Finite Group Databasebeta

← #27157150

Subgroups — #27157150

Order 12416.

222 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 56 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
2#20261679C211yesmaximal, characteristic, complemented
3#20261588C212yesmaximal, complemented
4#20261587C212yesmaximal
5#20261270C211yesmaximal, characteristic
6C2×C4⋊#87604—21no—
7C4⋊#11715897—21no—
8#15690867C2212yes—
9#15690865—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12#15690805—21no—
13#15690798C411yescharacteristic, complemented
14#15690795—21no—
15#15690790C2211yescharacteristic
16#15690785C2211yescharacteristic
17#15690775C411yescharacteristic, complemented
18#15690772—21no—
19#15690772C411yescharacteristic, complemented
20#15690768—21no—
21C42×D97C2211yescharacteristic
22#125×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25C4⋊#87604—42no—
26C2×Dic194C2×C411yescharacteristic
27C2×D388—21no—
28C2×D388C2×C411yescharacteristic
29C2×C4×D97—21no—
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C2×C4×D97C2311yescharacteristic
33C4⋊Dic97—21no—
34#13095490—21no—
35#13095488—21no—
36#13095488—21no—
37#13095488—21no—
38#13095488—21no—
39#13095488—21no—
40C4×Dic97—21no—
41C4×Dic97C2×C411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C4⋊C4×C97C2×C411yescharacteristic
48C42×C97C2×C411yescharacteristic
49C22×D97—21no—
50C22×D97C2×D411yescharacteristic
51C2×#87604—42no—
52C2×#87604—42no—
53C2×#87604—42no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22×C411yescharacteristic
59C2×Dic97—21no—
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63D388—41no—
64D388—41no—
65D388—41no—
66C4×D97—21no—
67C4×D97—21no—
68C4×D97—41no—
69C4×D97—41no—
70C4×D97—42no—
71Dic194—41no—
72#11715897—42no—
73C22×C97C2×C22⋊C411yescharacteristic
74C2×D97—21no—
75C2×D97—21no—
76C2×D97—41no—
77C2×D97—41no—
78C2×D97—41no—
79C2×D97AGL(1,ℤ/8)11yescharacteristic
80C2×D97#13711yescharacteristic
81#87604—82no—
82C4×C97—21no—
83C4×C97—21no—
84C4×C97—21no—
85C4×C97—41no—
86C4×C97—41no—
87Dic97—21no—
88Dic97—21no—
89Dic97—21no—
90Dic97—41no—
91C2×C97#41011yescharacteristic
92C2×C97#41711yescharacteristic
93C2×C97#42111yescharacteristic
94D97—21no—
95D97—21no—
96D97—81no—
97#1512—971nomaximal, Sylow
98C97#151211yescharacteristic, complemented, Sylow
99#529—971no—
100#432—971no—
101#382—971no—
102#328—972no—
103#327—972no—
104#126—1941no—
105#125#8760411yescharacteristic, complemented
106#124—971no—
107#123—1941no—
108C22⋊Q8—971no—
109#121—971no—
110C4×Q8—971no—
111C4×D4—971no—
112C4×D4—1941no—
113#117—1941no—
114C2×C4⋊C4—1941no—
115C2×C42—971no—
116C4⋊C8—1941no—
117C22⋊C8—972no—
118#97—1941no—
119#95—972no—
120#95—1941no—
121C2×Q8—971no—
122C2×D4—971no—
123C2×D4—1941no—
124C22×C4—971no—
125C22×C4—971no—
126C22×C4—971no—
127C22×C4—1941no—
128C22×C4—1942no—
129C2×C8—1942no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3882no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4C2×#8760411yescharacteristic
136C4⋊C4C2×#8760411yescharacteristic
137C22⋊C4—1941no—
138C22⋊C4—1941no—
139C22⋊C4—1941no—
140C22⋊C4—1941no—
141C22⋊C4—1941no—
142C42—971no—
143C42—971no—
144C42—971no—
145C42—1941no—
146C42C2×#8760411yescharacteristic
147C23—971no—
148C23—1941no—
149Q8—3881no—
150D4—3881no—
151D4—3881no—
152D4—3881no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—971no—
156C2×C4—971no—
157C2×C4—971no—
158C2×C4—3881no—
159C2×C4—3881no—
160C2×C4—3882no—
161C2×C4—3882no—
162C2×C4—3882no—
163C2×C4—3882no—
164C2×C4—1941no—
165C2×C4—1941no—
166C2×C4—1941no—
167C2×C4C22⋊#8760411yescharacteristic
168C2×C4C22⋊#8760411yescharacteristic
169C2×C4C22×#8760411yescharacteristic
170C8—3882no—
171C22—971no—
172C22—971no—
173C22—3881no—
174C22—3881no—
175C22—3881no—
176C22—1941no—
177C22—1941no—
178C22C2×C22⋊#8760411yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—41no—
184C4—7762no—
185C4—3881no—
186C4—1941no—
187C4—1941no—
188C4—1941no—
189C2—7761no—
190C2—1941no—
191C2—1941no—
192C2#2026171611yescharacteristic
193C2#2026172611yescharacteristic
194C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radical#15690785C22
cosocleC2×C4×D97C23
hypercenter#125#87604
hypocenterC97#1512
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #20261757C2semidirect
2. #20261679C2semidirect
3. #20261588C2semidirect
13. #15690798C4semidirect
17. #15690775C4semidirect
19. #15690772C4semidirect
22. #125×C97C4semidirect
98. C97#1512semidirect
105. #125#87604semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202616792
#202615882
#202615872
#202612702
#151297