Finite Group Databasebeta

← #30637875

Subgroups — #30637875

Order 16256.

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

normal subgroups only

This group has 218 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#22550255C211yesmaximal, characteristic, complemented
2#22550166C212yesmaximal, complemented
3#22549625C211yesmaximal, characteristic, complemented
4D4⋊(C127⋊C8)C212yesmaximal, complemented
5#555×C127C211yesmaximal, characteristic
6C2×#14523238—22no—
7#16861209C2211yescharacteristic
8#16861140—22no—
9C4⋊(C127⋊C8)C2212yes—
10#16861130—22no—
11C4×C127⋊C8C2211yescharacteristic
12C4⋊Q8×C127C2211yescharacteristic
13#124×C127—22no—
14#121×C127—22no—
15C4×D4×C127C2212yes—
16#117×C127—22no—
17C2×Dic254—21no—
18#14523238—44no—
19C2×C127⋊C8—21no—
20C2×C127⋊C8—22no—
21C4⋊Dic127—22no—
22C2×Q8×C127D412yes—
23C2×D4×C127—22no—
24C2×D4×C127D412yes—
25C22×C4×C127—22no—
26C4⋊C4×C127—22no—
27C4⋊C4×C127D412yes—
28C22⋊C4×C127—24no—
29C22⋊C4×C127—22no—
30C42×C127—22no—
31C42×C127C2311yescharacteristic
32Dic254—42no—
33C2×Dic127—41no—
34C127⋊C8—22no—
35C127⋊C8—42no—
36C23×C127—22no—
37Q8×C127—42no—
38D4×C127—22no—
39D4×C127—22no—
40D4×C127—82no—
41C2×C4×C127—22no—
42C2×C4×C127—22no—
43C2×C4×C127—42no—
44C2×C4×C127—42no—
45C2×C4×C127C2×D411yescharacteristic
46C2×C4×C127C2×D412yes—
47C22×C127—42no—
48C22×C127—42no—
49C22×C127—42no—
50C22×C127#12011yescharacteristic
51C4×C127—21no—
52C4×C127—42no—
53C4×C127—42no—
54C4×C127#13512yes—
55C4×C127#13712yes—
56Dic127—81no—
57C2×C127—82no—
58C2×C127#45212yes—
59C2×C127#45311yescharacteristic
60#1634—1271nomaximal, Sylow
61C127#163411yescharacteristic, complemented, Sylow
62#555D12711yescharacteristic
63#494—1271no—
64#476—1272no—
65#333—1271no—
66D4⋊C8—1272no—
67C2×Q16—2542no—
68C4⋊Q8—1271no—
69C4⋊Q8C2×D12711yescharacteristic
70#124—22no—
71#121—22no—
72C4×D4C2×D12712yes—
73#117—22no—
74#106—2542no—
75C4⋊C8—1272no—
76#102—2542no—
77C4×C8—1271no—
78C2×Q8—2541no—
79C2×Q8C22⋊D12712yes—
80C2×D4—22no—
81C2×D4C22⋊D12712yes—
82C22×C4—22no—
83Q16—5084no—
84C2×C8—2542no—
85C2×C8—2541no—
86C4⋊C4—22no—
87C4⋊C4—2542no—
88C4⋊C4C22⋊D12712yes—
89C22⋊C4—24no—
90C22⋊C4—22no—
91C42—22no—
92C42C22×D12711yescharacteristic
93C23—22no—
94Q8—42no—
95Q8—5082no—
96D4—22no—
97D4—22no—
98D4—82no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—42no—
102C2×C4—42no—
103C2×C4—5081no—
104C2×C4C2×C22⋊D12711yescharacteristic
105C2×C4C2×C22⋊D12712yes—
106C8—5082no—
107C8—2542no—
108C22—42no—
109C22—42no—
110C22—42no—
111C22#1686117811yescharacteristic
112C4—21no—
113C4—42no—
114C4—42no—
115C4—10161no—
116C4#1686122912yes—
117C4#1686123912yes—
118C2—82no—
119C2#2255003812yes—
120C2#2255004411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radical#555×C127C2
socleC22×C127#120
hypercenter#555D127
hypocenterC127#1634
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. #22550255C2semidirect
2. #22550166C2semidirect
3. #22549625C2semidirect
4. D4⋊(C127⋊C8)C2semidirect
61. C127#1634semidirect

Maximal subgroups

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

subgroupindex
#225502552
#225501662
#225496252
D4⋊(C127⋊C8)2
#555×C1272
#1634127