Finite Group Databasebeta

← #27140251

Subgroups — #27140251

Order 12416.

346 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 77 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×#15690784C211yesmaximal, characteristic
2#20260699C211yesmaximal, characteristic, complemented
3#20260468C214yesmaximal, complemented
4C4×C22⋊C4×C97C211yesmaximal, characteristic, complemented
5C22×D388C411yescharacteristic, complemented
6C22×C22⋊D97—21no—
7C2×#13095495—21no—
8#15690784—21no—
9#15690784—21no—
10#15690784C412yescomplemented
11#15690784C2212yes—
12C2×C4⋊Dic97C411yescharacteristic, complemented
13C2×#13095488—21no—
14C22⋊#11715895—21no—
15C22⋊#11715895C2212yescomplemented
16C23×C4×C97C2211yescharacteristic
17C2×C22⋊C4×C97—21no—
18C2×C42×C97—21no—
19#95×C97C2212yescomplemented
20C23×D97—21no—
21C2×D388—21no—
22C2×D388—42no—
23C2×D388C2×C412yes—
24C2×C22⋊D97—22no—
25C2×C22⋊D97—22no—
26C2×C22⋊D97—44no—
27C22×Dic97—21no—
28#13095495—42no—
29C2×#11715895—22no—
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C412yes—
32#13095488—22no—
33#13095488—22no—
34C24×C97D411yescharacteristic
35C22×C4×C97—22no—
36C22×C4×C97—42no—
37C22×C4×C97C2×C411yescharacteristic
38C22×C4×C97C2×C411yescharacteristic
39C22×C4×C97D411yescharacteristic
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D411yescharacteristic
42C22×C4×C97C2311yescharacteristic
43C22⋊C4×C97—24no—
44C42×C97—42no—
45C23×C97—22no—
46C23×C97—22no—
47C23×C97—41no—
48C23×C97C22⋊C411yescharacteristic
49C23×C97C22⋊C411yescharacteristic
50C23×C97C2×D411yescharacteristic
51C22×D97—22no—
52C22×D97—42no—
53C22×D97—42no—
54C22×D97—42no—
55C2×C4×C97—22no—
56C2×C4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97—24no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97—41no—
62C2×C4×C97—42no—
63C2×C4×C97—42no—
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22×C411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C22⋊D97—44no—
69C22⋊D97—44no—
70C2×Dic97—22no—
71C2×Dic97—42no—
72D388—42no—
73D388—42no—
74#11715895—42no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97—21no—
79C22×C97—21no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—44no—
83C22×C97#10412yes—
84C22×C97C2×C22⋊C411yescharacteristic
85C22×C97AGL(1,ℤ/8)12yes—
86C2×D97—42no—
87C2×D97—42no—
88C2×D97—42no—
89C2×D97—84no—
90C4×C97—21no—
91C4×C97—21no—
92C4×C97—21no—
93C4×C97—44no—
94C4×C97#9912yes—
95Dic97—42no—
96C2×C97—22no—
97C2×C97—22no—
98C2×C97—41no—
99C2×C97C2×#9911yescharacteristic
100C2×C97#41811yescharacteristic
101C2×C97C2×#10411yescharacteristic
102D97—82no—
103#1507—971nomaximal, Sylow
104C97#150711yescharacteristic, complemented, Sylow
105C2×#121—971no—
106#407—971no—
107C4×C22⋊C4D9711yescharacteristic, complemented
108#327—974no—
109C22×D4—971no—
110C22×D4—1941no—
111C23×C4C2×D9711yescharacteristic
112C2×M16—1941no—
113#121—972no—
114#121—972no—
115#121—1941no—
116#121—1941no—
117C2×C4⋊C4—971no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—1941no—
120C2×C42—21no—
121C22⋊C8—972no—
122C22⋊C8—1941no—
123#95C2×D9712yescomplemented
124C24—1941no—
125C24C22⋊D9711yescharacteristic
126C2×D4—972no—
127C2×D4—3882no—
128C2×D4—3884no—
129C2×D4—1942no—
130C2×D4—1942no—
131C2×D4—1941no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—1941no—
135C22×C4C4×D9711yescharacteristic
136C22×C4C4×D9711yescharacteristic
137C22×C4D38811yescharacteristic
138C22×C4D38811yescharacteristic
139C22×C4C22⋊D9711yescharacteristic
140C22×C4C22×D9711yescharacteristic
141M16—3882no—
142C2×C8—1942no—
143C4⋊C4—972no—
144C4⋊C4—1941no—
145C22⋊C4—24no—
146C22⋊C4—1942no—
147C22⋊C4—1942no—
148C42—42no—
149C23—22no—
150C23—22no—
151C23—41no—
152C23—3882no—
153C23—3882no—
154C23—3882no—
155C23—1942no—
156C23#1309548811yescharacteristic
157C23#1309548811yescharacteristic
158C23C2×C22⋊D9711yescharacteristic
159D4—3882no—
160D4—3884no—
161D4—3884no—
162D4—3882no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—3882no—
173C2×C4—1942no—
174C2×C4#1309548811yescharacteristic
175C2×C4#1309548811yescharacteristic
176C2×C4C2×C4×D9711yescharacteristic
177C2×C4C2×D38811yescharacteristic
178C8—3882no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—7764no—
188C22—3882no—
189C22—3882no—
190C22—3882no—
191C22#1569073812yes—
192C22C2×#1309548811yescharacteristic
193C22#1569083512yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—44no—
198C4—3882no—
199C4#1569072412yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—7762no—
204C2C2×#1569072411yescharacteristic
205C2#2026073911yescharacteristic
206C2C2×#1569073811yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC4×C22⋊C4×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×#15690784C2
cosocleC22×C4×C97C23
hypercenterC4×C22⋊C4D97
hypocenterC97#1507
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20260699C2semidirect
3. #20260468C2semidirect
4. C4×C22⋊C4×C97C2semidirect
5. C22×D388C4semidirect
10. #15690784C4semidirect
12. C2×C4⋊Dic97C4semidirect
15. C22⋊#11715895C22semidirect
19. #95×C97C22semidirect
104. C97#1507semidirect
107. C4×C22⋊C4D97semidirect
123. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907842
#202606992
#202604682
C4×C22⋊C4×C972
#150797