Finite Group Databasebeta

← #30640848

Subgroups — #30640848

Order 16256.

148 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C2×#16861150C211yesmaximal, characteristic, complemented
2#22550293C211yesmaximal, characteristic, complemented
3#22550146C211yesmaximal, characteristic, complemented
4#22549705C211yesmaximal, characteristic, complemented
5#22549689C211yesmaximal, characteristic, complemented
6#22549688C211yesmaximal, characteristic, complemented
7#472×C127C211yesmaximal, characteristic, complemented
8#16861234—21no—
9C2×D1016C2211yescharacteristic
10#16861150—21no—
11#16861150—21no—
12#16861149—21no—
13#16861149C2211yescharacteristic
14C2×C127⋊C16—21no—
15C2×C127⋊C16C2211yescharacteristic
16#16861145C2211yescharacteristic
17#16861125—21no—
18AGL(1,ℤ/8)×C127—21no—
19C2×D8×C127C2211yescharacteristic
20#131×C127C2211yescharacteristic
21#108×C127C2211yescharacteristic
22#100×C127—21no—
23C2×D508—21no—
24#14523235—41no—
25D1016—21no—
26D1016—21no—
27#14523233—41no—
28#14523232—21no—
29C127⋊C16—21no—
30C127⋊C16D411yescharacteristic
31C127⋊C16D411yescharacteristic
32#41×C127D411yescharacteristic
33C2×D4×C127—21no—
34SD16×C127—41no—
35D8×C127—21no—
36D8×C127—21no—
37D8×C127—41no—
38M16×C127—21no—
39M16×C127—21no—
40M16×C127D411yescharacteristic
41C2×C8×C127—21no—
42C2×C8×C127C2311yescharacteristic
43C22×D127—41no—
44D508—41no—
45D508—41no—
46C127⋊C8—41no—
47C23×C127—41no—
48Q8×C127D811yescharacteristic
49D4×C127—21no—
50D4×C127—41no—
51D4×C127—41no—
52D4×C127D811yescharacteristic
53C2×C4×C127—21no—
54C2×C4×C127C2×D411yescharacteristic
55C8×C127—21no—
56C8×C127—41no—
57C8×C127C2×D411yescharacteristic
58C8×C127#4111yescharacteristic
59C22×C127—21no—
60C22×C127—81no—
61C22×C127—161no—
62C22×C127#13511yescharacteristic
63C2×D127—81no—
64C2×D127—161no—
65C4×C127—21no—
66C4×C127#12111yescharacteristic
67C4×C127C2×D811yescharacteristic
68C2×C127—21no—
69C2×C127—41no—
70C2×C127—161no—
71C2×C127#46611yescharacteristic
72D127—161no—
73#2221—1271nomaximal, Sylow
74C127#222111yescharacteristic, complemented, Sylow
75C2×D16—1271no—
76#504—1271no—
77#472—1271no—
78#472D12711yescharacteristic, complemented
79#368—1271no—
80#361—1271no—
81#361—1271no—
82AGL(1,ℤ/8)—21no—
83AGL(1,ℤ/8)—2541no—
84C2×D8—1271no—
85C2×D8C2×D12711yescharacteristic
86#131C2×D12711yescharacteristic
87D16—2541no—
88D16—2541no—
89M32—1271no—
90M32—2541no—
91C2×C16—1271no—
92C2×C16—2541no—
93#108—1271no—
94#108C2×D12711yescharacteristic
95#100—21no—
96#100—2541no—
97#41C22⋊D12711yescharacteristic
98C2×D4—21no—
99C2×D4—2541no—
100SD16—41no—
101SD16—5081no—
102D8—21no—
103D8—21no—
104D8—41no—
105D8—5081no—
106D8—2541no—
107D8—2541no—
108M16—21no—
109M16—21no—
110M16—2541no—
111M16C22⋊D12711yescharacteristic
112C2×C8—21no—
113C2×C8C22×D12711yescharacteristic
114C16—1271no—
115C16—1271no—
116C16—2541no—
117C23—41no—
118C23—5081no—
119Q8#1452323311yescharacteristic
120D4—21no—
121D4—41no—
122D4—41no—
123D4—5081no—
124D4—5081no—
125D4#1452323311yescharacteristic
126C2×C4—21no—
127C2×C4C2×C22⋊D12711yescharacteristic
128C8—21no—
129C8—41no—
130C8—5081no—
131C8D4×D12711yescharacteristic
132C8#1452324711yescharacteristic
133C22—21no—
134C22—81no—
135C22—161no—
136C22—20321no—
137C22—10161no—
138C22#1686122911yescharacteristic
139C4—21no—
140C4#1686118111yescharacteristic
141C4C2×#1452323311yescharacteristic
142C2—21no—
143C2—41no—
144C2—161no—
145C2—20321no—
146C2#2255011011yescharacteristic
subgroupG/H
centerC2#22550110
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#472×C127C2
socleC2×C127#466
hypercenter#472D127
hypocenterC127#2221
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #472

Complements

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

NH ≅ G/Ntype
1. C2×#16861150C2semidirect
2. #22550293C2semidirect
3. #22550146C2semidirect
4. #22549705C2semidirect
5. #22549689C2semidirect
6. #22549688C2semidirect
7. #472×C127C2semidirect
74. C127#2221semidirect
78. #472D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611502
#225502932
#225501462
#225497052
#225496892
#225496882
#472×C1272
#2221127