Finite Group Databasebeta

← #30657381

Subgroups — #30657381

Order 16448.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#16861617C211yesmaximal, characteristic, complemented
2#22551579C211yesmaximal, characteristic, complemented
3#22551573C212yesmaximal, complemented
4#22551571C211yesmaximal, characteristic, complemented
5#22551564C212yesmaximal, complemented
6#22551563C211yesmaximal, characteristic, complemented
7#22551561C212yesmaximal, complemented
8D4×Dic257C211yesmaximal, characteristic, complemented
9C4×C22⋊D257C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic257C212yesmaximal, complemented
11C22×D4×C257C211yesmaximal, characteristic, complemented
12#16861617—24no—
13C2×Dic514C2211yescharacteristic, complemented
14C2×C22⋊D257C2211yescharacteristic
15C2×C22⋊D257C2212yescomplemented
16C2×C22⋊D257C2212yescomplemented
17C22×Dic257C2212yescomplemented
18C22×Dic257C2212yescomplemented
19C2×C4×D257C2211yescharacteristic, complemented
20C4⋊Dic257C2211yescharacteristic, complemented
21#16861597C2211yescharacteristic, complemented
22#16861597C2212yescomplemented
23C22⋊Dic257C2211yescharacteristic
24C22⋊Dic257C2212yes—
25C22⋊Dic257C2212yescomplemented
26C22⋊Dic257C2212yescomplemented
27C22⋊Dic257C2212yescomplemented
28C22⋊Dic257C2212yescomplemented
29#16861595C2211yescharacteristic, complemented
30C4×Dic257C2211yescharacteristic, complemented
31C24×C257C2212yes—
32C2×D4×C257—24no—
33C2×D4×C257C2211yescharacteristic
34C2×D4×C257C2212yes—
35C2×D4×C257C2211yescharacteristic
36C22×C4×C257C2211yescharacteristic
37C22×D257C2311yescharacteristic
38Dic514—22no—
39C22⋊D257—24no—
40C22⋊D257—22no—
41C22⋊D257—42no—
42C2×Dic257—24no—
43C2×Dic257—42no—
44C2×Dic257C2311yescharacteristic
45C2×Dic257C2311yescharacteristic
46C2×Dic257C2312yes—
47C2×Dic257C2311yescharacteristic
48C2×Dic257C2312yes—
49C4×D257—22no—
50C23×C257—24no—
51C23×C257—24no—
52C23×C257—42no—
53C23×C257C2312yes—
54C23×C257C2312yes—
55C23×C257C2311yescharacteristic
56D4×C257—24no—
57D4×C257—22no—
58D4×C257D414yescomplemented
59C2×C4×C257—22no—
60C2×C4×C257C2311yescharacteristic
61C2×C4×C257C2311yescharacteristic
62C22×C257—21no—
63C22×C257—22no—
64C22×C257—21no—
65C22×C257—22no—
66C22×C257—22no—
67C22×C257—22no—
68C22×C257—22no—
69C22×C257—48no—
70C22×C257C2×D414yes—
71C22×C257#4112yescomplemented
72C22×C257C2411yescharacteristic
73C2×D257—21no—
74C2×D257—21no—
75C2×D257—21no—
76C4×C257—21no—
77C4×C257C2×D412yes—
78Dic257—21no—
79Dic257—22no—
80Dic257—21no—
81Dic257—21no—
82Dic257—22no—
83C2×C257—22no—
84C2×C257—24no—
85C2×C257—42no—
86C2×C257C22×D411yescharacteristic
87C2×C257C2×#4111yescharacteristic
88C2×C257H(5,𝔽2)11yescharacteristic
89D257—41no—
90C257#54611yescharacteristic, complemented, Sylow
91#546—2571nomaximal, Sylow
92C2×#41—2571no—
93C22×D4D25711yescharacteristic, complemented
94#124—2571no—
95#123—2572no—
96C22⋊Q8—2571no—
97#121—2571no—
98#121—2572no—
99#120—2572no—
100C4×D4—2571no—
101C4×D4—2571no—
102C2×C22⋊C4—2572no—
103C24C2×D25712yes—
104#41—5144no—
105C2×Q8—2571no—
106C2×D4—2571no—
107C2×D4—2572no—
108C2×D4—2572no—
109C2×D4—24no—
110C2×D4C2×D25712yes—
111C2×D4C2×D25711yescharacteristic
112C2×D4C2×D25711yescharacteristic
113C22×C4—2572no—
114C22×C4—2572no—
115C22×C4—2571no—
116C22×C4C2×D25711yescharacteristic
117C4⋊C4—2571no—
118C4⋊C4—2571no—
119C4⋊C4—2572no—
120C22⋊C4—2571no—
121C22⋊C4—2572no—
122C22⋊C4—2572no—
123C22⋊C4—2572no—
124C22⋊C4—2572no—
125C22⋊C4—2572no—
126C22⋊C4—2571no—
127C42—2571no—
128C23—2571no—
129C23—24no—
130C23—24no—
131C23—42no—
132C23C22×D25712yes—
133C23C22×D25712yes—
134C23C22×D25711yescharacteristic
135Q8—5142no—
136D4—24no—
137D4—22no—
138D4—5144no—
139D4—5142no—
140D4—10282no—
141D4C22⋊D25714yescomplemented
142C2×C4—2571no—
143C2×C4—2571no—
144C2×C4—2572no—
145C2×C4—2571no—
146C2×C4—2572no—
147C2×C4—22no—
148C2×C4—5144no—
149C2×C4—5142no—
150C2×C4—10282no—
151C2×C4C22×D25711yescharacteristic
152C2×C4C22×D25711yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—5141no—
161C22—5141no—
162C22—5141no—
163C22—48no—
164C22C2×C22⋊D25714yes—
165C22#1686161712yescomplemented
166C22C23×D25711yescharacteristic
167C4—21no—
168C4—5141no—
169C4—5142no—
170C4—5141no—
171C4—5141no—
172C4—5142no—
173C4C2×C22⋊D25712yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—10281no—
178C2C22×C22⋊D25711yescharacteristic
179C2C2×#1686161711yescharacteristic
180C2#2255163711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
FittingC22×D4×C257C2
hypercenterC22×D4D257
hypocenterC257#546
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16861617C2semidirect
2. #22551579C2semidirect
3. #22551573C2semidirect
4. #22551571C2semidirect
5. #22551564C2semidirect
6. #22551563C2semidirect
7. #22551561C2semidirect
8. D4×Dic257C2semidirect
9. C4×C22⋊D257C2semidirect
10. C2×C22⋊Dic257C2semidirect
11. C22×D4×C257C2semidirect
13. C2×Dic514C22semidirect
15. C2×C22⋊D257C22semidirect
16. C2×C22⋊D257C22semidirect
17. C22×Dic257C22semidirect
18. C22×Dic257C22semidirect
19. C2×C4×D257C22semidirect
20. C4⋊Dic257C22semidirect
21. #16861597C22semidirect
22. #16861597C22semidirect
25. C22⋊Dic257C22semidirect
26. C22⋊Dic257C22semidirect
27. C22⋊Dic257C22semidirect
28. C22⋊Dic257C22semidirect
29. #16861595C22semidirect
30. C4×Dic257C22semidirect
58. D4×C257D4semidirect
71. C22×C257#41semidirect
90. C257#546semidirect
93. C22×D4D257semidirect
141. D4C22⋊D257semidirect
165. C22#16861617semidirect

Maximal subgroups

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

subgroupindex
C2×#168616172
#225515792
#225515732
#225515712
#225515642
#225515632
#225515612
D4×Dic2572
C4×C22⋊D2572
C2×C22⋊Dic2572
C22×D4×C2572
#546257