Finite Group Databasebeta

← #27157172

Subgroups — #27157172

Order 12416.

340 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261703C211yesmaximal, characteristic, complemented
2#20261677C211yesmaximal, characteristic, complemented
3#20261586C214yesmaximal, complemented
4C2×#15690781C211yesmaximal, characteristic
5C2×#13095519—21no—
6C2×C4⋊#87604—21no—
7C2×C22⋊#87604—21no—
8#15690867C2212yescomplemented
9C22⋊#11715897—21no—
10(C2×#87604)⋊C4C2212yescomplemented
11C22×C22⋊D97—21no—
12C22×C4×D97C2211yescharacteristic
13#15690781—21no—
14#15690781—21no—
15#15690781C412yescomplemented
16#15690781C2212yes—
17C2×C4⋊Dic97C411yescharacteristic, complemented
18C2×C22⋊Dic97—21no—
19C22×D4×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21#13095519—42no—
22C2×#11715897—22no—
23C4⋊#87604—42no—
24C22⋊#87604—42no—
25C23×D97D411yescharacteristic
26C2×C22⋊D97—22no—
27C2×C22⋊D97—22no—
28C2×C22⋊D97—42no—
29C2×C22⋊D97—42no—
30C22×Dic97—21no—
31C22×Dic97D411yescharacteristic
32C2×C4×D97—42no—
33C2×C4×D97C2×C411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97C2311yescharacteristic
37C4⋊Dic97—21no—
38C4⋊Dic97C2×C412yes—
39C22⋊Dic97—22no—
40C22⋊Dic97—22no—
41C24×C97—21no—
42C2×D4×C97—21no—
43C2×D4×C97—42no—
44C2×D4×C97C2×C412yes—
45C22×C4×C97C2×C411yescharacteristic
46C23×C97—22no—
47C23×C97—42no—
48C23×C97—42no—
49C23×C97—42no—
50C23×C97C22⋊C411yescharacteristic
51C22×D97—41no—
52C22×D97—41no—
53C22×D97C22⋊C411yescharacteristic
54C22×D97D812yes—
55C22×D97SD1612yes—
56C22×D97C2×D411yescharacteristic
57C2×#87604—42no—
58C2×#87604—42no—
59C2×#87604—42no—
60D4×C97—42no—
61D4×C97—42no—
62C2×C4×C97—22no—
63C2×C4×C97C22⋊C411yescharacteristic
64C2×C4×C97C22×C411yescharacteristic
65C22⋊D97—44no—
66C22⋊D97—42no—
67C22⋊D97—42no—
68C2×Dic97—22no—
69C2×Dic97—41no—
70C2×Dic97—42no—
71C2×Dic97C22⋊C411yescharacteristic
72C2×Dic97C2×D411yescharacteristic
73C4×D97—22no—
74C4×D97—22no—
75C4×D97—41no—
76C4×D97—41no—
77#11715897—42no—
78C22×C97—21no—
79C22×C97—21no—
80C22×C97—42no—
81C22×C97—42no—
82C22×C97—42no—
83C22×C97—84no—
84C22×C97#10212yes—
85C22×C97C2×C22⋊C411yescharacteristic
86C2×D97—21no—
87C2×D97—21no—
88C2×D97—21no—
89C2×D97—21no—
90C2×D97—42no—
91C2×D97—42no—
92C2×D97#10212yes—
93C2×D97C2×D811yescharacteristic
94C2×D97C2×SD1611yescharacteristic
95#87604—82no—
96C4×C97—21no—
97C4×C97#9912yes—
98Dic97—21no—
99Dic97—21no—
100Dic97—42no—
101C2×C97—22no—
102C2×C97—82no—
103C2×C97C2×#9911yescharacteristic
104C2×C97C2×#10211yescharacteristic
105C2×C97#42111yescharacteristic
106D97—21no—
107D97—22no—
108D97—21no—
109#1519—971nomaximal, Sylow
110C97#151911yescharacteristic, complemented, Sylow
111C2×#121—971no—
112C2×C22⋊C8—971no—
113#380—971no—
114#327—974no—
115C22×D4—1941no—
116C22×D4#8760411yescharacteristic, complemented
117C23×C4—971no—
118C22×C8—1941no—
119#121—972no—
120#121—972no—
121#121—1941no—
122#121—1941no—
123C2×C4⋊C4—971no—
124C2×C4⋊C4—1941no—
125C2×C22⋊C4—1941no—
126C2×C22⋊C4—1941no—
127C22⋊C8—972no—
128C22⋊C8—1941no—
129#95—972no—
130C24—21no—
131C24—971no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—3882no—
135C2×D4—3882no—
136C2×D4—1942no—
137C2×D4—1942no—
138C2×D4C2×#8760412yes—
139C22×C4—971no—
140C22×C4—971no—
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—3882no—
145C22×C4—1942no—
146C22×C4—1941no—
147C22×C4C2×#8760411yescharacteristic
148C2×C8—3882no—
149C2×C8—1942no—
150C4⋊C4—972no—
151C4⋊C4—3882no—
152C4⋊C4—1941no—
153C22⋊C4—3882no—
154C22⋊C4—1942no—
155C22⋊C4—1942no—
156C23—22no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—972no—
161C23—972no—
162C23—971no—
163C23—971no—
164C23—3881no—
165C23—3881no—
166C23C22⋊#8760411yescharacteristic
167D4—42no—
168D4—42no—
169D4—3882no—
170D4—3884no—
171D4—3882no—
172C2×C4—22no—
173C2×C4—971no—
174C2×C4—971no—
175C2×C4—3881no—
176C2×C4—3881no—
177C2×C4—3882no—
178C2×C4—3882no—
179C2×C4—3882no—
180C2×C4—3882no—
181C2×C4—3881no—
182C2×C4—1942no—
183C2×C4—1942no—
184C2×C4—1942no—
185C2×C4C22⋊#8760411yescharacteristic
186C2×C4C22×#8760411yescharacteristic
187C8—3882no—
188C22—21no—
189C22—21no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—84no—
194C22—971no—
195C22—972no—
196C22—971no—
197C22—3882no—
198C22—3882no—
199C22—1941no—
200C22—1941no—
201C22—1941no—
202C22—1941no—
203C22#1569087212yes—
204C22C2×C22⋊#8760411yescharacteristic
205C4—21no—
206C4—7762no—
207C4—3882no—
208C4—1941no—
209C4—1941no—
210C4#1569086812yes—
211C2—22no—
212C2—82no—
213C2—1941no—
214C2—1942no—
215C2—1941no—
216C2C2×#1569086811yescharacteristic
217C2C2×#1569087211yescharacteristic
218C2#2026173311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#15690781C2
cosocleC2×C4×D97C23
hypercenterC22×D4#87604
hypocenterC97#1519
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20261703C2semidirect
2. #20261677C2semidirect
3. #20261586C2semidirect
8. #15690867C22semidirect
10. (C2×#87604)⋊C4C22semidirect
15. #15690781C4semidirect
17. C2×C4⋊Dic97C4semidirect
19. C22×D4×C97C4semidirect
110. C97#1519semidirect
116. C22×D4#87604semidirect

Maximal subgroups

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

subgroupindex
#202617032
#202616772
#202615862
C2×#156907812
#151997