Finite Group Databasebeta

← #13180431

Subgroups — #13180431

Order 1728.

306 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 53 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#11724365C211yesmaximal, characteristic, complemented
2#11724344C211yesmaximal, characteristic
3C3×C4×AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
4#11724104C214yesmaximal, complemented
5#10596402—31nomaximal
6C2×C3⋊PSU(3,𝔽2)—21no—
7C2×#157⋊D3—21no—
8C2×#157⋊D3C2212yes—
9C2×C3×AGL(1,𝔽9)C2212yes—
10C4⋊#1113—21no—
11C4⋊(C3⋊#157)—21no—
12C3×C4×#157C2211yescharacteristic
13#88946C2212yes—
14C4⋊PSU(3,𝔽2)—31no—
15#64231—31no—
16C4×AGL(1,𝔽9)D311yescharacteristic, complemented
17GO+(4,𝔽2)⋊C4—34no—
18C2×#1113—22no—
19C2×C3⋊#157—22no—
20C2×C3×#157D411yescharacteristic
21C2×C3×#157C2311yescharacteristic
22C3⋊PSU(3,𝔽2)—42no—
23#157⋊D3—24no—
24#157⋊D3—42no—
25C3×AGL(1,𝔽9)—22no—
26C3×C4⋊(C3⋊D3)—41no—
27C3×C4×C3⋊D3D411yescharacteristic
28#6295—41no—
29#4694—91nomaximal
30C2×PSU(3,𝔽2)—61no—
31C2×GO+(4,𝔽2)—32no—
32C2×GO+(4,𝔽2)—61no—
33C2×AGL(1,𝔽9)C2×D312yes—
34#3802—61no—
35D3×D12—121no—
36C4⋊#157—61no—
37C4×#157C2×D311yescharacteristic
38#3777—32no—
39C2×C3×C3⋊D3—42no—
40C2×C3×C3⋊D3C2×D411yescharacteristic
41#1113—42no—
42#1113—42no—
43C3⋊#157—42no—
44C3×#157D812yes—
45C3×#157#4112yes—
46C33×C4SD1611yescharacteristic
47C3×C3⋊Dic3SD1611yescharacteristic
48#874—91no—
49#869—91no—
50C3×C4×C8—91no—
51#821—94no—
52C2×D32—62no—
53C2×D32—122no—
54C2×#157—62no—
55C2×#157D1211yescharacteristic
56C2×#157C22×D311yescharacteristic
57PSU(3,𝔽2)—122no—
58GO+(4,𝔽2)—64no—
59GO+(4,𝔽2)—122no—
60AGL(1,𝔽9)—22no—
61C4⋊(C3⋊D3)—41no—
62C4×C3⋊D3D1211yescharacteristic
63C3×D12—121no—
64C3×D12—121no—
65C3×D12—241no—
66C3×C4×D3—121no—
67#626—122no—
68#625—121no—
69#486—271noSylow
70C2×C33C2×SD1611yescharacteristic
71C3×C3⋊D3—82no—
72C3×C3⋊D3#12411yescharacteristic
73C3×C3⋊D3C2×D811yescharacteristic
74D3×D4—361no—
75C2×D12—92no—
76C2×D12—181no—
77C2×D12—181no—
78C2×Dic6—181no—
79C2×C3×C8—92no—
80C3×C42—91no—
81C4⋊Dic3—92no—
82C4⋊Dic3—181no—
83C2×C3⋊D3—42no—
84C2×C3⋊D3C2×D1211yescharacteristic
85C2×C3×D3—121no—
86C2×C3×D3—122no—
87C2×C3×D3—122no—
88C2×C3×D3—242no—
89D32—122no—
90D32—122no—
91D32—242no—
92D32—242no—
93#157—122no—
94#157D2412yes—
95#157#23512yes—
96C32×C4—41no—
97C32×C4—81no—
98C32×C4#20411yescharacteristic
99C3⋊Dic3#20411yescharacteristic
100C3×Dic3—121no—
101C4⋊Q8—271no—
102#127—271no—
103#102—274no—
104C4×C8—91no—
105C33#48611yescharacteristic, complemented, Sylow
106C22×D3—182no—
107C22×D3—362no—
108C3×D4—361no—
109C2×C3×C4—91no—
110C2×C3×C4—91no—
111C2×C3×C4—91no—
112#67—362no—
113C2×Dic3—182no—
114D12—121no—
115D12—121no—
116D12—184no—
117D12—241no—
118D12—361no—
119D12—362no—
120C4×D3—121no—
121Dic6—362no—
122C3×C8—182no—
123C2×C32—41no—
124C2×C32—81no—
125C2×C32C2×#20411yescharacteristic
126C3⋊D3—82no—
127C3⋊D3#87511yescharacteristic
128C3⋊D3C2×D2411yescharacteristic
129C3×D3—121no—
130C3×D3—121no—
131C3×D3—242no—
132C3×D3—242no—
133C3×D3—482no—
134C2×Q8—541no—
135C2×D4—272no—
136C2×D4—541no—
137C2×D4—541no—
138C2×C8—92no—
139C4⋊C4—272no—
140C4⋊C4—541no—
141C42—91no—
142C22×C3—91no—
143C22×C3—362no—
144C2×D3—121no—
145C2×D3—122no—
146C2×D3—122no—
147C2×D3—242no—
148C2×D3—362no—
149C2×D3—362no—
150C2×D3—722no—
151C2×D3—722no—
152C3×C4—41no—
153C3×C4—81no—
154C3×C4—92no—
155C3×C4—92no—
156C3×C4—91no—
157C3×C4AΓL(1,𝔽9)11yescharacteristic
158Dic3—121no—
159Dic3—362no—
160C32—41no—
161C32—81no—
162C32#469411yescharacteristic, complemented
163C23—542no—
164Q8—1082no—
165D4—361no—
166D4—544no—
167D4—1081no—
168D4—1082no—
169C2×C4—91no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—542no—
173C8—182no—
174C2×C3—41no—
175C2×C3—81no—
176C2×C3—91no—
177C2×C3—91no—
178C2×C3—722no—
179C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
180D3—121no—
181D3—121no—
182D3—242no—
183D3—242no—
184D3—482no—
185C22—91no—
186C22—362no—
187C22—1082no—
188C22—1082no—
189C4—92no—
190C4—92no—
191C4—91no—
192C4—1082no—
193C4AGL(1,𝔽9)⋊D311yescharacteristic
194C3—41no—
195C3—81no—
196C3#1059640211yescharacteristic, complemented
197C2—91no—
198C2—91no—
199C2—722no—
200C2C2×AGL(1,𝔽9)⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽9)⋊D3
commutator, cosocleC2×C3×#157C23
FittingC33×C4SD16
socleC2×C33C2×SD16
prime radical#11724344C2
hypercenterC4AGL(1,𝔽9)⋊D3
hypocenterC33#486
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #11724365C2semidirect
3. C3×C4×AGL(1,𝔽9)C2semidirect
4. #11724104C2semidirect
16. C4×AGL(1,𝔽9)D3semidirect
105. C33#486semidirect
162. C32#4694semidirect
196. C3#10596402semidirect

Maximal subgroups

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

subgroupindex
#117243652
#117243442
C3×C4×AGL(1,𝔽9)2
#117241042
#105964023
#46949