Finite Group Databasebeta

← #27150196

Subgroups — #27150196

Order 12416.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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
1C2×#15690855C211yesmaximal, characteristic
2#20261210C211yesmaximal, characteristic, complemented
3#20260708C214yesmaximal, complemented
4C2×#15690724C218yesmaximal, complemented
5#518×C97C211yesmaximal, characteristic, complemented
6#15690855—24no—
7#15690855—24no—
8C2×#13095511C2212yes—
9C2×#13095510C414yescomplemented
10C22×C22⋊D97C2214yes—
11C2×D4×D97C414yescomplemented
12D4×Dic97—24no—
13#15690769C2212yescomplemented
14C2×C22⋊Dic97C2214yescomplemented
15#15690724C22116yescomplemented
16C22×D4×C97C2211yescharacteristic
17C4×D4×C97—24no—
18#117×C97C2212yescomplemented
19C2×C22⋊C4×C97C2214yescomplemented
20C23×D97C2×C414yescomplemented
21#13095511—24no—
22#13095511—24no—
23#13095510—216no—
24C2×Dic194C2×C412yescomplemented
25C2×D388C2×C412yescomplemented
26C2×C22⋊D97—28no—
27C2×C22⋊D97—28no—
28C2×C22⋊D97C2×C418yescomplemented
29C2×C22⋊D97C2314yes—
30D4×D97—216no—
31C22×Dic97—24no—
32C22×Dic97C2×C414yescomplemented
33C2×C4×D97C2×C414yescomplemented
34C4⋊Dic97—22no—
35C22⋊Dic97—24no—
36C22⋊Dic97C2314yes—
37C4×Dic97—22no—
38C24×C97C2312yes—
39C2×D4×C97C2×C414yes—
40C2×D4×C97D414yes—
41C2×D4×C97D414yes—
42C22×C4×C97—24no—
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—22no—
45C22⋊C4×C97—24no—
46C22⋊C4×C97C2314yes—
47C42×C97—22no—
48C23×C97—82no—
49C23×C97C22×C414yes—
50C23×C97C2×D414yes—
51C23×C97C2×D414yes—
52C23×C97C2411yescharacteristic
53C22×D97—28no—
54C22×D97—84no—
55C22×D97C22×C414yes—
56D4×C97C22⋊C4116yes—
57C2×C4×C97—24no—
58C2×C4×C97—84no—
59C2×C4×C97C22×C412yes—
60C2×C4×C97C2×D412yes—
61C2×C4×C97C2×D412yes—
62C22⋊D97—216no—
63C22⋊D97—24no—
64C22⋊D97—24no—
65C2×Dic97—28no—
66C2×Dic97—24no—
67C2×Dic97—84no—
68C2×Dic97C22×C414yes—
69D388—24no—
70C4×D97—28no—
71Dic194—24no—
72C22×C97—44no—
73C22×C97—44no—
74C22×C97—44no—
75C22×C97—41no—
76C22×C97C2×C22⋊C418yes—
77C22×C97C23×C411yescharacteristic
78C22×C97C22×D411yescharacteristic
79C22×C97C22×D411yescharacteristic
80C2×D97—24no—
81C2×D97—44no—
82C2×D97—88no—
83C4×C97—44no—
84C4×C97C2×C22⋊C414yes—
85Dic97—24no—
86Dic97—44no—
87C2×C97—21no—
88C2×C97—21no—
89C2×C97—21no—
90C2×C97—28no—
91C2×C97C22×C22⋊C411yescharacteristic
92D97—44no—
93#2883—971nomaximal, Sylow
94C97#288311yescharacteristic, complemented, Sylow
95C2×H(5,𝔽2)—971no—
96#518—971no—
97#518D9711yescharacteristic, complemented
98#410—974no—
99C2×#99—978no—
100H(5,𝔽2)—1944no—
101H(5,𝔽2)—1944no—
102C2×#41—974no—
103C2×#41—972no—
104C22×D4—974no—
105C22×D4—974no—
106C22×D4C2×D9711yescharacteristic
107C4×D4—24no—
108C4×D4—1944no—
109#117—972no—
110#117C2×D9712yescomplemented
111C2×C22⋊C4—974no—
112C2×C22⋊C4C2×D9714yescomplemented
113#99—9716no—
114C24—974no—
115C24C22×D9712yes—
116#41—19416no—
117#41—1944no—
118#41—1944no—
119C2×Q8—972no—
120C2×D4—978no—
121C2×D4—974no—
122C2×D4—972no—
123C2×D4—1948no—
124C2×D4—1948no—
125C2×D4—19416no—
126C2×D4C4×D9714yes—
127C2×D4D38814yes—
128C2×D4C22⋊D9714yes—
129C22×C4—24no—
130C22×C4—974no—
131C22×C4—974no—
132C22×C4—1944no—
133C22×C4C22×D9711yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—1942no—
136C22⋊C4—24no—
137C22⋊C4—974no—
138C22⋊C4—1944no—
139C22⋊C4C22×D9714yes—
140C42—22no—
141C42—1942no—
142C23—82no—
143C23—7764no—
144C23—974no—
145C23—1948no—
146C23C2×C4×D9714yes—
147C23C2×C22⋊D9714yes—
148C23C2×D38814yes—
149C23C23×D9711yescharacteristic
150Q8—1944no—
151D4—19416no—
152D4—1944no—
153D4—1944no—
154D4—1944no—
155D4#13095488116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—7764no—
159C2×C4—974no—
160C2×C4—1944no—
161C2×C4—1948no—
162C2×C4—1948no—
163C2×C4C2×C4×D9712yes—
164C2×C4C2×C22⋊D9712yes—
165C2×C4C2×D38812yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—7768no—
171C22—3884no—
172C22—1944no—
173C22C2×#1309548818yes—
174C22C22×C4×D9711yescharacteristic
175C22C22×C22⋊D9711yescharacteristic
176C22C22×D38811yescharacteristic
177C4—44no—
178C4—3884no—
179C4—1944no—
180C4C2×#1309548814yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—3884no—
186C2C22×#1309548811yescharacteristic
subgroupG/H
centerC2C22×#13095488
commutatorC22×C97C23×C4
FrattiniC23C23×D97
Fitting#518×C97C2
socleC2×C97C22×C22⋊C4
prime radicalC2×#15690855C2
cosocleC23×C97C24
hypercenter#518D97
hypocenterC97#2883
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #20261210C2semidirect
3. #20260708C2semidirect
4. C2×#15690724C2semidirect
5. #518×C97C2semidirect
9. C2×#13095510C4semidirect
11. C2×D4×D97C4semidirect
13. #15690769C22semidirect
14. C2×C22⋊Dic97C22semidirect
15. #15690724C22semidirect
18. #117×C97C22semidirect
19. C2×C22⋊C4×C97C22semidirect
20. C23×D97C2×C4semidirect
24. C2×Dic194C2×C4semidirect
25. C2×D388C2×C4semidirect
28. C2×C22⋊D97C2×C4semidirect
32. C22×Dic97C2×C4semidirect
33. C2×C4×D97C2×C4semidirect
94. C97#2883semidirect
97. #518D97semidirect
110. #117C2×D97semidirect
112. C2×C22⋊C4C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908552
#202612102
#202607082
C2×#156907242
#518×C972
#288397