Finite Group Databasebeta

← #125×#272 #13154222

Subgroups — #125×#272 #13154222

Order 1728.

1243 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 1243 conjugacy classes of subgroups — too many to draw. Only its 221 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
2C4⋊C4×#272C214yesmaximal
3C4⋊C4×#272C212yesmaximal
4#11721758C212yesmaximal, complemented
5#11721757C214yesmaximal, complemented
6C42×#272C211yesmaximal, characteristic
7#11721716C211yesmaximal, characteristic, complemented
8C3×D3×#125—34nomaximal
9C2×C4×#272C2214yes—
10C2×C4×#272C2212yes—
11C2×C4×#272C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2214yes—
13C4⋊C4×H(3,𝔽3)C2212yes—
14C42×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2214yes—
16C4⋊#1084C2212yes—
17#88540C2214yes—
18#88540C2214yes—
19#88540C2214yes—
20C4×#1084C2212yes—
21C4×#1084C2211yescharacteristic
22C32×#125D314yescomplemented
23C3×D3×C4⋊C4—38no—
24C3×D3×C4⋊C4—316no—
25C3×#890—38no—
26C3×#889—316no—
27C3×C42×D3—34no—
28C3×#870—34no—
29C22×#272C2311yescharacteristic
30C2×C4×H(3,𝔽3)C2312yes—
31C2×C4×H(3,𝔽3)C2314yes—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084C2312yes—
34C2×#1084C2314yes—
35C2×#1084C2311yescharacteristic
36C4×#272—28no—
37C4×#272—24no—
38C4×#272Q814yes—
39C2×C3×#125—91no—
40D3×#125—34no—
41C2×C3×C4×D3—34no—
42C2×C3×C4×D3—316no—
43C2×C3×C4×D3—38no—
44C32×C4⋊C4C2×D3116yes—
45C32×C4⋊C4C2×D318yes—
46C32×C42C2×D314yes—
47C3×C4⋊Dic3—316no—
48C3×C4⋊Dic3—38no—
49C3×Dic3⋊C4—316no—
50C3×Dic3⋊C4—316no—
51C3×Dic3⋊C4—316no—
52C3×C4×Dic3—38no—
53C3×C4×Dic3—34no—
54C22×H(3,𝔽3)C2411yescharacteristic
55C2×#272C2×Q812yes—
56C2×#272#4114yes—
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)—24no—
59C4×H(3,𝔽3)C2×Q812yes—
60#1084—22no—
61#1084—24no—
62#1084C2×Q812yes—
63C3×#125—34no—
64C3×#125—94no—
65C3×#125—92no—
66C3×#125—91no—
67C3×#125C3⋊D311yescharacteristic
68C2×C3×C4⋊C4—92no—
69C2×C3×C4⋊C4—94no—
70C2×C3×C42—91no—
71D3×C4⋊C4—316no—
72D3×C4⋊C4—38no—
73#890—38no—
74#889—316no—
75C42×D3—34no—
76#870—34no—
77C22×C3×D3—34no—
78C2×C32×C4C22×D318yes—
79C2×C32×C4C22×D3116yes—
80C2×C32×C4C22×D314yes—
81C2×C3×Dic3—34no—
82C2×C3×Dic3—38no—
83C2×C3×Dic3—316no—
84C3×C4×D3—316no—
85C3×C4×D3—632no—
86C3×C4×D3—616no—
87C2×#125—91noSylow
88C2×H(3,𝔽3)C22×Q811yescharacteristic
89C2×H(3,𝔽3)C2×#4112yes—
90#272#12514yesdirect product factor
91C22×C3×C4—91no—
92C22×C3×C4—92no—
93C22×C3×C4—94no—
94C2×C4×D3—34no—
95C2×C4×D3—38no—
96C2×C4×D3—316no—
97C3×C4⋊C4—316no—
98C3×C4⋊C4—38no—
99C3×C4⋊C4—94no—
100C3×C4⋊C4—94no—
101C3×C4⋊C4—94no—
102C3×C4⋊C4—94no—
103C3×C4⋊C4—92no—
104C3×C4⋊C4C2×C3⋊D314yes—
105C3×C4⋊C4C2×C3⋊D312yes—
106C3×C42—34no—
107C3×C42—92no—
108C3×C42—91no—
109C3×C42C2×C3⋊D311yescharacteristic
110C4⋊Dic3—38no—
111C4⋊Dic3—316no—
112Dic3⋊C4—316no—
113Dic3⋊C4—316no—
114Dic3⋊C4—316no—
115C4×Dic3—38no—
116C4×Dic3—34no—
117C22×C32C23×D314yes—
118C2×C3×D3—38no—
119C2×C3×D3—316no—
120C32×C4—28no—
121C32×C4—216no—
122C32×C4D3×Q818yes—
123C3×Dic3—38no—
124C3×Dic3—68no—
125C3×Dic3—616no—
126#125—94no—
127#125—92no—
128#125—91no—
129#125#27211yescharacteristic, direct product factor
130C2×C4⋊C4—92no—
131C2×C4⋊C4—94no—
132C2×C42—91no—
133H(3,𝔽3)C2×#12511yescharacteristic, complemented, Sylow
134C23×C3—91no—
135C22×D3—34no—
136C2×C3×C4—34no—
137C2×C3×C4—38no—
138C2×C3×C4—316no—
139C2×C3×C4—91no—
140C2×C3×C4—94no—
141C2×C3×C4—92no—
142C2×C3×C4—94no—
143C2×C3×C4—184no—
144C2×C3×C4—188no—
145C2×C3×C4C22×C3⋊D312yes—
146C2×C3×C4C22×C3⋊D314yes—
147C2×C3×C4C22×C3⋊D311yescharacteristic
148C2×Dic3—34no—
149C2×Dic3—38no—
150C2×Dic3—316no—
151C4×D3—316no—
152C4×D3—616no—
153C4×D3—632no—
154C2×C32C2×D3×Q814yes—
155C2×C32D3×#4118yes—
156C3×D3—316no—
157C22×C4—91no—
158C22×C4—92no—
159C22×C4—94no—
160C4⋊C4—94no—
161C4⋊C4—94no—
162C4⋊C4—94no—
163C4⋊C4—94no—
164C4⋊C4—92no—
165C4⋊C4C2×#27214yes—
166C4⋊C4C2×#27212yes—
167C42—92no—
168C42—91no—
169C42C2×#27211yescharacteristic
170C22×C3—34no—
171C22×C3—92no—
172C22×C3—94no—
173C22×C3C23×C3⋊D311yescharacteristic
174C2×D3—38no—
175C2×D3—316no—
176C3×C4—22no—
177C3×C4—24no—
178C3×C4—38no—
179C3×C4—68no—
180C3×C4—616no—
181C3×C4—92no—
182C3×C4—182no—
183C3×C4—184no—
184C3×C4Q8×C3⋊D312yes—
185Dic3—38no—
186Dic3—68no—
187Dic3—616no—
188C32D3×#12514yescomplemented
189C23—91no—
190C2×C4—91no—
191C2×C4—94no—
192C2×C4—92no—
193C2×C4—94no—
194C2×C4—184no—
195C2×C4—188no—
196C2×C4C22×#27212yes—
197C2×C4C22×#27214yes—
198C2×C4C22×#27211yescharacteristic
199C2×C3—34no—
200C2×C3—38no—
201C2×C3—94no—
202C2×C3C2×Q8×C3⋊D311yescharacteristic
203C2×C3#41×C3⋊D312yes—
204D3—316no—
205C22—92no—
206C22—94no—
207C22C23×#27211yescharacteristic
208C4—22no—
209C4—24no—
210C4—92no—
211C4—182no—
212C4—184no—
213C4Q8×#27212yes—
214C3—34no—
215C3C3⋊D3×#12511yescharacteristic
216C2—94no—
217C2C2×Q8×#27211yescharacteristic
218C2#41×#27212yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
prime radicalC22×#272C23
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)C2×#125
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
4. #11721758C2semidirect
5. #11721757C2semidirect
7. #11721716C2semidirect
22. C32×#125D3semidirect
90. #272#125direct (N × H)
129. #125#272direct (N × H)
133. H(3,𝔽3)C2×#125semidirect
188. C32D3×#125semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
C4⋊C4×#2722
#117217582
#117217572
C42×#2722
#117217162
C3×D3×#1253