Finite Group Databasebeta

← #27140791

Subgroups — #27140791

Order 12416.

336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261370C211yesmaximal, characteristic, complemented
2#20261111C211yesmaximal, characteristic, complemented
3#20260931C212yesmaximal, complemented
4#20260462C212yesmaximal, complemented
5#331×C97C211yesmaximal, characteristic, complemented
6C2×D4×D97—22no—
7C2×D776—22no—
8C2×#13095496—22no—
9#15690807C2211yescharacteristic, complemented
10#15690781—22no—
11#15690778—22no—
12C4×D388C2212yes—
13#15690732—22no—
14C4×#11715895C2211yescharacteristic, complemented
15#127×C97C2211yescharacteristic
16C4⋊C8×C97C2212yescomplemented
17C23×D97—22no—
18C2×D388—21no—
19C2×D388—22no—
20C2×D388D412yes—
21C2×C22⋊D97—24no—
22D4×D97—44no—
23D4×D97—44no—
24C2×C4×D97—22no—
25D776—44no—
26#13095496—44no—
27C2×#11715895—21no—
28C4⋊Dic97D412yes—
29C22⋊Dic97—22no—
30#13095488—22no—
31C2×D4×C97—22no—
32C2×D4×C97D412yes—
33C2×C8×C97—22no—
34C42×C97C2311yescharacteristic
35C23×C97—22no—
36C22×D97—22no—
37C22×D97—48no—
38C22×D97—41no—
39C22×D97—82no—
40D4×C97—42no—
41D4×C97—42no—
42D4×C97—42no—
43C2×C4×C97C2×D412yes—
44C2×C4×C97C2×D411yescharacteristic
45C22⋊D97—48no—
46C2×Dic97—22no—
47D388—22no—
48D388—42no—
49D388—42no—
50D388—42no—
51D388D814yescomplemented
52C4×D97—42no—
53C4×D97—42no—
54C8×C97—42no—
55#11715895—22no—
56C22×C97—44no—
57C22×C97—42no—
58C22×C97#12011yescharacteristic
59C2×D97—24no—
60C2×D97—42no—
61C2×D97—42no—
62C2×D97—88no—
63C2×D97—82no—
64C2×D97—81no—
65C4×C97—21no—
66C4×C97C2×D812yes—
67C4×C97AGL(1,ℤ/8)12yes—
68Dic97—42no—
69C2×C97—82no—
70C2×C97#44712yes—
71C2×C97#45311yescharacteristic
72D97—44no—
73D97—161no—
74#1619—971nomaximal, Sylow
75C97#161911yescharacteristic, complemented, Sylow
76D42—971no—
77#493—971no—
78#459—972no—
79#331D9711yescharacteristic, complemented
80D4⋊C8—972no—
81C22×D4—1942no—
82C2×D8—1942no—
83C2×D8—1942no—
84#127—971no—
85#127C2×D9711yescharacteristic
86#121—1942no—
87#120—1942no—
88C4×D4—972no—
89C4⋊C8C2×D9712yescomplemented
90#102—1942no—
91C4×C8—971no—
92C24—1942no—
93C2×D4—22no—
94C2×D4—972no—
95C2×D4—3884no—
96C2×D4—3884no—
97C2×D4—1944no—
98C2×D4—1942no—
99C2×D4—1941no—
100C2×D4D38812yes—
101C22×C4—1942no—
102D8—3884no—
103D8—3884no—
104C2×C8—22no—
105C2×C8—1941no—
106C4⋊C4—972no—
107C22⋊C4—1942no—
108C22⋊C4—1942no—
109C42C22×D9711yescharacteristic
110C23—22no—
111C23—7762no—
112C23—3888no—
113C23—3881no—
114C23—1942no—
115D4—42no—
116D4—42no—
117D4—42no—
118D4—974no—
119D4—3882no—
120D4—3888no—
121D4—3882no—
122D4—3882no—
123D4—1942no—
124C2×C4—3882no—
125C2×C4—3882no—
126C2×C4—1942no—
127C2×C4D4×D9712yes—
128C2×C4C2×D38811yescharacteristic
129C8—42no—
130C8—1942no—
131C22—44no—
132C22—42no—
133C22—7768no—
134C22—7762no—
135C22—7761no—
136C22—3882no—
137C22—3882no—
138C22—1944no—
139C22#1569078011yescharacteristic
140C4—21no—
141C4—3882no—
142C4D8×D9712yes—
143C4#1569083712yes—
144C2—82no—
145C2—15521no—
146C2—3884no—
147C2#2026087012yes—
148C2#2026090511yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#331×C97C2
socleC22×C97#120
hypercenter#331D97
hypocenterC97#1619
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #20261370C2semidirect
2. #20261111C2semidirect
3. #20260931C2semidirect
4. #20260462C2semidirect
5. #331×C97C2semidirect
9. #15690807C22semidirect
14. C4×#11715895C22semidirect
16. C4⋊C8×C97C22semidirect
51. D388D8semidirect
75. C97#1619semidirect
79. #331D97semidirect
89. C4⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613702
#202611112
#202609312
#202604622
#331×C972
#161997