋊C4—2571no—
158C22⋊C4—2571no—
159C22⋊C4—2571no—
160C22⋊C4—2571no—
161C22⋊C4—2571no—
162C22⋊C4—2571no—
163C22⋊C4C2×D25711yescharacteristic
164C22⋊C4C2×D25711yescharacteristic
165C22⋊C4C2×D25711yescharacteristic
166C42—2571no—
167C42—2571no—
168C23—2571no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23C22×D25711yescharacteristic
173C23C22×D25711yescharacteristic
174C23C22×D25711yescharacteristic
175C23C22×D25711yescharacteristic
176Q8—10281no—
177D4—22no—
178D4—5142no—
179D4—41no—
180D4—41no—
181D4—10281no—
182D4—10281no—
183D4—10281no—
184C2×C4—2571no—
185C2×C4—2571no—
186C2×C4—2571no—
187C2×C4—2571no—
188C2×C4—2571no—
189C2×C4—2571no—
190C2×C4—2571no—
191C2×C4—10281no—
192C2×C4—10281no—
193C2×C4—10281no—
194C2×C4—10281no—
195C2×C4C22×D25711yescharacteristic
196C2×C4C22×D25711yescharacteristic
197C2×C4C22×D25711yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—5141no—
210C22—5141no—
211C22—5141no—
212C22—44no—
213C22#1686161712yes—
214C22C23×D25711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—5141no—
219C4—5141no—
220C4—5141no—
221C4—5141no—
222C4—5141no—
223C4—5141no—
224C4—5141no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—10281no—
230C2C2×#1686161711yescharacteristic
231C2#2255163711yescharacteristic
232C2#2255163711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
Fitting#120×C257C2
hypercenter#120D257
hypocenterC257#538
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #22551585C2semidirect
2. #22551585C2semidirect
3. #22551579C2semidirect
4. #22551578C2semidirect
5. #22551574C2semidirect
6. #22551573C2semidirect
7. #22551569C2semidirect
8. #22551564C2semidirect
9. #22551564C2semidirect
10. #22551563C2semidirect
11. #22551561C2semidirect
12. D4×Dic257C2semidirect
13. #22551552C2semidirect
14. C2×C22⋊Dic257C2semidirect
15. #120×C257C2semidirect
16. C2×Dic514C22semidirect
20. C2×C22⋊D257C22semidirect
23. C22×Dic257C22semidirect
24. C2×C4×D257C22semidirect
25. C4⋊Dic257C22semidirect
26. C4⋊Dic257C22semidirect
27. #16861597C22semidirect
28. #16861597C22semidirect
29. #16861597C22semidirect
30. #16861597C22semidirect
38. C22⋊Dic257C22semidirect
39. C22⋊Dic257C22semidirect
40. #16861595C22semidirect
41. #16861595C22semidirect
42. C4×Dic257C22semidirect
43. C4×Dic257C22semidirect
116. C257#538semidirect
129. #120D257semidirect

Maximal subgroups

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

subgroupindex
#225515852
#225515792
#225515782
#225515742
#225515732
#225515692
#225515642
#225515632
#225515612
D4×Dic2572
#225515522
C2×C22⋊Dic2572
#120×C2572
#538257