Finite Group Databasebeta

← C3×#10596408 #13180387

Subgroups — C3×#10596408 #13180387

Order 1728.

354 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C2×C3×AΓL(1,𝔽9)C214yesmaximal, complemented
2C3×C4⋊PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
3C3×#64231C211yesmaximal, characteristic
4C3×C4×AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
5#10596408C311yesmaximal, characteristic, direct product factor
6C2×C3×PSU(3,𝔽2)C2212yes—
7C2×C3×GO+(4,𝔽2)—21no—
8C2×C3×GO+(4,𝔽2)C2212yes—
9C2×C3×AGL(1,𝔽9)C2212yes—
10C3×AΓL(1,𝔽9)—28no—
11C3×#3802—21no—
12C3×C4⋊#157—21no—
13C3×C4×#157C2211yescharacteristic
14C3×#3777—21no—
15C2×AΓL(1,𝔽9)C2×C314yescomplemented
16C4⋊PSU(3,𝔽2)C2×C311yescharacteristic, complemented
17#64231C2×C311yescharacteristic
18C4×AGL(1,𝔽9)C2×C311yescharacteristic, complemented
19C2×C3×D32—22no—
20C2×C3×#157—22no—
21C2×C3×#157D411yescharacteristic
22C2×C3×#157C2311yescharacteristic
23C3×PSU(3,𝔽2)—24no—
24C3×GO+(4,𝔽2)—24no—
25C3×GO+(4,𝔽2)—42no—
26C3×AGL(1,𝔽9)D414yescomplemented
27C3×C4×C3⋊D3D411yescharacteristic
28C32×D12—41no—
29C3×#625—41no—
30C3×#492—91nomaximal
31C2×PSU(3,𝔽2)C22×C312yes—
32C2×GO+(4,𝔽2)—21no—
33C2×GO+(4,𝔽2)C22×C312yes—
34C2×AGL(1,𝔽9)C22×C312yes—
35AΓL(1,𝔽9)—28no—
36C3×D3×D4—121no—
37#3802—21no—
38C4⋊#157—21no—
39C4×#157C22×C311yescharacteristic
40#3777—21no—
41C2×C3×C3⋊D3C2×D411yescharacteristic
42C2×C32×D3—42no—
43C3×D32—42no—
44C3×D32—42no—
45C3×#157—42no—
46C3×#157SD1612yescomplemented
47C3×#157C2×D412yes—
48C33×C4SD1611yescharacteristic, complemented
49C3×C3⋊Dic3SD1611yescharacteristic, complemented
50C2×C3×SD16—94no—
51C3×C4⋊Q8—91no—
52C3×#127—91no—
53C3×C4×C8—91no—
54C22×C3×D3—122no—
55C2×D32—22no—
56C2×#157—22no—
57C2×#157C3×D411yescharacteristic
58C2×#157C23×C311yescharacteristic
59PSU(3,𝔽2)—24no—
60GO+(4,𝔽2)—24no—
61GO+(4,𝔽2)—42no—
62AGL(1,𝔽9)C3×D414yescomplemented
63C32×D4—121no—
64C4×C3⋊D3C3×D411yescharacteristic
65C3×#67—122no—
66C3×D12—41no—
67C3×D12—41no—
68C3×D12—81no—
69C3×C4×D3—121no—
70#625—41no—
71#492—91noSylow
72C2×C33C2×SD1611yescharacteristic
73C3×C3⋊D3#12711yescharacteristic
74C3×C3⋊D3C2×SD1611yescharacteristic
75C32×D3—82no—
76C2×C3×Q8—92no—
77C2×C3×D4—92no—
78C2×C3×D4—181no—
79C2×C3×D4—181no—
80D3×D4—121no—
81C3×SD16—188no—
82C2×C3×C8—92no—
83C3×C4⋊C4—181no—
84C3×C4⋊C4—181no—
85C3×C42—91no—
86C22×C32—122no—
87C2×C3⋊D3C2×C3×D411yescharacteristic
88C2×C3×D3—42no—
89C2×C3×D3—42no—
90C2×C3×D3—82no—
91C2×C3×D3—121no—
92C2×C3×D3—242no—
93C2×C3×D3—242no—
94D32—42no—
95D32—42no—
96#157—42no—
97#157C3×SD1612yescomplemented
98#157C2×C3×D412yes—
99C32×C4—41no—
100C32×C4—81no—
101C32×C4C3×SD1611yescharacteristic, complemented
102C3⋊Dic3C3×SD1611yescharacteristic, complemented
103C3×Dic3—121no—
104C2×SD16—94no—
105C4⋊Q8—91no—
106#127—91no—
107C4×C8—91no—
108C33#49211yescharacteristic, complemented, Sylow
109C23×C3—182no—
110C22×D3—122no—
111C3×Q8—184no—
112C3×D4—121no—
113C3×D4—121no—
114C3×D4—184no—
115C3×D4—241no—
116C3×D4—361no—
117C3×D4—362no—
118C2×C3×C4—91no—
119C2×C3×C4—91no—
120C2×C3×C4—91no—
121C2×C3×C4—182no—
122#67—122no—
123D12—41no—
124C4×D3—121no—
125C3×C8—94no—
126C2×C32—41no—
127C2×C32—81no—
128C2×C32—242no—
129C2×C32C2×C3×SD1611yescharacteristic
130C3⋊D3C3×#12711yescharacteristic
131C3⋊D3C2×C3×SD1611yescharacteristic
132C3×D3—82no—
133C3×D3—82no—
134C3×D3—121no—
135C3×D3—121no—
136C3×D3—162no—
137C2×Q8—92no—
138C2×D4—92no—
139C2×D4—181no—
140C2×D4—181no—
141SD16—188no—
142C2×C8—92no—
143C4⋊C4—181no—
144C4⋊C4—181no—
145C42—91no—
146C22×C3—91no—
147C22×C3—122no—
148C22×C3—122no—
149C22×C3—242no—
150C22×C3—362no—
151C22×C3—362no—
152C2×D3—42no—
153C2×D3—121no—
154C2×D3—242no—
155C2×D3—242no—
156C3×C4—41no—
157C3×C4—81no—
158C3×C4—92no—
159C3×C4—92no—
160C3×C4—91no—
161C3×C4—362no—
162C3×C4AΓL(1,𝔽9)11yescharacteristic, complemented
163Dic3—121no—
164C32—41no—
165C32—81no—
166C32C3×#49211yescharacteristic, complemented
167C23—182no—
168Q8—184no—
169D4—121no—
170D4—184no—
171D4—361no—
172D4—362no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—182no—
177C8—94no—
178C2×C3—41no—
179C2×C3—81no—
180C2×C3—91no—
181C2×C3—91no—
182C2×C3—242no—
183C2×C3—242no—
184C2×C3—482no—
185C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
186D3—82no—
187D3—121no—
188D3—121no—
189C22—91no—
190C22—122no—
191C22—362no—
192C22—362no—
193C4—92no—
194C4—92no—
195C4—91no—
196C4—362no—
197C4C3×AΓL(1,𝔽9)11yescharacteristic, complemented
198C3—41no—
199C3—81no—
200C3#1059640811yescharacteristic, direct product factor
201C2—91no—
202C2—91no—
203C2—242no—
204C2C2×C3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC2×C3C2×AΓL(1,𝔽9)
commutator, cosocleC2×#157C23×C3
FrattiniC2C2×C3×AΓL(1,𝔽9)
FittingC33×C4SD16
socleC2×C33C2×SD16
prime radicalC3×#64231C2
hypercenterC3×C4AΓL(1,𝔽9)
hypocenterC32C3×#492
LCSG ▷ C2×#157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×AΓL(1,𝔽9)C2semidirect
2. C3×C4⋊PSU(3,𝔽2)C2semidirect
4. C3×C4×AGL(1,𝔽9)C2semidirect
5. #10596408C3direct (N × H)
15. C2×AΓL(1,𝔽9)C2×C3semidirect
16. C4⋊PSU(3,𝔽2)C2×C3semidirect
18. C4×AGL(1,𝔽9)C2×C3semidirect
26. C3×AGL(1,𝔽9)D4semidirect
46. C3×#157SD16semidirect
48. C33×C4SD16semidirect
49. C3×C3⋊Dic3SD16semidirect
62. AGL(1,𝔽9)C3×D4semidirect
97. #157C3×SD16semidirect
101. C32×C4C3×SD16semidirect
102. C3⋊Dic3C3×SD16semidirect
108. C33#492semidirect
162. C3×C4AΓL(1,𝔽9)semidirect
166. C32C3×#492semidirect
197. C4C3×AΓL(1,𝔽9)semidirect
200. C3#10596408direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×AΓL(1,𝔽9)2
C3×C4⋊PSU(3,𝔽2)2
C3×#642312
C3×C4×AGL(1,𝔽9)2
#105964083
C3×#4929