Finite Group Databasebeta

← #30636663

Subgroups — #30636663

Order 16256.

134 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 37 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#22549781C211yesmaximal, characteristic
2#22549602C211yesmaximal, characteristic
3#324×C127C211yesmaximal, characteristic
4C22×Dic254C411yescharacteristic
5C2×C4×Dic127C411yescharacteristic
6C22⋊(C127⋊C8)—21no—
7C127⋊#95—21no—
8C127⋊#95—21no—
9C2×C4⋊C4×C127C2211yescharacteristic
10C22⋊C8×C127—21no—
11C2×Dic254—21no—
12C2×Dic254—41no—
13C2×Dic254—41no—
14C2×Dic254C812yescomplemented
15C22×Dic127—21no—
16C22×Dic127—21no—
17C2×C127⋊C8—41no—
18C4×Dic127—41no—
19C22×C4×C127C2×C411yescharacteristic
20C22×C4×C127D411yescharacteristic
21C22×C4×C127D411yescharacteristic
22C2×C8×C127—41no—
23C4⋊C4×C127—41no—
24Dic254—42no—
25Dic254—42no—
26C2×Dic127—22no—
27C2×Dic127—22no—
28C2×Dic127—41no—
29C2×Dic127—41no—
30C2×Dic127—41no—
31C2×Dic127—41no—
32C127⋊C8—81no—
33C23×C127C22⋊C411yescharacteristic
34C2×C4×C127—21no—
35C2×C4×C127—21no—
36C2×C4×C127—21no—
37C2×C4×C127—21no—
38C2×C4×C127—41no—
39C2×C4×C127—41no—
40C2×C4×C127C2×C811yescharacteristic
41C2×C4×C127M1611yescharacteristic
42C8×C127—81no—
43C22×C127—21no—
44C22×C127—21no—
45C22×C127C22⋊C811yescharacteristic
46C22×C127#9911yescharacteristic
47C22×C127#10011yescharacteristic
48C4×C127—21no—
49C4×C127—21no—
50C4×C127—41no—
51C4×C127—41no—
52Dic127—42no—
53Dic127—42no—
54C2×C127—21no—
55C2×C127—21no—
56C2×C127#32311yescharacteristic
57C2×C127#35411yescharacteristic
58C2×C127#35611yescharacteristic
59#1319—1271nomaximal, Sylow
60C127#131911yescharacteristic, complemented, Sylow
61#391—1271no—
62#324—1271no—
63#324D12711yescharacteristic
64C22×Q8—1271no—
65C2×C4⋊C4C2×D12711yescharacteristic
66C2×C42—1271no—
67C22⋊C8—21no—
68C22⋊C8—2541no—
69#95—2541no—
70#95—2541no—
71C2×Q8—1272no—
72C2×Q8—5081no—
73C2×Q8—5081no—
74C2×Q8—2541no—
75C22×C4—2541no—
76C22×C4—2541no—
77C22×C4C4×D12711yescharacteristic
78C22×C4C22⋊D12711yescharacteristic
79C22×C4D50811yescharacteristic
80C2×C8—41no—
81C2×C8—5081no—
82C4⋊C4—41no—
83C42—5081no—
84C23(C2×D127)⋊C411yescharacteristic
85Q8—5082no—
86Q8—5082no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—5081no—
94C2×C4—5081no—
95C2×C4—5081no—
96C2×C4—5081no—
97C2×C4—2542no—
98C2×C4—2542no—
99C2×C4C8×D12711yescharacteristic
100C2×C4#1452323111yescharacteristic
101C8—81no—
102C8—10161no—
103C22—21no—
104C22—21no—
105C22(C2×D127)⋊C811yescharacteristic
106C22#1686112311yescharacteristic
107C22#1686112511yescharacteristic
108C4—21no—
109C4—21no—
110C4—41no—
111C4—41no—
112C4—5082no—
113C4—5082no—
114C2—21no—
115C2—21no—
116C2#2254959911yescharacteristic
117C2#2254967011yescharacteristic
118C2#2254967411yescharacteristic
subgroupG/H
centerC22#16861123
commutatorC2×C4×C127C2×C8
FrattiniC2×C4⋊C4C2×D127
Fitting#324×C127C2
socleC22×C127#99
prime radicalC23×C127C22⋊C4
cosocleC2×C4⋊C4×C127C22
hypercenter#324D127
hypocenterC127#1319
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #324

Complements

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

NH ≅ G/Ntype
14. C2×Dic254C8semidirect
60. C127#1319semidirect

Maximal subgroups

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

subgroupindex
#225497812
#225496022
#324×C1272
#1319127