Finite Group Databasebeta

← #13154221

Subgroups — #13154221

Order 1728.

946 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 191 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic
2#11721758C214yesmaximal
3#11721757C214yesmaximal
4#11721757C214yesmaximal
5#11721716C212yesmaximal
6C3×C3⋊#564—34nomaximal
7C4⋊C4×H(3,𝔽3)C2212yes—
8C4⋊C4×H(3,𝔽3)C2214yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084C2212yes—
11C4⋊#1084C2214yes—
12C4⋊#1084C2212yes—
13#88540C2214yes—
14#88540C2214yes—
15#88540C2214yes—
16#88540C2214yes—
17C4×#1084C2214yes—
18C32×#125D314yes—
19C3×#890—316no—
20C3×#889—316no—
21C3×#889—316no—
22C3×#870—38no—
23C2×C4×H(3,𝔽3)C2314yes—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×C4×H(3,𝔽3)C2312yes—
26C2×#1084C2314yes—
27C2×#1084C2314yes—
28C3×#564—91no—
29C3⋊#564—34no—
30C32×C4⋊C4C2×D318yes—
31C32×C4⋊C4C2×D3116yes—
32C32×C42C2×D314yes—
33C3×C4⋊Dic3—38no—
34C3×C4⋊Dic3—316no—
35C3×C4⋊Dic3—38no—
36C3×Dic3⋊C4—316no—
37C3×Dic3⋊C4—316no—
38C3×Dic3⋊C4—316no—
39C3×Dic3⋊C4—316no—
40C3×C4×Dic3—316no—
41C22×H(3,𝔽3)C2411yescharacteristic
42C4×H(3,𝔽3)—21no—
43C4×H(3,𝔽3)—24no—
44C4×H(3,𝔽3)—22no—
45#1084—24no—
46#1084—24no—
47C3×#125—34no—
48C3×#125—94no—
49C3×#125—94no—
50C3×#125—94no—
51C3×#125—92no—
52C3×#125C3⋊D311yescharacteristic
53#890—316no—
54#889—316no—
55#889—316no—
56#870—38no—
57C2×C32×C4C22×D3116yes—
58C2×C32×C4C22×D314yes—
59C2×C32×C4C22×D318yes—
60C2×C3×Dic3—316no—
61C2×C3×Dic3—316no—
62#564—91noSylow
63C2×H(3,𝔽3)2−1+411yescharacteristic
64C2×H(3,𝔽3)2−1+412yes—
65C3×C4⋊C4—38no—
66C3×C4⋊C4—316no—
67C3×C4⋊C4—94no—
68C3×C4⋊C4—94no—
69C3×C4⋊C4—94no—
70C3×C4⋊C4—94no—
71C3×C4⋊C4—94no—
72C3×C4⋊C4—92no—
73C3×C4⋊C4—92no—
74C3×C4⋊C4C2×C3⋊D312yes—
75C3×C4⋊C4C2×C3⋊D314yes—
76C3×C42—34no—
77C3×C42—94no—
78C3×C42C2×C3⋊D311yescharacteristic
79C4⋊Dic3—38no—
80C4⋊Dic3—38no—
81C4⋊Dic3—316no—
82Dic3⋊C4—316no—
83Dic3⋊C4—316no—
84Dic3⋊C4—316no—
85Dic3⋊C4—316no—
86C4×Dic3—316no—
87C22×C32C23×D314yes—
88C32×C4—24no—
89C32×C4—216no—
90C32×C4—28no—
91C3×Dic3—616no—
92C3×Dic3—616no—
93#125—94no—
94#125—94no—
95#125—94no—
96#125—92no—
97#125#27211yescharacteristic
98H(3,𝔽3)#56411yescharacteristic, complemented, Sylow
99C2×C3×C4—316no—
100C2×C3×C4—34no—
101C2×C3×C4—38no—
102C2×C3×C4—94no—
103C2×C3×C4—94no—
104C2×C3×C4C22×C3⋊D314yes—
105C2×C3×C4C22×C3⋊D311yescharacteristic
106C2×C3×C4C22×C3⋊D312yes—
107C2×Dic3—316no—
108C2×Dic3—316no—
109C2×C32#100714yes—
110C2×C32#101018yes—
111C4⋊C4—94no—
112C4⋊C4—94no—
113C4⋊C4—94no—
114C4⋊C4—94no—
115C4⋊C4—94no—
116C4⋊C4—92no—
117C4⋊C4—92no—
118C4⋊C4C2×#27212yes—
119C4⋊C4C2×#27214yes—
120C42—94no—
121C42C2×#27211yescharacteristic
122C22×C3—34no—
123C22×C3C23×C3⋊D311yescharacteristic
124C3×C4—21no—
125C3×C4—24no—
126C3×C4—22no—
127C3×C4—64no—
128C3×C4—616no—
129C3×C4—68no—
130C3×C4—184no—
131C3×C4—184no—
132Dic3—616no—
133Dic3—616no—
134C32C3⋊#56414yescomplemented
135C2×C4—94no—
136C2×C4—94no—
137C2×C4C22×#27214yes—
138C2×C4C22×#27211yescharacteristic
139C2×C4C22×#27212yes—
140C2×C3—34no—
141C2×C3—38no—
142C2×C3#6436411yescharacteristic
143C2×C3#6436712yes—
144C22C23×#27211yescharacteristic
145C4—21no—
146C4—24no—
147C4—22no—
148C4—184no—
149C4—184no—
150C3—34no—
151C3#1059615711yescharacteristic
152C2#1172267011yescharacteristic
153C2#1172267612yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#564
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
98. H(3,𝔽3)#564semidirect
134. C32C3⋊#564semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217582
#117217572
#117217162
C3×C3⋊#5643