Finite Group Databasebeta

← #30637789

Subgroups — #30637789

Order 16256.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550252C211yesmaximal, characteristic
2#22550076C212yesmaximal, complemented
3#22549620C211yesmaximal, characteristic, complemented
4Q8⋊(C127⋊C8)C212yesmaximal, complemented
5#559×C127C211yesmaximal, characteristic, complemented
6C2×#14523235—22no—
7C2×#14523233—22no—
8#16861206C2211yescharacteristic
9C4⋊(C127⋊C8)C2212yes—
10#16861129—22no—
11C4×C127⋊C8C2211yescharacteristic
12#127×C127C2211yescharacteristic
13#124×C127—22no—
14#123×C127—22no—
15C4×Q8×C127C2212yes—
16C4×D4×C127—22no—
17C2×D508—21no—
18C2×D508—22no—
19#14523235—44no—
20#14523233—44no—
21C2×C127⋊C8—21no—
22C2×C127⋊C8—22no—
23C2×Q8×C127D412yes—
24C2×D4×C127—22no—
25C2×D4×C127D412yes—
26C22×C4×C127—22no—
27C4⋊C4×C127—22no—
28C4⋊C4×C127D412yes—
29C22⋊C4×C127—24no—
30C22⋊C4×C127—22no—
31C42×C127—22no—
32C42×C127C2311yescharacteristic
33C22×D127—41no—
34D508—42no—
35D508—42no—
36D508—42no—
37C127⋊C8—22no—
38C127⋊C8—42no—
39C23×C127—22no—
40Q8×C127—22no—
41Q8×C127—22no—
42D4×C127—42no—
43D4×C127—42no—
44D4×C127—42no—
45C2×C4×C127—22no—
46C2×C4×C127—22no—
47C2×C4×C127—82no—
48C2×C4×C127C2×D411yescharacteristic
49C2×C4×C127C2×D412yes—
50C22×C127—44no—
51C22×C127—42no—
52C22×C127#12011yescharacteristic
53C2×D127—82no—
54C2×D127—81no—
55C4×C127—21no—
56C4×C127—42no—
57C4×C127—42no—
58C4×C127#13512yes—
59C4×C127AGL(1,ℤ/8)12yes—
60C2×C127—82no—
61C2×C127#44912yes—
62C2×C127#45311yescharacteristic
63D127—161no—
64#1620—1271nomaximal, Sylow
65C127#162011yescharacteristic, complemented, Sylow
66#559D12711yescharacteristic, complemented
67#493—1271no—
68#460—1272no—
69#331—1271no—
70Q8⋊C8—1272no—
71C2×SD16—2542no—
72C2×D8—2542no—
73#127—1271no—
74#127C2×D12711yescharacteristic
75#124—22no—
76#123—22no—
77C4×Q8C2×D12712yes—
78C4×D4—22no—
79C4⋊C8—1272no—
80#102—2542no—
81C4×C8—1271no—
82C2×Q8C22⋊D12712yes—
83C2×D4—22no—
84C2×D4—2542no—
85C2×D4—2541no—
86C2×D4C22⋊D12712yes—
87C22×C4—22no—
88SD16—5084no—
89D8—5084no—
90C2×C8—2542no—
91C2×C8—2541no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D12712yes—
94C22⋊C4—24no—
95C22⋊C4—22no—
96C42—22no—
97C42C22×D12711yescharacteristic
98C23—22no—
99C23—5081no—
100Q8—22no—
101Q8—22no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—5082no—
106D4—5082no—
107D4—5082no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—82no—
111C2×C4C2×C22⋊D12711yescharacteristic
112C2×C4C2×C22⋊D12712yes—
113C8—5082no—
114C8—2542no—
115C22—44no—
116C22—42no—
117C22—10162no—
118C22—10161no—
119C22#1686117811yescharacteristic
120C4—21no—
121C4—42no—
122C4—42no—
123C4#1686122912yes—
124C4#1686123412yes—
125C2—82no—
126C2—20321no—
127C2#2255002112yes—
128C2#2255004411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#559×C127C2
socleC22×C127#120
prime radical#22550252C2
hypercenter#559D127
hypocenterC127#1620
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #22550076C2semidirect
3. #22549620C2semidirect
4. Q8⋊(C127⋊C8)C2semidirect
5. #559×C127C2semidirect
65. C127#1620semidirect
66. #559D127semidirect

Maximal subgroups

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

subgroupindex
#225502522
#225500762
#225496202
Q8⋊(C127⋊C8)2
#559×C1272
#1620127