Finite Group Databasebeta

← #41013072

Subgroups — #41013072

Order 27008.

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

normal subgroups only

This group has 176 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#27213551C211yesmaximal, characteristic
2#27213549C211yesmaximal, characteristic
3#27213525C211yesmaximal, characteristic
4#27213498C211yesmaximal, characteristic
5#27213496C211yesmaximal, characteristic
6#27212867C211yesmaximal, characteristic
7#498×C211C211yesmaximal, characteristic
8C2×#16809952—22no—
9#20269977C2211yescharacteristic
10#20269977C2211yescharacteristic
11#20269911—22no—
12#20269910—22no—
13#20269901—22no—
14#20269900—22no—
15C4×C211⋊C8C2211yescharacteristic
16C4×C211⋊C8C2211yescharacteristic
17C4⋊Q8×C211C2211yescharacteristic
18C4⋊Q8×C211C2211yescharacteristic
19#106×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×Q8×C211—21no—
32C2×C8×C211D412yes—
33C4⋊C4×C211—21no—
34C4⋊C4×C211—21no—
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—
44Q8×C211—42no—
45Q8×C211—42no—
46C2×C4×C211—41no—
47C2×C4×C211—41no—
48C2×C4×C211C2×D411yescharacteristic
49C2×C4×C211C2×D411yescharacteristic
50C2×C4×C211C2×D411yescharacteristic
51C8×C211Q1614yescomplemented
52C22×C211#12711yescharacteristic
53C4×C211—81no—
54C4×C211—81no—
55C4×C211C2×Q1612yes—
56C4×C211AGL(1,ℤ/8)12yes—
57C4×C211#13712yes—
58Dic211—81no—
59Dic211—81no—
60C2×C211#49411yescharacteristic
61C2×C211#49611yescharacteristic
62C2×C211#49711yescharacteristic
63C211#172311yescharacteristic, complemented, Sylow
64#1723—2111nomaximal, Sylow
65#500—2111no—
66#500—2111no—
67#498D21111yescharacteristic
68#494—2111no—
69#487—2111no—
70#487—2111no—
71#322—2111no—
72C2×Q16—4222no—
73C4⋊Q8—2111no—
74C4⋊Q8—2111no—
75C4⋊Q8C2×D21111yescharacteristic
76C4⋊Q8C2×D21111yescharacteristic
77#107—4222no—
78#107—4222no—
79#106—22no—
80#103—4222no—
81#103—4222no—
82C4×C8—2111no—
83C4×C8—2111no—
84C4×C8C2×D21111yescharacteristic
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—4221no—
88C2×Q8—4221no—
89Q16—8444no—
90C2×C8—2112no—
91C2×C8—2112no—
92C2×C8C22⋊D21112yes—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—4221no—
98C4⋊C4—4221no—
99C4⋊C4—4221no—
100C4⋊C4—4221no—
101C42C22×D21111yescharacteristic
102Q8—42no—
103Q8—42no—
104Q8—8442no—
105Q8—8442no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—8441no—
109C2×C4—8441no—
110C2×C4D4×D21111yescharacteristic
111C2×C4D4×D21111yescharacteristic
112C2×C4C2×C22⋊D21111yescharacteristic
113C8—4222no—
114C8—4222no—
115C8#1680995214yescomplemented
116C22#2026997311yescharacteristic
117C4—81no—
118C4—81no—
119C4—16881no—
120C4—16881no—
121C4C2×#1680995212yes—
122C4#2027000412yes—
123C4#2027000612yes—
124C2#2721352511yescharacteristic
125C2#2721353211yescharacteristic
126C2#2721353711yescharacteristic
subgroupG/H
centerC22#20269973
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#498×C211C2
socle, prime radicalC22×C211#127
hypercenter#498D211
hypocenterC211#1723
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
51. C8×C211Q16semidirect
63. C211#1723semidirect
115. C8#16809952semidirect

Maximal subgroups

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

subgroupindex
#272135512
#272135492
#272135252
#272134982
#272134962
#272128672
#498×C2112
#1723211