Finite Group Databasebeta

← #30636661

Subgroups — #30636661

Order 16256.

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

normal subgroups only

This group has 170 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#22549777C211yesmaximal, characteristic
2#22549602C211yesmaximal, characteristic, complemented
3#324×C127C211yesmaximal, characteristic, complemented
4C22×D508C411yescharacteristic
5C2×(C2×D127)⋊C4—21no—
6C2×(C2×D127)⋊C4—21no—
7C2×C4×Dic127C411yescharacteristic
8C22⋊(C127⋊C8)—21no—
9C2×C4⋊C4×C127C2211yescharacteristic
10C22⋊C8×C127—21no—
11C23×D127—21no—
12C2×D508—21no—
13C2×D508—41no—
14C2×D508—41no—
15C2×D508C812yescomplemented
16C22×Dic127—21no—
17C2×C127⋊C8—41no—
18(C2×D127)⋊C4—42no—
19(C2×D127)⋊C4—42no—
20C4×Dic127—41no—
21C22×C4×C127C2×C411yescharacteristic
22C22×C4×C127D411yescharacteristic
23C22×C4×C127D411yescharacteristic
24C2×C8×C127—41no—
25C4⋊C4×C127—41no—
26C22×D127—22no—
27C22×D127—41no—
28C22×D127—41no—
29C22×D127—41no—
30C22×D127—41no—
31C22×D127—41no—
32C22×D127—41no—
33D508—42no—
34D508—42no—
35C2×Dic127—22no—
36C2×Dic127—41no—
37C2×Dic127—41no—
38C127⋊C8—81no—
39C23×C127C22⋊C411yescharacteristic
40C2×C4×C127—21no—
41C2×C4×C127—21no—
42C2×C4×C127—21no—
43C2×C4×C127—21no—
44C2×C4×C127—41no—
45C2×C4×C127—41no—
46C2×C4×C127C2×C811yescharacteristic
47C2×C4×C127M1611yescharacteristic
48C8×C127—81no—
49C22×C127—21no—
50C22×C127—21no—
51C22×C127C22⋊C811yescharacteristic
52C22×C127#9911yescharacteristic
53C22×C127#10011yescharacteristic
54C2×D127—42no—
55C2×D127—42no—
56C2×D127—42no—
57C2×D127—81no—
58C2×D127—81no—
59C2×D127—81no—
60C2×D127—81no—
61C4×C127—21no—
62C4×C127—21no—
63C4×C127—41no—
64C4×C127—41no—
65Dic127—42no—
66C2×C127—21no—
67C2×C127—21no—
68C2×C127#32311yescharacteristic
69C2×C127#35311yescharacteristic
70C2×C127#35511yescharacteristic
71D127—82no—
72#1318—1271nomaximal, Sylow
73C127#131811yescharacteristic, complemented, Sylow
74#390—1271no—
75#324—1271no—
76#324D12711yescharacteristic, complemented
77C22×D4—1271no—
78C2×C4⋊C4C2×D12711yescharacteristic
79C2×C22⋊C4—2541no—
80C2×C22⋊C4—2541no—
81C2×C42—1271no—
82C22⋊C8—21no—
83C22⋊C8—2541no—
84C24—2541no—
85C2×D4—1272no—
86C2×D4—5081no—
87C2×D4—5081no—
88C2×D4—2541no—
89C22×C4—2541no—
90C22×C4C4×D12711yescharacteristic
91C22×C4C22⋊D12711yescharacteristic
92C22×C4D50811yescharacteristic
93C2×C8—41no—
94C2×C8—5081no—
95C4⋊C4—41no—
96C22⋊C4—5082no—
97C22⋊C4—5082no—
98C42—5081no—
99C23—5081no—
100C23—5081no—
101C23—5081no—
102C23—5081no—
103C23—5081no—
104C23—5081no—
105C23—2542no—
106C23(C2×D127)⋊C411yescharacteristic
107D4—5082no—
108D4—5082no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—5081no—
116C2×C4—5081no—
117C2×C4—2542no—
118C2×C4C8×D12711yescharacteristic
119C2×C4#1452323111yescharacteristic
120C8—81no—
121C8—10161no—
122C22—21no—
123C22—21no—
124C22—10161no—
125C22—10161no—
126C22—10161no—
127C22—10161no—
128C22—5082no—
129C22—5082no—
130C22—5082no—
131C22(C2×D127)⋊C811yescharacteristic
132C22#1686112311yescharacteristic
133C22#1686112511yescharacteristic
134C4—21no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—5082no—
139C2—21no—
140C2—21no—
141C2—10162no—
142C2#2254959911yescharacteristic
143C2#2254966811yescharacteristic
144C2#2254967211yescharacteristic
subgroupG/H
centerC22#16861123
commutatorC2×C4×C127C2×C8
FrattiniC2×C4⋊C4C2×D127
Fitting#324×C127C2
socleC22×C127#99
prime radicalC22×D508C4
cosocleC2×C4⋊C4×C127C22
hypercenter#324D127
hypocenterC127#1318
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
2. #22549602C2semidirect
3. #324×C127C2semidirect
15. C2×D508C8semidirect
73. C127#1318semidirect
76. #324D127semidirect

Maximal subgroups

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

subgroupindex
#225497772
#225496022
#324×C1272
#1318127