Finite Group Databasebeta

← #30637814

Subgroups — #30637814

Order 16256.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550592C211yesmaximal, characteristic, complemented
2#22550250C211yesmaximal, characteristic, complemented
3#22550072C212yesmaximal, complemented
4#22549603C212yesmaximal, complemented
5#333×C127C211yesmaximal, characteristic, complemented
6C2×#14523235—22no—
7C2×D1016—22no—
8#16861206C2211yescharacteristic
9#16861198—22no—
10#16861184—22no—
11C4×D508C2212yes—
12#16861167—22no—
13#16861131—22no—
14C4×C127⋊C8C2211yescharacteristic
15C4⋊Q8×C127C2211yescharacteristic
16C4⋊C8×C127C2212yes—
17C2×D508—22no—
18C2×D508—22no—
19C2×D508—21no—
20C2×D508D412yes—
21C2×C4×D127—22no—
22#14523235—44no—
23D1016—44no—
24C2×C127⋊C8—21no—
25C4⋊Dic127D412yes—
26(C2×D127)⋊C4—24no—
27(C2×D127)⋊C4—22no—
28C4×Dic127—22no—
29C2×Q8×C127D412yes—
30C2×C8×C127—22no—
31C4⋊C4×C127—22no—
32C42×C127C2311yescharacteristic
33C22×D127—22no—
34C22×D127—41no—
35D508—22no—
36D508—22no—
37D508—42no—
38D508—42no—
39D508—42no—
40D508—82no—
41C2×Dic127—22no—
42C4×D127—42no—
43C4×D127—42no—
44C127⋊C8—22no—
45Q8×C127—42no—
46C2×C4×C127—22no—
47C2×C4×C127C2×D412yes—
48C2×C4×C127C2×D411yescharacteristic
49C8×C127—42no—
50C22×C127#12011yescharacteristic
51C2×D127—42no—
52C2×D127—42no—
53C2×D127—42no—
54C2×D127—82no—
55C2×D127—81no—
56C4×C127—21no—
57C4×C127—42no—
58C4×C127#13512yes—
59C4×C127AGL(1,ℤ/8)12yes—
60Dic127—42no—
61C2×C127#44912yes—
62C2×C127#45311yescharacteristic
63D127—82no—
64D127—161no—
65#1624—1271nomaximal, Sylow
66C127#162411yescharacteristic, complemented, Sylow
67#555—1271no—
68#492—1271no—
69#459—1272no—
70#333D12711yescharacteristic, complemented
71D4⋊C8—1272no—
72C2×SD16—2542no—
73C2×D8—2542no—
74C4⋊Q8C2×D12711yescharacteristic
75#127—1271no—
76#124—2542no—
77#121—2542no—
78C4×D4—1272no—
79#117—2542no—
80C4⋊C8C2×D12712yes—
81#102—2542no—
82C4×C8—1271no—
83C2×Q8D50812yes—
84C2×D4—1272no—
85C2×D4—2542no—
86C2×D4—2542no—
87C2×D4—2541no—
88C22×C4—2542no—
89SD16—5084no—
90D8—5084no—
91C2×C8—22no—
92C2×C8—2541no—
93C4⋊C4—22no—
94C4⋊C4—1272no—
95C22⋊C4—2544no—
96C22⋊C4—2542no—
97C42—2542no—
98C42C22×D12711yescharacteristic
99C23—5081no—
100C23—2542no—
101Q8—42no—
102D4—10162no—
103D4—5082no—
104D4—5082no—
105D4—5082no—
106D4—2542no—
107D4—2542no—
108C2×C4—22no—
109C2×C4—5082no—
110C2×C4—5082no—
111C2×C4—2542no—
112C2×C4D4×D12712yes—
113C2×C4C2×D50811yescharacteristic
114C8—42no—
115C8—2542no—
116C22—10162no—
117C22—10161no—
118C22—5082no—
119C22—5082no—
120C22—5082no—
121C22#1686117911yescharacteristic
122C4—21no—
123C4—42no—
124C4—5082no—
125C4#1686123212yes—
126C4#1686123612yes—
127C2—20321no—
128C2—10162no—
129C2#2255002412yes—
130C2#2255004611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#333×C127C2
socleC22×C127#120
hypercenter#333D127
hypocenterC127#1624
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #22550592C2semidirect
2. #22550250C2semidirect
3. #22550072C2semidirect
4. #22549603C2semidirect
5. #333×C127C2semidirect
66. C127#1624semidirect
70. #333D127semidirect

Maximal subgroups

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

subgroupindex
#225505922
#225502502
#225500722
#225496032
#333×C1272
#1624127