Finite Group Databasebeta

← #30641307

Subgroups — #30641307

Order 16256.

1160 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C23×D508C211yesmaximal, characteristic
2C2×C22⋊C4×D127C212yesmaximal, complemented
3#22549754C218yesmaximal, complemented
4#22549739C211yesmaximal, characteristic, complemented
5#22549731C212yesmaximal, complemented
6#381×C127C211yesmaximal, characteristic, complemented
7C24×D127C2211yescharacteristic
8C24×D127C2211yescharacteristic
9C22×D508—28no—
10C22×D508C418yescomplemented
11C22×D508C2214yes—
12C22×C4×D127C2214yescomplemented
13C2×C22⋊Dic127C2212yes—
14C2×(C2×D127)⋊C4C2214yescomplemented
15C2×(C2×D127)⋊C4C2218yescomplemented
16C22⋊C4×D127—28no—
17C127⋊#95C2214yescomplemented
18C23×C4×C127C2211yescharacteristic
19C2×C22⋊C4×C127C2212yes—
20#95×C127C2214yescomplemented
21C23×D127—24no—
22C23×D127—44no—
23C23×D127—44no—
24C23×D127C2×C414yes—
25C23×D127C2×C414yescomplemented
26C23×D127D418yes—
27C23×D127C2312yes—
28C23×D127C2312yes—
29C2×D508—232no—
30C2×D508—28no—
31C2×D508—28no—
32C2×D508C2×C4116yescomplemented
33C22×Dic127C2314yes—
34C2×C4×D127—28no—
35C2×C4×D127—48no—
36C22⋊Dic127—24no—
37(C2×D127)⋊C4—28no—
38(C2×D127)⋊C4—48no—
39C24×C127C2311yescharacteristic
40C22×C4×C127—24no—
41C22×C4×C127C2×C414yescomplemented
42C22×C4×C127C2314yes—
43C22×C4×C127C2312yes—
44C22⋊C4×C127—24no—
45C22×D127—28no—
46C22×D127—28no—
47C22×D127—24no—
48C22×D127—28no—
49C22×D127—48no—
50C22×D127—48no—
51C22×D127—416no—
52C22×D127—416no—
53C22×D127C22⋊C4116yescomplemented
54C22×D127C22×C414yes—
55C22×D127C22×C414yes—
56C22×D127C2×D414yes—
57C22×D127C2×D414yes—
58D508—232no—
59C2×Dic127—24no—
60C2×Dic127—28no—
61C4×D127—416no—
62C23×C127—24no—
63C23×C127C22×C412yes—
64C23×C127C2×D414yes—
65C23×C127C2411yescharacteristic
66C2×C4×C127—28no—
67C2×C4×C127—24no—
68C2×C4×C127—24no—
69C2×C4×C127—28no—
70C2×C4×C127C22×C414yes—
71C22×C127—24no—
72C22×C127—28no—
73C22×C127C2×C22⋊C414yes—
74C22×C127C23×C411yescharacteristic
75C22×C127C22×D412yes—
76C22×C127H(5,𝔽2)14yes—
77C2×D127—28no—
78C2×D127—28no—
79C2×D127—28no—
80C2×D127—24no—
81C2×D127—28no—
82C2×D127—416no—
83C2×D127—416no—
84C2×D127—48no—
85C2×D127C2×C22⋊C418yes—
86C4×C127—24no—
87C4×C127—44no—
88Dic127—44no—
89C2×C127—24no—
90C2×C127C22×C22⋊C411yescharacteristic
91C2×C127#51812yes—
92C2×C127#53414yes—
93D127—28no—
94D127—44no—
95#2308—1271nomaximal, Sylow
96C127#230811yescharacteristic, complemented, Sylow
97C23×D4—1271no—
98C22×C22⋊C4—1272no—
99#386—1278no—
100#381—1271no—
101#381D12711yescharacteristic, complemented
102#379—1272no—
103C25—1271no—
104C25—1271no—
105C22×D4—1278no—
106C22×D4—1274no—
107C22×D4—2548no—
108C23×C4—1274no—
109C23×C4C2×D12711yescharacteristic
110C2×C22⋊C4—1278no—
111C2×C22⋊C4—1272no—
112C2×C22⋊C4—1274no—
113C2×C22⋊C4—2548no—
114C2×C22⋊C4C2×D12712yes—
115#95—1274no—
116#95C2×D12714yescomplemented
117C24—1274no—
118C24—1278no—
119C24—1272no—
120C24—1274no—
121C24—1272no—
122C24—5084no—
123C24—5084no—
124C24—2544no—
125C24C22×D12711yescharacteristic
126C2×D4—12716no—
127C2×D4—2548no—
128C2×D4—2548no—
129C2×D4—25432no—
130C22×C4—24no—
131C22×C4—1274no—
132C22×C4—5088no—
133C22×C4—2548no—
134C22×C4C4×D12714yescomplemented
135C22×C4C22×D12714yes—
136C22×C4C22×D12712yes—
137C22⋊C4—24no—
138C22⋊C4—5088no—
139C22⋊C4—2548no—
140C22⋊C4—2544no—
141C23—24no—
142C23—12716no—
143C23—1274no—
144C23—1274no—
145C23—1274no—
146C23—1274no—
147C23—50816no—
148C23—5088no—
149C23—5088no—
150C23—50816no—
151C23—2544no—
152C23—2548no—
153C23—2548no—
154C23—2548no—
155C23C2×C4×D12712yes—
156C23D4×D12714yes—
157C23C23×D12711yescharacteristic
158D4—25432no—
159C2×C4—24no—
160C2×C4—28no—
161C2×C4—24no—
162C2×C4—28no—
163C2×C4—50816no—
164C2×C4—2544no—
165C2×C4—2548no—
166C2×C4C2×C4×D12714yes—
167C22—24no—
168C22—28no—
169C22—1278no—
170C22—50816no—
171C22—50816no—
172C22—5088no—
173C22—2544no—
174C22—2548no—
175C22—2548no—
176C22—2548no—
177C22—2548no—
178C22C22⋊C4×D12714yes—
179C22C22×C4×D12711yescharacteristic
180C22C2×D4×D12712yes—
181C22#1686125514yes—
182C4—24no—
183C4—44no—
184C4—5084no—
185C2—24no—
186C2—5084no—
187C2—2548no—
188C2C2×C22⋊C4×D12711yescharacteristic
189C2#2255035012yes—
190C2#2255043714yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#381×C127C2
socle, cosocleC23×C127C24
prime radicalC23×D508C2
hypercenter#381D127
hypocenterC127#2308
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×D127C2semidirect
3. #22549754C2semidirect
4. #22549739C2semidirect
5. #22549731C2semidirect
6. #381×C127C2semidirect
10. C22×D508C4semidirect
12. C22×C4×D127C22semidirect
14. C2×(C2×D127)⋊C4C22semidirect
15. C2×(C2×D127)⋊C4C22semidirect
17. C127⋊#95C22semidirect
20. #95×C127C22semidirect
25. C23×D127C2×C4semidirect
32. C2×D508C2×C4semidirect
41. C22×C4×C127C2×C4semidirect
53. C22×D127C22⋊C4semidirect
96. C127#2308semidirect
101. #381D127semidirect
116. #95C2×D127semidirect
134. C22×C4C4×D127semidirect

Maximal subgroups

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

subgroupindex
C23×D5082
C2×C22⋊C4×D1272
#225497542
#225497392
#225497312
#381×C1272
#2308127