Finite Group Databasebeta

← C3×#10588566 #13142194

Subgroups — C3×#10588566 #13142194

Order 1728.

396 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3664C211yesmaximal, characteristic, complemented
2C2×C3×#3663C211yesmaximal, characteristic, complemented
3C3×#63459C211yesmaximal, characteristic, complemented
4C3×#63457C211yesmaximal, characteristic
5C3×C9×#107C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7C3×#63368C211yesmaximal, characteristic, complemented
8C3×#4886—31nomaximal
9#10588566C311yesmaximal, characteristic, direct product factor
10C2×C3×D36—21no—
11C2×C3×D36C2211yescharacteristic
12C2×C3×C4×D9C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C9×C4⋊C4C2211yescharacteristic
17C3×(C2×D9)⋊C4—21no—
18C3×(C2×D9)⋊C4—21no—
19C3×C4⋊Dic9—21no—
20C2×C3×C9⋊C8C2211yescharacteristic
21C3×#3664—22no—
22C3×#3663—22no—
23C2×C3×#204—31no—
24C2×C3×ΓL(1,𝔽25)—31no—
25C3×#897—31no—
26C3×#895—31no—
27C32×#107D311yescharacteristic
28C3×#810—31no—
29C3×#809—31no—
30C2×#3664C2×C311yescharacteristic, complemented
31C2×#3663C2×C311yescharacteristic, complemented
32#63459C2×C311yescharacteristic, complemented
33#63457C2×C311yescharacteristic
34C9×#107—21no—
35C9×#107C2×C311yescharacteristic, complemented
36#63369C2×C311yescharacteristic, complemented
37#63368C2×C311yescharacteristic, complemented
38C22×C3×D9—21no—
39C22×C3×D9D411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3×Dic9—21no—
44C2×C3×Dic9D411yescharacteristic
45C3×D36—21no—
46C3×D36—21no—
47C3×D36—42no—
48C3×C4×D9—21no—
49C3×C4×D9—21no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52C3×C8×C9D412yescomplemented
53C3×C9⋊C8—21no—
54C3×#468—91no—
55#4886—31no—
56C2×C3×D12—31no—
57C2×C3×D12—61no—
58C2×C3×C4×D3—31no—
59C2×C3×Dic6—31no—
60C2×C32×C8C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63C3×(C2×D3)⋊C4—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—61no—
66C2×C3×C3⋊C8—31no—
67C3×#204—62no—
68C3×ΓL(1,𝔽25)—62no—
69C2×D36—21no—
70C2×D36C22×C311yescharacteristic
71C2×C4×D9C22×C311yescharacteristic
72C2×Dic18C22×C311yescharacteristic
73C2×C8×C9—21no—
74C2×C8×C9C22×C311yescharacteristic
75C9×C4⋊C4—21no—
76C9×C4⋊C4—21no—
77C9×C4⋊C4C22×C311yescharacteristic
78C9×C4⋊C4C22×C311yescharacteristic
79(C2×D9)⋊C4—21no—
80(C2×D9)⋊C4—21no—
81C4⋊Dic9—21no—
82C2×C9⋊C8C22×C311yescharacteristic
83#3664—22no—
84#3663—22no—
85C22×C3×C9C2×D411yescharacteristic
86C2×C3×D9—21no—
87C2×C3×D9—21no—
88C2×C3×D9—21no—
89C2×C3×D9—41no—
90C2×C3×D9—41no—
91C2×C3×D9—41no—
92C3×C4×C9—41no—
93C3×C4×C9—41no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C3×Dic9—21no—
97C3×Dic9—41no—
98C2×C3×SD16—91no—
99C2×C3×M16—91no—
100C3×C22⋊Q8—91no—
101C3×#121—91no—
102C2×#204—31no—
103C2×ΓL(1,𝔽25)—31no—
104#897—31no—
105#895—31no—
106C3×#107—21no—
107C3×#107D911yescharacteristic, complemented
108C3×#107C3×D311yescharacteristic
109C3×#103—91no—
110C3×#102—91no—
111#810—31no—
112#809—31no—
113C22×C3×D3—31no—
114C22×C3×D3—61no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3×Dic3—31no—
119C2×C3×Dic3—61no—
120C3×D12—61no—
121C3×D12—61no—
122C3×D12—122no—
123C3×C4×D3—61no—
124C3×C4×D3—61no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127C22×D9—21no—
128C22×D9C3×D411yescharacteristic
129C32×C8D1212yes—
130C3×C3⋊C8—61no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—41no—
135C2×C4×C9—41no—
136C2×C4×C9C23×C311yescharacteristic
137C2×Dic9—21no—
138C2×Dic9C3×D411yescharacteristic
139D36—21no—
140D36—21no—
141D36—42no—
142C4×D9—21no—
143C4×D9—21no—
144Dic18—21no—
145Dic18—21no—
146C8×C9—22no—
147C8×C9C3×D412yescomplemented
148C9⋊C8—21no—
149#468—91noSylow
150C2×C3×C9#12111yescharacteristic
151C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
152C2×C3×C9#13711yescharacteristic
153C3×D9—41no—
154C3×D9—81no—
155C2×C3×Q8—91no—
156C2×C3×D4—91no—
157C2×C3×D4—181no—
158C22×C3×C4—91no—
159C2×D12—31no—
160C2×D12—61no—
161C2×C4×D3—31no—
162C2×Dic6—31no—
163C3×SD16—182no—
164C3×M16—182no—
165C2×C3×C8—21no—
166C2×C3×C8—91no—
167C2×C3×C8C2×D911yescharacteristic
168C2×C3×C8C2×C3×D311yescharacteristic
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—181no—
172C3×C4⋊C4C2×D911yescharacteristic
173C3×C4⋊C4C2×D911yescharacteristic
174C3×C4⋊C4C2×C3×D311yescharacteristic
175C3×C4⋊C4C2×C3×D311yescharacteristic
176C3×C22⋊C4—181no—
177C3×C22⋊C4—181no—
178(C2×D3)⋊C4—61no—
179(C2×D3)⋊C4—61no—
180C4⋊Dic3—61no—
181C2×C3⋊C8—31no—
182#204—62no—
183ΓL(1,𝔽25)—62no—
184C22×C32D3×D411yescharacteristic
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4C2×D1211yescharacteristic
194C32×C4#23911yescharacteristic
195C3×Dic3—61no—
196C3×Dic3—121no—
197C22×C9—21no—
198C22×C9C2×C3×D411yescharacteristic
199C2×D9—21no—
200C2×D9—21no—
201C2×D9—21no—
202C2×D9—41no—
203C2×D9—41no—
204C2×D9—41no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—41no—
208C4×C9—41no—
209C4×C9—81no—
210C4×C9—81no—
211C4×C9C2×C3×D411yescharacteristic
212C4×C9C3×#4111yescharacteristic
213Dic9—21no—
214Dic9—41no—
215C2×SD16—91no—
216C2×M16—91no—
217C22⋊Q8—91no—
218#121—91no—
219#107C3×D911yescharacteristic, complemented
220#103—91no—
221#102—91no—
222C3×C9#46811yescharacteristic, complemented, Sylow
223C23×C3—91no—
224C23×C3—181no—
225C22×D3—31no—
226C22×D3—61no—
227C3×Q8—181no—
228C3×Q8—181no—
229C3×D4—181no—
230C3×D4—181no—
231C3×D4—362no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—91no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C22×D911yescharacteristic
244C2×C3×C4C22×C3×D311yescharacteristic
245C2×Dic3—31no—
246C2×Dic3—61no—
247D12—61no—
248D12—61no—
249D12—122no—
250C4×D3—61no—
251C4×D3—61no—
252Dic6—61no—
253Dic6—61no—
254C3×C8—22no—
255C3×C8—181no—
256C3×C8D3612yescomplemented
257C3×C8C3×D1212yes—
258C3⋊C8—61no—
259C2×C32#89511yescharacteristic
260C2×C32#91111yescharacteristic
261C2×C32#91811yescharacteristic
262C3×D3—121no—
263C3×D3—241no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—21no—
267C2×C9C3×#12111yescharacteristic
268C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
269C2×C9C3×#13711yescharacteristic
270D9—41no—
271D9—81no—
272C2×Q8—91no—
273C2×D4—91no—
274C2×D4—181no—
275C22×C4—91no—
276SD16—182no—
277M16—182no—
278C2×C8—91no—
279C2×C8C2×C3×D911yescharacteristic
280C4⋊C4—181no—
281C4⋊C4C2×C3×D911yescharacteristic
282C4⋊C4C2×C3×D911yescharacteristic
283C22⋊C4—181no—
284C22⋊C4—181no—
285C22×C3—21no—
286C22×C3—181no—
287C22×C3—181no—
288C22×C3—181no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3—361no—
292C22×C3D4×D911yescharacteristic
293C22×C3C3×D3×D411yescharacteristic
294C2×D3—61no—
295C2×D3—61no—
296C2×D3—61no—
297C2×D3—121no—
298C2×D3—121no—
299C2×D3—121no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—81no—
307C3×C4—81no—
308C3×C4—181no—
309C3×C4—361no—
310C3×C4C2×D3611yescharacteristic
311C3×C4#370111yescharacteristic
312C3×C4C2×C3×D1211yescharacteristic
313C3×C4C3×#23911yescharacteristic
314Dic3—61no—
315Dic3—121no—
316C32#488611yescharacteristic
317C9—21no—
318C9C3×#46811yescharacteristic, complemented
319C23—91no—
320C23—181no—
321Q8—181no—
322Q8—181no—
323D4—181no—
324D4—181no—
325D4—362no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—91no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4C22×C3×D911yescharacteristic
333C8—181no—
334C8C3×D3612yescomplemented
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—361no—
339C2×C3—721no—
340C2×C3#6345711yescharacteristic
341C2×C3#6347311yescharacteristic
342C2×C3#6348011yescharacteristic
343C2×C3C3×#89511yescharacteristic
344C2×C3C3×#91111yescharacteristic
345C2×C3C3×#91811yescharacteristic
346D3—121no—
347D3—241no—
348C22—181no—
349C22—181no—
350C22—181no—
351C22—361no—
352C22—361no—
353C22—361no—
354C22C3×D4×D911yescharacteristic
355C4—41no—
356C4—41no—
357C4—181no—
358C4—361no—
359C4C2×C3×D3611yescharacteristic
360C4C3×#370111yescharacteristic
361C3—21no—
362C3#1058856611yescharacteristic, direct product factor
363C3C3×#488611yescharacteristic
364C2—361no—
365C2—721no—
366C2C3×#6345711yescharacteristic
367C2C3×#6347311yescharacteristic
368C2C3×#6348011yescharacteristic
subgroupG/H
centerC22×C3D4×D9
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#107C2
socleC22×C32D3×D4
prime radicalC3×#63457C2
hypercenterC3×#107D9
hypocenterC9C3×#468
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#107

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3664C2semidirect
2. C2×C3×#3663C2semidirect
3. C3×#63459C2semidirect
5. C3×C9×#107C2semidirect
6. C3×#63369C2semidirect
7. C3×#63368C2semidirect
9. #10588566C3direct (N × H)
30. C2×#3664C2×C3semidirect
31. C2×#3663C2×C3semidirect
32. #63459C2×C3semidirect
35. C9×#107C2×C3semidirect
36. #63369C2×C3semidirect
37. #63368C2×C3semidirect
52. C3×C8×C9D4semidirect
107. C3×#107D9semidirect
147. C8×C9C3×D4semidirect
219. #107C3×D9semidirect
222. C3×C9#468semidirect
256. C3×C8D36semidirect
318. C9C3×#468semidirect
334. C8C3×D36semidirect
362. C3#10588566direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36642
C2×C3×#36632
C3×#634592
C3×#634572
C3×C9×#1072
C3×#633692
C3×#633682
C3×#48863
#105885663