Finite Group Databasebeta

← #30637873

Subgroups — #30637873

Order 16256.

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

normal subgroups only

This group has 218 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
2#22550163C212yesmaximal, complemented
3#22549625C211yesmaximal, characteristic, complemented
4#22549605C212yesmaximal, complemented
5#494×C127C211yesmaximal, characteristic, complemented
6#16861198—22no—
7#16861184—22no—
8C4×D508C2212yes—
9#16861167—22no—
10#16861140—22no—
11C4⋊(C127⋊C8)C2212yes—
12#16861129—22no—
13C2×Q16×C127—22no—
14C4⋊Q8×C127C2211yescharacteristic
15C4⋊Q8×C127C2211yescharacteristic
16C4×C8×C127C2211yescharacteristic
17C2×D508—22no—
18C2×D508D412yes—
19C2×C4×D127—22no—
20C2×C127⋊C8—22no—
21C4⋊Dic127D412yes—
22(C2×D127)⋊C4—24no—
23(C2×D127)⋊C4—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
32C22×D127—22no—
33D508—22no—
34D508—22no—
35D508—82no—
36C2×Dic127—22no—
37C4×D127—42no—
38C4×D127—42no—
39C127⋊C8—42no—
40Q8×C127—42no—
41Q8×C127—42no—
42C2×C4×C127—22no—
43C2×C4×C127—41no—
44C2×C4×C127C2×D412yes—
45C2×C4×C127C2×D411yescharacteristic
46C8×C127—22no—
47C22×C127#12011yescharacteristic
48C2×D127—42no—
49C2×D127—42no—
50C2×D127—42no—
51C4×C127—21no—
52C4×C127—42no—
53C4×C127—81no—
54C4×C127#13512yes—
55C4×C127#13712yes—
56Dic127—42no—
57C2×C127#45212yes—
58C2×C127#45311yescharacteristic
59D127—82no—
60#1634—1271nomaximal, Sylow
61C127#163411yescharacteristic, complemented, Sylow
62#555—1271no—
63#494D12711yescharacteristic, complemented
64#476—1272no—
65#333—1271no—
66D4⋊C8—1272no—
67C2×Q16—22no—
68C4⋊Q8C2×D12711yescharacteristic
69C4⋊Q8C2×D12711yescharacteristic
70#124—2542no—
71#121—2542no—
72C4×D4—1272no—
73#117—2542no—
74#106—2542no—
75C4⋊C8—1272no—
76#102—2542no—
77C4×C8C2×D12711yescharacteristic
78C2×Q8—21no—
79C2×Q8C22⋊D12712yes—
80C2×D4—1272no—
81C2×D4—2542no—
82C22×C4—2542no—
83Q16—44no—
84C2×C8—21no—
85C2×C8—2542no—
86C4⋊C4—22no—
87C4⋊C4—22no—
88C4⋊C4—1272no—
89C22⋊C4—2544no—
90C22⋊C4—2542no—
91C42—2542no—
92C42C22×D12711yescharacteristic
93C23—2542no—
94Q8—42no—
95Q8—42no—
96D4—10162no—
97D4—2542no—
98D4—2542no—
99C2×C4—22no—
100C2×C4—41no—
101C2×C4—5082no—
102C2×C4—5082no—
103C2×C4—2542no—
104C2×C4D4×D12712yes—
105C2×C4C2×C22⋊D12711yescharacteristic
106C8—22no—
107C8—5082no—
108C22—5082no—
109C22—5082no—
110C22—5082no—
111C22#1686117711yescharacteristic
112C4—21no—
113C4—42no—
114C4—81no—
115C4—5082no—
116C4#1686123212yes—
117C4#1686124112yes—
118C2—10162no—
119C2#2255003712yes—
120C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#494×C127C2
socleC22×C127#120
prime radical#22550592C2
hypercenter#494D127
hypocenterC127#1634
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #22550163C2semidirect
3. #22549625C2semidirect
4. #22549605C2semidirect
5. #494×C127C2semidirect
61. C127#1634semidirect
63. #494D127semidirect

Maximal subgroups

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

subgroupindex
#225505922
#225501632
#225496252
#225496052
#494×C1272
#1634127