Finite Group Databasebeta

← #30657302

Subgroups — #30657302

Order 16448.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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
2C22×C22⋊D257C211yesmaximal, characteristic, complemented
3#22551588C212yesmaximal, complemented
4#22551578C212yesmaximal, complemented
5#22551565C212yesmaximal, complemented
6#22551564C212yesmaximal, complemented
7#22551562C211yesmaximal, characteristic, complemented
8#22551561C211yesmaximal, characteristic, complemented
9#22551560C211yesmaximal, characteristic, complemented
10#22551548C211yesmaximal, characteristic, complemented
11#120×C257C211yesmaximal, characteristic, complemented
12C23×D257C2211yescharacteristic
13#16861617—24no—
14C2×Dic514C2211yescharacteristic, complemented
15C2×D1028C2212yescomplemented
16C2×C22⋊D257—24no—
17C2×C22⋊D257C2211yescharacteristic
18C2×C22⋊D257C2212yes—
19C2×C22⋊D257C2212yes—
20C2×C22⋊D257C2211yescharacteristic, complemented
21C2×C22⋊D257C2212yescomplemented
22C2×C22⋊D257C2212yescomplemented
23C22×Dic257C2211yescharacteristic
24C22×Dic257C2211yescharacteristic, complemented
25C2×C4×D257C2211yescharacteristic, complemented
26#16861597C2212yescomplemented
27C22⋊Dic257C2211yescharacteristic
28C22⋊Dic257C2212yescomplemented
29#16861595C2212yescomplemented
30#16861595C2212yescomplemented
31C4×Dic257C2212yescomplemented
32C24×C257C2211yescharacteristic
33C2×D4×C257C2211yescharacteristic
34C2×D4×C257C2212yescomplemented
35C22⋊C4×C257C2211yescharacteristic
36C22⋊C4×C257C2212yescomplemented
37C22×D257—22no—
38C22×D257—42no—
39C22×D257C2312yes—
40C22×D257C2311yescharacteristic
41Dic514—22no—
42D1028—42no—
43C22⋊D257—22no—
44C22⋊D257—24no—
45C22⋊D257—24no—
46C22⋊D257—22no—
47C22⋊D257—42no—
48C22⋊D257—42no—
49C2×Dic257—22no—
50C2×Dic257D414yescomplemented
51C2×Dic257C2312yes—
52C2×Dic257C2311yescharacteristic
53C2×Dic257C2311yescharacteristic
54C2×Dic257C2311yescharacteristic
55C4×D257—22no—
56C23×C257—22no—
57C23×C257—42no—
58C23×C257C2312yes—
59C23×C257C2311yescharacteristic
60C23×C257C2311yescharacteristic
61D4×C257—22no—
62D4×C257—42no—
63C2×C4×C257C2312yes—
64C2×C4×C257C2311yescharacteristic
65C22×C257—22no—
66C22×C257—21no—
67C22×C257—22no—
68C22×C257—22no—
69C22×C257—22no—
70C22×C257—22no—
71C22×C257—44no—
72C22×C257C2×D412yes—
73C22×C257C2411yescharacteristic
74C2×D257—21no—
75C2×D257—22no—
76C2×D257—22no—
77C2×D257—22no—
78C2×D257—22no—
79C2×D257—44no—
80C4×C257—21no—
81C4×C257—22no—
82Dic257—21no—
83Dic257—22no—
84Dic257C2×D412yes—
85Dic257C2×D412yes—
86C2×C257—22no—
87C2×C257—41no—
88C2×C257—42no—
89C2×C257C22×D411yescharacteristic
90C2×C257H(5,𝔽2)12yes—
91D257—41no—
92D257—42no—
93C257#53511yescharacteristic, complemented, Sylow
94#535—2571nomaximal, Sylow
95C2×#41—2571no—
96C22×D4—2571no—
97#127—2572no—
98#124—2572no—
99#121—2572no—
100#121—2572no—
101#120—2571no—
102#120—2571no—
103#120—2571no—
104#120D25711yescharacteristic, complemented
105#117—2571no—
106C24—2571no—
107C24C2×D25711yescharacteristic
108#41—5144no—
109C2×Q8—2571no—
110C2×D4—2571no—
111C2×D4—2572no—
112C2×D4—2572no—
113C2×D4—2571no—
114C2×D4—2572no—
115C2×D4—2572no—
116C2×D4—2572no—
117C2×D4—5144no—
118C2×D4C2×D25711yescharacteristic
119C2×D4C2×D25712yescomplemented
120C22×C4—2571no—
121C22×C4—2571no—
122C22×C4—2571no—
123C4⋊C4—2572no—
124C22⋊C4—2571no—
125C22⋊C4—2572no—
126C22⋊C4—2572no—
127C22⋊C4—2572no—
128C22⋊C4C2×D25711yescharacteristic
129C22⋊C4C2×D25712yescomplemented
130C42—2572no—
131C23—2572no—
132C23—2571no—
133C23—22no—
134C23—5142no—
135C23—42no—
136C23—10282no—
137C23C22×D25712yes—
138C23C22×D25711yescharacteristic
139C23C22×D25711yescharacteristic
140Q8—5142no—
141D4—22no—
142D4—5144no—
143D4—5144no—
144D4—5142no—
145D4—5142no—
146D4—42no—
147D4—10282no—
148D4—10282no—
149D4—10282no—
150C2×C4—2574no—
151C2×C4—2571no—
152C2×C4—2571no—
153C2×C4—2572no—
154C2×C4—2571no—
155C2×C4—5142no—
156C2×C4—5142no—
157C2×C4C22×D25712yes—
158C2×C4C22×D25711yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—5141no—
166C22—5142no—
167C22—5142no—
168C22—5142no—
169C22—5142no—
170C22—44no—
171C22—10284no—
172C22D4×D25712yes—
173C22C23×D25711yescharacteristic
174C4—2572no—
175C4—2572no—
176C4—21no—
177C4—22no—
178C4—5141no—
179C4—5142no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—10282no—
184C2—10281no—
185C2C2×D4×D25711yescharacteristic
186C2#2255163712yes—
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
Fitting#120×C257C2
hypercenter#120D257
hypocenterC257#535
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#16861617C2semidirect
2. C22×C22⋊D257C2semidirect
3. #22551588C2semidirect
4. #22551578C2semidirect
5. #22551565C2semidirect
6. #22551564C2semidirect
7. #22551562C2semidirect
8. #22551561C2semidirect
9. #22551560C2semidirect
10. #22551548C2semidirect
11. #120×C257C2semidirect
14. C2×Dic514C22semidirect
15. C2×D1028C22semidirect
20. C2×C22⋊D257C22semidirect
21. C2×C22⋊D257C22semidirect
22. C2×C22⋊D257C22semidirect
24. C22×Dic257C22semidirect
25. C2×C4×D257C22semidirect
26. #16861597C22semidirect
28. C22⋊Dic257C22semidirect
29. #16861595C22semidirect
30. #16861595C22semidirect
31. C4×Dic257C22semidirect
34. C2×D4×C257C22semidirect
36. C22⋊C4×C257C22semidirect
50. C2×Dic257D4semidirect
93. C257#535semidirect
104. #120D257semidirect
119. C2×D4C2×D257semidirect
129. C22⋊C4C2×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#168616172
C22×C22⋊D2572
#225515882
#225515782
#225515652
#225515642
#225515622
#225515612
#225515602
#225515482
#120×C2572
#535257