Finite Group Databasebeta

← #30637239

Subgroups — #30637239

Order 16256.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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#22550426C211yesmaximal, characteristic
2#22549722C211yesmaximal, characteristic, complemented
3#22549611C214yesmaximal, complemented
4#432×C127C211yesmaximal, characteristic, complemented
5C2×#14523248—21no—
6#16861209C411yescharacteristic, complemented
7#16861206C411yescharacteristic, complemented
8#16861197—21no—
9#16861183—21no—
10#16861183C2212yes—
11C4×D508—21no—
12C4×D508C412yescomplemented
13C2×C4×Dic127—21no—
14C127⋊#95—21no—
15C127⋊#95C2212yes—
16C2×C42×C127C2211yescharacteristic
17C4⋊C8×C127—21no—
18C22⋊C8×C127C2212yes—
19#97×C127—21no—
20#14523248—44no—
21C2×Dic254—21no—
22C2×D508—21no—
23C2×D508—21no—
24C2×D508C2×C412yes—
25C2×C22⋊D127—22no—
26C22×Dic127—22no—
27C2×C4×D127—22no—
28C4⋊Dic127—21no—
29C4⋊Dic127C2×C412yes—
30(C2×D127)⋊C4—22no—
31(C2×D127)⋊C4—22no—
32C4×Dic127—24no—
33C22×C4×C127D411yescharacteristic
34C22×C4×C127D411yescharacteristic
35C22×C4×C127C2311yescharacteristic
36C2×C8×C127—22no—
37C42×C127C2×C411yescharacteristic
38C42×C127C2×C411yescharacteristic
39C42×C127D411yescharacteristic
40C42×C127D411yescharacteristic
41C22×D127—22no—
42Dic254—42no—
43D508—42no—
44D508—42no—
45D508—42no—
46C22⋊D127—44no—
47C2×Dic127—24no—
48C2×Dic127—22no—
49C2×Dic127—42no—
50C2×Dic127—42no—
51C4×D127—44no—
52C23×C127C2×D411yescharacteristic
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—41no—
56C2×C4×C127C22⋊C411yescharacteristic
57C2×C4×C127C22⋊C411yescharacteristic
58C2×C4×C127C22⋊C411yescharacteristic
59C2×C4×C127C22⋊C411yescharacteristic
60C2×C4×C127C22×C411yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C8×C127—42no—
63C22×C127—21no—
64C22×C127—21no—
65C22×C127C2×C22⋊C411yescharacteristic
66C22×C127AGL(1,ℤ/8)12yes—
67C2×D127—42no—
68C2×D127—42no—
69C2×D127—42no—
70C4×C127—21no—
71C4×C127—21no—
72C4×C127—21no—
73C4×C127—21no—
74C4×C127—21no—
75C4×C127#10412yes—
76Dic127—44no—
77Dic127—42no—
78C2×C127—22no—
79C2×C127#41011yescharacteristic
80C2×C127#41811yescharacteristic
81C2×C127C2×#10411yescharacteristic
82D127—82no—
83#1505—1271nomaximal, Sylow
84C127#150511yescharacteristic, complemented, Sylow
85#532—1271no—
86#432D12711yescharacteristic, complemented
87#376—1271no—
88#327—1274no—
89C2×#41—2541no—
90C4⋊Q8—1271no—
91#127—1271no—
92#124—2541no—
93#121—1272no—
94#121—2541no—
95C4×D4—1272no—
96C4×D4—2541no—
97C2×C42—2541no—
98C2×C42C2×D12711yescharacteristic
99C4⋊C8—21no—
100C22⋊C8C2×D12712yes—
101#97—21no—
102#95—1272no—
103#95—2541no—
104#41—5084no—
105C2×Q8—2541no—
106C2×D4—1272no—
107C2×D4—2542no—
108C2×D4—2541no—
109C2×D4—2541no—
110C22×C4—2542no—
111C22×C4—2542no—
112C22×C4D50811yescharacteristic
113C22×C4D50811yescharacteristic
114C22×C4C22×D12711yescharacteristic
115C2×C8—22no—
116C4⋊C4—1272no—
117C4⋊C4—2541no—
118C22⋊C4—2542no—
119C22⋊C4—2542no—
120C42—2544no—
121C42C4×D12711yescharacteristic
122C42C4×D12711yescharacteristic
123C42C22⋊D12711yescharacteristic
124C42C22⋊D12711yescharacteristic
125C23—2542no—
126C23C2×D50811yescharacteristic
127Q8—5082no—
128D4—5082no—
129D4—5082no—
130D4—5084no—
131D4—5082no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—5084no—
136C2×C4—5082no—
137C2×C4—5082no—
138C2×C4—2544no—
139C2×C4—2542no—
140C2×C4(C2×D127)⋊C411yescharacteristic
141C2×C4(C2×D127)⋊C411yescharacteristic
142C2×C4(C2×D127)⋊C411yescharacteristic
143C2×C4(C2×D127)⋊C411yescharacteristic
144C2×C4C2×C4×D12711yescharacteristic
145C2×C4C2×C22⋊D12711yescharacteristic
146C8—42no—
147C22—21no—
148C22—21no—
149C22—5082no—
150C22—5082no—
151C22—5082no—
152C22C2×(C2×D127)⋊C411yescharacteristic
153C22#1686123712yes—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—5084no—
160C4—5082no—
161C4#1686113512yes—
162C2—22no—
163C2—10162no—
164C2#2254984911yescharacteristic
165C2#2254988211yescharacteristic
166C2C2×#1686113511yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#432×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550426C2
cosocleC22×C4×C127C23
hypercenter#432D127
hypocenterC127#1505
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #22549722C2semidirect
3. #22549611C2semidirect
4. #432×C127C2semidirect
6. #16861209C4semidirect
7. #16861206C4semidirect
12. C4×D508C4semidirect
84. C127#1505semidirect
86. #432D127semidirect

Maximal subgroups

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

subgroupindex
#225504262
#225497222
#225496112
#432×C1272
#1505127