Finite Group Databasebeta

← #30640810

Subgroups — #30640810

Order 16256.

150 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 43 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#22550319C211yesmaximal, characteristic
2#22550317C211yesmaximal, characteristic
3#22550254C211yesmaximal, characteristic, complemented
4#22549992C211yesmaximal, characteristic
5#22549696C211yesmaximal, characteristic, complemented
6#22549691C211yesmaximal, characteristic, complemented
7#362×C127C211yesmaximal, characteristic, complemented
8#16861231C2211yescharacteristic
9C2×Dic508C2211yescharacteristic
10C2×#14523238—21no—
11#16861217—21no—
12#16861207—21no—
13Dic1016—21no—
14#16861155—21no—
15#16861154—21no—
16#16861152—21no—
17#16861149C2211yescharacteristic
18#16861135—21no—
19C8×Dic127C2211yescharacteristic
20C2×Q16×C127C2211yescharacteristic
21M32×C127C2211yescharacteristic
22#108×C127C2211yescharacteristic
23#14523248—21no—
24C2×Dic254—21no—
25Dic508—21no—
26Dic508—21no—
27Dic508D411yescharacteristic
28#14523238—41no—
29#14523238—41no—
30#14523237—21no—
31D1016D411yescharacteristic
32#14523231—41no—
33C2×C127⋊C8D411yescharacteristic
34C8×D127—41no—
35Dic127⋊C4—42no—
36C4×Dic127D411yescharacteristic
37C127⋊C16—21no—
38C2×Q8×C127—21no—
39Q16×C127—21no—
40Q16×C127—21no—
41M16×C127—21no—
42C2×C8×C127C2311yescharacteristic
43C16×C127—21no—
44Dic254—21no—
45Dic254—41no—
46Dic254—41no—
47D508—21no—
48C22⋊D127—41no—
49C2×Dic127—21no—
50C2×Dic127—41no—
51C4×D127—41no—
52C127⋊C8D811yescharacteristic
53C127⋊C8D811yescharacteristic
54Q8×C127—41no—
55Q8×C127—41no—
56C2×C4×C127—41no—
57C2×C4×C127C2×D411yescharacteristic
58C8×C127—41no—
59C8×C127C2×D411yescharacteristic
60C8×C127#4111yescharacteristic
61C22×C127AGL(1,ℤ/8)11yescharacteristic
62C2×D127—41no—
63C4×C127—81no—
64C4×C127#12111yescharacteristic
65C4×C127C2×D811yescharacteristic
66Dic127—22no—
67Dic127—41no—
68Dic127—81no—
69C2×C127—21no—
70C2×C127#45911yescharacteristic
71D127—81no—
72#2214—1271nomaximal, Sylow
73C127#221411yescharacteristic, complemented, Sylow
74#510—1271no—
75#510—1271no—
76#494—1271no—
77#443—1271no—
78#364—1271no—
79#362—1271no—
80#362D12711yescharacteristic, complemented
81#135—1271no—
82C2×Q16—1271no—
83C2×Q16—2541no—
84C2×Q16C2×D12711yescharacteristic
85#131—2541no—
86C4⋊Q8—2541no—
87Q32—2541no—
88Q32—2541no—
89SD32—2541no—
90SD32—2541no—
91M32—1271no—
92M32C2×D12711yescharacteristic
93#108C2×D12711yescharacteristic
94#104—2541no—
95C4×C8—1271no—
96#41—2541no—
97C2×Q8—21no—
98C2×Q8—2541no—
99Q16—21no—
100Q16—21no—
101Q16—1271no—
102Q16—5081no—
103Q16—5081no—
104Q16—2541no—
105Q16—2541no—
106SD16—2541no—
107D8—1271no—
108M16—21no—
109M16—5081no—
110C2×C8—1271no—
111C2×C8—5081no—
112C2×C8C22×D12711yescharacteristic
113C4⋊C4—5082no—
114C42—1271no—
115C16—21no—
116C16—2541no—
117Q8—41no—
118Q8—41no—
119Q8—5081no—
120Q8—5081no—
121Q8—2541no—
122D4—5081no—
123D4—2541no—
124C2×C4—41no—
125C2×C4—5081no—
126C2×C4—5081no—
127C2×C4—2541no—
128C2×C4D4×D12711yescharacteristic
129C8—41no—
130C8—1271no—
131C8—1271no—
132C8D4×D12711yescharacteristic
133C8#1452324811yescharacteristic
134C22—5081no—
135C22#1686123611yescharacteristic
136C4—81no—
137C4—10161no—
138C4—5081no—
139C4—2542no—
140C4#1686118211yescharacteristic
141C4D8×D12711yescharacteristic
142C2—21no—
143C2—10161no—
144C2#2255007011yescharacteristic
subgroupG/H
centerC2#22550070
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#362×C127C2
socleC2×C127#459
prime radical#16861231C22
hypercenter#362D127
hypocenterC127#2214
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #362

Complements

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

NH ≅ G/Ntype
3. #22550254C2semidirect
5. #22549696C2semidirect
6. #22549691C2semidirect
7. #362×C127C2semidirect
73. C127#2214semidirect
80. #362D127semidirect

Maximal subgroups

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

subgroupindex
#225503192
#225503172
#225502542
#225499922
#225496962
#225496912
#362×C1272
#2214127