Finite Group Databasebeta

← #13147223

Subgroups — #13147223

Order 1728.

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

normal subgroups only

This group has 356 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#120⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11720769C212yesmaximal, complemented
3#10596065C311yesmaximal, characteristic, complemented
4C3⋊#5450—31nomaximal
5C32⋊(C22⋊A4)C2211yescharacteristic, complemented
6C2×(C3⋊D3)⋊A4—41nomaximal
7C32×#120C2×C311yescharacteristic, complemented
8#64162C2×C312yescomplemented
9#64146—31no—
10#64145—31no—
11#64094—32no—
12C3×#120⋊C3—21no—
13C3×#120⋊C3D311yescharacteristic, complemented
14C3⋊(#120⋊C3)—32no—
15C2×C22⋊H(3,𝔽3)—41no—
16(C3⋊D3)⋊A4—42no—
17#5594—31no—
18#5594—91no—
19#5450—91no—
20C24×C32C22×C311yescharacteristic, complemented
21C23×C3⋊D3A411yescharacteristic, complemented
22C3×C22⋊A4—21no—
23C3×C22⋊A4C2×D311yescharacteristic, complemented
24C2×D3×A4—121no—
25C2×C32×D4—31no—
26C2×#638—32no—
27C2×#634—31no—
28C32×C22⋊C4—31no—
29#3757—31no—
30#3757—32no—
31#3752—32no—
32C4×C3⋊Dic3—31no—
33C2×(C3⋊D3)⋊C3—161no—
34C22⋊H(3,𝔽3)—41no—
35C22⋊H(3,𝔽3)—161no—
36C3×#120—31no—
37C3×#120C3×D311yescharacteristic
38#953—32no—
39#953—92no—
40#937—91no—
41#937—93no—
42#936—93no—
43#936—91no—
44#885—96no—
45#885—92no—
46#120⋊C3—31no—
47#120⋊C3—61no—
48#120⋊C3—92no—
49C23×C32—31no—
50C23×C32—121no—
51C23×C32C2×A411yescharacteristic
52C22×C3⋊D3—121no—
53C22×C3⋊D3C2×A412yes—
54C2×C3×A4—41no—
55C2×C3×A4—81no—
56D3×A4—122no—
57C32×D4—121no—
58C2×C32×C4—31no—
59#638—122no—
60C2×C3⋊Dic3—32no—
61#634—121no—
62H(3,𝔽4)—91noSylow
63C2×H(3,𝔽3)—161no—
64(C3⋊D3)⋊C3—162no—
65C24×C3—31no—
66C24×C3C2×C3×D311yescharacteristic
67C23×D3—31no—
68C23×D3—91no—
69C22⋊A4—31no—
70C22⋊A4—61no—
71C22×A4—361no—
72C2×C3×D4—33no—
73C2×C3×D4—31no—
74C2×#67—92no—
75C2×#67—96no—
76C2×Dic6—93no—
77C2×Dic6—91no—
78C3×C22⋊C4—33no—
79C3×C22⋊C4—31no—
80#217—96no—
81#217—93no—
82#217—91no—
83#217—92no—
84(C2×D3)⋊C4—92no—
85(C2×D3)⋊C4—96no—
86C4×Dic3—93no—
87C4×Dic3—91no—
88C22×C32—41no—
89C22×C32—61no—
90C22×C32—62no—
91C22×C32—61no—
92C22×C32—121no—
93C22×C32C22×A411yescharacteristic
94C2×C3⋊D3—41no—
95C2×C3⋊D3—62no—
96C2×C3⋊D3—121no—
97C2×C3×D3—481no—
98C3×A4—41no—
99C3×A4—81no—
100C3×A4—161no—
101C3×A4—162no—
102C32×C4—61no—
103C3⋊Dic3—62no—
104#124—272no—
105#124—271no—
106#120—92no—
107#120—271no—
108#120(C3⋊D3)⋊C311yescharacteristic, complemented
109H(3,𝔽3)—161noSylow
110C23×C3—33no—
111C23×C3—31no—
112C23×C3—31no—
113C23×C3—121no—
114C23×C3—123no—
115C23×C3D3×A411yescharacteristic
116C22×D3—32no—
117C22×D3—92no—
118C22×D3—361no—
119C22×D3—363no—
120C2×A4—121no—
121C2×A4—241no—
122C2×A4—362no—
123C3×D4—121no—
124C3×D4—123no—
125C2×C3×C4—33no—
126C2×C3×C4—31no—
127#67—362no—
128#67—366no—
129C2×Dic3—96no—
130C2×Dic3—92no—
131Dic6—363no—
132Dic6—361no—
133C2×C32—31no—
134C2×C32—41no—
135C2×C32—121no—
136C2×C32—161no—
137C2×C32—321no—
138C3⋊D3—42no—
139C3×D3—482no—
140C24—91no—
141C24C2×(C3⋊D3)⋊C311yescharacteristic, complemented
142C2×Q8—271no—
143C2×D4—31no—
144C2×D4—272no—
145C22⋊C4—31no—
146C22⋊C4—272no—
147C22⋊C4—271no—
148C22⋊C4—272no—
149C42—271no—
150C22×C3—31no—
151C22×C3—41no—
152C22×C3—61no—
153C22×C3—66no—
154C22×C3—62no—
155C22×C3—63no—
156C22×C3—63no—
157C22×C3—61no—
158C22×C3—121no—
159C22×C3—121no—
160C22×C3—123no—
161C22×C3—1441no—
162C22×C3C2×D3×A411yescharacteristic
163C2×D3—121no—
164C2×D3—182no—
165C2×D3—186no—
166C2×D3—361no—
167C2×D3—361no—
168C2×D3—363no—
169A4—121no—
170A4—241no—
171A4—481no—
172A4—482no—
173C3×C4—63no—
174C3×C4—61no—
175Dic3—186no—
176Dic3—182no—
177C32—161no—
178C32—321no—
179C32#545011yescharacteristic, complemented
180C23—31no—
181C23—92no—
182C23—121no—
183C23—1081no—
184C23(C3⋊D3)⋊A411yescharacteristic
185Q8—1081no—
186D4—121no—
187D4—1082no—
188C2×C4—31no—
189C2×C4—272no—
190C2×C3—33no—
191C2×C3—31no—
192C2×C3—41no—
193C2×C3—121no—
194C2×C3—123no—
195C2×C3—121no—
196C2×C3—481no—
197C2×C3—961no—
198C2×C3—1442no—
199D3—122no—
200D3—362no—
201C22—41no—
202C22—61no—
203C22—61no—
204C22—62no—
205C22—121no—
206C22—361no—
207C22—542no—
208C22—1081no—
209C22C2×(C3⋊D3)⋊A411yescharacteristic
210C4—61no—
211C4—542no—
212C3—31no—
213C3—481no—
214C3—961no—
215C3C3⋊#545011yescharacteristic
216C2—31no—
217C2—41no—
218C2—121no—
219C2—362no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC24×C32C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC32×#120C2×C3
LCSG ▷ C24×C32
UCSC1

Complements

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

NH ≅ G/Ntype
1. #120⋊H(3,𝔽3)C2semidirect
2. #11720769C2semidirect
3. #10596065C3semidirect
5. C32⋊(C22⋊A4)C22semidirect
7. C32×#120C2×C3semidirect
8. #64162C2×C3semidirect
13. C3×#120⋊C3D3semidirect
20. C24×C32C22×C3semidirect
21. C23×C3⋊D3A4semidirect
23. C3×C22⋊A4C2×D3semidirect
108. #120(C3⋊D3)⋊C3semidirect
141. C24C2×(C3⋊D3)⋊C3semidirect
179. C32#5450semidirect

Maximal subgroups

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

subgroupindex
#120⋊H(3,𝔽3)2
#117207692
#105960653
C3⋊#54503
C2×(C3⋊D3)⋊A44