Finite Group Databasebeta

← #41013059

Subgroups — #41013059

Order 27008.

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

normal subgroups only

This group has 186 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#27213542C211yesmaximal, characteristic, complemented
2#27213525C211yesmaximal, characteristic, complemented
3#27213497C211yesmaximal, characteristic, complemented
4#27213492C211yesmaximal, characteristic
5#27213492C211yesmaximal, characteristic
6#27212867C211yesmaximal, characteristic, complemented
7#486×C211C211yesmaximal, characteristic
8C2×#16809952—22no—
9#20269977C2211yescharacteristic
10#20269977C2211yescharacteristic
11#20269908—22no—
12#20269902—22no—
13#20269898—22no—
14#20269898—22no—
15C4×C211⋊C8C2211yescharacteristic
16C4×C211⋊C8C2211yescharacteristic
17C4⋊Q8×C211C2211yescharacteristic
18#127×C211C2211yescharacteristic
19#102×C211—22no—
20C4×C8×C211C2211yescharacteristic
21C2×Dic422—21no—
22C2×Dic422—21no—
23#16809952—44no—
24C2×C211⋊C8D412yes—
25C2×C211⋊C8D412yes—
26C4⋊Dic211—21no—
27C4⋊Dic211—21no—
28C4⋊Dic211—21no—
29C4⋊Dic211—21no—
30C2×Q8×C211—21no—
31C2×D4×C211—21no—
32C2×D4×C211—21no—
33C2×D4×C211—21no—
34C2×C8×C211D412yes—
35C4⋊C4×C211—21no—
36C4⋊C4×C211—21no—
37C42×C211C2311yescharacteristic
38Dic422—42no—
39Dic422—42no—
40C2×Dic211—41no—
41C2×Dic211—41no—
42C211⋊C8—22no—
43C211⋊C8—22no—
44C23×C211—41no—
45Q8×C211—42no—
46D4×C211—42no—
47D4×C211—42no—
48D4×C211—42no—
49C2×C4×C211—41no—
50C2×C4×C211C2×D411yescharacteristic
51C2×C4×C211C2×D411yescharacteristic
52C2×C4×C211C2×D411yescharacteristic
53C8×C211—22no—
54C22×C211—81no—
55C22×C211—81no—
56C22×C211—81no—
57C22×C211#12711yescharacteristic
58C4×C211—81no—
59C4×C211#13512yes—
60C4×C211AGL(1,ℤ/8)12yes—
61C4×C211#13712yes—
62Dic211—81no—
63Dic211—81no—
64C2×C211—161no—
65C2×C211#49511yescharacteristic
66C2×C211#49611yescharacteristic
67C2×C211#49711yescharacteristic
68C211#172011yescharacteristic, complemented, Sylow
69#1720—2111nomaximal, Sylow
70#498—2111no—
71#494—2111no—
72#487—2111no—
73#486—2111no—
74#486—2111no—
75#486D21111yescharacteristic
76#322—2111no—
77C2×Q16—4222no—
78C4⋊Q8—2111no—
79C4⋊Q8—2111no—
80C4⋊Q8C2×D21111yescharacteristic
81#127C2×D21111yescharacteristic
82#106—4222no—
83#103—4222no—
84#102—22no—
85#102—4222no—
86#102—4222no—
87C4×C8—2111no—
88C4×C8—2111no—
89C4×C8C2×D21111yescharacteristic
90C2×Q8—21no—
91C2×Q8—4221no—
92C2×Q8—4221no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96Q16—8444no—
97C2×C8—2112no—
98C2×C8—2112no—
99C2×C8C22⋊D21112yes—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—4221no—
103C4⋊C4—4221no—
104C4⋊C4—4221no—
105C4⋊C4—4221no—
106C42C22×D21111yescharacteristic
107C23—41no—
108Q8—42no—
109Q8—8442no—
110Q8—8442no—
111D4—42no—
112D4—42no—
113D4—42no—
114C2×C4—41no—
115C2×C4—8441no—
116C2×C4—8441no—
117C2×C4D4×D21111yescharacteristic
118C2×C4D4×D21111yescharacteristic
119C2×C4C2×C22⋊D21111yescharacteristic
120C8—22no—
121C8—4222no—
122C8—4222no—
123C22—81no—
124C22—81no—
125C22—81no—
126C22#2026997311yescharacteristic
127C4—81no—
128C4—16881no—
129C4—16881no—
130C4#2026999712yes—
131C4#2027000112yes—
132C4#2027000612yes—
133C2—161no—
134C2#2721352811yescharacteristic
135C2#2721353411yescharacteristic
136C2#2721353711yescharacteristic
subgroupG/H
centerC22#20269973
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#486×C211C2
socleC22×C211#127
prime radical#127×C211C22
hypercenter#486D211
hypocenterC211#1720
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #27213542C2semidirect
2. #27213525C2semidirect
3. #27213497C2semidirect
6. #27212867C2semidirect
68. C211#1720semidirect

Maximal subgroups

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

subgroupindex
#272135422
#272135252
#272134972
#272134922
#272128672
#486×C2112
#1720211