Finite Group Databasebeta

← #13147231

Subgroups — #13147231

Order 1728.

339 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 28 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#127⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11720758C212yesmaximal, complemented
3#10596191C311yesmaximal, characteristic, complemented
4#5720⋊C3—31nomaximal
5C2×(C3⋊D3)⋊A4—41nomaximal
6C32⋊(C42⋊C3)C2211yescharacteristic, complemented
7C32×#127C2×C311yescharacteristic, complemented
8#64148—32no—
9#64146—31no—
10#64144—31no—
11#64085C2×C312yescomplemented
12C3×#127⋊C3—21no—
13C3×#127⋊C3D311yescharacteristic, complemented
14C3⋊(#126⋊C3)—32no—
15C2×C22⋊H(3,𝔽3)—41no—
16(C3⋊D3)⋊A4—42no—
17#5720—31no—
18#5720—91no—
19#560⋊C3—91no—
20C23×C3⋊D3A411yescharacteristic
21C2×D3×A4—121no—
22C2×C32×D4—31no—
23C2×C32×D4—31no—
24C2×#638—32no—
25C22×C3⋊Dic3—31no—
26C32×C42C22×C311yescharacteristic, complemented
27#3757—31no—
28#3757—31no—
29#3752—32no—
30#3750—32no—
31C3×C42⋊C3—21no—
32C3×C42⋊C3C2×D311yescharacteristic, complemented
33C2×(C3⋊D3)⋊C3—161no—
34C22⋊H(3,𝔽3)—41no—
35C3×#127—31no—
36C3×#127C3×D311yescharacteristic
37#939—96no—
38#939—92no—
39#937—91no—
40#937—93no—
41#935—93no—
42#935—91no—
43#876—32no—
44#876—92no—
45#127⋊C3—31no—
46#127⋊C3—61no—
47#126⋊C3—92no—
48C23×C32—31no—
49C23×C32C2×A411yescharacteristic
50C22×C3⋊D3—121no—
51C22×C3⋊D3C2×A412yes—
52C2×C3×A4—41no—
53C2×C3×A4—81no—
54D3×A4—122no—
55C32×D4—121no—
56C32×D4—121no—
57C2×C32×C4—31no—
58#638—122no—
59C2×C3⋊Dic3—32no—
60C2×C3⋊Dic3—121no—
61#560—91noSylow
62C2×H(3,𝔽3)—161no—
63(C3⋊D3)⋊C3—162no—
64C23×D3—31no—
65C23×D3—91no—
66C22×A4—361no—
67C2×C3×D4—31no—
68C2×C3×D4—33no—
69C2×C3×D4—33no—
70C2×C3×D4—31no—
71C2×#67—92no—
72C2×#67—96no—
73C22×Dic3—93no—
74C22×Dic3—91no—
75C3×C42—31no—
76C3×C42C2×C3×D311yescharacteristic
77#217—93no—
78#217—93no—
79#217—91no—
80#217—91no—
81(C2×D3)⋊C4—92no—
82(C2×D3)⋊C4—96no—
83Dic3⋊C4—96no—
84Dic3⋊C4—92no—
85C42⋊C3—31no—
86C42⋊C3—61no—
87C22×C32—61no—
88C22×C32—61no—
89C22×C32—62no—
90C22×C32C22×A411yescharacteristic
91C2×C3⋊D3—41no—
92C2×C3⋊D3—62no—
93C2×C3⋊D3—121no—
94C2×C3×D3—481no—
95C3×A4—41no—
96C3×A4—81no—
97C32×C4—61no—
98C3⋊Dic3—62no—
99#127(C3⋊D3)⋊C311yescharacteristic, complemented
100#126—92no—
101#123—271no—
102#121—272no—
103#120—271no—
104H(3,𝔽3)—161noSylow
105C23×C3—33no—
106C23×C3—31no—
107C23×C3—31no—
108C23×C3D3×A411yescharacteristic
109C22×D3—32no—
110C22×D3—92no—
111C22×D3—361no—
112C22×D3—363no—
113C2×A4—121no—
114C2×A4—241no—
115C2×A4—362no—
116C3×D4—121no—
117C3×D4—123no—
118C3×D4—121no—
119C3×D4—123no—
120C2×C3×C4—33no—
121C2×C3×C4—31no—
122#67—362no—
123#67—366no—
124C2×Dic3—96no—
125C2×Dic3—92no—
126C2×Dic3—363no—
127C2×Dic3—361no—
128C2×C32—31no—
129C2×C32—41no—
130C2×C32—121no—
131C2×C32—161no—
132C2×C32—321no—
133C3⋊D3—42no—
134C3×D3—482no—
135C24—91no—
136C2×D4—31no—
137C2×D4—31no—
138C2×D4—272no—
139C22×C4—271no—
140C4⋊C4—272no—
141C22⋊C4—272no—
142C22⋊C4—271no—
143C22⋊C4—271no—
144C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
145C22×C3—31no—
146C22×C3—61no—
147C22×C3—63no—
148C22×C3—61no—
149C22×C3—66no—
150C22×C3—62no—
151C22×C3—63no—
152C22×C3—1441no—
153C22×C3C2×D3×A411yescharacteristic
154C2×D3—121no—
155C2×D3—182no—
156C2×D3—186no—
157C2×D3—361no—
158C2×D3—361no—
159C2×D3—363no—
160A4—121no—
161A4—241no—
162C3×C4—63no—
163C3×C4—61no—
164Dic3—186no—
165Dic3—182no—
166C32—161no—
167C32—321no—
168C32#560⋊C311yescharacteristic, complemented
169C23—31no—
170C23—92no—
171C23—1081no—
172C23(C3⋊D3)⋊A411yescharacteristic
173D4—121no—
174D4—121no—
175D4—1082no—
176C2×C4—31no—
177C2×C4—272no—
178C2×C4—1081no—
179C2×C3—33no—
180C2×C3—31no—
181C2×C3—41no—
182C2×C3—121no—
183C2×C3—123no—
184C2×C3—121no—
185C2×C3—481no—
186C2×C3—961no—
187C2×C3—1442no—
188D3—122no—
189D3—362no—
190C22—61no—
191C22—61no—
192C22—62no—
193C22—361no—
194C22—542no—
195C22—1081no—
196C22C2×(C3⋊D3)⋊A411yescharacteristic
197C4—61no—
198C4—542no—
199C3—31no—
200C3—481no—
201C3—961no—
202C3#5720⋊C311yescharacteristic
203C2—31no—
204C2—41no—
205C2—121no—
206C2—362no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC32×C42C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC32×#127C2×C3
LCSG ▷ C32×C42
UCSC1

Complements

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

NH ≅ G/Ntype
1. #127⋊H(3,𝔽3)C2semidirect
2. #11720758C2semidirect
3. #10596191C3semidirect
6. C32⋊(C42⋊C3)C22semidirect
7. C32×#127C2×C3semidirect
11. #64085C2×C3semidirect
13. C3×#127⋊C3D3semidirect
26. C32×C42C22×C3semidirect
32. C3×C42⋊C3C2×D3semidirect
99. #127(C3⋊D3)⋊C3semidirect
144. C42C2×(C3⋊D3)⋊C3semidirect
168. C32#560⋊C3semidirect

Maximal subgroups

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

subgroupindex
#127⋊H(3,𝔽3)2
#117207582
#105961913
#5720⋊C33
C2×(C3⋊D3)⋊A44