Finite Group Databasebeta

← #30640809

Subgroups — #30640809

Order 16256.

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

normal subgroups only

This group has 150 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
2#22550254C211yesmaximal, characteristic, complemented
3#22549993C211yesmaximal, characteristic
4#22549696C211yesmaximal, characteristic, complemented
5#22549693C211yesmaximal, characteristic, complemented
6#22549692C211yesmaximal, characteristic, complemented
7#510×C127C211yesmaximal, characteristic
8C2×Dic508C2211yescharacteristic
9C2×#14523238—21no—
10#16861216—21no—
11#16861207—21no—
12#16861155—21no—
13#16861153—21no—
14#16861149C2211yescharacteristic
15#16861145C2211yescharacteristic
16#16861136—21no—
17C8×Dic127C2211yescharacteristic
18#135×C127C2211yescharacteristic
19C2×Q16×C127C2211yescharacteristic
20Q32×C127—21no—
21SD32×C127—21no—
22M32×C127C2211yescharacteristic
23C2×Dic254—21no—
24Dic508—21no—
25Dic508—21no—
26#14523238—41no—
27#14523238—41no—
28#14523232—21no—
29#14523232—41no—
30C2×C127⋊C8—41no—
31C2×C127⋊C8D411yescharacteristic
32Dic127⋊C4—42no—
33C4×Dic127D411yescharacteristic
34C127⋊C16—21no—
35#41×C127—21no—
36C2×Q8×C127—21no—
37Q16×C127—21no—
38Q16×C127—21no—
39Q16×C127D411yescharacteristic
40SD16×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
51Q8×C127—21no—
52Q8×C127—41no—
53Q8×C127—41no—
54D4×C127—21no—
55D4×C127—41no—
56C2×C4×C127—41no—
57C2×C4×C127—41no—
58C2×C4×C127C2×D411yescharacteristic
59C8×C127C2×D411yescharacteristic
60C8×C127#4111yescharacteristic
61C22×C127—41no—
62C22×C127AGL(1,ℤ/8)11yescharacteristic
63C4×C127—41no—
64C4×C127—81no—
65C4×C127#12111yescharacteristic
66C4×C127C2×D811yescharacteristic
67Dic127—22no—
68Dic127—81no—
69C2×C127—21no—
70C2×C127—81no—
71C2×C127#45911yescharacteristic
72#2214—1271nomaximal, Sylow
73C127#221411yescharacteristic, complemented, Sylow
74#510—1271no—
75#510D12711yescharacteristic
76#494—1271no—
77#443—1271no—
78#364—1271no—
79#362—1271no—
80#362—1271no—
81#135C2×D12711yescharacteristic
82C2×Q16—1271no—
83C2×Q16—2541no—
84C2×Q16C2×D12711yescharacteristic
85#131—2541no—
86C4⋊Q8—2541no—
87Q32—21no—
88Q32—2541no—
89SD32—21no—
90SD32—2541no—
91M32—1271no—
92M32C2×D12711yescharacteristic
93#108—1271no—
94#104—2541no—
95C4×C8—1271no—
96#41—21no—
97C2×Q8—21no—
98C2×Q8—2541no—
99Q16—21no—
100Q16—21no—
101Q16—5081no—
102Q16—5081no—
103Q16—2541no—
104Q16—2541no—
105Q16C22⋊D12711yescharacteristic
106SD16—21no—
107D8C22⋊D12711yescharacteristic
108M16—5081no—
109M16—2541no—
110C2×C8—1271no—
111C2×C8—5081no—
112C2×C8C22×D12711yescharacteristic
113C4⋊C4—5082no—
114C42—1271no—
115C16—21no—
116C16—2541no—
117Q8—21no—
118Q8—41no—
119Q8—41no—
120Q8—5081no—
121Q8—5081no—
122D4—21no—
123D4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—5081no—
127C2×C4—2541no—
128C2×C4D4×D12711yescharacteristic
129C8—1271no—
130C8—1271no—
131C8—5081no—
132C8C2×C22⋊D12711yescharacteristic
133C8#1452324711yescharacteristic
134C22—41no—
135C22#1686123311yescharacteristic
136C4—41no—
137C4—81no—
138C4—10161no—
139C4—2542no—
140C4#1686118111yescharacteristic
141C4D8×D12711yescharacteristic
142C2—21no—
143C2—81no—
144C2#2255006911yescharacteristic
subgroupG/H
centerC2#22550069
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#510×C127C2
socleC2×C127#459
prime radical#135×C127C22
hypercenter#510D127
hypocenterC127#2214
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. #22550254C2semidirect
4. #22549696C2semidirect
5. #22549693C2semidirect
6. #22549692C2semidirect
73. C127#2214semidirect

Maximal subgroups

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

subgroupindex
#225503162
#225502542
#225499932
#225496962
#225496932
#225496922
#510×C1272
#2214127