Finite Group Databasebeta

← #13147124

Subgroups — #13147124

Order 1728.

213 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 30 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
1C9×#127⋊C3C211yesmaximal, characteristic, complemented
2#11720745C212yesmaximal, complemented
3#5720⋊C3—31nomaximal
4#10589822C311yesmaximal, characteristic, complemented
5C2×A4×D9—41nomaximal
6C9×C42⋊C3C2211yescharacteristic, complemented
7C3×#127⋊C3D311yescharacteristic
8C3⋊(#126⋊C3)—32no—
9C9×#127C2×C311yescharacteristic, complemented
10#63501—32no—
11#63499—31no—
12#63497—31no—
13#63438C2×C312yescomplemented
14#127⋊C9—21no—
15C2×C9×A4—41no—
16A4×D9—42no—
17#5720—31no—
18#560⋊C3—91no—
19C2×D3×A4—121no—
20C23×D9A411yescharacteristic
21C3×C42⋊C3C2×D311yescharacteristic
22C2×D4×C9—31no—
23C2×D4×C9—31no—
24C2×#611—32no—
25C22×Dic9—31no—
26C42×C9C22×C311yescharacteristic, complemented
27#3676—31no—
28#3676—31no—
29(C2×D9)⋊C4—32no—
30Dic9⋊C4—32no—
31C42⋊C9—21no—
32C2×C3×D9—161no—
33C9×A4—41no—
34C3×#127C3×D311yescharacteristic
35#939—92no—
36#937—91no—
37#935—91no—
38#876—32no—
39#127⋊C3—21no—
40#127⋊C3D911yescharacteristic, complemented
41#126⋊C3—92no—
42C2×C3×A4—41no—
43D3×A4—122no—
44C23×C9—31no—
45C23×C9C2×A411yescharacteristic
46C22×D9—121no—
47C22×D9C2×A412yes—
48C2×C22⋊C9—81no—
49D4×C9—121no—
50D4×C9—121no—
51C2×C4×C9—31no—
52#611—122no—
53C2×Dic9—32no—
54C2×Dic9—121no—
55#560—91noSylow
56C2×C3×C9—161no—
57C3×D9—162no—
58C23×D3—31no—
59C22×A4—361no—
60C2×C3×D4—31no—
61C2×C3×D4—31no—
62C2×#67—92no—
63C22×Dic3—91no—
64C3×C42C2×C3×D311yescharacteristic
65#217—91no—
66#217—91no—
67(C2×D3)⋊C4—92no—
68Dic3⋊C4—92no—
69C42⋊C3—21no—
70C42⋊C3C2×D911yescharacteristic, complemented
71C2×C3×D3—481no—
72C3×A4—41no—
73C22×C9—61no—
74C22×C9—61no—
75C22×C9—62no—
76C22×C9C22×A411yescharacteristic
77C2×D9—41no—
78C2×D9—62no—
79C2×D9—121no—
80C22⋊C9—81no—
81C4×C9—61no—
82Dic9—62no—
83#127C3×D911yescharacteristic, complemented
84#126—92no—
85#123—271no—
86#121—272no—
87#120—271no—
88C3×C9—161noSylow
89C23×C3—31no—
90C23×C3D3×A411yescharacteristic
91C22×D3—32no—
92C22×D3—361no—
93C2×A4—41no—
94C2×A4—81no—
95C2×A4—362no—
96C3×D4—121no—
97C3×D4—121no—
98C2×C3×C4—31no—
99#67—362no—
100C2×Dic3—92no—
101C2×Dic3—361no—
102C2×C32—161no—
103C3×D3—482no—
104C2×C9—31no—
105C2×C9—41no—
106C2×C9—121no—
107C2×C9—321no—
108D9—42no—
109C24—91no—
110C2×D4—31no—
111C2×D4—31no—
112C2×D4—272no—
113C22×C4—271no—
114C4⋊C4—272no—
115C22⋊C4—272no—
116C22⋊C4—271no—
117C22⋊C4—271no—
118C42C2×C3×D911yescharacteristic, complemented
119C22×C3—61no—
120C22×C3—61no—
121C22×C3—62no—
122C22×C3—1441no—
123C22×C3C2×D3×A411yescharacteristic
124C2×D3—121no—
125C2×D3—182no—
126C2×D3—361no—
127A4—41no—
128A4—81no—
129C3×C4—61no—
130Dic3—182no—
131C32—161no—
132C9—321no—
133C9#560⋊C311yescharacteristic, complemented
134C23—31no—
135C23—92no—
136C23—1081no—
137C23A4×D911yescharacteristic
138D4—121no—
139D4—121no—
140D4—1082no—
141C2×C4—31no—
142C2×C4—272no—
143C2×C4—1081no—
144C2×C3—31no—
145C2×C3—41no—
146C2×C3—121no—
147C2×C3—161no—
148C2×C3—321no—
149C2×C3—1442no—
150D3—122no—
151C22—61no—
152C22—61no—
153C22—62no—
154C22—361no—
155C22—542no—
156C22—1081no—
157C22C2×A4×D911yescharacteristic
158C4—61no—
159C4—542no—
160C3—161no—
161C3—321no—
162C3#5720⋊C311yescharacteristic
163C2—31no—
164C2—41no—
165C2—121no—
166C2—362no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC42×C9C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC9×#127C2×C3
LCSG ▷ C42×C9
UCSC1

Complements

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

NH ≅ G/Ntype
1. C9×#127⋊C3C2semidirect
2. #11720745C2semidirect
4. #10589822C3semidirect
6. C9×C42⋊C3C22semidirect
9. C9×#127C2×C3semidirect
13. #63438C2×C3semidirect
26. C42×C9C22×C3semidirect
40. #127⋊C3D9semidirect
70. C42⋊C3C2×D9semidirect
83. #127C3×D9semidirect
118. C42C2×C3×D9semidirect
133. C9#560⋊C3semidirect

Maximal subgroups

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

subgroupindex
C9×#127⋊C32
#117207452
#5720⋊C33
#105898223
C2×A4×D94