Finite Group Databasebeta

← #30637464

Subgroups — #30637464

Order 16256.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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#22550427C211yesmaximal, characteristic
2#22549898C212yesmaximal, complemented
3#22549625C211yesmaximal, characteristic, complemented
4#22549619C212yesmaximal, complemented
5#333×C127C211yesmaximal, characteristic, complemented
6C2×#14523249C412yes—
7C2×#14523231—22no—
8#16861206C2211yescharacteristic
9#16861198C412yes—
10#16861184—22no—
11#16861171—22no—
12C42×D127C2211yescharacteristic
13C4⋊(C127⋊C8)C2212yes—
14Dic127⋊C8—22no—
15C4⋊Q8×C127C2211yescharacteristic
16C4⋊C8×C127C2212yes—
17#14523249—44no—
18C2×D508—22no—
19C2×D508—22no—
20C2×D508C2×C412yes—
21C2×C4×D127—22no—
22C2×C4×D127C2×C411yescharacteristic
23C2×C4×D127D412yes—
24#14523231—44no—
25C2×C127⋊C8—22no—
26(C2×D127)⋊C4—24no—
27C4×Dic127C2×C411yescharacteristic
28C4×Dic127D412yes—
29C2×Q8×C127C2×C412yes—
30C2×C8×C127—22no—
31C4⋊C4×C127—22no—
32C42×C127C2311yescharacteristic
33C22×D127—22no—
34C22×D127C22⋊C411yescharacteristic
35D508—42no—
36D508—44no—
37D508—42no—
38D508—42no—
39C2×Dic127C22⋊C412yes—
40C2×Dic127C22⋊C411yescharacteristic
41C4×D127—22no—
42C4×D127—22no—
43C4×D127—22no—
44C4×D127—44no—
45C127⋊C8—42no—
46Q8×C127—42no—
47C2×C4×C127—22no—
48C2×C4×C127C22×C411yescharacteristic
49C2×C4×C127C2×D412yes—
50C8×C127—42no—
51C22×C127C2×C22⋊C411yescharacteristic
52C2×D127—21no—
53C2×D127—22no—
54C2×D127—42no—
55C2×D127—44no—
56C4×C127—21no—
57C4×C127—42no—
58C4×C127AGL(1,ℤ/8)12yes—
59C4×C127#13712yes—
60Dic127—22no—
61Dic127#10012yes—
62C2×C127C2×#10011yescharacteristic
63C2×C127#41712yes—
64D127—41no—
65D127—82no—
66#1560—1271nomaximal, Sylow
67C127#156011yescharacteristic, complemented, Sylow
68#532—1271no—
69#423—1272no—
70#333—1271no—
71#333D12711yescharacteristic, complemented
72#331—1272no—
73C2×#41—1272no—
74C2×M16—2542no—
75C4⋊Q8C2×D12711yescharacteristic
76#127—1271no—
77#124—1272no—
78#121—2542no—
79C4×D4—2542no—
80C2×C42—1271no—
81C4⋊C8—1272no—
82C4⋊C8—2542no—
83C4⋊C8C2×D12712yes—
84#41—5084no—
85C2×Q8C4×D12712yes—
86C2×D4—1272no—
87C2×D4—2542no—
88C2×D4—2542no—
89C22×C4—1271no—
90C22×C4—1272no—
91C22×C4—2542no—
92M16—5084no—
93C2×C8—22no—
94C2×C8—2542no—
95C4⋊C4—22no—
96C22⋊C4—2544no—
97C42—1271no—
98C42—1272no—
99C42C22×D12711yescharacteristic
100C23—1271no—
101C23—2542no—
102Q8—42no—
103D4—5082no—
104D4—5084no—
105D4—5082no—
106D4—5082no—
107C2×C4—22no—
108C2×C4—1271no—
109C2×C4—1272no—
110C2×C4—5084no—
111C2×C4—2542no—
112C2×C4—2542no—
113C2×C4—2542no—
114C2×C4C2×C4×D12711yescharacteristic
115C2×C4D4×D12712yes—
116C8—42no—
117C8—5082no—
118C22—5082no—
119C22—5084no—
120C22—2541no—
121C22—2542no—
122C22C22⋊C4×D12711yescharacteristic
123C4—21no—
124C4—42no—
125C4—1272no—
126C4—2542no—
127C4#1686123612yes—
128C4#1686124112yes—
129C2—10162no—
130C2—5081no—
131C2#100×D12711yescharacteristic
132C2#2254987612yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#333×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550427C2
cosocleC42×C127C23
hypercenter#333D127
hypocenterC127#1560
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #22549898C2semidirect
3. #22549625C2semidirect
4. #22549619C2semidirect
5. #333×C127C2semidirect
67. C127#1560semidirect
71. #333D127semidirect

Maximal subgroups

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

subgroupindex
#225504272
#225498982
#225496252
#225496192
#333×C1272
#1560127