Finite Group Databasebeta

← #30640405

Subgroups — #30640405

Order 16256.

238 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 73 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#22549850C212yesmaximal
2#22549671C214yesmaximal, complemented
3#536×C127C211yesmaximal, characteristic
4#16861168—22no—
5#16861124—22no—
6#16861124C2214yes—
7C2×#41×C127C2211yescharacteristic
8C22×Q8×C127C411yescharacteristic, complemented
9C4⋊Q8×C127C412yescomplemented
10#124×C127C2212yes—
11#123×C127—22no—
12C22⋊Q8×C127—22no—
13#117×C127C411yescharacteristic, complemented
14C4⋊Dic127—42no—
15C22⋊Dic127—24no—
16C4×Dic127—42no—
17#41×C127—42no—
18C2×Q8×C127—21no—
19C2×Q8×C127—42no—
20C2×Q8×C127C2×C412yes—
21C2×Q8×C127C2×C411yescharacteristic
22C2×D4×C127D412yes—
23C2×D4×C127C2311yescharacteristic
24C22×C4×C127—21no—
25C22×C4×C127C2×C411yescharacteristic
26C22×C4×C127D412yes—
27C4⋊C4×C127—21no—
28C4⋊C4×C127—24no—
29C22⋊C4×C127—21no—
30C22⋊C4×C127—24no—
31C42×C127C2×C412yes—
32C2×Dic127—44no—
33C23×C127C22⋊C411yescharacteristic
34C23×C127C2×D412yes—
35Q8×C127—41no—
36Q8×C127—42no—
37Q8×C127—42no—
38D4×C127—42no—
39D4×C127—41no—
40C2×C4×C127—22no—
41C2×C4×C127—22no—
42C2×C4×C127—22no—
43C2×C4×C127—22no—
44C2×C4×C127—22no—
45C2×C4×C127—42no—
46C2×C4×C127C22⋊C412yes—
47C2×C4×C127C22⋊C411yescharacteristic
48C2×C4×C127C22×C411yescharacteristic
49C22×C127—21no—
50C22×C127—22no—
51C22×C127—41no—
52C22×C127—42no—
53C22×C127C2×C22⋊C411yescharacteristic
54C4×C127—22no—
55C4×C127—21no—
56C4×C127—42no—
57C4×C127—42no—
58C4×C127#9912yes—
59Dic127—84no—
60C2×C127—21no—
61C2×C127—42no—
62C2×C127—41no—
63C2×C127C2×#9911yescharacteristic
64#2129—1271nomaximal, Sylow
65C127#212911yescharacteristic, complemented, Sylow
66#536D12711yescharacteristic
67#410—1272no—
68#354—1274no—
69C2×#41C2×D12711yescharacteristic
70C22×Q8Dic12711yescharacteristic, complemented
71C4⋊Q8Dic12712yescomplemented
72#124C2×D12712yes—
73#123—22no—
74C22⋊Q8—22no—
75#117—2542no—
76#117Dic12711yescharacteristic, complemented
77#99—1274no—
78#99—2542no—
79#41—42no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×Q8C2×Dic12712yes—
83C2×Q8C2×Dic12711yescharacteristic
84C2×D4C22⋊D12712yes—
85C2×D4C22×D12711yescharacteristic
86C22×C4—21no—
87C22×C4C2×Dic12711yescharacteristic
88C22×C4C22⋊D12712yes—
89C4⋊C4—24no—
90C4⋊C4—21no—
91C4⋊C4—5082no—
92C22⋊C4—21no—
93C22⋊C4—24no—
94C22⋊C4—2544no—
95C42—5082no—
96C42C2×Dic12712yes—
97C23C22⋊Dic12711yescharacteristic
98C23C2×C22⋊D12712yes—
99Q8—41no—
100Q8—42no—
101Q8—42no—
102D4—42no—
103D4—41no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—5084no—
111C2×C4C22⋊Dic12712yes—
112C2×C4C22⋊Dic12711yescharacteristic
113C2×C4C22×Dic12711yescharacteristic
114C22—21no—
115C22—22no—
116C22—41no—
117C22—42no—
118C22C2×C22⋊Dic12711yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C4—10164no—
124C4#1686112412yes—
125C2—21no—
126C2—42no—
127C2—41no—
128C2C2×#1686112411yescharacteristic
subgroupG/H
centerC2C2×#16861124
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#536×C127C2
socleC2×C127C2×#99
prime radicalC2×#41×C127C22
cosocleC2×D4×C127C23
hypercenter#536D127
hypocenterC127#2129
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #22549671C2semidirect
8. C22×Q8×C127C4semidirect
9. C4⋊Q8×C127C4semidirect
13. #117×C127C4semidirect
65. C127#2129semidirect
70. C22×Q8Dic127semidirect
71. C4⋊Q8Dic127semidirect
76. #117Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498502
#225496712
#536×C1272
#2129127