Finite Group Databasebeta

← #30637868

Subgroups — #30637868

Order 16256.

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

normal subgroups only

This group has 206 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#22550606C211yesmaximal, characteristic
2#22550168C212yesmaximal
3#22549625C211yesmaximal, characteristic
4#22549608C212yesmaximal
5#494×C127C211yesmaximal, characteristic
6#16861208—22no—
7#16861207—22no—
8C4×Dic254C2212yes—
9#16861174—22no—
10#16861142—22no—
11C4⋊(C127⋊C8)C2212yes—
12#16861132—22no—
13C2×Q16×C127—22no—
14C4⋊Q8×C127C2211yescharacteristic
15C4⋊Q8×C127C2211yescharacteristic
16C4×C8×C127C2211yescharacteristic
17C2×Dic254—22no—
18C2×Dic254D412yes—
19C2×C127⋊C8—22no—
20C4⋊Dic127D412yes—
21Dic127⋊C4—24no—
22Dic127⋊C4—22no—
23C4×Dic127—22no—
24C4×Dic127—22no—
25C2×Q8×C127—21no—
26C2×Q8×C127D412yes—
27Q16×C127—44no—
28C2×C8×C127—21no—
29C4⋊C4×C127—22no—
30C4⋊C4×C127—22no—
31C42×C127C2311yescharacteristic
32Dic254—22no—
33Dic254—82no—
34Dic254Q1614yescomplemented
35C2×Dic127—22no—
36C2×Dic127—22no—
37C127⋊C8—42no—
38Q8×C127—42no—
39Q8×C127—42no—
40C2×C4×C127—22no—
41C2×C4×C127—41no—
42C2×C4×C127C2×D412yes—
43C2×C4×C127C2×D411yescharacteristic
44C8×C127—22no—
45C22×C127#12011yescharacteristic
46C4×C127—21no—
47C4×C127—42no—
48C4×C127—81no—
49C4×C127C2×Q1612yes—
50C4×C127#13712yes—
51Dic127—24no—
52Dic127—42no—
53C2×C127#45112yes—
54C2×C127#45311yescharacteristic
55#1633—1271nomaximal, Sylow
56C127#163311yescharacteristic, complemented, Sylow
57Q82—1271no—
58#494D12711yescharacteristic
59#477—1272no—
60#333—1271no—
61Q8⋊C8—1272no—
62C2×Q16—22no—
63C4⋊Q8—2542no—
64C4⋊Q8—2542no—
65C4⋊Q8C2×D12711yescharacteristic
66C4⋊Q8C2×D12711yescharacteristic
67C4×Q8—1272no—
68C4×Q8—2542no—
69#107—2542no—
70C4⋊C8—1272no—
71#103—2542no—
72C4×C8C2×D12711yescharacteristic
73C2×Q8—21no—
74C2×Q8—1272no—
75C2×Q8—2542no—
76C2×Q8C22⋊D12712yes—
77Q16—44no—
78C2×C8—21no—
79C2×C8—2542no—
80C4⋊C4—22no—
81C4⋊C4—22no—
82C4⋊C4—1272no—
83C4⋊C4—2544no—
84C4⋊C4—2542no—
85C42—2542no—
86C42—2542no—
87C42C22×D12711yescharacteristic
88Q8—42no—
89Q8—42no—
90Q8—1274no—
91Q8—10162no—
92Q8—2542no—
93C2×C4—22no—
94C2×C4—41no—
95C2×C4—2542no—
96C2×C4—2542no—
97C2×C4D4×D12712yes—
98C2×C4C2×C22⋊D12711yescharacteristic
99C8—22no—
100C8—5082no—
101C22#1686117711yescharacteristic
102C4—21no—
103C4—42no—
104C4—81no—
105C4—5082no—
106C4—2544no—
107C4Q16×D12712yes—
108C4#1686124112yes—
109C2#2255003012yes—
110C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#494×C127C2
socle, prime radicalC22×C127#120
hypercenter#494D127
hypocenterC127#1633
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
34. Dic254Q16semidirect
56. C127#1633semidirect

Maximal subgroups

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

subgroupindex
#225506062
#225501682
#225496252
#225496082
#494×C1272
#1633127