Finite Group Databasebeta

← #27156636

Subgroups — #27156636

Order 12416.

200 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 46 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#20261706C211yesmaximal, characteristic
2#20261677C211yesmaximal, characteristic
3#20260698C211yesmaximal, characteristic
4C2×#13095521—21no—
5C2×C4⋊#87604—21no—
6C2×C22⋊#87604C412yes—
7#15690867—21no—
8C22⋊#11715897C412yescomplemented
9(C2×#87604)⋊C4—21no—
10C22×C4×D97C2211yescharacteristic
11C2×#13095495C411yescharacteristic, complemented
12#15690722—21no—
13#15690722—21no—
14C2×M16×C97C411yescharacteristic, complemented
15C22×#87604—22no—
16#13095521—41no—
17#13095521—41no—
18C2×#11715897—22no—
19C4⋊#87604—41no—
20C4⋊#87604—41no—
21C22⋊#87604—42no—
22C23×D97C2×C411yescharacteristic
23C22×Dic97C2×C411yescharacteristic
24C2×C4×D97—41no—
25C2×C4×D97—41no—
26C2×C4×D97D411yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×C4×D97Q811yescharacteristic
30#13095495—41no—
31C2×#11715895—21no—
32C22×C4×C97C2×C411yescharacteristic
33M16×C97—41no—
34C2×C8×C97—21no—
35C23×C97C4211yescharacteristic
36C22×D97—21no—
37C22×D97—21no—
38C22×D97—41no—
39C22×D97—41no—
40C22×D97C22⋊C411yescharacteristic
41C22×D97C4⋊C411yescharacteristic
42C2×#87604—42no—
43C2×#87604—42no—
44C2×#87604—42no—
45C2×C4×C97—21no—
46C2×C4×C97—21no—
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C4⋊C411yescharacteristic
49C2×Dic97—41no—
50C2×Dic97C22⋊C411yescharacteristic
51C2×Dic97C4⋊C411yescharacteristic
52C4×D97—21no—
53C4×D97—21no—
54C4×D97—21no—
55C4×D97—21no—
56C4×D97—41no—
57C4×D97—41no—
58#11715897—42no—
59C8×C97—41no—
60#11715895—41no—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97#9511yescharacteristic
65C2×D97—21no—
66C2×D97—21no—
67C2×D97—21no—
68C2×D97—21no—
69C2×D97—21no—
70C2×D97—42no—
71C2×D97—42no—
72C2×D97#10011yescharacteristic
73C2×D97#10111yescharacteristic
74#87604—82no—
75C4×C97—21no—
76C4×C97#9911yescharacteristic
77C4×C97#9911yescharacteristic
78Dic97—21no—
79Dic97—21no—
80C2×C97—41no—
81C2×C97#34211yescharacteristic
82C2×C97#34311yescharacteristic
83C2×C97#34411yescharacteristic
84D97—21no—
85D97—21no—
86D97—41no—
87#1305—971nomaximal, Sylow
88C97#130511yescharacteristic, complemented, Sylow
89#407—971no—
90#407—971no—
91#380—971no—
92C23×C4—971no—
93C2×M16—971no—
94C2×M16—1941no—
95C2×M16#8760411yescharacteristic, complemented
96C2×C4⋊C4—1941no—
97C2×C22⋊C4—972no—
98C22⋊C8—972no—
99C22⋊C8—1941no—
100C22⋊C8—1941no—
101C22⋊C8—1941no—
102#95—1941no—
103C24—971no—
104C22×C4—971no—
105C22×C4—971no—
106C22×C4—971no—
107C22×C4—971no—
108C22×C4—971no—
109C22×C4—3881no—
110C22×C4—3881no—
111C22×C4—1942no—
112C22×C4C2×#8760411yescharacteristic
113M16—41no—
114M16—3881no—
115M16—3881no—
116M16—3881no—
117C2×C8—21no—
118C2×C8—1942no—
119C2×C8—1941no—
120C4⋊C4—3881no—
121C4⋊C4—3881no—
122C22⋊C4—3882no—
123C23—971no—
124C23—971no—
125C23—3881no—
126C23—3881no—
127C23—1941no—
128C23—1941no—
129C23C4×#8760411yescharacteristic
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—971no—
133C2×C4—971no—
134C2×C4—3881no—
135C2×C4—3881no—
136C2×C4—3882no—
137C2×C4—3882no—
138C2×C4—3882no—
139C2×C4—3881no—
140C2×C4—1941no—
141C2×C4—1941no—
142C2×C4—1941no—
143C2×C4—1941no—
144C2×C4C22⋊#8760411yescharacteristic
145C2×C4C4⋊#8760411yescharacteristic
146C8—41no—
147C8—3882no—
148C8—3881no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—971no—
153C22—971no—
154C22—3882no—
155C22—3882no—
156C22—1941no—
157C22—1941no—
158C22—1941no—
159C22—1941no—
160C22—1941no—
161C22(C2×#87604)⋊C411yescharacteristic
162C4—21no—
163C4—7762no—
164C4—1941no—
165C4—1941no—
166C4#1569086811yescharacteristic
167C4#1569086911yescharacteristic
168C2—41no—
169C2—3881no—
170C2—1941no—
171C2—1941no—
172C2#2026161311yescharacteristic
173C2#2026161411yescharacteristic
174C2#2026161811yescharacteristic
subgroupG/H
centerC22(C2×#87604)⋊C4
commutatorC23×C97C42
FrattiniC22×C4C2×#87604
FittingC2×M16×C97C4
socleC22×C97#95
prime radicalC23×D97C2×C4
cosocleC22×C4×D97C22
hypercenterC2×M16#87604
hypocenterC97#1305
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
8. C22⋊#11715897C4semidirect
11. C2×#13095495C4semidirect
14. C2×M16×C97C4semidirect
88. C97#1305semidirect
95. C2×M16#87604semidirect

Maximal subgroups

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

subgroupindex
#202617062
#202616772
#202606982
#130597