Finite Group Databasebeta

← #27150028

Subgroups — #27150028

Order 12416.

1102 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 213 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×#15690793C212yesmaximal, complemented
2C2×#15690779C211yesmaximal, characteristic
3#20260614C214yesmaximal, complemented
4#20260608C214yesmaximal, complemented
5#20260598C212yesmaximal, complemented
6#20260591C211yesmaximal, characteristic, complemented
7C24×C4×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97C2212yes—
9C22×C22⋊D97C2211yescharacteristic
10#15690793—28no—
11#15690779—24no—
12C2×#13095490C2214yescomplemented
13C2×C22⋊Dic97C2212yes—
14C2×C22⋊Dic97C2214yes—
15C2×C22⋊Dic97C2211yescharacteristic
16C2×C22⋊Dic97C2212yes—
17C2×#13095488C2214yescomplemented
18#15690717C2218yescomplemented
19C25×C97C2211yescharacteristic
20C23×C4×C97—24no—
21C23×C4×C97—28no—
22C23×C4×C97C2214yes—
23C23×C4×C97C2212yes—
24C23×D97C2311yescharacteristic
25C2×C22⋊D97—22no—
26C2×C22⋊D97—24no—
27C2×C22⋊D97—24no—
28C2×C22⋊D97—28no—
29C22×Dic97C2314yes—
30C22×Dic97C2311yescharacteristic
31C22×Dic97C2312yes—
32#13095490—28no—
33C22⋊Dic97—22no—
34C22⋊Dic97—24no—
35C22⋊Dic97—24no—
36C22⋊Dic97—28no—
37#13095488—28no—
38C24×C97—24no—
39C24×C97—28no—
40C24×C97D414yes—
41C24×C97C2311yescharacteristic
42C24×C97C2312yes—
43C22×C4×C97—232no—
44C22×C4×C97—28no—
45C22×C4×C97—24no—
46C22×C4×C97—24no—
47C22×C4×C97—216no—
48C22×C4×C97D418yescomplemented
49C22×C4×C97C2314yes—
50C23×C97—232no—
51C23×C97—28no—
52C23×C97—216no—
53C23×C97—24no—
54C23×C97—24no—
55C23×C97—24no—
56C23×C97C2×D412yes—
57C23×C97C2×D414yes—
58C23×C97C2×D414yes—
59C23×C97#4118yes—
60C23×C97C2411yescharacteristic
61C22×D97—22no—
62C22×D97—21no—
63C22×D97—24no—
64C2×C4×C97—232no—
65C2×C4×C97—216no—
66C2×C4×C97—24no—
67C2×C4×C97—24no—
68C2×C4×C97C2×D418yes—
69C22⋊D97—44no—
70C22⋊D97—48no—
71C2×Dic97—24no—
72C2×Dic97—21no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—28no—
76C2×Dic97—24no—
77C22×C97—232no—
78C22×C97—24no—
79C22×C97—24no—
80C22×C97—24no—
81C22×C97—28no—
82C22×C97—216no—
83C22×C97#12014yes—
84C22×C97#12318yescomplemented
85C22×C97C22×D411yescharacteristic
86C22×C97C22×D412yes—
87C22×C97C2×#4114yes—
88C2×D97—41no—
89C2×D97—42no—
90C2×D97—44no—
91C4×C97—28no—
92Dic97—41no—
93Dic97—42no—
94Dic97—44no—
95C2×C97—24no—
96C2×C97—28no—
97C2×C97C2×#12011yescharacteristic
98C2×C97C2×#12312yes—
99C2×C97#52514yes—
100D97—81no—
101#2849—971nomaximal, Sylow
102C97#284911yescharacteristic, complemented, Sylow
103C24×C4D9711yescharacteristic, complemented
104C2×#123—972no—
105C2×#120—971no—
106#386—974no—
107#385—974no—
108#381—972no—
109#379—971no—
110C25C2×D9711yescharacteristic
111C22×D4—971no—
112C22×D4—972no—
113C23×C4—24no—
114C23×C4—28no—
115C23×C4C2×D9714yes—
116C23×C4C2×D9712yes—
117#123—1948no—
118#120—1944no—
119C2×C4⋊C4—974no—
120C2×C22⋊C4—971no—
121C2×C22⋊C4—972no—
122C2×C22⋊C4—972no—
123C2×C22⋊C4—974no—
124C2×C22⋊C4—974no—
125#95—978no—
126C24—24no—
127C24—28no—
128C24—971no—
129C24C22⋊D9714yes—
130C24C22×D9711yescharacteristic
131C24C22×D9712yes—
132C2×D4—1942no—
133C2×D4—1944no—
134C2×D4—1944no—
135C2×D4—1948no—
136C22×C4—232no—
137C22×C4—28no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—216no—
141C22×C4—972no—
142C22×C4—974no—
143C22×C4—971no—
144C22×C4C22⋊D9718yescomplemented
145C22×C4C22×D9714yes—
146C4⋊C4—1948no—
147C22⋊C4—1942no—
148C22⋊C4—1944no—
149C22⋊C4—1944no—
150C22⋊C4—1948no—
151C22⋊C4—1948no—
152C23—232no—
153C23—28no—
154C23—216no—
155C23—24no—
156C23—24no—
157C23—24no—
158C23—1942no—
159C23—1941no—
160C23—1944no—
161C23C2×C22⋊D9712yes—
162C23C2×C22⋊D9714yes—
163C23C2×C22⋊D9714yes—
164C23#1309551118yes—
165C23C23×D9711yescharacteristic
166D4—3884no—
167D4—3888no—
168C2×C4—232no—
169C2×C4—216no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—1944no—
173C2×C4—1941no—
174C2×C4—1942no—
175C2×C4—1942no—
176C2×C4—1948no—
177C2×C4—1944no—
178C2×C4C2×C22⋊D9718yes—
179C22—232no—
180C22—24no—
181C22—24no—
182C22—24no—
183C22—28no—
184C22—216no—
185C22—3881no—
186C22—3882no—
187C22—3884no—
188C22#1569077914yes—
189C22#1569079318yescomplemented
190C22C22×C22⋊D9711yescharacteristic
191C22C22×C22⋊D9712yes—
192C22C2×#1309551114yes—
193C4—28no—
194C4—3881no—
195C4—3882no—
196C4—3884no—
197C2—24no—
198C2—28no—
199C2—7761no—
200C2C2×#1569077911yescharacteristic
201C2C2×#1569079312yes—
202C2#2026124614yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC24×C4×C97C2
prime radicalC2×#15690779C2
hypercenterC24×C4D97
hypocenterC97#2849
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. C2×#15690793C2semidirect
3. #20260614C2semidirect
4. #20260608C2semidirect
5. #20260598C2semidirect
6. #20260591C2semidirect
7. C24×C4×C97C2semidirect
12. C2×#13095490C22semidirect
17. C2×#13095488C22semidirect
18. #15690717C22semidirect
48. C22×C4×C97D4semidirect
84. C22×C97#123semidirect
102. C97#2849semidirect
103. C24×C4D97semidirect
144. C22×C4C22⋊D97semidirect
189. C22#15690793semidirect

Maximal subgroups

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

subgroupindex
C2×#156907932
C2×#156907792
#202606142
#202606082
#202605982
#202605912
C24×C4×C972
#284997