Finite Group Databasebeta

← #13154234

Subgroups — #13154234

Order 1728.

1210 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1210 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, complemented
2#11721764C214yesmaximal, complemented
3#11721763C214yesmaximal, complemented
4#11721762C214yesmaximal, complemented
5#11721721C212yesmaximal, complemented
6C3×#5700—34nomaximal
7C2×C4⋊#272C2214yes—
8C2×C4⋊#272C2212yes—
9C2×C4×#272C2214yes—
10C2×#6229C2212yes—
11C4⋊C4×H(3,𝔽3)C2212yes—
12C4⋊C4×H(3,𝔽3)C2214yes—
13C42×H(3,𝔽3)C2211yescharacteristic
14#88542C2214yes—
15#88542C2214yes—
16#88542C2214yes—
17#88540C2214yes—
18C32×#125D314yescomplemented
19C3×#896—316no—
20C3×#895—316no—
21C3×#894—316no—
22C3×#875—38no—
23C22×#272C2314yes—
24C2×C4×H(3,𝔽3)C2312yes—
25C2×C4×H(3,𝔽3)C2314yes—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084C2314yes—
28C4⋊#272—42no—
29C4⋊#272—44no—
30C4×#272—44no—
31#6229—42no—
32C3×#562—91no—
33#5700—34no—
34C2×C3×D12—316no—
35C2×C3×D12—38no—
36C2×C3×C4×D3—316no—
37C2×C3×Dic6—38no—
38C32×C4⋊C4C2×D318yes—
39C32×C4⋊C4C2×D3116yes—
40C32×C42C2×D314yes—
41C3×(C2×D3)⋊C4—316no—
42C3×(C2×D3)⋊C4—316no—
43C3×(C2×D3)⋊C4—316no—
44C3×Dic3⋊C4—316no—
45C22×H(3,𝔽3)C2411yescharacteristic
46C2×#272—24no—
47C2×#272—24no—
48C2×#272—24no—
49C4×H(3,𝔽3)—22no—
50C4×H(3,𝔽3)—24no—
51C4×H(3,𝔽3)—21no—
52#1084—24no—
53C3×#125—34no—
54C3×#125C3⋊D311yescharacteristic
55C3×#124—92no—
56C3×#123—94no—
57C3×C22⋊Q8—94no—
58C3×#121—94no—
59#896—316no—
60#895—316no—
61#894—316no—
62#875—38no—
63C22×C3×D3—316no—
64C2×C32×C4C22×D3116yes—
65C2×C32×C4C22×D314yes—
66C2×C32×C4C22×D318yes—
67C2×C3×Dic3—316no—
68C3×D12—128no—
69C3×D12—1216no—
70C3×C4×D3—1216no—
71C3×Dic6—128no—
72#562—91noSylow
73C2×H(3,𝔽3)H(5,𝔽2)12yes—
74C2×H(3,𝔽3)2−1+411yescharacteristic
75#272—44no—
76C2×C3×Q8—92no—
77C2×C3×D4—94no—
78C2×C3×D4—92no—
79C22×C3×C4—94no—
80C2×D12—38no—
81C2×D12—316no—
82C2×C4×D3—316no—
83C2×Dic6—38no—
84C3×C4⋊C4—38no—
85C3×C4⋊C4—316no—
86C3×C4⋊C4—94no—
87C3×C4⋊C4C2×C3⋊D314yes—
88C3×C4⋊C4C2×C3⋊D312yes—
89C3×C22⋊C4—94no—
90C3×C22⋊C4—94no—
91C3×C22⋊C4—94no—
92C3×C42—34no—
93C3×C42C2×C3⋊D311yescharacteristic
94(C2×D3)⋊C4—316no—
95(C2×D3)⋊C4—316no—
96(C2×D3)⋊C4—316no—
97Dic3⋊C4—316no—
98C22×C32C23×D314yes—
99C2×C3×D3—616no—
100C2×C3×D3—616no—
101C2×C3×D3—616no—
102C32×C4—24no—
103C32×C4—28no—
104C32×C4—216no—
105C3×Dic3—616no—
106#125#27211yescharacteristic, complemented
107#124—92no—
108#123—94no—
109C22⋊Q8—94no—
110#121—94no—
111H(3,𝔽3)#56211yescharacteristic, complemented, Sylow
112C23×C3—94no—
113C22×D3—316no—
114C3×Q8—362no—
115C3×D4—364no—
116C3×D4—362no—
117C2×C3×C4—316no—
118C2×C3×C4—34no—
119C2×C3×C4—38no—
120C2×C3×C4—94no—
121C2×C3×C4—364no—
122C2×C3×C4C22×C3⋊D311yescharacteristic
123C2×C3×C4C22×C3⋊D312yes—
124C2×C3×C4C22×C3⋊D314yes—
125C2×Dic3—316no—
126D12—128no—
127D12—1216no—
128C4×D3—1216no—
129Dic6—128no—
130C2×C32#100714yes—
131C2×C32#100918yes—
132C3×D3—1216no—
133C2×Q8—92no—
134C2×D4—94no—
135C2×D4—92no—
136C22×C4—94no—
137C4⋊C4—94no—
138C4⋊C4C2×#27214yes—
139C4⋊C4C2×#27212yes—
140C22⋊C4—94no—
141C22⋊C4—94no—
142C22⋊C4—94no—
143C42C2×#27211yescharacteristic
144C22×C3—34no—
145C22×C3—184no—
146C22×C3—184no—
147C22×C3—184no—
148C22×C3C23×C3⋊D311yescharacteristic
149C2×D3—616no—
150C2×D3—616no—
151C2×D3—616no—
152C3×C4—21no—
153C3×C4—24no—
154C3×C4—22no—
155C3×C4—68no—
156C3×C4—616no—
157C3×C4—64no—
158C3×C4—184no—
159Dic3—616no—
160C32#570014yescomplemented
161C23—94no—
162Q8—362no—
163D4—364no—
164D4—362no—
165C2×C4—94no—
166C2×C4—364no—
167C2×C4C22×#27211yescharacteristic
168C2×C4C22×#27212yes—
169C2×C4C22×#27214yes—
170C2×C3—34no—
171C2×C3—38no—
172C2×C3—364no—
173C2×C3#6436411yescharacteristic
174C2×C3#6436612yes—
175D3—1216no—
176C22—184no—
177C22—184no—
178C22—184no—
179C22C23×#27211yescharacteristic
180C4—21no—
181C4—24no—
182C4—22no—
183C4—184no—
184C3—34no—
185C3#1059617111yescharacteristic
186C2—364no—
187C2#1172267011yescharacteristic
188C2#1172267512yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#562
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
1. H(3,𝔽3)×#125C2semidirect
2. #11721764C2semidirect
3. #11721763C2semidirect
4. #11721762C2semidirect
5. #11721721C2semidirect
18. C32×#125D3semidirect
106. #125#272semidirect
111. H(3,𝔽3)#562semidirect
160. C32#5700semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217642
#117217632
#117217622
#117217212
C3×#57003