Finite Group Databasebeta

← #30640847

Subgroups — #30640847

Order 16256.

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

normal subgroups only

This group has 148 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#22550292C211yesmaximal, characteristic, complemented
2#22550143C212yesmaximal, complemented
3#22549706C211yesmaximal, characteristic, complemented
4#22549689C212yesmaximal, complemented
5C2×D16×C127C211yesmaximal, characteristic, complemented
6#16861235—22no—
7#16861215C2211yescharacteristic
8#16861149C2211yescharacteristic
9#16861148—21no—
10C16×D127—21no—
11#16861145C2212yes—
12#16861126—22no—
13C2×D8×C127C2212yes—
14D16×C127—21no—
15D16×C127—21no—
16C2×C16×C127C2211yescharacteristic
17#14523248D411yescharacteristic
18#14523236—42no—
19#14523233—42no—
20#14523232—22no—
21#14523232D411yescharacteristic
22#14523231—21no—
23C8×D127—21no—
24C127⋊C16—21no—
25C2×D4×C127—22no—
26D8×C127—22no—
27D8×C127—22no—
28C2×C8×C127C2311yescharacteristic
29C16×C127D411yescharacteristic
30C16×C127D411yescharacteristic
31Dic254D811yescharacteristic
32D508D811yescharacteristic
33C22⋊D127—21no—
34C4×D127—21no—
35C127⋊C8—21no—
36C127⋊C8—42no—
37C23×C127—42no—
38D4×C127—42no—
39D4×C127—42no—
40C2×C4×C127C2×D411yescharacteristic
41C8×C127C2×D411yescharacteristic
42C8×C127#4111yescharacteristic
43C22×C127—82no—
44C22×C127—162no—
45C22×C127#13511yescharacteristic
46C2×D127—21no—
47C4×C127#12111yescharacteristic
48C4×C127C2×D811yescharacteristic
49Dic127—21no—
50C2×C127—21no—
51C2×C127—162no—
52C2×C127#46611yescharacteristic
53D127—41no—
54#2221—1271nomaximal, Sylow
55C127#222111yescharacteristic, complemented, Sylow
56C2×D16D12711yescharacteristic, complemented
57#504—1271no—
58#472—1272no—
59#368—1271no—
60#361—1272no—
61AGL(1,ℤ/8)—2542no—
62C2×D8C2×D12712yes—
63#131—1271no—
64D16—21no—
65D16—21no—
66M32—1271no—
67M32—2541no—
68C2×C16—2541no—
69C2×C16C2×D12711yescharacteristic
70#108—1272no—
71#100—2542no—
72#41—1271no—
73C2×D4—22no—
74SD16—5082no—
75D8—22no—
76D8—22no—
77D8—5082no—
78M16—1271no—
79M16—2542no—
80M16—2541no—
81C2×C8—2541no—
82C2×C8C22×D12711yescharacteristic
83C16—2541no—
84C16C22⋊D12711yescharacteristic
85C16C22⋊D12711yescharacteristic
86C23—42no—
87Q8—1271no—
88D4—42no—
89D4—42no—
90D4—1271no—
91D4—2541no—
92C2×C4—2541no—
93C2×C4D4×D12711yescharacteristic
94C8—5082no—
95C8—2541no—
96C8C2×C22⋊D12711yescharacteristic
97C8#1452324711yescharacteristic
98C22—82no—
99C22—162no—
100C22—2541no—
101C22#1686122811yescharacteristic
102C4—2541no—
103C4#1686118011yescharacteristic
104C4D8×D12711yescharacteristic
105C2—21no—
106C2—162no—
107C2—5081no—
108C2#2255010911yescharacteristic
subgroupG/H
centerC2#22550109
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
FittingC2×D16×C127C2
socleC2×C127#466
hypercenterC2×D16D127
hypocenterC127#2221
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #22550292C2semidirect
2. #22550143C2semidirect
3. #22549706C2semidirect
4. #22549689C2semidirect
5. C2×D16×C127C2semidirect
55. C127#2221semidirect
56. C2×D16D127semidirect

Maximal subgroups

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

subgroupindex
#225502922
#225501432
#225497062
#225496892
C2×D16×C1272
#2221127