Finite Group Databasebeta

← #30640797

Subgroups — #30640797

Order 16256.

154 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550316C211yesmaximal, characteristic, complemented
2#22550259C211yesmaximal, characteristic, complemented
3#22549993C211yesmaximal, characteristic, complemented
4#22549696C211yesmaximal, characteristic, complemented
5#22549693C211yesmaximal, characteristic, complemented
6#22549689C211yesmaximal, characteristic, complemented
7#509×C127C211yesmaximal, characteristic
8C2×Dic508C2211yescharacteristic
9C2×#14523236—21no—
10#16861216—21no—
11#16861196—21no—
12#16861155—21no—
13#16861153—21no—
14#16861149C2211yescharacteristic
15#16861145C2211yescharacteristic
16#16861136—21no—
17C8×Dic127C2211yescharacteristic
18#135×C127C2211yescharacteristic
19C2×D8×C127C2211yescharacteristic
20SD32×C127—21no—
21D16×C127—21no—
22M32×C127C2211yescharacteristic
23C2×Dic254—21no—
24Dic508—21no—
25Dic508—21no—
26#14523236—41no—
27#14523236—41no—
28#14523232—21no—
29#14523232—41no—
30C2×C127⋊C8—41no—
31C2×C127⋊C8D411yescharacteristic
32C22⋊Dic127—42no—
33C4×Dic127D411yescharacteristic
34C127⋊C16—21no—
35#41×C127—21no—
36C2×D4×C127—21no—
37Q16×C127D411yescharacteristic
38SD16×C127—21no—
39D8×C127—21no—
40D8×C127—21no—
41D8×C127D411yescharacteristic
42C2×C8×C127C2311yescharacteristic
43C16×C127—21no—
44Dic254—41no—
45Dic254—41no—
46C2×Dic127—21no—
47C2×Dic127—41no—
48C127⋊C8—41no—
49C127⋊C8D811yescharacteristic
50C127⋊C8D811yescharacteristic
51C23×C127—41no—
52Q8×C127—21no—
53D4×C127—21no—
54D4×C127—41no—
55D4×C127—41no—
56D4×C127—41no—
57C2×C4×C127—41no—
58C2×C4×C127C2×D411yescharacteristic
59C8×C127C2×D411yescharacteristic
60C8×C127#4111yescharacteristic
61C22×C127—41no—
62C22×C127—82no—
63C22×C127—81no—
64C22×C127AGL(1,ℤ/8)11yescharacteristic
65C4×C127—41no—
66C4×C127#12111yescharacteristic
67C4×C127C2×D811yescharacteristic
68Dic127—41no—
69Dic127—81no—
70C2×C127—21no—
71C2×C127—81no—
72C2×C127—161no—
73C2×C127#45911yescharacteristic
74#2212—1271nomaximal, Sylow
75C127#221211yescharacteristic, complemented, Sylow
76#510—1271no—
77#509D12711yescharacteristic
78#495—1271no—
79#443—1271no—
80#364—1271no—
81#362—1271no—
82#361—1271no—
83#135C2×D12711yescharacteristic
84C2×Q16—1271no—
85C2×SD16—2541no—
86C2×D8C2×D12711yescharacteristic
87#131—2541no—
88#124—2541no—
89Q32—2541no—
90SD32—21no—
91SD32—2541no—
92D16—21no—
93M32—1271no—
94M32C2×D12711yescharacteristic
95#108—1271no—
96#104—2541no—
97C4×C8—1271no—
98#41—21no—
99C2×Q8—2541no—
100C2×D4—21no—
101Q16—2541no—
102Q16—2541no—
103Q16C22⋊D12711yescharacteristic
104SD16—21no—
105SD16—5081no—
106SD16—5081no—
107D8—21no—
108D8—21no—
109D8C22⋊D12711yescharacteristic
110M16—5081no—
111M16—2541no—
112C2×C8—1271no—
113C2×C8—5081no—
114C2×C8C22×D12711yescharacteristic
115C22⋊C4—5082no—
116C42—1271no—
117C16—21no—
118C16—2541no—
119C23—41no—
120Q8—21no—
121Q8—5081no—
122Q8—5081no—
123D4—21no—
124D4—41no—
125D4—41no—
126D4—41no—
127C2×C4—41no—
128C2×C4—5081no—
129C2×C4—2541no—
130C2×C4D4×D12711yescharacteristic
131C8—1271no—
132C8—1271no—
133C8—5081no—
134C8C2×C22⋊D12711yescharacteristic
135C8#1452324711yescharacteristic
136C22—41no—
137C22—82no—
138C22—81no—
139C22#1686123311yescharacteristic
140C4—41no—
141C4—10161no—
142C4—5081no—
143C4#1686118111yescharacteristic
144C4D8×D12711yescharacteristic
145C2—21no—
146C2—81no—
147C2—161no—
148C2#2255006911yescharacteristic
subgroupG/H
centerC2#22550069
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting, prime radical#509×C127C2
socleC2×C127#459
hypercenter#509D127
hypocenterC127#2212
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22550316C2semidirect
2. #22550259C2semidirect
3. #22549993C2semidirect
4. #22549696C2semidirect
5. #22549693C2semidirect
6. #22549689C2semidirect
75. C127#2212semidirect

Maximal subgroups

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

subgroupindex
#225503162
#225502592
#225499932
#225496962
#225496932
#225496892
#509×C1272
#2212127