Finite Group Databasebeta

← #41012515

Subgroups — #41012515

Order 27008.

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

normal subgroups only

This group has 314 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#27213781C211yesmaximal, characteristic
2#27213518C211yesmaximal, characteristic, complemented
3#27213437C212yesmaximal, complemented
4#27212873C212yesmaximal, complemented
5#331×C211C211yesmaximal, characteristic, complemented
6C2×D4×D211—22no—
7C2×#16809950—22no—
8#20269977C2211yescharacteristic, complemented
9#20269948—22no—
10#20269945—22no—
11C4×D844C2212yes—
12#20269909—22no—
13#20269899—22no—
14C4×C211⋊C8C2211yescharacteristic, complemented
15#127×C211C2211yescharacteristic
16C4⋊C8×C211C2212yescomplemented
17C23×D211—22no—
18C2×Dic422—21no—
19C2×D844D412yes—
20C2×C22⋊D211—24no—
21D4×D211—44no—
22D4×D211—44no—
23C2×C4×D211—22no—
24#16809950—44no—
25C2×C211⋊C8—21no—
26C4⋊Dic211—22no—
27C4⋊Dic211D412yes—
28C22⋊Dic211—22no—
29(C2×D211)⋊C4—22no—
30C2×D4×C211—22no—
31C2×D4×C211D412yes—
32C2×C8×C211—22no—
33C42×C211C2311yescharacteristic
34C22×D211—22no—
35C22×D211—48no—
36C22×D211—82no—
37Dic422—42no—
38D844—22no—
39D844SD1614yescomplemented
40C22⋊D211—48no—
41C2×Dic211—22no—
42C2×Dic211—41no—
43C4×D211—42no—
44C4×D211—42no—
45C211⋊C8—22no—
46C23×C211—22no—
47D4×C211—42no—
48D4×C211—42no—
49D4×C211—42no—
50C2×C4×C211C2×D412yes—
51C2×C4×C211C2×D411yescharacteristic
52C8×C211—42no—
53C22×C211—44no—
54C22×C211—42no—
55C22×C211#12011yescharacteristic
56C2×D211—24no—
57C2×D211—42no—
58C2×D211—42no—
59C2×D211—88no—
60C4×C211—21no—
61C4×C211C2×SD1612yes—
62C4×C211AGL(1,ℤ/8)12yes—
63Dic211—42no—
64Dic211—81no—
65C2×C211—82no—
66C2×C211#45012yes—
67C2×C211#45311yescharacteristic
68D211—44no—
69C211#162911yescharacteristic, complemented, Sylow
70#1629—2111nomaximal, Sylow
71D42—2111no—
72#492—2111no—
73#476—2112no—
74#331D21111yescharacteristic, complemented
75D4⋊C8—2112no—
76C22×D4—4222no—
77C2×SD16—4222no—
78C4⋊Q8—2111no—
79#127C2×D21111yescharacteristic
80#121—4222no—
81#120—4222no—
82C4×D4—2112no—
83#106—4222no—
84C4⋊C8C2×D21112yescomplemented
85#102—4222no—
86C4×C8—2111no—
87C24—4222no—
88C2×Q8—4221no—
89C2×D4—22no—
90C2×D4—8444no—
91C2×D4—8444no—
92C2×D4—4224no—
93C2×D4—2112no—
94C2×D4D84412yes—
95C22×C4—4222no—
96SD16—8444no—
97C2×C8—22no—
98C2×C8—4221no—
99C4⋊C4—4222no—
100C4⋊C4—2112no—
101C22⋊C4—4222no—
102C22⋊C4—4222no—
103C42C22×D21111yescharacteristic
104C23—22no—
105C23—8448no—
106C23—16882no—
107C23—4222no—
108Q8—8442no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—8448no—
113D4—4222no—
114D4—2114no—
115C2×C4—8442no—
116C2×C4—8442no—
117C2×C4—8441no—
118C2×C4—4222no—
119C2×C4D4×D21112yes—
120C2×C4C2×D84411yescharacteristic
121C8—42no—
122C8—4222no—
123C22—42no—
124C22—44no—
125C22—8442no—
126C22—8442no—
127C22—16888no—
128C22—4224no—
129C22#2026994711yescharacteristic
130C4—21no—
131C4—8442no—
132C4—16881no—
133C4SD16×D21112yes—
134C4#2027000112yes—
135C2—82no—
136C2—8444no—
137C2#2721329812yes—
138C2#2721331611yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#331×C211C2
socleC22×C211#120
prime radical#27213781C2
hypercenter#331D211
hypocenterC211#1629
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27213518C2semidirect
3. #27213437C2semidirect
4. #27212873C2semidirect
5. #331×C211C2semidirect
8. #20269977C22semidirect
14. C4×C211⋊C8C22semidirect
16. C4⋊C8×C211C22semidirect
39. D844SD16semidirect
69. C211#1629semidirect
74. #331D211semidirect
84. C4⋊C8C2×D211semidirect

Maximal subgroups

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

subgroupindex
#272137812
#272135182
#272134372
#272128732
#331×C2112
#1629211