Finite Group Databasebeta

← #27157143

Subgroups — #27157143

Order 12416.

326 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 60 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
1#20261707C211yesmaximal, characteristic, complemented
2#20261678C211yesmaximal, characteristic, complemented
3#20261587C214yesmaximal, complemented
4C2×#15690785C211yesmaximal, characteristic
5C2×#13095520—21no—
6C2×C22⋊#87604—21no—
7#15690867—21no—
8#15690867C2212yescomplemented
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4C2212yescomplemented
11C22×D388—21no—
12C22×D388C411yescharacteristic, complemented
13C22×C4×D97C2211yescharacteristic
14#15690785—21no—
15#15690785—21no—
16#15690785C412yescomplemented
17#15690785C2212yes—
18C2×#13095488—21no—
19C2×C4⋊C4×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21#13095520—42no—
22C2×#11715897—22no—
23C22⋊#87604—42no—
24C23×D97—21no—
25C23×D97D411yescharacteristic
26C2×D388—22no—
27C2×D388—22no—
28C2×D388—21no—
29C2×D388—42no—
30C2×D388—44no—
31C2×D388C2×C412yes—
32C22×Dic97D411yescharacteristic
33C2×C4×D97—42no—
34C2×C4×D97C2×C411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97C2311yescharacteristic
38#13095488—22no—
39#13095488—22no—
40C22×C4×C97—21no—
41C22×C4×C97C2×C411yescharacteristic
42C4⋊C4×C97—21no—
43C4⋊C4×C97C2×C412yes—
44C23×C97C22⋊C411yescharacteristic
45C22×D97—22no—
46C22×D97—22no—
47C22×D97—22no—
48C22×D97—41no—
49C22×D97—42no—
50C22×D97—42no—
51C22×D97—42no—
52C22×D97C22⋊C411yescharacteristic
53C22×D97C2×D411yescharacteristic
54C2×#87604—42no—
55C2×#87604—42no—
56C2×#87604—42no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97—42no—
60C2×C4×C97C22⋊C411yescharacteristic
61C2×C4×C97C22×C411yescharacteristic
62C2×Dic97—22no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65D388—42no—
66D388—42no—
67D388—44no—
68D388—44no—
69C4×D97—41no—
70C4×D97—41no—
71C4×D97—41no—
72C4×D97—41no—
73#11715897—42no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97#9912yes—
77C22×C97C2×C22⋊C411yescharacteristic
78C2×D97—21no—
79C2×D97—21no—
80C2×D97—21no—
81C2×D97—21no—
82C2×D97—21no—
83C2×D97—44no—
84C2×D97—42no—
85C2×D97—42no—
86C2×D97—42no—
87C2×D97—84no—
88C2×D97AGL(1,ℤ/8)12yes—
89#87604—82no—
90C4×C97—21no—
91C4×C97—21no—
92C4×C97—42no—
93Dic97—21no—
94Dic97—21no—
95C2×C97—22no—
96C2×C97C2×#9911yescharacteristic
97C2×C97#41811yescharacteristic
98C2×C97#42111yescharacteristic
99D97—22no—
100D97—41no—
101D97—82no—
102#1510—971nomaximal, Sylow
103C97#151011yescharacteristic, complemented, Sylow
104C2×#121—971no—
105#407—971no—
106#381—971no—
107#327—974no—
108C22×D4—971no—
109C22×D4—1941no—
110C23×C4—971no—
111C2×M16—1941no—
112#121—972no—
113#121—972no—
114#121—1941no—
115#121—1941no—
116C2×C4⋊C4#8760411yescharacteristic, complemented
117C2×C22⋊C4—1941no—
118C2×C22⋊C4—1941no—
119C22⋊C8—972no—
120C22⋊C8—1941no—
121#95—972no—
122#95—1941no—
123C24—971no—
124C24—1941no—
125C2×D4—972no—
126C2×D4—3882no—
127C2×D4—3884no—
128C2×D4—1942no—
129C2×D4—1942no—
130C2×D4—1941no—
131C22×C4—21no—
132C22×C4—971no—
133C22×C4—971no—
134C22×C4—971no—
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—3882no—
138C22×C4—1942no—
139C22×C4C2×#8760411yescharacteristic
140M16—3882no—
141C2×C8—1942no—
142C4⋊C4—21no—
143C4⋊C4C2×#8760412yes—
144C22⋊C4—3882no—
145C22⋊C4—1942no—
146C22⋊C4—1942no—
147C23—971no—
148C23—971no—
149C23—3882no—
150C23—3882no—
151C23—3881no—
152C23—3882no—
153C23—1942no—
154C23—1942no—
155C23—1942no—
156C23C22⋊#8760411yescharacteristic
157D4—3882no—
158D4—3882no—
159D4—3884no—
160D4—3884no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—971no—
165C2×C4—971no—
166C2×C4—3881no—
167C2×C4—3881no—
168C2×C4—3881no—
169C2×C4—3881no—
170C2×C4—3882no—
171C2×C4—3882no—
172C2×C4—3882no—
173C2×C4—1942no—
174C2×C4C22⋊#8760411yescharacteristic
175C2×C4C22×#8760411yescharacteristic
176C8—3882no—
177C22—21no—
178C22—21no—
179C22—7764no—
180C22—972no—
181C22—3884no—
182C22—3882no—
183C22—3882no—
184C22—3882no—
185C22—1941no—
186C22—1941no—
187C22—1941no—
188C22—1941no—
189C22—1941no—
190C22#1569086912yes—
191C22C2×C22⋊#8760411yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—7762no—
196C4—1941no—
197C4—1941no—
198C2—22no—
199C2—7762no—
200C2—3881no—
201C2—1942no—
202C2C2×#1569086911yescharacteristic
203C2#2026172711yescharacteristic
204C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#15690785C2
cosocleC2×C4×D97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1510
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261707C2semidirect
2. #20261678C2semidirect
3. #20261587C2semidirect
8. #15690867C22semidirect
10. (C2×#87604)⋊C4C22semidirect
12. C22×D388C4semidirect
16. #15690785C4semidirect
19. C2×C4⋊C4×C97C4semidirect
103. C97#1510semidirect
116. C2×C4⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202617072
#202616782
#202615872
C2×#156907852
#151097