Finite Group Databasebeta

← #30639906

Subgroups — #30639906

Order 16256.

278 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 67 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×#16861123C213yesmaximal
2#22549642C213yesmaximal, complemented
3#543×C127C211yesmaximal, characteristic, complemented
4C22×C22⋊D127C2211yescharacteristic
5C2×C22⋊Dic127C2213yes—
6#16861123—26no—
7C22⋊Q8×C127—26no—
8C2×C22⋊C4×C127C2213yes—
9C23×D127Q811yescharacteristic, complemented
10C2×C22⋊D127—23no—
11C2×C22⋊D127—23no—
12C22×Dic127—23no—
13C22×Dic127Q811yescharacteristic, complemented
14C22⋊Dic127—23no—
15C22⋊Dic127—23no—
16C24×C127C2311yescharacteristic
17C2×Q8×C127—22no—
18C22×C4×C127—23no—
19C4⋊C4×C127—26no—
20C22⋊C4×C127—23no—
21C22⋊C4×C127D416yes—
22C22×D127—23no—
23C22×D127—81no—
24C22⋊D127—41no—
25C22⋊D127—43no—
26C2×Dic127—23no—
27C2×Dic127—43no—
28C2×Dic127—83no—
29C23×C127—81no—
30C23×C127C2×D413yes—
31C23×C127C2×Q811yescharacteristic
32C23×C127#4113yes—
33Q8×C127—82no—
34C2×C4×C127—26no—
35C2×C4×C127—83no—
36C22×C127—43no—
37C22×C127—41no—
38C22×C127—43no—
39C22×C127#12011yescharacteristic
40C22×C127C22⋊Q813yes—
41C22×C127#12413yes—
42C2×D127—41no—
43C2×D127—83no—
44C4×C127—46no—
45Dic127—41no—
46Dic127—83no—
47C2×C127—23no—
48C2×C127—21no—
49C2×C127—23no—
50C2×C127#39311yescharacteristic
51D127—81no—
52#2032—1271nomaximal, Sylow
53C127#203211yescharacteristic, complemented, Sylow
54#543D12711yescharacteristic, complemented
55C2×#99—1273no—
56#342—1273no—
57C22×D4—1271no—
58C22⋊Q8—26no—
59C2×C22⋊C4—1273no—
60C2×C22⋊C4C2×D12713yes—
61#99—2546no—
62C24—1271no—
63C24C22×D12711yescharacteristic
64C2×Q8—22no—
65C2×D4—2543no—
66C2×D4—2543no—
67C22×C4—23no—
68C22×C4—1271no—
69C22×C4—2543no—
70C4⋊C4—26no—
71C22⋊C4—23no—
72C22⋊C4—2543no—
73C22⋊C4—2543no—
74C22⋊C4C22⋊D12716yes—
75C23—81no—
76C23—10161no—
77C23—2543no—
78C23C2×C22⋊D12713yes—
79C23Q8×D12711yescharacteristic
80C23#1452324913yes—
81Q8—82no—
82D4—5083no—
83D4—5081no—
84C2×C4—26no—
85C2×C4—83no—
86C2×C4—10163no—
87C2×C4—5083no—
88C2×C4—2543no—
89C22—43no—
90C22—43no—
91C22—41no—
92C22—10163no—
93C22—5081no—
94C22#1686117811yescharacteristic
95C22#1686118513yes—
96C22#1686119813yes—
97C4—46no—
98C4—10163no—
99C4—5081no—
100C2—21no—
101C2—23no—
102C2—23no—
103C2—10161no—
104C2#2254978711yescharacteristic
subgroupG/H
centerC2#22549787
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#543×C127C2
socleC2×C127#393
prime radicalC22×C22⋊D127C22
hypercenter#543D127
hypocenterC127#2032
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C24 ◁ #543

Complements

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

NH ≅ G/Ntype
2. #22549642C2semidirect
3. #543×C127C2semidirect
9. C23×D127Q8semidirect
13. C22×Dic127Q8semidirect
53. C127#2032semidirect
54. #543D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611232
#225496422
#543×C1272
#2032127