Finite Group Databasebeta

← #13143807

Subgroups — #13143807

Order 1728.

495 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 109 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#11721716C211yesmaximal, characteristic
2#11721715C211yesmaximal, characteristic
3H(3,𝔽3)×#97C211yesmaximal, characteristic
4#11720549C214yesmaximal
5C3×#4703—34nomaximal
6C2×#6229C2212yes—
7C2×C8×H(3,𝔽3)C2212yes—
8C42×H(3,𝔽3)C2211yescharacteristic
9C4⋊#1084—22no—
10C4⋊#1084C2212yes—
11#88540—22no—
12C3×#870—34no—
13C3×#869—34no—
14C32×#97D314yes—
15C3×#816—316no—
16C2×C4×H(3,𝔽3)D412yes—
17C2×C4×H(3,𝔽3)C2311yescharacteristic
18C2×#1084—22no—
19C2×#1084—22no—
20#6229—24no—
21C8×H(3,𝔽3)—22no—
22C3×#491—91no—
23#4703—34no—
24C2×C3×Dic6—38no—
25C2×C32×C8C2×D318yes—
26C32×C42C2×D314yes—
27C3×C4⋊Dic3—38no—
28C3×C4⋊Dic3—68no—
29C3×Dic3⋊C4—68no—
30C22×H(3,𝔽3)C2×D411yescharacteristic
31C4×H(3,𝔽3)—22no—
32C4×H(3,𝔽3)#4112yes—
33#1084—42no—
34#1084—42no—
35C3×C4⋊Q8—91no—
36C3×#125—91no—
37#870—34no—
38#869—34no—
39C3×#103—94no—
40C3×#97—34no—
41C3×#97C3⋊D311yescharacteristic
42#816—316no—
43C2×C32×C4D1218yes—
44C2×C32×C4C22×D314yes—
45C2×C3×Dic3—68no—
46C2×C3×Dic3—68no—
47C3×Dic6—616no—
48C32×C8—28no—
49#491—91noSylow
50C2×H(3,𝔽3)#12411yescharacteristic
51C2×H(3,𝔽3)#13712yes—
52C2×C3×Q8—92no—
53C2×Dic6—38no—
54C2×C3×C8—38no—
55C2×C3×C8C2×C3⋊D312yes—
56C3×C4⋊C4—92no—
57C3×C4⋊C4—182no—
58C3×C4⋊C4—182no—
59C3×C42—34no—
60C3×C42C2×C3⋊D311yescharacteristic
61C4⋊Dic3—38no—
62C4⋊Dic3—68no—
63Dic3⋊C4—68no—
64C22×C32C2×D1214yes—
65C32×C4—28no—
66C32×C4#23518yes—
67C3×Dic3—128no—
68C3×Dic3—128no—
69C4⋊Q8—91no—
70#125—91no—
71#103—94no—
72#97#27211yescharacteristic
73H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
74C3×Q8—184no—
75C2×C3×C4—38no—
76C2×C3×C4—34no—
77C2×C3×C4—182no—
78C2×C3×C4—182no—
79C2×C3×C4C4⋊(C3⋊D3)12yes—
80C2×C3×C4C22×C3⋊D311yescharacteristic
81C2×Dic3—68no—
82C2×Dic3—68no—
83Dic6—616no—
84C3×C8—22no—
85C3×C8—68no—
86C2×C32#87514yes—
87C2×C32#90918yes—
88C2×Q8—92no—
89C2×C8C2×#27212yes—
90C4⋊C4—92no—
91C4⋊C4—182no—
92C4⋊C4—182no—
93C42C2×#27211yescharacteristic
94C22×C3—34no—
95C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
96C3×C4—22no—
97C3×C4—38no—
98C3×C4—68no—
99C3×C4—362no—
100C3×C4—362no—
101C3×C4#382812yes—
102Dic3—128no—
103Dic3—128no—
104C32#470314yescomplemented
105Q8—184no—
106C2×C4—182no—
107C2×C4—182no—
108C2×C4C4⋊#27212yes—
109C2×C4C22×#27211yescharacteristic
110C8—22no—
111C2×C3—38no—
112C2×C3—34no—
113C2×C3#6408411yescharacteristic
114C2×C3#6411812yes—
115C22C2×C4⋊#27211yescharacteristic
116C4—22no—
117C4—362no—
118C4—362no—
119C4#8874012yes—
120C3—34no—
121C3#1059240211yescharacteristic
122C2#1172172111yescharacteristic
123C2#1172178812yes—
subgroupG/H
center, socleC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#97C2
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC3×#97C3⋊D3
hypocenterH(3,𝔽3)#491
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
73. H(3,𝔽3)#491semidirect
104. C32#4703semidirect

Maximal subgroups

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

subgroupindex
#117217162
#117217152
H(3,𝔽3)×#972
#117205492
C3×#47033