Finite Group Databasebeta

← #30637813

Subgroups — #30637813

Order 16256.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550250C211yesmaximal, characteristic, complemented
2#22550071C212yesmaximal, complemented
3#22549625C211yesmaximal, characteristic, complemented
4D4⋊(C127⋊C8)C212yesmaximal, complemented
5#555×C127C211yesmaximal, characteristic, complemented
6C2×#14523235—22no—
7C2×#14523233—22no—
8#16861206C2211yescharacteristic
9C4⋊(C127⋊C8)C2212yes—
10#16861129—22no—
11C4×C127⋊C8C2211yescharacteristic
12C4⋊Q8×C127C2211yescharacteristic
13#124×C127—22no—
14#121×C127—22no—
15C4×D4×C127C2212yes—
16#117×C127—22no—
17C2×D508—22no—
18C2×D508—21no—
19#14523235—44no—
20#14523233—44no—
21C2×C127⋊C8—21no—
22C2×C127⋊C8—22no—
23C2×Q8×C127D412yes—
24C2×D4×C127—22no—
25C2×D4×C127D412yes—
26C22×C4×C127—22no—
27C4⋊C4×C127—22no—
28C4⋊C4×C127D412yes—
29C22⋊C4×C127—24no—
30C22⋊C4×C127—22no—
31C42×C127—22no—
32C42×C127C2311yescharacteristic
33C22×D127—41no—
34D508—42no—
35D508—42no—
36D508—42no—
37C127⋊C8—22no—
38C127⋊C8—42no—
39C23×C127—22no—
40Q8×C127—42no—
41D4×C127—22no—
42D4×C127—22no—
43D4×C127—82no—
44C2×C4×C127—22no—
45C2×C4×C127—22no—
46C2×C4×C127—42no—
47C2×C4×C127—42no—
48C2×C4×C127C2×D411yescharacteristic
49C2×C4×C127C2×D412yes—
50C22×C127—42no—
51C22×C127—42no—
52C22×C127—42no—
53C22×C127#12011yescharacteristic
54C2×D127—82no—
55C2×D127—81no—
56C4×C127—21no—
57C4×C127—42no—
58C4×C127—42no—
59C4×C127#13512yes—
60C4×C127AGL(1,ℤ/8)12yes—
61C2×C127—82no—
62C2×C127#44912yes—
63C2×C127#45311yescharacteristic
64D127—161no—
65#1624—1271nomaximal, Sylow
66C127#162411yescharacteristic, complemented, Sylow
67#555D12711yescharacteristic, complemented
68#492—1271no—
69#459—1272no—
70#333—1271no—
71D4⋊C8—1272no—
72C2×SD16—2542no—
73C2×D8—2542no—
74C4⋊Q8C2×D12711yescharacteristic
75#127—1271no—
76#124—22no—
77#121—22no—
78C4×D4C2×D12712yes—
79#117—22no—
80C4⋊C8—1272no—
81#102—2542no—
82C4×C8—1271no—
83C2×Q8C22⋊D12712yes—
84C2×D4—22no—
85C2×D4—2542no—
86C2×D4—2541no—
87C2×D4C22⋊D12712yes—
88C22×C4—22no—
89SD16—5084no—
90D8—5084no—
91C2×C8—2542no—
92C2×C8—2541no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D12712yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D12711yescharacteristic
99C23—22no—
100C23—5081no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—5082no—
106D4—5082no—
107D4—5082no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D12711yescharacteristic
113C2×C4C2×C22⋊D12712yes—
114C8—5082no—
115C8—2542no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—10162no—
120C22—10161no—
121C22#1686117811yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#1686122912yes—
126C4#1686123412yes—
127C2—82no—
128C2—20321no—
129C2#2255002112yes—
130C2#2255004411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#555×C127C2
socleC22×C127#120
hypercenter#555D127
hypocenterC127#1624
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #22550250C2semidirect
2. #22550071C2semidirect
3. #22549625C2semidirect
4. D4⋊(C127⋊C8)C2semidirect
5. #555×C127C2semidirect
66. C127#1624semidirect
67. #555D127semidirect

Maximal subgroups

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

subgroupindex
#225502502
#225500712
#225496252
D4⋊(C127⋊C8)2
#555×C1272
#1624127