Finite Group Databasebeta

← #13180154

Subgroups — #13180154

Order 1728.

286 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 46 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#11724154C211yesmaximal, characteristic, complemented
2#11724111C211yesmaximal, characteristic
3#11724104C211yesmaximal, characteristic
4#11724101C211yesmaximal, characteristic
5#11724100C211yesmaximal, characteristic, complemented
6C3×AGL(1,𝔽9)⋊C4C211yesmaximal, characteristic, complemented
7Dic3⋊AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
8GO+(4,𝔽2)⋊Q8—31nomaximal
9C2×C3⋊PSU(3,𝔽2)—21no—
10C2×#157⋊D3C2211yescharacteristic
11C2×C3⋊AGL(1,𝔽9)C2211yescharacteristic
12C2×C3×AGL(1,𝔽9)C2211yescharacteristic
13#88949—21no—
14#88946C2211yescharacteristic
15C3×#3777C2211yescharacteristic
16#88932—21no—
17C3×#3773C2211yescharacteristic
18Dic3⋊#157—21no—
19Dic3×#157C2211yescharacteristic
20C4⋊PSU(3,𝔽2)—31no—
21C4×GO+(4,𝔽2)—31no—
22C4⋊AGL(1,𝔽9)—31no—
23AGL(1,𝔽9)⋊C4—31no—
24AGL(1,𝔽9)⋊C4D311yescharacteristic, complemented
25GO+(4,𝔽2)⋊C4—31no—
26GO+(4,𝔽2)⋊C4—31no—
27C2×#1113—21no—
28C2×C3⋊#157—21no—
29C2×C3⋊#157D411yescharacteristic
30C2×C3×#157—21no—
31C2×C3×#157C2311yescharacteristic
32C3⋊PSU(3,𝔽2)—42no—
33#157⋊D3—21no—
34#157⋊D3Q812yescomplemented
35C3⋊AGL(1,𝔽9)—21no—
36C3×AGL(1,𝔽9)—21no—
37Dic3×C3⋊D3—41no—
38Dic3×C3⋊D3D411yescharacteristic
39C3×#624—21no—
40C3×#624—41no—
41#4870—91nomaximal
42C2×PSU(3,𝔽2)—61no—
43C2×GO+(4,𝔽2)—31no—
44C2×AGL(1,𝔽9)—31no—
45C2×AGL(1,𝔽9)C2×D311yescharacteristic
46C4×D32—61no—
47C4×D32—121no—
48C4⋊#157—61no—
49C4×#157—31no—
50#3777—31no—
51#3777C2×D311yescharacteristic
52#3773C2×D311yescharacteristic
53D32⋊C4—61no—
54C2×C3×C3⋊D3—41no—
55C2×C3×C3⋊D3C2×D411yescharacteristic
56#1113—22no—
57#1113—41no—
58C3⋊#157—21no—
59C3⋊#157—41no—
60C3×#157—41no—
61C3×#157C2×Q811yescharacteristic
62C3×#157#4111yescharacteristic
63C3⋊(C3⋊Dic3)SD1611yescharacteristic
64C32×Dic3—41no—
65C32×Dic3SD1611yescharacteristic
66#893—91no—
67#888—91no—
68C3×#106—91no—
69#821—91no—
70Dic3⋊C8—91no—
71#809—91no—
72#808—91no—
73C2×D32—61no—
74C2×D32—121no—
75C2×#157—21no—
76C2×#157—31no—
77C2×#157—61no—
78C2×#157C22×D311yescharacteristic
79PSU(3,𝔽2)—122no—
80GO+(4,𝔽2)—32no—
81GO+(4,𝔽2)—61no—
82AGL(1,𝔽9)—21no—
83AGL(1,𝔽9)—61no—
84C4×C3⋊D3—31no—
85C4×C3⋊D3—121no—
86C3×C4×D3—121no—
87C3×C4×D3—121no—
88C3×C4×D3—121no—
89#624—21no—
90#624—41no—
91D3×Dic3—121no—
92D3×Dic3—121no—
93D3×Dic3—121no—
94D3×Dic3—241no—
95#476—271noSylow
96C2×C33C2×SD1611yescharacteristic
97C3×C3⋊D3—42no—
98C3×C3⋊D3C22⋊Q811yescharacteristic
99C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
100C2×D12—91no—
101C2×C4×D3—181no—
102C2×C4×D3—361no—
103C2×Dic6—181no—
104C2×C3×C8—91no—
105C3×C4⋊C4—91no—
106C3×C4⋊C4—91no—
107(C2×D3)⋊C4—181no—
108C4⋊Dic3—91no—
109Dic3⋊C4—181no—
110C4×Dic3—91no—
111C2×C3⋊C8—91no—
112C2×C3⋊D3—41no—
113C2×C3⋊D3D3×D411yescharacteristic
114C2×C3×D3—121no—
115C2×C3×D3—121no—
116C2×C3×D3—121no—
117C2×C3×D3—241no—
118D32—62no—
119D32—122no—
120D32—122no—
121D32—121no—
122#157—41no—
123#157—61no—
124#157—121no—
125#157#23511yescharacteristic
126#157D3×Q811yescharacteristic
127C32×C4—31no—
128C32×C4—121no—
129C3⋊Dic3—31no—
130C3⋊Dic3—121no—
131C3⋊Dic3—241no—
132C3×Dic3—41no—
133C3×Dic3—41no—
134C3×Dic3—41no—
135C3×Dic3—81no—
136C4⋊Q8—271no—
137C4×D4—271no—
138#106—91no—
139#106—271no—
140C4⋊C8—271no—
141#102—271no—
142#102—271no—
143C33#47611yescharacteristic, complemented, Sylow
144C22×D3—181no—
145C22×D3—361no—
146C2×C3×C4—91no—
147C2×C3×C4—181no—
148C2×C3×C4—181no—
149C2×C3×C4—361no—
150C2×Dic3—91no—
151C2×Dic3—91no—
152C2×Dic3—181no—
153C2×Dic3—361no—
154D12—92no—
155D12—181no—
156C4×D3—121no—
157C4×D3—121no—
158C4×D3—121no—
159C4×D3—361no—
160C4×D3—361no—
161C4×D3—361no—
162C4×D3—721no—
163Dic6—362no—
164C3×C8—181no—
165C3⋊C8—181no—
166C2×C32—41no—
167C2×C32—81no—
168C2×C32D3×SD1611yescharacteristic
169C3⋊D3—42no—
170C3⋊D3#89611yescharacteristic
171C3⋊D3#91411yescharacteristic
172C3×D3—121no—
173C3×D3—121no—
174C3×D3—122no—
175C3×D3—122no—
176C3×D3—242no—
177C2×Q8—541no—
178C2×D4—271no—
179C22×C4—541no—
180C2×C8—91no—
181C2×C8—271no—
182C4⋊C4—91no—
183C4⋊C4—91no—
184C4⋊C4—271no—
185C4⋊C4—541no—
186C22⋊C4—541no—
187C42—271no—
188C22×C3—91no—
189C22×C3—361no—
190C2×D3—121no—
191C2×D3—121no—
192C2×D3—121no—
193C2×D3—182no—
194C2×D3—241no—
195C2×D3—362no—
196C2×D3—361no—
197C2×D3—362no—
198C3×C4—91no—
199C3×C4—91no—
200C3×C4—121no—
201C3×C4—121no—
202C3×C4—121no—
203C3×C4—241no—
204C3×C4—361no—
205Dic3—41no—
206Dic3—91no—
207Dic3—181no—
208Dic3—361no—
209Dic3—361no—
210Dic3—721no—
211Dic3AΓL(1,𝔽9)11yescharacteristic
212C32—41no—
213C32—81no—
214C32#487011yescharacteristic, complemented
215C23—541no—
216Q8—1082no—
217D4—272no—
218D4—541no—
219C2×C4—91no—
220C2×C4—181no—
221C2×C4—181no—
222C2×C4—271no—
223C2×C4—271no—
224C2×C4—361no—
225C2×C4—541no—
226C2×C4—1081no—
227C8—181no—
228C8—541no—
229C2×C3—41no—
230C2×C3—81no—
231C2×C3—91no—
232C2×C3—91no—
233C2×C3—362no—
234C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
235D3—121no—
236D3—121no—
237D3—122no—
238D3—122no—
239D3—242no—
240C22—91no—
241C22—361no—
242C22—542no—
243C22—1081no—
244C4—31no—
245C4—91no—
246C4—91no—
247C4—121no—
248C4—271no—
249C4—361no—
250C4—541no—
251C4—1081no—
252C3—41no—
253C3—81no—
254C3GO+(4,𝔽2)⋊Q811yescharacteristic, complemented
255C2—91no—
256C2—91no—
257C2—362no—
258C2D3×AΓL(1,𝔽9)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×AΓL(1,𝔽9)
commutator, cosocleC2×C3×#157C23
Fitting, socleC2×C33C2×SD16
prime radicalC2×#157⋊D3C22
hypocenterC33#476
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #11724154C2semidirect
5. #11724100C2semidirect
6. C3×AGL(1,𝔽9)⋊C4C2semidirect
7. Dic3⋊AGL(1,𝔽9)C2semidirect
24. AGL(1,𝔽9)⋊C4D3semidirect
34. #157⋊D3Q8semidirect
143. C33#476semidirect
214. C32#4870semidirect
254. C3GO+(4,𝔽2)⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#117241542
#117241112
#117241042
#117241012
#117241002
C3×AGL(1,𝔽9)⋊C42
Dic3⋊AGL(1,𝔽9)2
GO+(4,𝔽2)⋊Q83
#48709