Finite Group Databasebeta

← C3×#10588684 #13142679

Subgroups — C3×#10588684 #13142679

Order 1728.

394 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#117C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic
3C3×#63389C211yesmaximal, characteristic, complemented
4C3×#63369C211yesmaximal, characteristic, complemented
5C3×#63369C211yesmaximal, characteristic, complemented
6C3×#63367C211yesmaximal, characteristic, complemented
7C3×#63366C211yesmaximal, characteristic, complemented
8C3×#5003—31nomaximal
9#10588684C311yesmaximal, characteristic, direct product factor
10C22×C3×C4×C9C2211yescharacteristic
11C2×C3×#611—21no—
12C2×C3×D36C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C3×C9×C22⋊C4—21no—
16C3×C42×C9—21no—
17C3×(C2×D9)⋊C4—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C32×#117D311yescharacteristic
22C3×#930—31no—
23C3×#830—31no—
24C3×#809—31no—
25C3×#809—31no—
26C3×#808—31no—
27C3×#807—31no—
28C9×#117—21no—
29C9×#117C2×C311yescharacteristic, complemented
30#63492C2×C311yescharacteristic
31#63389C2×C311yescharacteristic, complemented
32#63369C2×C311yescharacteristic, complemented
33#63369C2×C311yescharacteristic, complemented
34#63367C2×C311yescharacteristic, complemented
35#63366C2×C311yescharacteristic, complemented
36C23×C3×C9D411yescharacteristic
37C22×C3×D9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9D411yescharacteristic
43C2×C3×C4×C9C2311yescharacteristic
44C3×#611—42no—
45C2×C3×Dic9—21no—
46C3×D36—21no—
47C3×D36—21no—
48C3×C9⋊C8—21no—
49C3×C9⋊C8—21no—
50C3×#482—91no—
51#5003—31no—
52C22×C32×C4C2×D311yescharacteristic
53C2×C3×#67—61no—
54C2×C3×D12—31no—
55C32×C4⋊C4C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57C32×C22⋊C4—21no—
58C32×C42—21no—
59C3×(C2×D3)⋊C4—61no—
60C3×C4⋊Dic3—31no—
61C2×C3×C3⋊C8—31no—
62C2×C3×C3⋊C8—31no—
63C22×C4×C9—21no—
64C22×C4×C9C22×C311yescharacteristic
65C2×#611—21no—
66C2×D36C22×C311yescharacteristic
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C22×C311yescharacteristic
70C9×C4⋊C4C22×C311yescharacteristic
71C9×C22⋊C4—22no—
72C9×C22⋊C4—21no—
73C42×C9—22no—
74C42×C9—21no—
75(C2×D9)⋊C4—21no—
76C4⋊Dic9C22×C311yescharacteristic
77C2×C9⋊C8C22×C311yescharacteristic
78C2×C9⋊C8C22×C311yescharacteristic
79C22×C3×C9—21no—
80C22×C3×C9—21no—
81C22×C3×C9—21no—
82C22×C3×C9C2×D411yescharacteristic
83C2×C3×D9—41no—
84C2×C3×D9—41no—
85C2×C3×D9—41no—
86C3×C4×C9—21no—
87C3×C4×C9—41no—
88C3×C4×C9—41no—
89C3×C4×C9#4111yescharacteristic
90C3×C4×C9#4111yescharacteristic
91C3×Dic9—41no—
92C3×#121—91no—
93C3×#117—21no—
94C3×#117D911yescharacteristic, complemented
95C3×#117C3×D311yescharacteristic
96#930—31no—
97C3×#107—91no—
98C3×#106—91no—
99C3×#102—91no—
100C3×#102—91no—
101C3×C22⋊C8—91no—
102#830—31no—
103#809—31no—
104#809—31no—
105#808—31no—
106#807—31no—
107C23×C32#6711yescharacteristic
108C22×C3×D3—61no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C3×#67—122no—
116C2×C3×Dic3—61no—
117C3×D12—61no—
118C3×D12—61no—
119C23×C9—21no—
120C23×C9C3×D411yescharacteristic
121C22×D9—21no—
122C3×C3⋊C8—61no—
123C3×C3⋊C8—61no—
124C2×C4×C9—22no—
125C2×C4×C9—22no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—22no—
129C2×C4×C9—22no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9C3×D411yescharacteristic
135C2×C4×C9C23×C311yescharacteristic
136#611—42no—
137C2×Dic9—21no—
138D36—21no—
139D36—21no—
140C9⋊C8—21no—
141C9⋊C8—21no—
142#482—91noSylow
143C2×C3×C9—41no—
144C2×C3×C9#12311yescharacteristic
145C2×C3×C9#13511yescharacteristic
146C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
147C3×D9—81no—
148C2×C3×D4—91no—
149C2×C3×D4—181no—
150C22×C3×C4—21no—
151C22×C3×C4C2×D911yescharacteristic
152C22×C3×C4C2×C3×D311yescharacteristic
153C2×#67—61no—
154C2×D12—31no—
155C2×C3×C8—91no—
156C2×C3×C8—91no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4C2×D911yescharacteristic
161C3×C4⋊C4C2×D911yescharacteristic
162C3×C4⋊C4C2×C3×D311yescharacteristic
163C3×C4⋊C4C2×C3×D311yescharacteristic
164C3×C22⋊C4—22no—
165C3×C22⋊C4—21no—
166C3×C22⋊C4—21no—
167C3×C22⋊C4—181no—
168C3×C42—22no—
169C3×C42—21no—
170C3×C42—21no—
171(C2×D3)⋊C4—61no—
172C4⋊Dic3—31no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32C2×#6711yescharacteristic
179C2×C3×D3—121no—
180C2×C3×D3—121no—
181C2×C3×D3—121no—
182C32×C4—21no—
183C32×C4—41no—
184C32×C4—41no—
185C32×C4#23511yescharacteristic
186C32×C4#23511yescharacteristic
187C3×Dic3—121no—
188C22×C9—22no—
189C22×C9—21no—
190C22×C9—21no—
191C22×C9—21no—
192C22×C9—21no—
193C22×C9—41no—
194C22×C9—41no—
195C22×C9C2×C3×D411yescharacteristic
196C2×D9—41no—
197C2×D9—41no—
198C2×D9—41no—
199C4×C9—22no—
200C4×C9—21no—
201C4×C9—21no—
202C4×C9—21no—
203C4×C9—42no—
204C4×C9—42no—
205C4×C9—41no—
206C4×C9—41no—
207C4×C9C3×#4111yescharacteristic
208C4×C9C3×#4111yescharacteristic
209Dic9—41no—
210#121—91no—
211#117C3×D911yescharacteristic, complemented
212#107—91no—
213#106—91no—
214#102—91no—
215#102—91no—
216C22⋊C8—91no—
217C3×C9#48211yescharacteristic, complemented, Sylow
218C23×C3—21no—
219C23×C3—181no—
220C23×C3#61111yescharacteristic
221C23×C3C3×#6711yescharacteristic
222C22×D3—61no—
223C3×D4—181no—
224C3×D4—181no—
225C3×D4—362no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—181no—
241C2×C3×C4#61111yescharacteristic
242C2×C3×C4C22×D911yescharacteristic
243C2×C3×C4C3×#6711yescharacteristic
244C2×C3×C4C22×C3×D311yescharacteristic
245#67—122no—
246C2×Dic3—61no—
247D12—61no—
248D12—61no—
249C3×C8—181no—
250C3×C8—181no—
251C3⋊C8—61no—
252C3⋊C8—61no—
253C2×C32—41no—
254C2×C32#92911yescharacteristic
255C2×C32#94911yescharacteristic
256C2×C32#95011yescharacteristic
257C3×D3—241no—
258C2×C9—21no—
259C2×C9—21no—
260C2×C9—21no—
261C2×C9—42no—
262C2×C9—41no—
263C2×C9C3×#12311yescharacteristic
264C2×C9C3×#13511yescharacteristic
265C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
266D9—81no—
267C2×D4—91no—
268C2×D4—181no—
269C22×C4C2×C3×D911yescharacteristic
270C2×C8—91no—
271C2×C8—91no—
272C4⋊C4—91no—
273C4⋊C4C2×C3×D911yescharacteristic
274C4⋊C4C2×C3×D911yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—181no—
277C42—21no—
278C22×C3—22no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—361no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3C2×#61111yescharacteristic
292C22×C3C2×C3×#6711yescharacteristic
293C2×D3—121no—
294C2×D3—121no—
295C2×D3—121no—
296C3×C4—22no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—42no—
302C3×C4—42no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—361no—
308C3×C4#369711yescharacteristic
309C3×C4#369711yescharacteristic
310C3×C4C3×#23511yescharacteristic
311C3×C4C3×#23511yescharacteristic
312Dic3—121no—
313C32#500311yescharacteristic
314C9—21no—
315C9C3×#48211yescharacteristic, complemented
316C23—181no—
317C23C3×#61111yescharacteristic
318D4—181no—
319D4—181no—
320D4—362no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—181no—
326C2×C4C3×#61111yescharacteristic
327C2×C4C22×C3×D911yescharacteristic
328C8—181no—
329C8—181no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—42no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—721no—
337C2×C3#6349111yescharacteristic
338C2×C3#6351211yescharacteristic
339C2×C3#6351311yescharacteristic
340C2×C3C3×#92911yescharacteristic
341C2×C3C3×#94911yescharacteristic
342C2×C3C3×#95011yescharacteristic
343D3—241no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—361no—
348C22—361no—
349C22—361no—
350C22C2×C3×#61111yescharacteristic
351C4—21no—
352C4—41no—
353C4—41no—
354C4—361no—
355C4C3×#369711yescharacteristic
356C4C3×#369711yescharacteristic
357C3—21no—
358C3#1058868411yescharacteristic, direct product factor
359C3C3×#500311yescharacteristic
360C2—41no—
361C2—721no—
362C2C3×#6349111yescharacteristic
363C2C3×#6351211yescharacteristic
364C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#117C2
socleC22×C32C2×#67
prime radicalC3×#63492C2
hypercenterC3×#117D9
hypocenterC9C3×#482
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
1. C3×C9×#117C2semidirect
3. C3×#63389C2semidirect
4. C3×#63369C2semidirect
5. C3×#63369C2semidirect
6. C3×#63367C2semidirect
7. C3×#63366C2semidirect
9. #10588684C3direct (N × H)
29. C9×#117C2×C3semidirect
31. #63389C2×C3semidirect
32. #63369C2×C3semidirect
33. #63369C2×C3semidirect
34. #63367C2×C3semidirect
35. #63366C2×C3semidirect
94. C3×#117D9semidirect
211. #117C3×D9semidirect
217. C3×C9#482semidirect
315. C9C3×#482semidirect
358. C3#10588684direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1172
C3×#634922
C3×#633892
C3×#633692
C3×#633672
C3×#633662
C3×#50033
#105886843