Finite Group Databasebeta

← #30637354

Subgroups — #30637354

Order 16256.

248 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 77 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#22549933C212yesmaximal, complemented
2#22549616C214yesmaximal, complemented
3C2×#124×C127C211yesmaximal, characteristic
4C4⋊(C127⋊C8)—22no—
5C22⋊(C127⋊C8)—22no—
6#16861120C2214yescomplemented
7C22×Q8×C127C411yescharacteristic
8C22×D4×C127C411yescharacteristic
9#124×C127—22no—
10#124×C127C412yes—
11#124×C127C2212yes—
12C2×C22⋊C4×C127—22no—
13C2×C42×C127C2211yescharacteristic
14C2×C127⋊C8—24no—
15C24×C127—21no—
16C2×Q8×C127—21no—
17C2×Q8×C127—42no—
18C2×Q8×C127C2×C412yes—
19C2×D4×C127—21no—
20C2×D4×C127—42no—
21C2×D4×C127C2×C412yes—
22C22×C4×C127—21no—
23C22×C4×C127C2×C411yescharacteristic
24C22×C4×C127D412yes—
25C22⋊C4×C127—24no—
26C22⋊C4×C127—24no—
27C42×C127C2×C411yescharacteristic
28C42×C127D412yes—
29C42×C127C2311yescharacteristic
30C127⋊C8—44no—
31C23×C127—22no—
32C23×C127—44no—
33C23×C127—42no—
34C23×C127C22⋊C411yescharacteristic
35Q8×C127—42no—
36Q8×C127—42no—
37D4×C127—42no—
38D4×C127—42no—
39C2×C4×C127—22no—
40C2×C4×C127—22no—
41C2×C4×C127—42no—
42C2×C4×C127—42no—
43C2×C4×C127C22⋊C412yes—
44C2×C4×C127C22⋊C411yescharacteristic
45C2×C4×C127C22×C411yescharacteristic
46C2×C4×C127C2×D412yes—
47C22×C127—22no—
48C22×C127—42no—
49C22×C127—44no—
50C22×C127—84no—
51C22×C127#10012yes—
52C22×C127C2×C22⋊C411yescharacteristic
53C4×C127—21no—
54C4×C127—21no—
55C4×C127—22no—
56C4×C127—22no—
57C4×C127—42no—
58C2×C127—22no—
59C2×C127—82no—
60C2×C127C2×#10011yescharacteristic
61C2×C127#42112yes—
62#1533—1271nomaximal, Sylow
63C127#153311yescharacteristic, complemented, Sylow
64C2×#124D12711yescharacteristic
65#432—1272no—
66#329—1274no—
67C22×Q8Dic12711yescharacteristic
68C22×D4Dic12711yescharacteristic
69#124—22no—
70#124Dic12712yes—
71#124C2×D12712yes—
72C2×C22⋊C4—22no—
73C2×C42C2×D12711yescharacteristic
74C4⋊C8—2542no—
75C22⋊C8—2542no—
76#97—1274no—
77C24—21no—
78C2×Q8—21no—
79C2×Q8—42no—
80C2×Q8C2×Dic12712yes—
81C2×D4—21no—
82C2×D4—42no—
83C2×D4C2×Dic12712yes—
84C22×C4—21no—
85C22×C4C2×Dic12711yescharacteristic
86C22×C4C22⋊D12712yes—
87C2×C8—2544no—
88C22⋊C4—24no—
89C22⋊C4—24no—
90C42C2×Dic12711yescharacteristic
91C42C22⋊D12712yes—
92C42C22×D12711yescharacteristic
93C23—22no—
94C23—44no—
95C23—42no—
96C23C22⋊Dic12711yescharacteristic
97Q8—42no—
98Q8—42no—
99D4—42no—
100D4—42no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4C22⋊Dic12712yes—
106C2×C4C22⋊Dic12711yescharacteristic
107C2×C4C22×Dic12711yescharacteristic
108C2×C4C2×C22⋊D12712yes—
109C8—5084no—
110C22—22no—
111C22—42no—
112C22—44no—
113C22—84no—
114C22#1686112612yes—
115C22C2×C22⋊Dic12711yescharacteristic
116C4—21no—
117C4—21no—
118C4—22no—
119C4—22no—
120C4—42no—
121C2—22no—
122C2—82no—
123C2C2×#1686112611yescharacteristic
124C2#2254989312yes—
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingC2×#124×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D4×C127C4
cosocleC42×C127C23
hypercenterC2×#124D127
hypocenterC127#1533
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #22549933C2semidirect
2. #22549616C2semidirect
6. #16861120C22semidirect
63. C127#1533semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225496162
C2×#124×C1272
#1533127