Finite Group Databasebeta

← #13180446

Subgroups — #13180446

Order 1728.

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

normal subgroups only

This group has 318 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
1C22×C3×AGL(1,𝔽9)C211yesmaximal, characteristic, complemented
2C2×AGL(1,𝔽9)⋊D3C211yesmaximal, characteristic, complemented
3#11724369C211yesmaximal, characteristic, complemented
4#11724355C211yesmaximal, characteristic
5AGL(1,𝔽9)⋊Dic3C211yesmaximal, characteristic, complemented
6#11724105C211yesmaximal, characteristic, complemented
7#11724104C211yesmaximal, characteristic, complemented
8#10596416—31nomaximal
9C22×C3×#157C2211yescharacteristic
10C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
11C2×#157⋊D3C2211yescharacteristic
12C2×C3×AGL(1,𝔽9)—22no—
13C2×C3×AGL(1,𝔽9)C2211yescharacteristic
14C2×C3×AGL(1,𝔽9)C2211yescharacteristic
15AGL(1,𝔽9)⋊D3—22no—
16#89049—21no—
17#88994—21no—
18#88946C2211yescharacteristic
19#88945—21no—
20#88940C2211yescharacteristic
21#88939—21no—
22C22×AGL(1,𝔽9)D311yescharacteristic, complemented
23C2×AΓL(1,𝔽9)—31no—
24C22⋊PSU(3,𝔽2)—31no—
25#64242—31no—
26AGL(1,𝔽9)⋊C4—31no—
27PSU(3,𝔽2)⋊C4—31no—
28GO+(4,𝔽2)⋊C4—31no—
29C22×C3×C3⋊D3D411yescharacteristic
30C2×#1113—21no—
31C2×C3⋊#157—21no—
32C2×C3⋊#157—21no—
33C2×C3×#157—21no—
34C2×C3×#157—21no—
35C2×C3×#157D411yescharacteristic
36C2×C3×#157C2311yescharacteristic
37C3⋊PSU(3,𝔽2)—21no—
38C3⋊PSU(3,𝔽2)—21no—
39#157⋊D3—21no—
40#157⋊D3—21no—
41C3×AGL(1,𝔽9)—21no—
42C3×AGL(1,𝔽9)D412yescomplemented
43C3×#638—41no—
44#6295—41no—
45#6294—21no—
46#5114—91nomaximal
47C22×#157C2×D311yescharacteristic
48C2×PSU(3,𝔽2)—31no—
49C2×GO+(4,𝔽2)—31no—
50C2×AGL(1,𝔽9)—22no—
51C2×AGL(1,𝔽9)C2×D311yescharacteristic
52C2×AGL(1,𝔽9)C2×D311yescharacteristic
53AΓL(1,𝔽9)—62no—
54#3811—61no—
55D3×#67—121no—
56C22⋊#157—61no—
57#3777—31no—
58#3777—61no—
59#3773—31no—
60D32⋊C4—61no—
61C22×C33SD1611yescharacteristic, complemented
62C2×C3×C3⋊D3—21no—
63C2×C3×C3⋊D3—21no—
64C2×C3×C3⋊D3—41no—
65C2×C3×C3⋊D3SD1611yescharacteristic, complemented
66C2×C3×C3⋊D3C2×D411yescharacteristic
67#1113—41no—
68#1113—41no—
69C3⋊#157—41no—
70C3⋊#157—41no—
71C3×#157—21no—
72C3×#157C2×D411yescharacteristic
73C3×#157#4111yescharacteristic
74C3×C3⋊Dic3—41no—
75C22×C3×C8—91no—
76#930—91no—
77#924—91no—
78C2×#204—91no—
79#821—91no—
80#817—91no—
81#816—91no—
82C22×C3⋊D3D1211yescharacteristic
83C22×C3×D3—121no—
84C2×D32—61no—
85C2×D32—121no—
86C2×#157—21no—
87C2×#157—21no—
88C2×#157—61no—
89C2×#157—61no—
90C2×#157D1211yescharacteristic
91C2×#157C22×D311yescharacteristic
92PSU(3,𝔽2)—61no—
93PSU(3,𝔽2)—61no—
94GO+(4,𝔽2)—61no—
95GO+(4,𝔽2)—61no—
96AGL(1,𝔽9)—21no—
97AGL(1,𝔽9)#6712yescomplemented
98#638—41no—
99C3×#67—121no—
100C3×#67—121no—
101C3×#67—241no—
102#626—121no—
103#626—121no—
104#625—121no—
105#624—61no—
106D3×Dic3—121no—
107#465—271noSylow
108C2×C33—21no—
109C2×C33C2×SD1611yescharacteristic
110C3×C3⋊D3—21no—
111C3×C3⋊D3—81no—
112C3×C3⋊D3#12111yescharacteristic
113C3×C3⋊D3#13511yescharacteristic
114C22×C3×C4—91no—
115C2×#67—181no—
116D3×D4—361no—
117C2×D12—91no—
118C2×Dic6—91no—
119C2×C3×C8—91no—
120C2×C3×C8—91no—
121C2×C3×C8—182no—
122#217—181no—
123(C2×D3)⋊C4—181no—
124C4⋊Dic3—91no—
125C4⋊Dic3—91no—
126Dic3⋊C4—181no—
127#204—182no—
128C22×C32—41no—
129C22×C32—81no—
130C22×C32#20411yescharacteristic, complemented
131C2×C3⋊D3—21no—
132C2×C3⋊D3—21no—
133C2×C3⋊D3—41no—
134C2×C3⋊D3#20411yescharacteristic, complemented
135C2×C3⋊D3C2×D1211yescharacteristic
136C2×C3×D3—121no—
137C2×C3×D3—121no—
138C2×C3×D3—121no—
139C2×C3×D3—121no—
140C2×C3×D3—241no—
141C2×C3×D3—241no—
142C2×C3×D3—241no—
143D32—121no—
144D32—121no—
145D32—241no—
146D32—241no—
147#157—21no—
148#157—121no—
149#157—121no—
150#157#23511yescharacteristic
151#157C2×#6711yescharacteristic
152C3⋊Dic3—41no—
153C3×Dic3—121no—
154C3×Dic3—121no—
155C3×Dic3—241no—
156C2×SD16—271no—
157C22×C8—91no—
158C22⋊Q8—271no—
159#121—271no—
160#106—271no—
161#103—271no—
162#102—271no—
163C33#46511yescharacteristic, complemented, Sylow
164C23×C3—91no—
165C22×D3—121no—
166C22×D3—181no—
167C22×D3—361no—
168C3×D4—361no—
169C2×C3×C4—91no—
170C2×C3×C4—91no—
171C2×C3×C4—181no—
172C2×C3×C4—181no—
173#67—121no—
174#67—121no—
175#67—241no—
176#67—361no—
177#67—361no—
178C2×Dic3—181no—
179C2×Dic3—181no—
180C2×Dic3—181no—
181D12—181no—
182D12—181no—
183D12—361no—
184C4×D3—361no—
185Dic6—181no—
186Dic6—181no—
187C3×C8—92no—
188C3×C8—181no—
189C2×C32—21no—
190C2×C32—41no—
191C2×C32—81no—
192C2×C32—81no—
193C2×C32—161no—
194C2×C32C2×#20411yescharacteristic
195C3⋊D3—21no—
196C3⋊D3—81no—
197C3⋊D3#90411yescharacteristic
198C3⋊D3#93011yescharacteristic
199C3×D3—121no—
200C3×D3—121no—
201C3×D3—241no—
202C3×D3—241no—
203C3×D3—241no—
204C3×D3—481no—
205C2×Q8—271no—
206C2×D4—271no—
207C2×D4—541no—
208C22×C4—91no—
209SD16—542no—
210C2×C8—91no—
211C2×C8—91no—
212C2×C8—182no—
213C4⋊C4—271no—
214C4⋊C4—271no—
215C4⋊C4—541no—
216C22⋊C4—541no—
217C22⋊C4—541no—
218C22×C3—41no—
219C22×C3—81no—
220C22×C3—91no—
221C22×C3—91no—
222C22×C3—181no—
223C22×C3—181no—
224C22×C3—361no—
225C22×C3AΓL(1,𝔽9)11yescharacteristic, complemented
226C2×D3—121no—
227C2×D3—121no—
228C2×D3—121no—
229C2×D3—121no—
230C2×D3—241no—
231C2×D3—241no—
232C2×D3—241no—
233C2×D3—361no—
234C2×D3—361no—
235C2×D3—721no—
236C2×D3—721no—
237C3×C4—91no—
238C3×C4—91no—
239C3×C4—181no—
240C3×C4—361no—
241Dic3—121no—
242Dic3—121no—
243Dic3—241no—
244Dic3—361no—
245Dic3—361no—
246C32—41no—
247C32—81no—
248C32#511411yescharacteristic, complemented
249C23—91no—
250C23—541no—
251Q8—541no—
252Q8—541no—
253D4—361no—
254D4—541no—
255D4—541no—
256D4—1081no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4—541no—
262C2×C4—541no—
263C2×C4—541no—
264C8—92no—
265C8—181no—
266C2×C3—21no—
267C2×C3—41no—
268C2×C3—81no—
269C2×C3—81no—
270C2×C3—91no—
271C2×C3—91no—
272C2×C3—161no—
273C2×C3—181no—
274C2×C3—721no—
275C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
276D3—121no—
277D3—121no—
278D3—241no—
279D3—241no—
280D3—241no—
281D3—481no—
282C22—91no—
283C22—91no—
284C22—181no—
285C22—181no—
286C22—361no—
287C22—1081no—
288C22—1081no—
289C22AGL(1,𝔽9)⋊D311yescharacteristic, complemented
290C4—91no—
291C4—91no—
292C4—181no—
293C4—361no—
294C4—1081no—
295C4—1081no—
296C3—41no—
297C3—81no—
298C3#1059641611yescharacteristic, complemented
299C2—21no—
300C2—91no—
301C2—91no—
302C2—181no—
303C2—721no—
304C2C2×AGL(1,𝔽9)⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽9)⋊D3
commutator, cosocleC2×C3×#157C23
FittingC22×C33SD16
socleC2×C33C2×SD16
prime radical#11724355C2
hypercenterC22AGL(1,𝔽9)⋊D3
hypocenterC33#465
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×AGL(1,𝔽9)C2semidirect
2. C2×AGL(1,𝔽9)⋊D3C2semidirect
3. #11724369C2semidirect
5. AGL(1,𝔽9)⋊Dic3C2semidirect
6. #11724105C2semidirect
7. #11724104C2semidirect
22. C22×AGL(1,𝔽9)D3semidirect
42. C3×AGL(1,𝔽9)D4semidirect
61. C22×C33SD16semidirect
65. C2×C3×C3⋊D3SD16semidirect
97. AGL(1,𝔽9)#67semidirect
130. C22×C32#204semidirect
134. C2×C3⋊D3#204semidirect
163. C33#465semidirect
225. C22×C3AΓL(1,𝔽9)semidirect
248. C32#5114semidirect
289. C22AGL(1,𝔽9)⋊D3semidirect
298. C3#10596416semidirect

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,𝔽9)2
C2×AGL(1,𝔽9)⋊D32
#117243692
#117243552
AGL(1,𝔽9)⋊Dic32
#117241052
#117241042
#105964163
#51149