Finite Group Databasebeta

← #30657384

Subgroups — #30657384

Order 16448.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16861618C211yesmaximal, characteristic, complemented
2C2×D4×D257C211yesmaximal, characteristic, complemented
3#22551579C211yesmaximal, characteristic, complemented
4#22551574C212yesmaximal, complemented
5#22551572C211yesmaximal, characteristic, complemented
6#22551563C212yesmaximal, complemented
7#22551560C212yesmaximal, complemented
8#22551554C211yesmaximal, characteristic, complemented
9C4×C22⋊D257C211yesmaximal, characteristic, complemented
10C22⋊C4×D257C212yesmaximal, complemented
11#121×C257C211yesmaximal, characteristic, complemented
12C23×D257C2212yes—
13#16861618—24no—
14C2×Dic514C2211yescharacteristic, complemented
15C2×D1028C2211yescharacteristic
16C2×C22⋊D257C2211yescharacteristic
17C2×C22⋊D257C2212yes—
18C2×C22⋊D257C2212yescomplemented
19D4×D257—24no—
20C2×C4×D257C2211yescharacteristic
21C2×C4×D257C2212yescomplemented
22C2×C4×D257C2212yescomplemented
23C4⋊Dic257C2212yescomplemented
24#16861597C2211yescharacteristic, complemented
25C22⋊Dic257C2211yescharacteristic, complemented
26C22⋊Dic257C2212yescomplemented
27C22⋊Dic257C2212yescomplemented
28#16861595C2211yescharacteristic
29#16861595C2212yes—
30#16861595C2212yescomplemented
31C4×Dic257C2211yescharacteristic, complemented
32C2×D4×C257C2211yescharacteristic
33C2×D4×C257C2212yescomplemented
34C22×C4×C257C2211yescharacteristic, complemented
35C4⋊C4×C257C2211yescharacteristic, complemented
36C22⋊C4×C257C2212yescomplemented
37C22×D257—24no—
38C22×D257—24no—
39C22×D257—42no—
40C22×D257C2312yes—
41C22×D257C2311yescharacteristic
42Dic514—22no—
43D1028D414yescomplemented
44C22⋊D257—24no—
45C22⋊D257—24no—
46C22⋊D257—22no—
47C2×Dic257C2311yescharacteristic
48C2×Dic257C2311yescharacteristic
49C2×Dic257C2312yes—
50C2×Dic257C2311yescharacteristic
51C4×D257—24no—
52C4×D257—22no—
53C4×D257—42no—
54C23×C257C2312yes—
55C23×C257C2311yescharacteristic
56D4×C257—22no—
57D4×C257—42no—
58C2×C4×C257—22no—
59C2×C4×C257C2311yescharacteristic
60C2×C4×C257C2312yes—
61C2×C4×C257C2311yescharacteristic
62C22×C257—21no—
63C22×C257—22no—
64C22×C257—21no—
65C22×C257—22no—
66C22×C257—21no—
67C22×C257—22no—
68C22×C257C2411yescharacteristic
69C2×D257—21no—
70C2×D257—22no—
71C2×D257—22no—
72C2×D257—21no—
73C2×D257—48no—
74C2×D257C2×D414yes—
75C2×D257#4112yescomplemented
76C4×C257—21no—
77C4×C257—22no—
78C4×C257C2×D412yes—
79Dic257—21no—
80Dic257—22no—
81Dic257—21no—
82Dic257—21no—
83C2×C257—42no—
84C2×C257—41no—
85C2×C257C22×D411yescharacteristic
86C2×C257C2×#4111yescharacteristic
87C2×C257H(5,𝔽2)11yescharacteristic
88D257—22no—
89D257—24no—
90C257#54611yescharacteristic, complemented, Sylow
91#546—2571nomaximal, Sylow
92C2×#41—2571no—
93C22×D4—2571no—
94#124—2571no—
95#123—2572no—
96C22⋊Q8—2571no—
97#121—2572no—
98#121D25711yescharacteristic, complemented
99#120—2572no—
100C4×D4—2571no—
101C4×D4—2571no—
102C2×C22⋊C4—2572no—
103C24—2572no—
104#41—5144no—
105C2×Q8—2571no—
106C2×D4—2571no—
107C2×D4—2572no—
108C2×D4—2572no—
109C2×D4—2571no—
110C2×D4—5144no—
111C2×D4C2×D25711yescharacteristic
112C2×D4C2×D25712yescomplemented
113C22×C4—2572no—
114C22×C4—2571no—
115C22×C4—2572no—
116C22×C4C2×D25711yescharacteristic, complemented
117C4⋊C4—2571no—
118C4⋊C4—2572no—
119C4⋊C4C2×D25711yescharacteristic, complemented
120C22⋊C4—2572no—
121C22⋊C4—2571no—
122C22⋊C4—2572no—
123C22⋊C4—2571no—
124C22⋊C4—2572no—
125C22⋊C4—2572no—
126C22⋊C4C2×D25712yescomplemented
127C42—2571no—
128C23—2571no—
129C23—2572no—
130C23—5144no—
131C23—5144no—
132C23—10282no—
133C23C22×D25712yes—
134C23C22×D25711yescharacteristic
135Q8—5142no—
136D4—2574no—
137D4—22no—
138D4—5144no—
139D4—5144no—
140D4—5142no—
141D4—42no—
142C2×C4—2571no—
143C2×C4—2571no—
144C2×C4—2572no—
145C2×C4—2571no—
146C2×C4—22no—
147C2×C4—5144no—
148C2×C4—5142no—
149C2×C4—10282no—
150C2×C4C22×D25712yes—
151C2×C4C22×D25711yescharacteristic
152C2×C4C22×D25711yescharacteristic
153C22—2574no—
154C22—2572no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—21no—
160C22—22no—
161C22—5141no—
162C22—5142no—
163C22—5142no—
164C22—5141no—
165C22—10288no—
166C22C23×D25711yescharacteristic
167C4—21no—
168C4—22no—
169C4—5141no—
170C4—5142no—
171C4—5141no—
172C4—5141no—
173C4D4×D25712yes—
174C2—5142no—
175C2—5144no—
176C2—42no—
177C2—41no—
178C2C2×D4×D25711yescharacteristic
179C2#41×D25711yescharacteristic
180C2#2255163711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
Fitting#121×C257C2
hypercenter#121D257
hypocenterC257#546
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16861618C2semidirect
2. C2×D4×D257C2semidirect
3. #22551579C2semidirect
4. #22551574C2semidirect
5. #22551572C2semidirect
6. #22551563C2semidirect
7. #22551560C2semidirect
8. #22551554C2semidirect
9. C4×C22⋊D257C2semidirect
10. C22⋊C4×D257C2semidirect
11. #121×C257C2semidirect
14. C2×Dic514C22semidirect
18. C2×C22⋊D257C22semidirect
21. C2×C4×D257C22semidirect
22. C2×C4×D257C22semidirect
23. C4⋊Dic257C22semidirect
24. #16861597C22semidirect
25. C22⋊Dic257C22semidirect
26. C22⋊Dic257C22semidirect
27. C22⋊Dic257C22semidirect
30. #16861595C22semidirect
31. C4×Dic257C22semidirect
33. C2×D4×C257C22semidirect
34. C22×C4×C257C22semidirect
35. C4⋊C4×C257C22semidirect
36. C22⋊C4×C257C22semidirect
43. D1028D4semidirect
75. C2×D257#41semidirect
90. C257#546semidirect
98. #121D257semidirect
112. C2×D4C2×D257semidirect
116. C22×C4C2×D257semidirect
119. C4⋊C4C2×D257semidirect
126. C22⋊C4C2×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#168616182
C2×D4×D2572
#225515792
#225515742
#225515722
#225515632
#225515602
#225515542
C4×C22⋊D2572
C22⋊C4×D2572
#121×C2572
#546257