Finite Group Databasebeta

← C3×#10588528 #13142022

Subgroups — C3×#10588528 #13142022

Order 1728.

410 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D72C211yesmaximal, characteristic
2C2×C3×#3664C211yesmaximal, characteristic, complemented
3C3×#63437C211yesmaximal, characteristic, complemented
4C3×C4×D36C211yesmaximal, characteristic, complemented
5C3×C9×C4⋊C8C211yesmaximal, characteristic, complemented
6C3×#63380C211yesmaximal, characteristic, complemented
7C3×#63376C211yesmaximal, characteristic, complemented
8C3×#4844—31nomaximal
9#10588528C311yesmaximal, characteristic, direct product factor
10C2×C3×D36C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C3×C4×D9—21no—
13C2×C3×Dic18C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C2×C3×C8×C9C2211yescharacteristic
16C3×C42×C9C2211yescharacteristic
17C3×(C2×D9)⋊C4—21no—
18C3×(C2×D9)⋊C4—21no—
19C3×(C2×D9)⋊C4—21no—
20C3×C4⋊Dic9C2211yescharacteristic
21C3×D72—22no—
22C3×#3664—22no—
23C2×C3×D24—31no—
24C2×C3×#204—31no—
25C3×#875—31no—
26C3×C4×D12—31no—
27C32×C4⋊C8D311yescharacteristic
28C3×#821—31no—
29C3×#816—31no—
30C2×D72C2×C311yescharacteristic
31C2×#3664C2×C311yescharacteristic, complemented
32#63437C2×C311yescharacteristic, complemented
33C4×D36C2×C311yescharacteristic, complemented
34C9×C4⋊C8—21no—
35C9×C4⋊C8C2×C311yescharacteristic, complemented
36#63380C2×C311yescharacteristic, complemented
37#63376C2×C311yescharacteristic, complemented
38C22×C3×D9—21no—
39C22×C3×D9—21no—
40C2×C3×C4×C9D411yescharacteristic
41C2×C3×C4×C9D411yescharacteristic
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3×Dic9—21no—
44C2×C3×Dic9—21no—
45C3×D36—21no—
46C3×D36—21no—
47C3×D36—21no—
48C3×D36D412yes—
49C3×C4×D9—42no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52C3×C8×C9—21no—
53C3×C8×C9—21no—
54C3×#463—91no—
55#4844—31no—
56C2×C3×D12—31no—
57C2×C3×D12—31no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—31no—
60C2×C32×C8C2×D311yescharacteristic
61C2×C32×C8C2×D311yescharacteristic
62C32×C42C2×D311yescharacteristic
63C3×(C2×D3)⋊C4—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×(C2×D3)⋊C4—61no—
66C3×C4⋊Dic3—31no—
67C3×D24—62no—
68C3×#204—62no—
69C2×D36C22×C311yescharacteristic
70C2×D36C22×C311yescharacteristic
71C2×C4×D9—21no—
72C2×Dic18C22×C311yescharacteristic
73C2×C8×C9—21no—
74C2×C8×C9—21no—
75C2×C8×C9C22×C311yescharacteristic
76C2×C8×C9C22×C311yescharacteristic
77C42×C9—21no—
78C42×C9C22×C311yescharacteristic
79(C2×D9)⋊C4—21no—
80(C2×D9)⋊C4—21no—
81(C2×D9)⋊C4—21no—
82C4⋊Dic9C22×C311yescharacteristic
83D72—22no—
84#3664—22no—
85C22×C3×C9C2×D411yescharacteristic
86C2×C3×D9—22no—
87C2×C3×D9—41no—
88C2×C3×D9—41no—
89C2×C3×D9—41no—
90C2×C3×D9—41no—
91C2×C3×D9—41no—
92C3×C4×C9—21no—
93C3×C4×C9—21no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C3×Dic9—41no—
97C3×Dic9—41no—
98C2×C3×SD16—91no—
99C2×C3×D8—91no—
100C3×#124—91no—
101C3×C4×D4—91no—
102C2×D24—31no—
103C2×#204—31no—
104#875—31no—
105C4×D12—31no—
106C3×C4⋊C8—21no—
107C3×C4⋊C8D911yescharacteristic, complemented
108C3×C4⋊C8C3×D311yescharacteristic
109C3×#103—91no—
110C3×#102—91no—
111#821—31no—
112#816—31no—
113C22×C3×D3—61no—
114C22×C3×D3—61no—
115C2×C32×C4D1211yescharacteristic
116C2×C32×C4D1211yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C3×D12—32no—
121C3×D12—61no—
122C3×D12—61no—
123C3×D12—61no—
124C3×C4×D3—122no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127C22×D9—21no—
128C22×D9—21no—
129C32×C8—21no—
130C32×C8—21no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9C3×D411yescharacteristic
135C2×C4×C9C3×D411yescharacteristic
136C2×C4×C9C23×C311yescharacteristic
137C2×Dic9—21no—
138C2×Dic9—21no—
139D36—21no—
140D36—21no—
141D36—21no—
142D36C3×D412yes—
143C4×D9—42no—
144Dic18—21no—
145Dic18—21no—
146C8×C9—21no—
147C8×C9—21no—
148C8×C9—41no—
149C8×C9—41no—
150#463—91noSylow
151C2×C3×C9#12111yescharacteristic
152C2×C3×C9#13511yescharacteristic
153C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
154C3×D9—42no—
155C3×D9—81no—
156C2×C3×Q8—91no—
157C2×C3×D4—91no—
158C2×C3×D4—91no—
159C22×C3×C4—181no—
160C2×D12—31no—
161C2×D12—31no—
162C2×C4×D3—61no—
163C2×Dic6—31no—
164C3×SD16—182no—
165C3×D8—182no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8C2×D911yescharacteristic
169C2×C3×C8C2×D911yescharacteristic
170C2×C3×C8C2×C3×D311yescharacteristic
171C2×C3×C8C2×C3×D311yescharacteristic
172C3×C4⋊C4—91no—
173C3×C22⋊C4—181no—
174C3×C22⋊C4—181no—
175C3×C22⋊C4—181no—
176C3×C42—21no—
177C3×C42C2×D911yescharacteristic
178C3×C42C2×C3×D311yescharacteristic
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—61no—
182C4⋊Dic3—31no—
183D24—62no—
184#204—62no—
185C22×C32C2×D1211yescharacteristic
186C2×C3×D3—62no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4D3×D411yescharacteristic
195C32×C4#23911yescharacteristic
196C3×Dic3—121no—
197C3×Dic3—121no—
198C22×C9—21no—
199C22×C9C2×C3×D411yescharacteristic
200C2×D9—22no—
201C2×D9—41no—
202C2×D9—41no—
203C2×D9—41no—
204C2×D9—41no—
205C2×D9—41no—
206C4×C9—22no—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—21no—
210C4×C9—21no—
211C4×C9—41no—
212C4×C9C2×C3×D411yescharacteristic
213C4×C9C3×#4111yescharacteristic
214Dic9—41no—
215Dic9—41no—
216C2×SD16—91no—
217C2×D8—91no—
218#124—91no—
219C4×D4—91no—
220C4⋊C8C3×D911yescharacteristic, complemented
221#103—91no—
222#102—91no—
223C3×C9#46311yescharacteristic, complemented, Sylow
224C23×C3—181no—
225C23×C3—181no—
226C22×D3—61no—
227C22×D3—61no—
228C3×Q8—181no—
229C3×Q8—181no—
230C3×D4—92no—
231C3×D4—181no—
232C3×D4—181no—
233C3×D4—181no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—362no—
240C2×C3×C4D3611yescharacteristic
241C2×C3×C4D3611yescharacteristic
242C2×C3×C4C22×D911yescharacteristic
243C2×C3×C4C3×D1211yescharacteristic
244C2×C3×C4C3×D1211yescharacteristic
245C2×C3×C4C22×C3×D311yescharacteristic
246C2×Dic3—61no—
247C2×Dic3—61no—
248D12—32no—
249D12—61no—
250D12—61no—
251D12—61no—
252C4×D3—122no—
253Dic6—61no—
254Dic6—61no—
255C3×C8—21no—
256C3×C8—21no—
257C3×C8—21no—
258C3×C8—21no—
259C3×C8—41no—
260C3×C8—41no—
261C2×C32#89511yescharacteristic
262C2×C32#90411yescharacteristic
263C2×C32#90811yescharacteristic
264C3×D3—122no—
265C3×D3—241no—
266C2×C9—21no—
267C2×C9—21no—
268C2×C9—21no—
269C2×C9C3×#12111yescharacteristic
270C2×C9C3×#13511yescharacteristic
271C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
272D9—42no—
273D9—81no—
274C2×Q8—91no—
275C2×D4—91no—
276C2×D4—91no—
277C22×C4—181no—
278SD16—182no—
279D8—182no—
280C2×C8C2×C3×D911yescharacteristic
281C2×C8C2×C3×D911yescharacteristic
282C4⋊C4—91no—
283C22⋊C4—181no—
284C22⋊C4—181no—
285C22⋊C4—181no—
286C42C2×C3×D911yescharacteristic
287C22×C3—21no—
288C22×C3—182no—
289C22×C3—361no—
290C22×C3—361no—
291C22×C3—361no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3C2×D3611yescharacteristic
295C22×C3C2×C3×D1211yescharacteristic
296C2×D3—62no—
297C2×D3—121no—
298C2×D3—121no—
299C2×D3—121no—
300C2×D3—121no—
301C2×D3—121no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—41no—
310C3×C4—361no—
311C3×C4—361no—
312C3×C4D4×D911yescharacteristic
313C3×C4#370111yescharacteristic
314C3×C4C3×D3×D411yescharacteristic
315C3×C4C3×#23911yescharacteristic
316Dic3—121no—
317Dic3—121no—
318C32#484411yescharacteristic
319C9—21no—
320C9C3×#46311yescharacteristic, complemented
321C23—181no—
322C23—181no—
323Q8—181no—
324Q8—181no—
325D4—92no—
326D4—181no—
327D4—181no—
328D4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—362no—
332C2×C4C3×D3611yescharacteristic
333C2×C4C3×D3611yescharacteristic
334C2×C4C22×C3×D911yescharacteristic
335C8—21no—
336C8—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—362no—
341C2×C3—721no—
342C2×C3#6345711yescharacteristic
343C2×C3#6346711yescharacteristic
344C2×C3#6347011yescharacteristic
345C2×C3C3×#89511yescharacteristic
346C2×C3C3×#90411yescharacteristic
347C2×C3C3×#90811yescharacteristic
348D3—122no—
349D3—241no—
350C22—182no—
351C22—361no—
352C22—361no—
353C22—361no—
354C22—361no—
355C22—361no—
356C22C2×C3×D3611yescharacteristic
357C4—21no—
358C4—21no—
359C4—361no—
360C4—361no—
361C4C3×D4×D911yescharacteristic
362C4C3×#370111yescharacteristic
363C3—21no—
364C3#1058852811yescharacteristic, direct product factor
365C3C3×#484411yescharacteristic
366C2—362no—
367C2—721no—
368C2C3×#6345711yescharacteristic
369C2C3×#6346711yescharacteristic
370C2C3×#6347011yescharacteristic
subgroupG/H
centerC22×C3C2×D36
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C4⋊C8C2
socleC22×C32C2×D12
prime radicalC2×C3×D72C2
hypercenterC3×C4⋊C8D9
hypocenterC9C3×#463
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3664C2semidirect
3. C3×#63437C2semidirect
4. C3×C4×D36C2semidirect
5. C3×C9×C4⋊C8C2semidirect
6. C3×#63380C2semidirect
7. C3×#63376C2semidirect
9. #10588528C3direct (N × H)
31. C2×#3664C2×C3semidirect
32. #63437C2×C3semidirect
33. C4×D36C2×C3semidirect
35. C9×C4⋊C8C2×C3semidirect
36. #63380C2×C3semidirect
37. #63376C2×C3semidirect
107. C3×C4⋊C8D9semidirect
220. C4⋊C8C3×D9semidirect
223. C3×C9#463semidirect
320. C9C3×#463semidirect
364. C3#10588528direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D722
C2×C3×#36642
C3×#634372
C3×C4×D362
C3×C9×C4⋊C82
C3×#633802
C3×#633762
C3×#48443
#105885283