Finite Group Databasebeta

← #30657247

Subgroups — #30657247

Order 16448.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×C4×D257C211yesmaximal, characteristic, complemented
3#22551574C212yesmaximal, complemented
4#22551572C211yesmaximal, characteristic, complemented
5#22551571C211yesmaximal, characteristic, complemented
6#22551563C211yesmaximal, characteristic, complemented
7#22551560C212yesmaximal, complemented
8D4×Dic257C212yesmaximal, complemented
9#22551552C212yesmaximal, complemented
10#22551550C211yesmaximal, characteristic, complemented
11#121×C257C211yesmaximal, characteristic, complemented
12C23×D257C2211yescharacteristic
13#16861617—24no—
14C2×Dic514C2211yescharacteristic, complemented
15C2×C22⋊D257C2212yes—
16C2×C22⋊D257C2212yes—
17C22×Dic257C2211yescharacteristic
18C22×Dic257C2212yescomplemented
19C2×C4×D257—24no—
20C2×C4×D257C2211yescharacteristic
21C2×C4×D257C2211yescharacteristic
22C2×C4×D257C2211yescharacteristic
23C2×C4×D257C2211yescharacteristic
24C4⋊Dic257C2212yescomplemented
25C4⋊Dic257C2211yescharacteristic, complemented
26#16861597C2212yescomplemented
27C22⋊Dic257C2212yes—
28C22⋊Dic257C2212yescomplemented
29#16861595C2212yes—
30#16861595C2212yes—
31C4×Dic257C2212yescomplemented
32C2×D4×C257C2212yes—
33C2×D4×C257C2211yescharacteristic, complemented
34C22×C4×C257C2211yescharacteristic
35C4⋊C4×C257C2211yescharacteristic, complemented
36C22⋊C4×C257C2212yescomplemented
37C22×D257—22no—
38C22×D257—22no—
39C22×D257—22no—
40C22×D257C2311yescharacteristic
41C22×D257C2311yescharacteristic
42Dic514—22no—
43C22⋊D257—24no—
44C22⋊D257—24no—
45C2×Dic257—22no—
46C2×Dic257—24no—
47C2×Dic257C2312yes—
48C2×Dic257C2312yes—
49C2×Dic257C2311yescharacteristic
50C2×Dic257C2311yescharacteristic
51C4×D257—22no—
52C4×D257—22no—
53C4×D257—22no—
54C4×D257D414yescomplemented
55C23×C257C2312yes—
56C23×C257C2311yescharacteristic
57D4×C257—24no—
58D4×C257—22no—
59C2×C4×C257—22no—
60C2×C4×C257C2311yescharacteristic
61C2×C4×C257C2312yes—
62C2×C4×C257C2311yescharacteristic
63C22×C257—21no—
64C22×C257—21no—
65C22×C257—22no—
66C22×C257—22no—
67C22×C257—22no—
68C22×C257#4112yescomplemented
69C22×C257C2411yescharacteristic
70C2×D257—28no—
71C2×D257—21no—
72C2×D257—21no—
73C2×D257—21no—
74C2×D257—21no—
75C2×D257C2×D412yes—
76C2×D257#4112yescomplemented
77C4×C257—21no—
78C4×C257—22no—
79C4×C257C2×D412yes—
80Dic257—21no—
81Dic257—22no—
82Dic257—22no—
83Dic257C2×D412yes—
84C2×C257—22no—
85C2×C257—42no—
86C2×C257C22×D411yescharacteristic
87C2×C257C2×#4111yescharacteristic
88C2×C257C2×#4111yescharacteristic
89D257—22no—
90D257—22no—
91C257#52511yescharacteristic, complemented, Sylow
92#525—2571nomaximal, Sylow
93C2×#41—2571no—
94C23×C4—2571no—
95#123—2572no—
96C22⋊Q8—2571no—
97C22⋊Q8—2571no—
98#121—2571no—
99#121D25711yescharacteristic, complemented
100#120—2572no—
101C4×D4—2572no—
102C4×D4—2572no—
103#117—2571no—
104C24—2571no—
105#41—5144no—
106C2×Q8—2571no—
107C2×D4—2572no—
108C2×D4—2572no—
109C2×D4C2×D25712yes—
110C2×D4C2×D25711yescharacteristic, complemented
111C22×C4—2571no—
112C22×C4—2571no—
113C22×C4—2572no—
114C22×C4—2571no—
115C22×C4—2571no—
116C22×C4—2571no—
117C22×C4—5144no—
118C22×C4C2×D25711yescharacteristic
119C4⋊C4—2571no—
120C4⋊C4—2572no—
121C4⋊C4—2572no—
122C4⋊C4C2×D25711yescharacteristic, complemented
123C22⋊C4—2572no—
124C22⋊C4—2572no—
125C22⋊C4—2572no—
126C22⋊C4—2572no—
127C22⋊C4C2×D25712yescomplemented
128C42—2572no—
129C23—2571no—
130C23—2571no—
131C23—5142no—
132C23—5142no—
133C23—5142no—
134C23C22×D25712yes—
135C23C22×D25711yescharacteristic
136Q8—5142no—
137D4—24no—
138D4—22no—
139D4—5144no—
140D4—5144no—
141C2×C4—2571no—
142C2×C4—2574no—
143C2×C4—2572no—
144C2×C4—2571no—
145C2×C4—2572no—
146C2×C4—22no—
147C2×C4—5142no—
148C2×C4—5142no—
149C2×C4—5144no—
150C2×C4—5142no—
151C2×C4—5142no—
152C2×C4C22×D25712yes—
153C2×C4C22×D25711yescharacteristic
154C2×C4C22×D25711yescharacteristic
155C22—2572no—
156C22—2572no—
157C22—21no—
158C22—21no—
159C22—22no—
160C22—22no—
161C22—22no—
162C22—5148no—
163C22—5141no—
164C22—5141no—
165C22—5141no—
166C22—5141no—
167C22#1686161712yescomplemented
168C22C23×D25711yescharacteristic
169C4—2572no—
170C4—21no—
171C4—22no—
172C4—5141no—
173C4—5142no—
174C4—5142no—
175C4D4×D25712yes—
176C2—22no—
177C2—5142no—
178C2—5142no—
179C2—42no—
180C2C2×D4×D25711yescharacteristic
181C2#41×D25711yescharacteristic
182C2C2×#1686161711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
Fitting#121×C257C2
hypercenter#121D257
hypocenterC257#525
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16861617C2semidirect
2. C22×C4×D257C2semidirect
3. #22551574C2semidirect
4. #22551572C2semidirect
5. #22551571C2semidirect
6. #22551563C2semidirect
7. #22551560C2semidirect
8. D4×Dic257C2semidirect
9. #22551552C2semidirect
10. #22551550C2semidirect
11. #121×C257C2semidirect
14. C2×Dic514C22semidirect
18. C22×Dic257C22semidirect
24. C4⋊Dic257C22semidirect
25. C4⋊Dic257C22semidirect
26. #16861597C22semidirect
28. C22⋊Dic257C22semidirect
31. C4×Dic257C22semidirect
33. C2×D4×C257C22semidirect
35. C4⋊C4×C257C22semidirect
36. C22⋊C4×C257C22semidirect
54. C4×D257D4semidirect
68. C22×C257#41semidirect
76. C2×D257#41semidirect
91. C257#525semidirect
99. #121D257semidirect
110. C2×D4C2×D257semidirect
122. C4⋊C4C2×D257semidirect
127. C22⋊C4C2×D257semidirect
167. C22#16861617semidirect

Maximal subgroups

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

subgroupindex
C2×#168616172
C22×C4×D2572
#225515742
#225515722
#225515712
#225515632
#225515602
D4×Dic2572
#225515522
#225515502
#121×C2572
#525257