Finite Group Databasebeta

← #27140790

Subgroups — #27140790

Order 12416.

336 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261111C211yesmaximal, characteristic, complemented
2#20260930C212yesmaximal, complemented
3#20260479C211yesmaximal, characteristic, complemented
4D4⋊#11715895C212yesmaximal, complemented
5D42×C97C211yesmaximal, characteristic, complemented
6C2×#13095496—22no—
7C2×#13095496—22no—
8#15690807C2211yescharacteristic, complemented
9C4⋊#11715895C2212yescomplemented
10#15690730—22no—
11C4×#11715895C2211yescharacteristic, complemented
12C22×D4×C97—22no—
13#127×C97C2211yescharacteristic
14#121×C97—22no—
15#120×C97—22no—
16C4×D4×C97C2212yes—
17C2×D388—21no—
18C2×D388—22no—
19#13095496—44no—
20#13095496—44no—
21C2×#11715895—21no—
22C2×#11715895—22no—
23C24×C97—22no—
24C2×D4×C97—24no—
25C2×D4×C97—22no—
26C2×D4×C97—44no—
27C2×D4×C97—44no—
28C2×D4×C97D412yes—
29C2×D4×C97D412yes—
30C22×C4×C97—22no—
31C4⋊C4×C97D412yes—
32C22⋊C4×C97—22no—
33C22⋊C4×C97—22no—
34C42×C97C2311yescharacteristic
35C23×C97—22no—
36C23×C97—22no—
37C23×C97—48no—
38C23×C97—82no—
39C22×D97—41no—
40D4×C97—22no—
41D4×C97—42no—
42D4×C97—48no—
43D4×C97—42no—
44D4×C97—42no—
45D4×C97D814yescomplemented
46C2×C4×C97—22no—
47C2×C4×C97—42no—
48C2×C4×C97—42no—
49C2×C4×C97C2×D411yescharacteristic
50C2×C4×C97C2×D412yes—
51D388—42no—
52D388—42no—
53D388—42no—
54#11715895—22no—
55#11715895—42no—
56C22×C97—24no—
57C22×C97—42no—
58C22×C97—42no—
59C22×C97—44no—
60C22×C97—42no—
61C22×C97—88no—
62C22×C97#12011yescharacteristic
63C2×D97—82no—
64C2×D97—81no—
65C4×C97—21no—
66C4×C97—42no—
67C4×C97C2×D812yes—
68C4×C97AGL(1,ℤ/8)12yes—
69C2×C97—44no—
70C2×C97—82no—
71C2×C97#44712yes—
72C2×C97#45311yescharacteristic
73D97—161no—
74#1619—971nomaximal, Sylow
75C97#161911yescharacteristic, complemented, Sylow
76D42D9711yescharacteristic, complemented
77#493—971no—
78#459—972no—
79#331—971no—
80D4⋊C8—972no—
81C22×D4—22no—
82C2×D8—1942no—
83C2×D8—1942no—
84#127—971no—
85#127C2×D9711yescharacteristic
86#121—22no—
87#120—22no—
88C4×D4C2×D9712yes—
89C4⋊C8—972no—
90#102—1942no—
91C4×C8—971no—
92C24—22no—
93C2×D4—24no—
94C2×D4—22no—
95C2×D4—44no—
96C2×D4—44no—
97C2×D4—1942no—
98C2×D4—1941no—
99C2×D4C22⋊D9712yes—
100C2×D4C22⋊D9712yes—
101C22×C4—22no—
102D8—3884no—
103D8—3884no—
104C2×C8—1942no—
105C2×C8—1941no—
106C4⋊C4C22⋊D9712yes—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C42C22×D9711yescharacteristic
110C23—22no—
111C23—22no—
112C23—48no—
113C23—82no—
114C23—3881no—
115D4—22no—
116D4—42no—
117D4—48no—
118D4—42no—
119D4—42no—
120D4—3882no—
121D4—3882no—
122D4—3882no—
123D4#1309549614yescomplemented
124C2×C4—22no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4C2×C22⋊D9711yescharacteristic
128C2×C4C2×C22⋊D9712yes—
129C8—3882no—
130C8—1942no—
131C22—24no—
132C22—42no—
133C22—42no—
134C22—44no—
135C22—42no—
136C22—88no—
137C22—7762no—
138C22—7761no—
139C22#1569077911yescharacteristic
140C4—21no—
141C4—42no—
142C4C2×#1309549612yes—
143C4#1569083612yes—
144C2—44no—
145C2—82no—
146C2—15521no—
147C2#2026086912yes—
148C2#2026090311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD42×C97C2
socleC22×C97#120
hypercenterD42D97
hypocenterC97#1619
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #20261111C2semidirect
2. #20260930C2semidirect
3. #20260479C2semidirect
4. D4⋊#11715895C2semidirect
5. D42×C97C2semidirect
8. #15690807C22semidirect
9. C4⋊#11715895C22semidirect
11. C4×#11715895C22semidirect
45. D4×C97D8semidirect
75. C97#1619semidirect
76. D42D97semidirect
123. D4#13095496semidirect

Maximal subgroups

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

subgroupindex
#202611112
#202609302
#202604792
D4⋊#117158952
D42×C972
#161997