Finite Group Databasebeta

← #27156714

Subgroups — #27156714

Order 12416.

222 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 30 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#20261613C211yesmaximal, characteristic, complemented
2#20261579C211yesmaximal, characteristic, complemented
3#20260643C211yesmaximal, characteristic
4C2×C22⋊#87604—21no—
5C22⋊#11715897—21no—
6C22×C22⋊D97—21no—
7C2×C22⋊Dic97—21no—
8C2×#13095488C2211yescharacteristic
9#15690717C411yescharacteristic, complemented
10C22×D4×C97C411yescharacteristic, complemented
11C22×#87604—41no—
12C2×#11715897—41no—
13C22⋊#87604—22no—
14C22⋊#87604—41no—
15C23×D97D411yescharacteristic
16C2×C22⋊D97—42no—
17C2×C22⋊D97—42no—
18C2×C22⋊D97—41no—
19C2×C22⋊D97—41no—
20C22×Dic97—21no—
21C22×Dic97D411yescharacteristic
22C22⋊Dic97—42no—
23#13095488—41no—
24C24×C97—21no—
25C2×D4×C97—41no—
26C2×D4×C97—42no—
27C22×C4×C97C2×C411yescharacteristic
28C23×C97—22no—
29C23×C97—41no—
30C23×C97—41no—
31C23×C97—41no—
32C23×C97—41no—
33C23×C97—41no—
34C23×C97—41no—
35C23×C97C22⋊C411yescharacteristic
36C22×D97—21no—
37C22×D97—21no—
38C22×D97—41no—
39C22×D97—41no—
40C22×D97D811yescharacteristic
41C22×D97SD1611yescharacteristic
42C2×#87604—42no—
43C2×#87604—81no—
44C2×#87604—81no—
45D4×C97—42no—
46D4×C97—42no—
47C2×C4×C97—21no—
48C2×C4×C97—41no—
49C22⋊D97—44no—
50C22⋊D97—44no—
51C2×Dic97—22no—
52C2×Dic97—21no—
53C2×Dic97—41no—
54C2×Dic97—41no—
55C2×Dic97—41no—
56#11715897—81no—
57C22×C97—21no—
58C22×C97—21no—
59C22×C97—42no—
60C22×C97—42no—
61C22×C97—42no—
62C22×C97—81no—
63C22×C97—81no—
64C22×C97—81no—
65C22×C97—81no—
66C22×C97#9911yescharacteristic
67C22×C97#10211yescharacteristic
68C22×C97#10411yescharacteristic
69C2×D97—21no—
70C2×D97—21no—
71C2×D97—41no—
72C2×D97—41no—
73C2×D97—41no—
74C2×D97—41no—
75C2×D97—81no—
76#87604—82no—
77C4×C97—41no—
78Dic97—42no—
79Dic97—41no—
80C2×C97—21no—
81C2×C97—21no—
82C2×C97—82no—
83C2×C97#32711yescharacteristic
84C2×C97AΣL(1,𝔽16)11yescharacteristic
85C2×C97#35311yescharacteristic
86D97—41no—
87D97—41no—
88#1343—971nomaximal, Sylow
89C97#134311yescharacteristic, complemented, Sylow
90#392—971no—
91#342—971no—
92#323—971no—
93C22×D4—1941no—
94C22×D4#8760411yescharacteristic, complemented
95C2×C22⋊C4—971no—
96C2×C22⋊C4—1941no—
97C2×C22⋊C4—1941no—
98C22⋊C8—1941no—
99#95—971no—
100C24—21no—
101C24—971no—
102C2×D4—41no—
103C2×D4—42no—
104C2×D4—3882no—
105C2×D4—3881no—
106C2×D4—3881no—
107C2×D4—3882no—
108C22×C4—971no—
109C22×C4—3881no—
110C22×C4—1941no—
111C22×C4C2×#8760411yescharacteristic
112C2×C8—3881no—
113C22⋊C4—3881no—
114C22⋊C4—3881no—
115C22⋊C4—3882no—
116C22⋊C4—1942no—
117C23—22no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—971no—
125C23—971no—
126C23—3881no—
127C23—3881no—
128C23—1941no—
129C23—1941no—
130C23C22⋊#8760411yescharacteristic
131D4—42no—
132D4—42no—
133D4—3884no—
134D4—3884no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—7761no—
138C2×C4—7761no—
139C2×C4—3882no—
140C2×C4—3881no—
141C2×C4—3881no—
142C2×C4—3881no—
143C2×C4—1942no—
144C2×C4—1941no—
145C8—7761no—
146C22—21no—
147C22—21no—
148C22—42no—
149C22—42no—
150C22—42no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22—81no—
155C22—7761no—
156C22—3881no—
157C22—3881no—
158C22—3881no—
159C22—3881no—
160C22—1941no—
161C22—1941no—
162C22#1569086811yescharacteristic
163C22#1569087211yescharacteristic
164C22#1569087411yescharacteristic
165C4—41no—
166C4—7762no—
167C4—3882no—
168C4—3881no—
169C2—21no—
170C2—21no—
171C2—82no—
172C2—3881no—
173C2—3881no—
174C2#2026158611yescharacteristic
175C2#2026163511yescharacteristic
176C2#2026163911yescharacteristic
subgroupG/H
centerC22#15690868
commutatorC22×C4×C97C2×C4
FrattiniC22×C4C2×#87604
FittingC22×D4×C97C4
socleC22×C97#99
prime radical#20260643C2
cosocleC2×#13095488C22
hypercenterC22×D4#87604
hypocenterC97#1343
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20261613C2semidirect
2. #20261579C2semidirect
9. #15690717C4semidirect
10. C22×D4×C97C4semidirect
89. C97#1343semidirect
94. C22×D4#87604semidirect

Maximal subgroups

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

subgroupindex
#202616132
#202615792
#202606432
#134397