Finite Group Databasebeta

← #30637462

Subgroups — #30637462

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#22550424C211yesmaximal, characteristic
2#22549898C212yesmaximal, complemented
3#22549624C212yesmaximal, complemented
4#22549620C211yesmaximal, characteristic, complemented
5#331×C127C211yesmaximal, characteristic, complemented
6C2×#14523247C412yes—
7C2×#14523231—22no—
8#16861209C2211yescharacteristic
9#16861196C412yes—
10#16861180—22no—
11D4×Dic127—22no—
12C42×D127C2211yescharacteristic
13C4⋊(C127⋊C8)C2212yes—
14Dic127⋊C8—22no—
15#127×C127C2211yescharacteristic
16C4⋊C8×C127C2212yes—
17#14523247—44no—
18C2×Dic254C2×C412yes—
19C2×C22⋊D127—22no—
20C22×Dic127—22no—
21C2×C4×D127C2×C411yescharacteristic
22C2×C4×D127D412yes—
23#14523231—44no—
24C2×C127⋊C8—22no—
25C4⋊Dic127—22no—
26C22⋊Dic127—24no—
27C4×Dic127C2×C411yescharacteristic
28C4×Dic127D412yes—
29C2×D4×C127—22no—
30C2×D4×C127C2×C412yes—
31C2×C8×C127—22no—
32C42×C127C2311yescharacteristic
33C22×D127C22⋊C411yescharacteristic
34Dic254—42no—
35C22⋊D127—44no—
36C2×Dic127—22no—
37C2×Dic127—44no—
38C2×Dic127C22⋊C412yes—
39C2×Dic127C22⋊C411yescharacteristic
40C4×D127—22no—
41C4×D127—22no—
42C4×D127—22no—
43C127⋊C8—42no—
44C23×C127—22no—
45D4×C127—42no—
46D4×C127—42no—
47D4×C127—42no—
48C2×C4×C127C22×C411yescharacteristic
49C2×C4×C127C2×D412yes—
50C8×C127—42no—
51C22×C127—42no—
52C22×C127—44no—
53C22×C127C2×C22⋊C411yescharacteristic
54C2×D127—21no—
55C2×D127—22no—
56C4×C127—21no—
57C4×C127AGL(1,ℤ/8)12yes—
58C4×C127#13712yes—
59Dic127—22no—
60Dic127—42no—
61Dic127#10012yes—
62C2×C127—82no—
63C2×C127C2×#10011yescharacteristic
64C2×C127#41712yes—
65D127—41no—
66#1560—1271nomaximal, Sylow
67C127#156011yescharacteristic, complemented, Sylow
68#532—1271no—
69#423—1272no—
70#333—1272no—
71#331—1271no—
72#331D12711yescharacteristic, complemented
73C2×#41—1272no—
74C2×M16—2542no—
75C4⋊Q8—1271no—
76#127C2×D12711yescharacteristic
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×Q8—1272no—
86C2×D4—22no—
87C2×D4—2542no—
88C2×D4C4×D12712yes—
89C22×C4—1271no—
90C22×C4—1272no—
91C22×C4—2542no—
92M16—5084no—
93C2×C8—22no—
94C2×C8—2542no—
95C4⋊C4—2542no—
96C22⋊C4—2544no—
97C42—1271no—
98C42—1272no—
99C42C22×D12711yescharacteristic
100C23—22no—
101C23—1271no—
102Q8—5082no—
103D4—42no—
104D4—42no—
105D4—42no—
106D4—5084no—
107C2×C4—1271no—
108C2×C4—1272no—
109C2×C4—5084no—
110C2×C4—2542no—
111C2×C4—2542no—
112C2×C4—2542no—
113C2×C4—2542no—
114C2×C4C2×C4×D12711yescharacteristic
115C2×C4D4×D12712yes—
116C8—42no—
117C8—5082no—
118C22—42no—
119C22—44no—
120C22—2541no—
121C22—2542no—
122C22C22⋊C4×D12711yescharacteristic
123C4—21no—
124C4—1272no—
125C4—5082no—
126C4—2542no—
127C4#1686123312yes—
128C4#1686123812yes—
129C2—82no—
130C2—5081no—
131C2#100×D12711yescharacteristic
132C2#2254987412yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550424C2
cosocleC42×C127C23
hypercenter#331D127
hypocenterC127#1560
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22549898C2semidirect
3. #22549624C2semidirect
4. #22549620C2semidirect
5. #331×C127C2semidirect
67. C127#1560semidirect
72. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225504242
#225498982
#225496242
#225496202
#331×C1272
#1560127