Finite Group Databasebeta

← #13146608

Subgroups — #13146608

Order 1728.

319 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 49 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
1C2×#88589C214yesmaximal
2#11721117C212yesmaximal
3C4×#6188C211yesmaximal, characteristic
4C2×#6196C2214yes—
5C2×#6188C2212yes—
6#88589—28no—
7C4×#1084C2211yescharacteristic
8C4⋊((C3⋊Dic3)⋊C3)—22no—
9#88449—22no—
10#63918—62no—
11C2×C4×H(3,𝔽3)D411yescharacteristic
12C2×#1084D411yescharacteristic
13C2×#1084C2311yescharacteristic
14C2×(C3⋊Dic3)⋊C3—24no—
15#6196—28no—
16#6188D414yes—
17#4699—91nomaximal
18C2×#627—64no—
19C4⋊(C3⋊Dic3)—22no—
20C3×C4⋊Dic3—62no—
21C3×C4×Dic3—62no—
22Dic3⋊Dic3—64no—
23C22×H(3,𝔽3)C2×D411yescharacteristic
24C4×H(3,𝔽3)Q1612yescomplemented
25#1084Q1612yescomplemented
26#1084C2×D412yes—
27(C3⋊Dic3)⋊C3—44no—
28C2×Dic12—94no—
29#888—182no—
30#869—92no—
31C3×C4×C8—91no—
32C2×C32×C4—22no—
33C2×C3⋊Dic3—24no—
34C2×C3×Dic3—62no—
35C2×C3×Dic3—62no—
36C2×C3×Dic3—64no—
37#627—128no—
38#494—271noSylow
39C2×H(3,𝔽3)#12711yescharacteristic
40C2×H(3,𝔽3)C2×Q1611yescharacteristic
41C2×H(3,𝔽3)C2×Q1611yescharacteristic
42C2×Dic6—94no—
43C2×Dic6—184no—
44C2×C3×C8—92no—
45C3×C4⋊C4—182no—
46C3×C42—91no—
47C4⋊Dic3—62no—
48C4⋊Dic3—182no—
49C4⋊Dic3—182no—
50Dic3⋊C4—184no—
51C4×Dic3—62no—
52Dic12—188no—
53C22×C32—22no—
54C32×C4—24no—
55C3⋊Dic3—44no—
56C3×Dic3—64no—
57C3×Dic3—64no—
58C3×Dic3—124no—
59C2×Q16—274no—
60C4⋊Q8—272no—
61C4×C8—91no—
62H(3,𝔽3)#49411yescharacteristic, complemented, Sylow
63C2×C3×C4—62no—
64C2×C3×C4—91no—
65C2×C3×C4—91no—
66C2×C3×C4—184no—
67C2×C3×C4GO+(4,𝔽2)11yescharacteristic
68C2×Dic3—62no—
69C2×Dic3—62no—
70C2×Dic3—64no—
71C2×Dic3—184no—
72Dic6—188no—
73Dic6—368no—
74C3×C8—94no—
75C2×C32—22no—
76C2×C32—22no—
77C2×C32—22no—
78C2×Q8—274no—
79Q16—548no—
80C2×C8—92no—
81C4⋊C4—182no—
82C4⋊C4—542no—
83C42—91no—
84C22×C3—62no—
85C22×C3C2×GO+(4,𝔽2)11yescharacteristic
86C3×C4—64no—
87C3×C4—92no—
88C3×C4—92no—
89C3×C4—364no—
90C3×C4#377612yes—
91Dic3—64no—
92Dic3—64no—
93Dic3—124no—
94Dic3—364no—
95C32—22no—
96Q8—548no—
97C2×C4—91no—
98C2×C4—91no—
99C2×C4—184no—
100C2×C4#1088⋊C211yescharacteristic
101C8—94no—
102C2×C3—62no—
103C2×C3—62no—
104C2×C3—62no—
105C2×C3#6422811yescharacteristic
106C2×C3#6423111yescharacteristic
107C2×C3C2×#377611yescharacteristic
108C22C2×#1088⋊C211yescharacteristic
109C4—92no—
110C4—92no—
111C4—364no—
112C4#8858912yescomplemented
113C3—62no—
114C3#1059345011yescharacteristic
115C2#1172229911yescharacteristic
116C2#1172230511yescharacteristic
117C2C2×#8858911yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocleC2×#1084C23
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#494
LCSG ▷ C2×#1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
24. C4×H(3,𝔽3)Q16semidirect
25. #1084Q16semidirect
62. H(3,𝔽3)#494semidirect
112. C4#88589semidirect

Maximal subgroups

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

subgroupindex
C2×#885892
#117211172
C4×#61882
#46999