Finite Group Databasebeta

← #30638053

Subgroups — #30638053

Order 16256.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550246C211yesmaximal, characteristic, complemented
2#22550235C212yesmaximal, complemented
3#22549618C211yesmaximal, characteristic, complemented
4#22549616C212yesmaximal, complemented
5#562×C127C211yesmaximal, characteristic
6#16861209C2211yescharacteristic
7#16861133—22no—
8#16861132—22no—
9#16861130—22no—
10#16861120C2211yescharacteristic
11#16861120C2212yes—
12#125×C127C2211yescharacteristic
13#124×C127C2212yes—
14#123×C127—22no—
15C22⋊Q8×C127—22no—
16#121×C127—22no—
17C2×Dic254—21no—
18C2×C127⋊C8—22no—
19C2×C127⋊C8—21no—
20C4⋊Dic127—22no—
21C2×Q8×C127D412yes—
22C2×D4×C127—22no—
23C2×D4×C127D412yes—
24C22×C4×C127—22no—
25C4⋊C4×C127—22no—
26C4⋊C4×C127—22no—
27C4⋊C4×C127D412yes—
28C22⋊C4×C127—22no—
29C22⋊C4×C127—22no—
30C22⋊C4×C127—22no—
31C42×C127C2311yescharacteristic
32Dic254—42no—
33C2×Dic127—41no—
34C127⋊C8—41no—
35C127⋊C8—42no—
36C23×C127—22no—
37Q8×C127—42no—
38D4×C127—42no—
39D4×C127—82no—
40C2×C4×C127—22no—
41C2×C4×C127—22no—
42C2×C4×C127—82no—
43C2×C4×C127C2×D411yescharacteristic
44C2×C4×C127C2×D412yes—
45C22×C127—42no—
46C22×C127—42no—
47C22×C127—42no—
48C22×C127#12011yescharacteristic
49C4×C127—21no—
50C4×C127—22no—
51C4×C127—42no—
52C4×C127—42no—
53Dic127—81no—
54C2×C127—82no—
55C2×C127#45411yescharacteristic
56C2×C127#45512yes—
57#1662—1271nomaximal, Sylow
58C127#166211yescharacteristic, complemented, Sylow
59#562D12711yescharacteristic
60#491—1271no—
61#489—1272no—
62#330—1271no—
63#329—1272no—
64C4⋊Q8—1271no—
65#125C2×D12711yescharacteristic
66#124C2×D12712yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—2542no—
71#103—2542no—
72#102—2542no—
73#97—1271no—
74#97—1272no—
75C2×Q8—2541no—
76C2×Q8C22⋊D12712yes—
77C2×D4—22no—
78C2×D4C22⋊D12712yes—
79C22×C4—22no—
80C2×C8—2542no—
81C2×C8—2541no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—2542no—
85C4⋊C4C22⋊D12712yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D12711yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—5082no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—5081no—
99C2×C4C2×C22⋊D12711yescharacteristic
100C2×C4C2×C22⋊D12712yes—
101C8—5081no—
102C8—5082no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#1686117811yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—10161no—
112C2—82no—
113C2#2255004911yescharacteristic
114C2#2255005412yes—
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radical#562×C127C2
socleC22×C127#120
hypercenter#562D127
hypocenterC127#1662
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #22550246C2semidirect
2. #22550235C2semidirect
3. #22549618C2semidirect
4. #22549616C2semidirect
58. C127#1662semidirect

Maximal subgroups

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

subgroupindex
#225502462
#225502352
#225496182
#225496162
#562×C1272
#1662127