Finite Group Databasebeta

← C22×#15690785 #27154959

Subgroups — C22×#15690785 #27154959

Order 12416.

2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 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×D388C212yesmaximal, complemented
2C23×D388C211yesmaximal, characteristic, complemented
3C23×C4×D97C211yesmaximal, characteristic, complemented
4C2×#15690785C2124yesmaximal, direct product factor
5C22×#13095488C212yesmaximal, complemented
6C22×C4⋊C4×C97C211yesmaximal, characteristic, complemented
7C24×D97C2212yes—
8C24×D97C2211yescharacteristic
9C22×D388—216no—
10C22×D388—28no—
11C22×D388C22124yescomplemented
12C22×D388C22112yescomplemented
13C23×Dic97C2211yescharacteristic
14C22×C4×D97—28no—
15C22×C4×D97C22112yescomplemented
16#15690785C22164yesdirect product factor
17C2×#13095488C22124yescomplemented
18C23×C4×C97C2212yes—
19C23×C4×C97C2211yescharacteristic
20C2×C4⋊C4×C97C22112yescomplemented
21C23×D97—24no—
22C23×D97—24no—
23C23×D97—28no—
24C23×D97—28no—
25C23×D97—28no—
26C23×D97D418yescomplemented
27C23×D97C23112yes—
28C23×D97C2316yes—
29C2×D388—296no—
30C2×D388—248no—
31C2×D388C23132yescomplemented
32C2×D388C23116yescomplemented
33C22×Dic97—24no—
34C22×Dic97C2316yes—
35C2×C4×D97—248no—
36C2×C4×D97C23116yescomplemented
37#13095488C23132yescomplemented
38C24×C97C2311yescharacteristic
39C22×C4×C97—28no—
40C22×C4×C97D418yescomplemented
41C22×C4×C97C23112yes—
42C22×C4×C97C2316yes—
43C4⋊C4×C97C23116yescomplemented
44C23×C97C2×D414yes—
45C23×C97C2×D414yes—
46C23×C97#4114yes—
47C23×C97C2413yes—
48C22×D97—232no—
49C22×D97—212no—
50C22×D97—212no—
51C22×D97—224no—
52C22×D97—224no—
53C22×D97—224no—
54C22×D97—432no—
55C22×D97C2×D4124yescomplemented
56C22×D97C2418yes—
57C22×D97C2414yes—
58C2×C4×C97—224no—
59C2×C4×C97C2×D4124yescomplemented
60C2×C4×C97C2418yes—
61C2×C4×C97C2414yes—
62C2×Dic97—212no—
63C2×Dic97C2414yes—
64D388—264no—
65D388—232no—
66C4×D97—232no—
67C22×C97#121116yescomplemented
68C22×C97C22×D416yes—
69C22×C97C22×D416yes—
70C22×C97C2×#4116yes—
71C22×C97C2511yescharacteristic
72C2×D97—248no—
73C2×D97—24no—
74C2×D97—24no—
75C2×D97—28no—
76C2×D97—28no—
77C2×D97—28no—
78C2×D97—448no—
79C2×D97C22×D418yescomplemented
80C4×C97—28no—
81C4×C97C22×D418yescomplemented
82Dic97—24no—
83C2×C97C2×#121112yescomplemented
84C2×C97C23×D411yescharacteristic
85C2×C97C23×D411yescharacteristic
86C2×C97C22×#4111yescharacteristic
87D97—28no—
88D97—48no—
89C22×#121—971nomaximal, Sylow
90C97C22×#12111yescharacteristic, complemented, Sylow
91C23×D4—972no—
92C23×D4—971no—
93C24×C4—971no—
94C2×#121—9724no—
95C22×C4⋊C4D9711yescharacteristic, complemented
96C22×C22⋊C4—972no—
97C25—971no—
98C25—972no—
99C22×D4—9724no—
100C22×D4—9712no—
101C22×D4—19416no—
102C22×D4—1948no—
103C23×C4—971no—
104C23×C4—9712no—
105C23×C4—1948no—
106C23×C4C2×D9712yes—
107C23×C4C2×D9711yescharacteristic
108#121—9764no—
109C2×C4⋊C4C2×D97112yescomplemented
110C2×C22⋊C4—9724no—
111C24—978no—
112C24—976no—
113C24—9712no—
114C24—1948no—
115C24—1948no—
116C24—1948no—
117C24—1944no—
118C24—1944no—
119C24C22×D9711yescharacteristic
120C2×D4—9732no—
121C2×D4—9716no—
122C2×D4—19496no—
123C2×D4—19448no—
124C22×C4—28no—
125C22×C4—976no—
126C22×C4—9716no—
127C22×C4—1944no—
128C22×C4—19448no—
129C22×C4D38818yescomplemented
130C22×C4C22×D97112yes—
131C22×C4C22×D9716yes—
132C4⋊C4C22×D97116yescomplemented
133C22⋊C4—9732no—
134C23—9724no—
135C23—974no—
136C23—978no—
137C23—38832no—
138C23—19424no—
139C23—19424no—
140C23—19424no—
141C23—19432no—
142C23—19412no—
143C23—19412no—
144C23D4×D9714yes—
145C23C2×D38814yes—
146C23#1309551214yes—
147C23C23×D9713yes—
148D4—19464no—
149D4—19432no—
150C2×C4—224no—
151C2×C4—974no—
152C2×C4—19412no—
153C2×C4—19432no—
154C2×C4C2×D388124yescomplemented
155C2×C4C23×D9718yes—
156C2×C4C23×D9714yes—
157C22—978no—
158C22—38848no—
159C22—1948no—
160C22—1948no—
161C22—1948no—
162C22—19448no—
163C22—1944no—
164C22—1944no—
165C22#15690785116yesdirect product factor
166C22C2×D4×D9716yes—
167C22C22×D38816yes—
168C22C2×#1309551216yes—
169C22C24×D9711yescharacteristic
170C4—28no—
171C4—1944no—
172C4C22×D38818yescomplemented
173C2—3888no—
174C2—1948no—
175C2C2×#15690785112yesdirect product factor
176C2C22×D4×D9711yescharacteristic
177C2C23×D38811yescharacteristic
178C2C22×#1309551211yescharacteristic
subgroupG/H
centerC24C22×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC22×C4⋊C4×C97C2
socleC24×C97C23
hypercenterC22×C4⋊C4D97
hypocenterC97C22×#121
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×D388C2semidirect
2. C23×D388C2semidirect
3. C23×C4×D97C2semidirect
4. C2×#15690785C2direct (N × H)
5. C22×#13095488C2semidirect
6. C22×C4⋊C4×C97C2semidirect
11. C22×D388C22semidirect
12. C22×D388C22semidirect
15. C22×C4×D97C22semidirect
16. #15690785C22direct (N × H)
17. C2×#13095488C22semidirect
20. C2×C4⋊C4×C97C22semidirect
26. C23×D97D4semidirect
31. C2×D388C23semidirect
32. C2×D388C23semidirect
36. C2×C4×D97C23semidirect
37. #13095488C23semidirect
40. C22×C4×C97D4semidirect
43. C4⋊C4×C97C23semidirect
55. C22×D97C2×D4semidirect
59. C2×C4×C97C2×D4semidirect
67. C22×C97#121semidirect
79. C2×D97C22×D4semidirect
81. C4×C97C22×D4semidirect
83. C2×C97C2×#121semidirect
90. C97C22×#121semidirect
95. C22×C4⋊C4D97semidirect
109. C2×C4⋊C4C2×D97semidirect
129. C22×C4D388semidirect
132. C4⋊C4C22×D97semidirect
154. C2×C4C2×D388semidirect
165. C22#15690785direct (N × H)
172. C4C22×D388semidirect
175. C2C2×#15690785direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3882
C23×C4×D972
C2×#156907852
C22×#130954882
C22×C4⋊C4×C972
C22×#12197