Finite Group Databasebeta

← #27157138

Subgroups — #27157138

Order 12416.

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

normal subgroups only

This group has 242 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#20261757C211yesmaximal, characteristic, complemented
2C4×C4⋊#87604C211yesmaximal, characteristic, complemented
3#20261588C212yesmaximal, complemented
4#20261586C212yesmaximal
5#20261267C211yesmaximal, characteristic
6C2×C4⋊#87604—21no—
7C2×C4×#87604—21no—
8C4⋊#11715897—21no—
9#15690867C2212yes—
10#15690865—21no—
11(C2×#87604)⋊C4C2212yes—
12#15690803—21no—
13#15690801C411yescharacteristic, complemented
14#15690797C411yescharacteristic, complemented
15#15690792—21no—
16#15690790C2211yescharacteristic
17#15690781C2211yescharacteristic
18C4×Dic194C411yescharacteristic, complemented
19C4×C22⋊D97—21no—
20#15690769—21no—
21C42×D97C2211yescharacteristic
22C4×D4×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25C4⋊#87604—24no—
26C4×#87604—42no—
27C2×Dic194C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97C2311yescharacteristic
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97C2×C411yescharacteristic
35#13095490—21no—
36#13095490—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39C22⋊Dic97—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C4×Dic97C2×C411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C2×D4×C97C2×C411yescharacteristic
46C22×C4×C97—21no—
47C4⋊C4×C97C2×C411yescharacteristic
48C22⋊C4×C97—21no—
49C42×C97C2×C411yescharacteristic
50C23×C97—21no—
51C22×D97C2×D411yescharacteristic
52C2×#87604—24no—
53C2×#87604—42no—
54C2×#87604—42no—
55D4×C97—41no—
56C2×C4×C97—21no—
57C2×C4×C97—42no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22⋊C411yescharacteristic
60C2×C4×C97C22×C411yescharacteristic
61C22⋊D97—41no—
62C22⋊D97—41no—
63C2×Dic97—21no—
64C2×Dic97—21no—
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C2×D411yescharacteristic
68C4×D97—21no—
69C4×D97—21no—
70C4×D97—22no—
71C4×D97—41no—
72Dic194—41no—
73#11715897—42no—
74C22×C97—41no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97C2×C22⋊C411yescharacteristic
78C2×D97—21no—
79C2×D97—21no—
80C2×D97AGL(1,ℤ/8)11yescharacteristic
81C2×D97#13711yescharacteristic
82#87604—44no—
83C4×C97—21no—
84C4×C97—21no—
85C4×C97—41no—
86C4×C97#10412yes—
87Dic97—21no—
88Dic97—21no—
89Dic97—21no—
90Dic97—41no—
91Dic97—41no—
92C2×C97—81no—
93C2×C97#41011yescharacteristic
94C2×C97#41711yescharacteristic
95C2×C97C2×#10411yescharacteristic
96D97—21no—
97D97—21no—
98#1509—971nomaximal, Sylow
99C97#150911yescharacteristic, complemented, Sylow
100#529—971no—
101#432—971no—
102C4×C4⋊C4—971no—
103#328—972no—
104#327—972no—
105#126—1941no—
106#125—971no—
107#124—971no—
108#123—1941no—
109C22⋊Q8—971no—
110#121—971no—
111C4×Q8—971no—
112C4×D4—1941no—
113C4×D4#8760411yescharacteristic, complemented
114#117—1941no—
115C2×C4⋊C4—1941no—
116C2×C42—971no—
117C2×C42—1941no—
118C4⋊C8—1941no—
119C22⋊C8—972no—
120#97—1941no—
121#95—972no—
122C2×Q8—971no—
123C2×D4—1941no—
124C2×D4C2×#8760411yescharacteristic
125C22×C4—21no—
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—971no—
129C22×C4—1942no—
130C2×C8—1942no—
131C4⋊C4—971no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4—1944no—
136C4⋊C4—1941no—
137C4⋊C4C2×#8760411yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—1941no—
140C22⋊C4—1941no—
141C22⋊C4—1941no—
142C22⋊C4—1941no—
143C42—971no—
144C42—971no—
145C42—971no—
146C42—3882no—
147C42—1941no—
148C42C2×#8760411yescharacteristic
149C23—21no—
150C23—971no—
151Q8—3881no—
152D4—41no—
153D4—3881no—
154D4—3881no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—971no—
158C2×C4—971no—
159C2×C4—971no—
160C2×C4—3881no—
161C2×C4—3882no—
162C2×C4—3882no—
163C2×C4—1941no—
164C2×C4—1941no—
165C2×C4—1944no—
166C2×C4—1941no—
167C2×C4—1941no—
168C2×C4—1942no—
169C2×C4C22⋊#8760411yescharacteristic
170C2×C4C22⋊#8760411yescharacteristic
171C2×C4C22×#8760411yescharacteristic
172C8—3882no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—971no—
177C22—971no—
178C22—1941no—
179C22—1941no—
180C22C2×C22⋊#8760411yescharacteristic
181C4—21no—
182C4—21no—
183C4—41no—
184C4—3884no—
185C4—3881no—
186C4—3881no—
187C4—1941no—
188C4—1941no—
189C4—1941no—
190C4#1569087412yes—
191C2—81no—
192C2—1941no—
193C2—1941no—
194C2#2026171411yescharacteristic
195C2#2026172611yescharacteristic
196C2C2×#1569087411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radical#15690781C22
cosocleC2×C4×D97C23
hypercenterC4×D4#87604
hypocenterC97#1509
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261757C2semidirect
2. C4×C4⋊#87604C2semidirect
3. #20261588C2semidirect
13. #15690801C4semidirect
14. #15690797C4semidirect
18. C4×Dic194C4semidirect
22. C4×D4×C97C4semidirect
99. C97#1509semidirect
113. C4×D4#87604semidirect

Maximal subgroups

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

subgroupindex
#202617572
C4×C4⋊#876042
#202615882
#202615862
#202612672
#150997