Finite Group Databasebeta

← #30637468

Subgroups — #30637468

Order 16256.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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#22549900C212yesmaximal
2#22549622C214yesmaximal, complemented
3#533×C127C211yesmaximal, characteristic
4C2×#14523232—22no—
5C4⋊(C127⋊C8)—22no—
6C4⋊(C127⋊C8)C2214yes—
7C4⋊Q8×C127C2212yes—
8#125×C127C412yes—
9C22⋊Q8×C127—22no—
10C4×Q8×C127—22no—
11#117×C127C412yes—
12C2×C42×C127C2211yescharacteristic
13#14523232—44no—
14C2×C127⋊C8—24no—
15C2×Q8×C127—22no—
16C22×C4×C127C2×C411yescharacteristic
17C22×C4×C127D412yes—
18C4⋊C4×C127—24no—
19C4⋊C4×C127—22no—
20C4⋊C4×C127C2×C414yes—
21C22⋊C4×C127—22no—
22C42×C127—22no—
23C42×C127C2×C411yescharacteristic
24C42×C127D412yes—
25C42×C127C2311yescharacteristic
26C127⋊C8—44no—
27C23×C127C22⋊C411yescharacteristic
28Q8×C127—42no—
29Q8×C127—42no—
30C2×C4×C127—22no—
31C2×C4×C127—22no—
32C2×C4×C127—22no—
33C2×C4×C127—24no—
34C2×C4×C127C22⋊C412yes—
35C2×C4×C127C22⋊C411yescharacteristic
36C2×C4×C127C22×C411yescharacteristic
37C2×C4×C127C2×D412yes—
38C22×C127—21no—
39C22×C127—22no—
40C22×C127C2×C22⋊C411yescharacteristic
41C4×C127—21no—
42C4×C127—22no—
43C4×C127—44no—
44C4×C127#10112yes—
45C4×C127AGL(1,ℤ/8)12yes—
46C4×C127#13712yes—
47C2×C127—41no—
48C2×C127C2×#10111yescharacteristic
49C2×C127#41712yes—
50#1561—1271nomaximal, Sylow
51C127#156111yescharacteristic, complemented, Sylow
52#533D12711yescharacteristic
53#423—1272no—
54#332—1274no—
55C2×M16—2542no—
56C4⋊Q8C2×D12712yes—
57#125Dic12712yes—
58C22⋊Q8—22no—
59C4×Q8—22no—
60#117Dic12712yes—
61C2×C42C2×D12711yescharacteristic
62C4⋊C8—1274no—
63C4⋊C8—2542no—
64C2×Q8—22no—
65C22×C4C2×Dic12711yescharacteristic
66C22×C4C22⋊D12712yes—
67M16—5084no—
68C2×C8—2544no—
69C4⋊C4—24no—
70C4⋊C4—22no—
71C4⋊C4C2×Dic12714yes—
72C22⋊C4—22no—
73C42—22no—
74C42C2×Dic12711yescharacteristic
75C42C22⋊D12712yes—
76C42C22×D12711yescharacteristic
77C23C22⋊Dic12711yescharacteristic
78Q8—42no—
79Q8—42no—
80C2×C4—22no—
81C2×C4—22no—
82C2×C4—22no—
83C2×C4—24no—
84C2×C4C22⋊Dic12712yes—
85C2×C4C22⋊Dic12711yescharacteristic
86C2×C4C22×Dic12711yescharacteristic
87C2×C4C2×C22⋊D12712yes—
88C8—5084no—
89C22—21no—
90C22—22no—
91C22C2×C22⋊Dic12711yescharacteristic
92C4—21no—
93C4—22no—
94C4—44no—
95C4#1686112812yes—
96C4#1686123412yes—
97C4#1686123912yes—
98C2—41no—
99C2C2×#1686112811yescharacteristic
100C2#2254987712yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#533×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C22⋊C4
cosocleC42×C127C23
hypercenter#533D127
hypocenterC127#1561
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #22549622C2semidirect
51. C127#1561semidirect

Maximal subgroups

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

subgroupindex
#225499002
#225496222
#533×C1272
#1561127