Finite Group Databasebeta

← #27157155

Subgroups — #27157155

Order 12416.

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

normal subgroups only

This group has 252 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#20261680C211yesmaximal, characteristic, complemented
3#20261587C212yesmaximal, complemented
4#20261586C212yesmaximal, complemented
5#20261282C211yesmaximal, characteristic
6C4⋊#11715897—21no—
7#15690867C2212yes—
8#15690865—21no—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12C2×#13095511—21no—
13#15690809C411yescharacteristic, complemented
14#15690806C411yescharacteristic, complemented
15#15690796—21no—
16#15690785C2211yescharacteristic
17#15690783—21no—
18#15690781C2211yescharacteristic
19C4×D388C411yescharacteristic, complemented
20C4×C22⋊D97—21no—
21C42×D97C2211yescharacteristic
22C4×D4×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25#13095511—44no—
26C2×Dic194—21no—
27C2×D388—21no—
28C2×D388C2×C411yescharacteristic
29C2×C22⋊D97—21no—
30C2×C22⋊D97—21no—
31C2×C22⋊D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97D411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97C2311yescharacteristic
36C4⋊Dic97C2×C411yescharacteristic
37#13095490—21no—
38C22⋊Dic97—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97C2×C411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C2×D4×C97C2×C411yescharacteristic
45C22×C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C22⋊C4×C97—21no—
48C42×C97C2×C411yescharacteristic
49C23×C97—21no—
50C22×D97—21no—
51C22×D97C2×D411yescharacteristic
52C2×#87604—42no—
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—42no—
62C22⋊D97—42no—
63C22⋊D97—42no—
64C2×Dic97—21no—
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C2×D411yescharacteristic
68D388—41no—
69D388—42no—
70C4×D97—22no—
71C4×D97—41no—
72C4×D97—41no—
73C4×D97—42no—
74Dic194—42no—
75#11715897—42no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97C2×C22⋊C411yescharacteristic
80C2×D97—21no—
81C2×D97—21no—
82C2×D97—41no—
83C2×D97—41no—
84C2×D97—41no—
85C2×D97AGL(1,ℤ/8)12yes—
86#87604—82no—
87C4×C97—21no—
88C4×C97—21no—
89C4×C97—21no—
90C4×C97—41no—
91Dic97—21no—
92Dic97—21no—
93Dic97—21no—
94Dic97—41no—
95C2×C97—81no—
96C2×C97#41011yescharacteristic
97C2×C97#41811yescharacteristic
98C2×C97#42111yescharacteristic
99D97—22no—
100D97—81no—
101#1514—971nomaximal, Sylow
102C97#151411yescharacteristic, complemented, Sylow
103#532—971no—
104#432—971no—
105#383—971no—
106#327—972no—
107#327—972no—
108C2×#41—1941no—
109C4⋊Q8—971no—
110#127—971no—
111#124—1941no—
112#121—971no—
113#121—971no—
114#121—1941no—
115C4×D4—971no—
116C4×D4—1941no—
117C4×D4#8760411yescharacteristic, complemented
118C2×C42—971no—
119C4⋊C8—1941no—
120C22⋊C8—972no—
121#97—1941no—
122#95—972no—
123#95—1941no—
124#95—1941no—
125#41—3884no—
126C2×Q8—1941no—
127C2×D4—971no—
128C2×D4—1941no—
129C2×D4—1941no—
130C2×D4—1941no—
131C2×D4—1941no—
132C2×D4C2×#8760411yescharacteristic
133C22×C4—21no—
134C22×C4—971no—
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—1942no—
138C22×C4—1941no—
139C2×C8—1942no—
140C4⋊C4—971no—
141C4⋊C4—1941no—
142C4⋊C4C2×#8760411yescharacteristic
143C22⋊C4—21no—
144C22⋊C4—1941no—
145C22⋊C4—1941no—
146C22⋊C4—1941no—
147C42—971no—
148C42—971no—
149C42—971no—
150C42C2×#8760411yescharacteristic
151C23—21no—
152C23—971no—
153C23—1941no—
154Q8—3882no—
155D4—41no—
156D4—3881no—
157D4—3882no—
158D4—3882no—
159D4—3882no—
160D4—3882no—
161C2×C4—21no—
162C2×C4—42no—
163C2×C4—971no—
164C2×C4—971no—
165C2×C4—971no—
166C2×C4—3882no—
167C2×C4—3881no—
168C2×C4—3881no—
169C2×C4—3882no—
170C2×C4—3882no—
171C2×C4—3882no—
172C2×C4—1942no—
173C2×C4—1941no—
174C2×C4C22⋊#8760411yescharacteristic
175C2×C4C22⋊#8760411yescharacteristic
176C2×C4C22×#8760411yescharacteristic
177C8—3882no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—972no—
182C22—3881no—
183C22—3881no—
184C22—3881no—
185C22—1941no—
186C22—1941no—
187C22C2×C22⋊#8760411yescharacteristic
188C4—21no—
189C4—21no—
190C4—21no—
191C4—41no—
192C4—7762no—
193C4—3881no—
194C4—1941no—
195C4—1941no—
196C4—1941no—
197C2—81no—
198C2—7761no—
199C2—1942no—
200C2#2026171411yescharacteristic
201C2#2026172711yescharacteristic
202C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261282C2
cosocleC2×C4×D97C23
hypercenterC4×D4#87604
hypocenterC97#1514
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. #20261680C2semidirect
3. #20261587C2semidirect
4. #20261586C2semidirect
13. #15690809C4semidirect
14. #15690806C4semidirect
19. C4×D388C4semidirect
22. C4×D4×C97C4semidirect
102. C97#1514semidirect
117. C4×D4#87604semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202616802
#202615872
#202615862
#202612822
#151497