Finite Group Databasebeta

← #27144313

Subgroups — #27144313

Order 12416.

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×D388C211yesmaximal, characteristic
2C2×C22⋊C4×D97C212yesmaximal, complemented
3#20260613C218yesmaximal, complemented
4#20260598C211yesmaximal, characteristic, complemented
5#20260590C212yesmaximal, complemented
6#381×C97C211yesmaximal, characteristic, complemented
7C24×D97C2211yescharacteristic
8C24×D97C2211yescharacteristic
9C22×D388—28no—
10C22×D388C418yescomplemented
11C22×D388C2214yes—
12C22×C4×D97C2214yescomplemented
13C2×C22⋊Dic97C2212yes—
14C2×#13095488C2214yescomplemented
15C2×#13095488C2218yescomplemented
16C22⋊C4×D97—28no—
17#15690717C2214yescomplemented
18C23×C4×C97C2211yescharacteristic
19C2×C22⋊C4×C97C2212yes—
20#95×C97C2214yescomplemented
21C23×D97—24no—
22C23×D97—44no—
23C23×D97—44no—
24C23×D97C2×C414yes—
25C23×D97C2×C414yescomplemented
26C23×D97D418yes—
27C23×D97C2312yes—
28C23×D97C2312yes—
29C2×D388—232no—
30C2×D388—28no—
31C2×D388—28no—
32C2×D388C2×C4116yescomplemented
33C22×Dic97C2314yes—
34C2×C4×D97—28no—
35C2×C4×D97—48no—
36C22⋊Dic97—24no—
37#13095488—28no—
38#13095488—48no—
39C24×C97C2311yescharacteristic
40C22×C4×C97—24no—
41C22×C4×C97C2×C414yescomplemented
42C22×C4×C97C2314yes—
43C22×C4×C97C2312yes—
44C22⋊C4×C97—24no—
45C23×C97—24no—
46C23×C97C22×C412yes—
47C23×C97C2×D414yes—
48C23×C97C2411yescharacteristic
49C22×D97—28no—
50C22×D97—28no—
51C22×D97—24no—
52C22×D97—28no—
53C22×D97—48no—
54C22×D97—48no—
55C22×D97—416no—
56C22×D97—416no—
57C22×D97C22⋊C4116yescomplemented
58C22×D97C22×C414yes—
59C22×D97C22×C414yes—
60C22×D97C2×D414yes—
61C22×D97C2×D414yes—
62C2×C4×C97—28no—
63C2×C4×C97—24no—
64C2×C4×C97—24no—
65C2×C4×C97—28no—
66C2×C4×C97C22×C414yes—
67C2×Dic97—24no—
68C2×Dic97—28no—
69D388—232no—
70C4×D97—416no—
71C22×C97—24no—
72C22×C97—28no—
73C22×C97C2×C22⋊C414yes—
74C22×C97C23×C411yescharacteristic
75C22×C97C22×D412yes—
76C22×C97H(5,𝔽2)14yes—
77C2×D97—28no—
78C2×D97—28no—
79C2×D97—28no—
80C2×D97—24no—
81C2×D97—28no—
82C2×D97—416no—
83C2×D97—416no—
84C2×D97—48no—
85C2×D97C2×C22⋊C418yes—
86C4×C97—24no—
87C4×C97—44no—
88Dic97—44no—
89C2×C97—24no—
90C2×C97C22×C22⋊C411yescharacteristic
91C2×C97#51812yes—
92C2×C97#53414yes—
93D97—28no—
94D97—44no—
95#2308—971nomaximal, Sylow
96C97#230811yescharacteristic, complemented, Sylow
97C23×D4—971no—
98C22×C22⋊C4—972no—
99#386—978no—
100#381—971no—
101#381D9711yescharacteristic, complemented
102#379—972no—
103C25—971no—
104C25—971no—
105C22×D4—978no—
106C22×D4—974no—
107C22×D4—1948no—
108C23×C4—974no—
109C23×C4C2×D9711yescharacteristic
110C2×C22⋊C4—978no—
111C2×C22⋊C4—972no—
112C2×C22⋊C4—974no—
113C2×C22⋊C4—1948no—
114C2×C22⋊C4C2×D9712yes—
115#95—974no—
116#95C2×D9714yescomplemented
117C24—974no—
118C24—978no—
119C24—972no—
120C24—974no—
121C24—972no—
122C24—3884no—
123C24—3884no—
124C24—1944no—
125C24C22×D9711yescharacteristic
126C2×D4—9716no—
127C2×D4—19432no—
128C2×D4—1948no—
129C2×D4—1948no—
130C22×C4—24no—
131C22×C4—974no—
132C22×C4—3888no—
133C22×C4—1948no—
134C22×C4C4×D9714yescomplemented
135C22×C4C22×D9714yes—
136C22×C4C22×D9712yes—
137C22⋊C4—24no—
138C22⋊C4—3888no—
139C22⋊C4—1948no—
140C22⋊C4—1944no—
141C23—24no—
142C23—9716no—
143C23—974no—
144C23—974no—
145C23—974no—
146C23—974no—
147C23—38816no—
148C23—3888no—
149C23—3888no—
150C23—38816no—
151C23—1948no—
152C23—1944no—
153C23—1948no—
154C23—1948no—
155C23C2×C4×D9712yes—
156C23D4×D9714yes—
157C23C23×D9711yescharacteristic
158D4—19432no—
159C2×C4—28no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—28no—
163C2×C4—38816no—
164C2×C4—1944no—
165C2×C4—1948no—
166C2×C4C2×C4×D9714yes—
167C22—24no—
168C22—28no—
169C22—978no—
170C22—38816no—
171C22—38816no—
172C22—3888no—
173C22—1944no—
174C22—1948no—
175C22—1948no—
176C22—1948no—
177C22—1948no—
178C22C22⋊C4×D9714yes—
179C22C22×C4×D9711yescharacteristic
180C22C2×D4×D9712yes—
181C22#1569085614yes—
182C4—24no—
183C4—44no—
184C4—3884no—
185C2—24no—
186C2—3884no—
187C2—1948no—
188C2C2×C22⋊C4×D9711yescharacteristic
189C2#2026120912yes—
190C2#2026129614yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#381×C97C2
socle, cosocleC23×C97C24
prime radicalC23×D388C2
hypercenter#381D97
hypocenterC97#2308
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×D97C2semidirect
3. #20260613C2semidirect
4. #20260598C2semidirect
5. #20260590C2semidirect
6. #381×C97C2semidirect
10. C22×D388C4semidirect
12. C22×C4×D97C22semidirect
14. C2×#13095488C22semidirect
15. C2×#13095488C22semidirect
17. #15690717C22semidirect
20. #95×C97C22semidirect
25. C23×D97C2×C4semidirect
32. C2×D388C2×C4semidirect
41. C22×C4×C97C2×C4semidirect
57. C22×D97C22⋊C4semidirect
96. C97#2308semidirect
101. #381D97semidirect
116. #95C2×D97semidirect
134. C22×C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C23×D3882
C2×C22⋊C4×D972
#202606132
#202605982
#202605902
#381×C972
#230897