Finite Group Databasebeta

← #30637811

Subgroups — #30637811

Order 16256.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550592C211yesmaximal, characteristic, complemented
2#22550068C212yesmaximal, complemented
3#22549625C211yesmaximal, characteristic, complemented
4#22549605C212yesmaximal, complemented
5#492×C127C211yesmaximal, characteristic, complemented
6C2×#14523233—22no—
7#16861198—22no—
8#16861184—22no—
9C4×D508C2212yes—
10#16861167—22no—
11C4⋊(C127⋊C8)C2212yes—
12#16861130—22no—
13C2×SD16×C127—22no—
14C4⋊Q8×C127C2211yescharacteristic
15#127×C127C2211yescharacteristic
16C4×C8×C127C2211yescharacteristic
17C2×D508—22no—
18C2×D508D412yes—
19C2×C4×D127—22no—
20#14523233—44no—
21C2×C127⋊C8—22no—
22C4⋊Dic127D412yes—
23(C2×D127)⋊C4—24no—
24(C2×D127)⋊C4—22no—
25C4×Dic127—22no—
26C2×Q8×C127D412yes—
27C2×D4×C127—22no—
28C2×D4×C127—21no—
29SD16×C127—44no—
30C2×C8×C127—21no—
31C4⋊C4×C127—22no—
32C42×C127C2311yescharacteristic
33C22×D127—22no—
34D508—22no—
35D508—22no—
36D508—82no—
37C2×Dic127—22no—
38C4×D127—42no—
39C4×D127—42no—
40C127⋊C8—42no—
41C23×C127—41no—
42Q8×C127—42no—
43D4×C127—42no—
44D4×C127—42no—
45D4×C127—42no—
46C2×C4×C127—22no—
47C2×C4×C127C2×D412yes—
48C2×C4×C127C2×D411yescharacteristic
49C8×C127—22no—
50C22×C127—82no—
51C22×C127—81no—
52C22×C127#12011yescharacteristic
53C2×D127—42no—
54C2×D127—42no—
55C2×D127—42no—
56C4×C127—21no—
57C4×C127—42no—
58C4×C127#13512yes—
59C4×C127AGL(1,ℤ/8)12yes—
60Dic127—42no—
61C2×C127—161no—
62C2×C127#44912yes—
63C2×C127#45311yescharacteristic
64D127—82no—
65#1624—1271nomaximal, Sylow
66C127#162411yescharacteristic, complemented, Sylow
67#555—1271no—
68#492D12711yescharacteristic, complemented
69#459—1272no—
70#333—1271no—
71D4⋊C8—1272no—
72C2×SD16—22no—
73C2×D8—2542no—
74C4⋊Q8C2×D12711yescharacteristic
75#127C2×D12711yescharacteristic
76#124—2542no—
77#121—2542no—
78C4×D4—1272no—
79#117—2542no—
80C4⋊C8—1272no—
81#102—2542no—
82C4×C8C2×D12711yescharacteristic
83C2×Q8C22⋊D12712yes—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—1272no—
87C2×D4—2542no—
88C22×C4—2542no—
89SD16—44no—
90D8—5084no—
91C2×C8—21no—
92C2×C8—2542no—
93C4⋊C4—22no—
94C4⋊C4—1272no—
95C22⋊C4—2544no—
96C22⋊C4—2542no—
97C42—2542no—
98C42C22×D12711yescharacteristic
99C23—41no—
100C23—2542no—
101Q8—42no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—10162no—
106D4—2542no—
107D4—2542no—
108C2×C4—22no—
109C2×C4—5082no—
110C2×C4—5082no—
111C2×C4—2542no—
112C2×C4D4×D12712yes—
113C2×C4C2×C22⋊D12711yescharacteristic
114C8—22no—
115C8—5082no—
116C22—82no—
117C22—81no—
118C22—5082no—
119C22—5082no—
120C22—5082no—
121C22#1686117711yescharacteristic
122C4—21no—
123C4—42no—
124C4—5082no—
125C4#1686123012yes—
126C4#1686123612yes—
127C2—161no—
128C2—10162no—
129C2#2255002012yes—
130C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#492×C127C2
socleC22×C127#120
hypercenter#492D127
hypocenterC127#1624
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #22550592C2semidirect
2. #22550068C2semidirect
3. #22549625C2semidirect
4. #22549605C2semidirect
5. #492×C127C2semidirect
66. C127#1624semidirect
68. #492D127semidirect

Maximal subgroups

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

subgroupindex
#225505922
#225500682
#225496252
#225496052
#492×C1272
#1624127