Finite Group Databasebeta

← #30637447

Subgroups — #30637447

Order 16256.

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

normal subgroups only

This group has 228 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#22549933C211yesmaximal, characteristic
2#22549933C211yesmaximal, characteristic
3#22549622C214yesmaximal, complemented
4C2×C4⋊Q8×C127C211yesmaximal, characteristic
5C4⋊(C127⋊C8)C2212yes—
6C4⋊(C127⋊C8)C2212yes—
7C22⋊(C127⋊C8)—21no—
8C22⋊(C127⋊C8)—21no—
9#16861120—21no—
10#16861120—21no—
11C22×Q8×C127—21no—
12C4⋊Q8×C127—21no—
13C4⋊Q8×C127—21no—
14C4⋊Q8×C127C412yes—
15C4⋊Q8×C127C2212yes—
16C2×C4⋊C4×C127—21no—
17C2×C4⋊C4×C127C412yes—
18C2×C42×C127C2211yescharacteristic
19C2×C127⋊C8—22no—
20C2×C127⋊C8—22no—
21C2×Q8×C127—22no—
22C2×Q8×C127—22no—
23C2×Q8×C127—44no—
24C22×C4×C127—22no—
25C22×C4×C127C2×C411yescharacteristic
26C22×C4×C127D411yescharacteristic
27C22×C4×C127D411yescharacteristic
28C4⋊C4×C127—22no—
29C4⋊C4×C127—22no—
30C4⋊C4×C127—22no—
31C4⋊C4×C127C2×C414yes—
32C42×C127C2×C411yescharacteristic
33C42×C127D411yescharacteristic
34C42×C127D411yescharacteristic
35C42×C127C2311yescharacteristic
36C127⋊C8—42no—
37C127⋊C8—42no—
38C23×C127C22⋊C411yescharacteristic
39Q8×C127—44no—
40Q8×C127—44no—
41C2×C4×C127—22no—
42C2×C4×C127—22no—
43C2×C4×C127—22no—
44C2×C4×C127—24no—
45C2×C4×C127—44no—
46C2×C4×C127C22⋊C411yescharacteristic
47C2×C4×C127C22⋊C411yescharacteristic
48C2×C4×C127C22⋊C411yescharacteristic
49C2×C4×C127C22×C411yescharacteristic
50C2×C4×C127C2×D411yescharacteristic
51C2×C4×C127C2×D411yescharacteristic
52C22×C127—21no—
53C22×C127—21no—
54C22×C127#10112yes—
55C22×C127C2×C22⋊C411yescharacteristic
56C4×C127—21no—
57C4×C127—21no—
58C4×C127—21no—
59C4×C127—21no—
60C4×C127—44no—
61C4×C127AGL(1,ℤ/8)12yes—
62C4×C127#13712yes—
63C2×C127—22no—
64C2×C127C2×#10111yescharacteristic
65C2×C127#41811yescharacteristic
66C2×C127#41911yescharacteristic
67#1556—1271nomaximal, Sylow
68C127#155611yescharacteristic, complemented, Sylow
69C2×C4⋊Q8D12711yescharacteristic
70#432—1271no—
71#432—1271no—
72#332—1274no—
73C22×Q8—21no—
74C4⋊Q8—21no—
75C4⋊Q8—21no—
76C4⋊Q8Dic12712yes—
77C4⋊Q8C2×D12712yes—
78C2×C4⋊C4—21no—
79C2×C4⋊C4Dic12712yes—
80C2×C42C2×D12711yescharacteristic
81C4⋊C8—1272no—
82C4⋊C8—1272no—
83C22⋊C8—2541no—
84C22⋊C8—2541no—
85#97—2541no—
86#97—2541no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—44no—
90C22×C4—22no—
91C22×C4C2×Dic12711yescharacteristic
92C22×C4C22⋊D12711yescharacteristic
93C22×C4C22⋊D12711yescharacteristic
94C2×C8—2542no—
95C2×C8—2542no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4C2×Dic12714yes—
100C42C2×Dic12711yescharacteristic
101C42C22⋊D12711yescharacteristic
102C42C22⋊D12711yescharacteristic
103C42C22×D12711yescharacteristic
104C23C22⋊Dic12711yescharacteristic
105Q8—44no—
106Q8—44no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—24no—
111C2×C4—44no—
112C2×C4C22⋊Dic12711yescharacteristic
113C2×C4C22⋊Dic12711yescharacteristic
114C2×C4C22⋊Dic12711yescharacteristic
115C2×C4C22×Dic12711yescharacteristic
116C2×C4C2×C22⋊D12711yescharacteristic
117C2×C4C2×C22⋊D12711yescharacteristic
118C8—5082no—
119C8—5082no—
120C22—21no—
121C22—21no—
122C22#1686112812yes—
123C22C2×C22⋊Dic12711yescharacteristic
124C4—21no—
125C4—21no—
126C4—21no—
127C4—21no—
128C4—44no—
129C4#1686123512yes—
130C4#1686124012yes—
131C2—22no—
132C2C2×#1686112811yescharacteristic
133C2#2254988111yescharacteristic
134C2#2254988511yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingC2×C4⋊Q8×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×C127C22⋊C4
cosocleC42×C127C23
hypercenterC2×C4⋊Q8D127
hypocenterC127#1556
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22549622C2semidirect
68. C127#1556semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225496222
C2×C4⋊Q8×C1272
#1556127