Finite Group Databasebeta

← C3×#10588529 #13142011

Subgroups — C3×#10588529 #13142011

Order 1728.

374 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3664C211yesmaximal, characteristic
2C2×C3×Dic36C211yesmaximal, characteristic, complemented
3C3×#63437C211yesmaximal, characteristic
4C3×C4×Dic18C211yesmaximal, characteristic, complemented
5C3×C9×C4⋊C8C211yesmaximal, characteristic, complemented
6C3×#63380C211yesmaximal, characteristic
7C3×#63376C211yesmaximal, characteristic, complemented
8C3×#4845—31nomaximal
9#10588529C311yesmaximal, characteristic, direct product factor
10C2×C3×D36C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C42×C9C2211yescharacteristic
16C3×(C2×D9)⋊C4—21no—
17C3×(C2×D9)⋊C4—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C3×Dic9⋊C4—21no—
20C3×C4×Dic9—21no—
21C3×#3664—22no—
22C3×Dic36—22no—
23C2×C3×Dic12—31no—
24C2×C3×#204—31no—
25C3×#875—31no—
26C3×C4×Dic6—31no—
27C32×C4⋊C8D311yescharacteristic
28C3×#821—31no—
29C3×#816—31no—
30C2×#3664C2×C311yescharacteristic
31C2×Dic36C2×C311yescharacteristic, complemented
32#63437C2×C311yescharacteristic
33C4×Dic18C2×C311yescharacteristic, complemented
34C9×C4⋊C8—21no—
35C9×C4⋊C8C2×C311yescharacteristic, complemented
36#63380C2×C311yescharacteristic
37#63376C2×C311yescharacteristic, complemented
38C22×C3×D9—21no—
39C2×C3×C4×C9D411yescharacteristic
40C2×C3×C4×C9D411yescharacteristic
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3×Dic9—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9—21no—
45C3×D36—21no—
46C3×D36—21no—
47C3×Dic18—21no—
48C3×Dic18—21no—
49C3×Dic18—21no—
50C3×Dic18D412yes—
51C3×C8×C9—21no—
52C3×C8×C9—21no—
53C3×#464—91no—
54#4845—31no—
55C2×C3×D12—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—31no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C2×D311yescharacteristic
60C32×C42C2×D311yescharacteristic
61C3×(C2×D3)⋊C4—61no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—31no—
64C3×Dic3⋊C4—61no—
65C3×C4×Dic3—61no—
66C3×Dic12—62no—
67C3×#204—62no—
68C2×D36C22×C311yescharacteristic
69C2×Dic18C22×C311yescharacteristic
70C2×Dic18C22×C311yescharacteristic
71C2×C8×C9—21no—
72C2×C8×C9—21no—
73C2×C8×C9C22×C311yescharacteristic
74C2×C8×C9C22×C311yescharacteristic
75C42×C9—21no—
76C42×C9C22×C311yescharacteristic
77(C2×D9)⋊C4—21no—
78(C2×D9)⋊C4—21no—
79C4⋊Dic9C22×C311yescharacteristic
80Dic9⋊C4—21no—
81C4×Dic9—21no—
82#3664—22no—
83Dic36—22no—
84C22×C3×C9C2×D411yescharacteristic
85C2×C3×D9—41no—
86C2×C3×D9—41no—
87C2×C3×D9—41no—
88C3×C4×C9—21no—
89C3×C4×C9—21no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C3×Dic9—22no—
93C3×Dic9—41no—
94C3×Dic9—41no—
95C2×C3×Q16—91no—
96C2×C3×SD16—91no—
97C3×#124—91no—
98C3×C4×Q8—91no—
99C2×Dic12—31no—
100C2×#204—31no—
101#875—31no—
102C4×Dic6—31no—
103C3×C4⋊C8—21no—
104C3×C4⋊C8D911yescharacteristic, complemented
105C3×C4⋊C8C3×D311yescharacteristic
106C3×#103—91no—
107C3×#102—91no—
108#821—31no—
109#816—31no—
110C22×C3×D3—61no—
111C2×C32×C4D1211yescharacteristic
112C2×C32×C4D1211yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C3×D12—61no—
118C3×D12—61no—
119C3×Dic6—32no—
120C3×Dic6—61no—
121C3×Dic6—61no—
122C3×Dic6—61no—
123C22×D9—21no—
124C32×C8—21no—
125C32×C8—21no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9C3×D411yescharacteristic
130C2×C4×C9C3×D411yescharacteristic
131C2×C4×C9C23×C311yescharacteristic
132C2×Dic9—21no—
133C2×Dic9—21no—
134C2×Dic9—21no—
135D36—21no—
136D36—21no—
137Dic18—21no—
138Dic18—21no—
139Dic18—21no—
140Dic18C3×D412yes—
141C8×C9—21no—
142C8×C9—21no—
143C8×C9—41no—
144C8×C9—41no—
145#464—91noSylow
146C2×C3×C9#12111yescharacteristic
147C2×C3×C9#13511yescharacteristic
148C2×C3×C9#13711yescharacteristic
149C3×D9—81no—
150C2×C3×Q8—91no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C2×D12—31no—
154C2×Dic6—31no—
155C2×Dic6—31no—
156C3×Q16—182no—
157C3×SD16—182no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8C2×D911yescharacteristic
161C2×C3×C8C2×D911yescharacteristic
162C2×C3×C8C2×C3×D311yescharacteristic
163C2×C3×C8C2×C3×D311yescharacteristic
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C22⋊C4—181no—
167C3×C22⋊C4—181no—
168C3×C42—21no—
169C3×C42—181no—
170C3×C42C2×D911yescharacteristic
171C3×C42C2×C3×D311yescharacteristic
172(C2×D3)⋊C4—61no—
173(C2×D3)⋊C4—61no—
174C4⋊Dic3—31no—
175Dic3⋊C4—61no—
176C4×Dic3—61no—
177Dic12—62no—
178#204—62no—
179C22×C32C2×D1211yescharacteristic
180C2×C3×D3—121no—
181C2×C3×D3—121no—
182C2×C3×D3—121no—
183C32×C4—21no—
184C32×C4—21no—
185C32×C4D3×D411yescharacteristic
186C32×C4#23911yescharacteristic
187C3×Dic3—62no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C22×C9—21no—
191C22×C9C2×C3×D411yescharacteristic
192C2×D9—41no—
193C2×D9—41no—
194C2×D9—41no—
195C4×C9—22no—
196C4×C9—21no—
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—21no—
200C4×C9—41no—
201C4×C9C2×C3×D411yescharacteristic
202C4×C9C3×#4111yescharacteristic
203Dic9—22no—
204Dic9—41no—
205Dic9—41no—
206C2×Q16—91no—
207C2×SD16—91no—
208#124—91no—
209C4×Q8—91no—
210C4⋊C8C3×D911yescharacteristic, complemented
211#103—91no—
212#102—91no—
213C3×C9#46411yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—61no—
216C3×Q8—92no—
217C3×Q8—181no—
218C3×Q8—181no—
219C3×Q8—181no—
220C3×D4—181no—
221C3×D4—181no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4D3611yescharacteristic
229C2×C3×C4D3611yescharacteristic
230C2×C3×C4C22×D911yescharacteristic
231C2×C3×C4C3×D1211yescharacteristic
232C2×C3×C4C3×D1211yescharacteristic
233C2×C3×C4C22×C3×D311yescharacteristic
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237D12—61no—
238D12—61no—
239Dic6—32no—
240Dic6—61no—
241Dic6—61no—
242Dic6—61no—
243C3×C8—21no—
244C3×C8—21no—
245C3×C8—21no—
246C3×C8—21no—
247C3×C8—41no—
248C3×C8—41no—
249C2×C32#89511yescharacteristic
250C2×C32#90411yescharacteristic
251C2×C32#90911yescharacteristic
252C3×D3—241no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9C3×#12111yescharacteristic
257C2×C9C3×#13511yescharacteristic
258C2×C9C3×#13711yescharacteristic
259D9—81no—
260C2×Q8—91no—
261C2×Q8—91no—
262C2×D4—91no—
263Q16—182no—
264SD16—182no—
265C2×C8C2×C3×D911yescharacteristic
266C2×C8C2×C3×D911yescharacteristic
267C4⋊C4—91no—
268C4⋊C4—181no—
269C22⋊C4—181no—
270C22⋊C4—181no—
271C42—181no—
272C42C2×C3×D911yescharacteristic
273C22×C3—21no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3—361no—
277C22×C3C2×D3611yescharacteristic
278C22×C3C2×C3×D1211yescharacteristic
279C2×D3—121no—
280C2×D3—121no—
281C2×D3—121no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—182no—
291C3×C4—361no—
292C3×C4—361no—
293C3×C4D4×D911yescharacteristic
294C3×C4#370111yescharacteristic
295C3×C4C3×D3×D411yescharacteristic
296C3×C4C3×#23911yescharacteristic
297Dic3—62no—
298Dic3—121no—
299Dic3—121no—
300C32#484511yescharacteristic
301C9—21no—
302C9C3×#46411yescharacteristic, complemented
303C23—181no—
304Q8—92no—
305Q8—181no—
306Q8—181no—
307Q8—181no—
308D4—181no—
309D4—181no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—181no—
313C2×C4C3×D3611yescharacteristic
314C2×C4C3×D3611yescharacteristic
315C2×C4C22×C3×D911yescharacteristic
316C8—21no—
317C8—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—721no—
322C2×C3#6345711yescharacteristic
323C2×C3#6346711yescharacteristic
324C2×C3#6347111yescharacteristic
325C2×C3C3×#89511yescharacteristic
326C2×C3C3×#90411yescharacteristic
327C2×C3C3×#90911yescharacteristic
328D3—241no—
329C22—361no—
330C22—361no—
331C22—361no—
332C22C2×C3×D3611yescharacteristic
333C4—21no—
334C4—21no—
335C4—182no—
336C4—361no—
337C4—361no—
338C4C3×D4×D911yescharacteristic
339C4C3×#370111yescharacteristic
340C3—21no—
341C3#1058852911yescharacteristic, direct product factor
342C3C3×#484511yescharacteristic
343C2—721no—
344C2C3×#6345711yescharacteristic
345C2C3×#6346711yescharacteristic
346C2C3×#6347111yescharacteristic
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×D36C22
hypercenterC3×C4⋊C8D9
hypocenterC9C3×#464
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×Dic36C2semidirect
4. C3×C4×Dic18C2semidirect
5. C3×C9×C4⋊C8C2semidirect
7. C3×#63376C2semidirect
9. #10588529C3direct (N × H)
31. C2×Dic36C2×C3semidirect
33. C4×Dic18C2×C3semidirect
35. C9×C4⋊C8C2×C3semidirect
37. #63376C2×C3semidirect
104. C3×C4⋊C8D9semidirect
210. C4⋊C8C3×D9semidirect
213. C3×C9#464semidirect
302. C9C3×#464semidirect
341. C3#10588529direct (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×Dic362
C3×#634372
C3×C4×Dic182
C3×C9×C4⋊C82
C3×#633802
C3×#633762
C3×#48453
#105885293