Finite Group Databasebeta

← C3×#10589606 #13152402

Subgroups — C3×#10589606 #13152402

Order 1728.

974 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×C4×#611C214yesmaximal, complemented
4C3×#63488C211yesmaximal, characteristic, complemented
5C3×#63455C212yesmaximal, complemented
6C3×#63454C212yesmaximal, complemented
7C3×C4⋊C4×D9C212yesmaximal, complemented
8C3×#63449C212yesmaximal, complemented
9C3×#5504—31nomaximal
10#10589606C311yesmaximal, characteristic, direct product factor
11C22×C3×C4×C9C2212yes—
12C22×C3×C4×C9C2211yescharacteristic
13C2×C3×#611C2212yes—
14C3×#3697C418yescomplemented
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2214yes—
17C2×C3×C4×D9C2212yes—
18C2×C3×Dic18C2211yescharacteristic
19C3×C9×C4⋊C4C2212yes—
20C3×C9×C4⋊C4C2212yes—
21C3×#3676C2212yes—
22C3×(C2×D9)⋊C4C2214yes—
23C3×C4⋊Dic9C2211yescharacteristic
24C3×C4⋊Dic9C2211yescharacteristic
25C3×Dic9⋊C4C2214yes—
26C3×C4×Dic9C2212yes—
27C3×C4×Dic9C2214yes—
28C2×C3×#235—31no—
29C2×C32×C4⋊C4D311yescharacteristic
30C3×C4×#67—34no—
31C3×#926—31no—
32C3×#893—32no—
33C3×#892—32no—
34C3×D3×C4⋊C4—32no—
35C3×#887—32no—
36C2×#3697C2×C311yescharacteristic
37C2×C9×C4⋊C4—21no—
38C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
39C4×#611C2×C314yescomplemented
40#63488C2×C311yescharacteristic, complemented
41#63455C2×C312yescomplemented
42#63454C2×C312yescomplemented
43C4⋊C4×D9C2×C312yescomplemented
44#63449C2×C312yescomplemented
45C23×C3×C9C2311yescharacteristic
46C22×C3×D9C2312yes—
47C2×C3×C4×C9—42no—
48C2×C3×C4×C9C2×C414yescomplemented
49C2×C3×C4×C9C2314yes—
50C2×C3×C4×C9C2311yescharacteristic
51C2×C3×C4×C9C2311yescharacteristic
52C3×#611C2×C418yescomplemented
53C2×C3×Dic9C2312yes—
54C2×C3×Dic9C2312yes—
55C2×C3×Dic9C2312yes—
56C3×D36C2×C414yescomplemented
57C3×C4×D9—44no—
58C3×C4×D9C2×C418yescomplemented
59C3×Dic18C2×C414yescomplemented
60C3×#520—91no—
61#5504—31no—
62C22×C32×C4C2×D312yes—
63C22×C32×C4C2×D311yescharacteristic
64C2×C3×#67—32no—
65C3×#235—38no—
66C2×C3×D12—31no—
67C2×C3×C4×D3—32no—
68C2×C3×C4×D3—34no—
69C2×C3×Dic6—31no—
70C32×C4⋊C4C2×D312yes—
71C32×C4⋊C4C2×D312yes—
72C3×#217—32no—
73C3×(C2×D3)⋊C4—34no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×Dic3⋊C4—34no—
77C3×C4×Dic3—34no—
78C3×C4×Dic3—32no—
79C22×C4×C9—21no—
80C22×C4×C9—22no—
81C22×C4×C9C22×C312yes—
82C22×C4×C9C22×C311yescharacteristic
83C2×#611C22×C312yes—
84#3697C3×C418yescomplemented
85C2×D36C22×C311yescharacteristic
86C2×C4×D9C22×C314yes—
87C2×C4×D9C22×C312yes—
88C2×Dic18C22×C311yescharacteristic
89C9×C4⋊C4—22no—
90C9×C4⋊C4—22no—
91C9×C4⋊C4C22×C312yes—
92C9×C4⋊C4C22×C312yes—
93#3676C22×C312yes—
94(C2×D9)⋊C4C22×C314yes—
95C4⋊Dic9C22×C311yescharacteristic
96C4⋊Dic9C22×C311yescharacteristic
97Dic9⋊C4C22×C314yes—
98C4×Dic9C22×C312yes—
99C4×Dic9C22×C314yes—
100C22×C3×C9—21no—
101C22×C3×C9—21no—
102C22×C3×C9C22×C412yes—
103C22×C3×C9C2411yescharacteristic
104C2×C3×D9—22no—
105C2×C3×D9—22no—
106C2×C3×D9C22×C414yes—
107C3×C4×C9—24no—
108C3×C4×C9C22×C412yes—
109C3×C4×C9C22×C412yes—
110C3×Dic9—22no—
111C3×Dic9—22no—
112C3×Dic9C22×C414yes—
113C2×C3×#41—91no—
114C2×#235—31no—
115C3×C4×Q8—92no—
116C3×C4×D4—94no—
117C3×C4×D4—92no—
118C3×#117—91no—
119C3×#117—92no—
120C2×C3×C4⋊C4—21no—
121C2×C3×C4⋊C4—92no—
122C2×C3×C4⋊C4D911yescharacteristic, complemented
123C2×C3×C4⋊C4C3×D311yescharacteristic
124C4×#67—34no—
125#926—31no—
126#893—32no—
127#892—32no—
128D3×C4⋊C4—32no—
129#887—32no—
130C23×C32C22×D311yescharacteristic
131C22×C3×D3—32no—
132C2×C32×C4—42no—
133C2×C32×C4C4×D314yes—
134C2×C32×C4C22×D314yes—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C3×#67—38no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—32no—
140C2×C3×Dic3—32no—
141C3×D12—34no—
142C3×C4×D3—38no—
143C3×C4×D3—124no—
144C3×Dic6—34no—
145C23×C9—21no—
146C23×C9C23×C311yescharacteristic
147C22×D9C23×C312yes—
148C2×C4×C9—24no—
149C2×C4×C9—21no—
150C2×C4×C9—21no—
151C2×C4×C9—24no—
152C2×C4×C9—42no—
153C2×C4×C9—44no—
154C2×C4×C9C2×C3×C414yescomplemented
155C2×C4×C9C23×C314yes—
156C2×C4×C9C23×C311yescharacteristic
157C2×C4×C9C23×C311yescharacteristic
158#611C2×C3×C418yescomplemented
159C2×Dic9C23×C312yes—
160C2×Dic9C23×C312yes—
161C2×Dic9C23×C312yes—
162D36C2×C3×C414yescomplemented
163C4×D9—44no—
164C4×D9C2×C3×C418yescomplemented
165Dic18C2×C3×C414yescomplemented
166#520—91noSylow
167C2×C3×C9—22no—
168C2×C3×C9C23×C411yescharacteristic
169C2×C3×C9H(5,𝔽2)11yescharacteristic
170C2×C3×C92−1+411yescharacteristic
171C3×D9—24no—
172C3×#41—98no—
173C2×C3×Q8—91no—
174C2×C3×D4—92no—
175C2×C3×D4—91no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—92no—
179C22×C3×C4—94no—
180C22×C3×C4C2×D912yes—
181C22×C3×C4C2×D911yescharacteristic
182C22×C3×C4C2×C3×D312yes—
183C22×C3×C4C2×C3×D311yescharacteristic
184C2×#67—32no—
185#235—38no—
186C2×D12—31no—
187C2×C4×D3—32no—
188C2×C4×D3—34no—
189C2×Dic6—31no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—94no—
193C3×C4⋊C4—91no—
194C3×C4⋊C4—91no—
195C3×C4⋊C4C2×D912yes—
196C3×C4⋊C4C2×D912yes—
197C3×C4⋊C4C2×C3×D312yes—
198C3×C4⋊C4C2×C3×D312yes—
199C3×C22⋊C4—92no—
200C3×C22⋊C4—94no—
201C3×C42—94no—
202C3×C42—92no—
203#217—32no—
204(C2×D3)⋊C4—34no—
205C4⋊Dic3—31no—
206C4⋊Dic3—31no—
207Dic3⋊C4—34no—
208C4×Dic3—34no—
209C4×Dic3—32no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32C2×C4×D312yes—
213C22×C32C23×D311yescharacteristic
214C2×C3×D3—34no—
215C2×C3×D3—62no—
216C2×C3×D3—62no—
217C32×C4—24no—
218C32×C4C2×C4×D312yes—
219C32×C4C2×C4×D312yes—
220C3×Dic3—34no—
221C3×Dic3—62no—
222C3×Dic3—62no—
223C22×C9—22no—
224C22×C9—22no—
225C22×C9—21no—
226C22×C9—21no—
227C22×C9—21no—
228C22×C9—22no—
229C22×C9C22×C3×C412yes—
230C22×C9C24×C311yescharacteristic
231C2×D9—22no—
232C2×D9—22no—
233C2×D9C22×C3×C414yes—
234C4×C9—24no—
235C4×C9—22no—
236C4×C9—22no—
237C4×C9—44no—
238C4×C9C22×C3×C412yes—
239C4×C9C22×C3×C412yes—
240Dic9—22no—
241Dic9—22no—
242Dic9C22×C3×C414yes—
243C2×#41—91no—
244C4×Q8—92no—
245C4×D4—94no—
246C4×D4—92no—
247#117—91no—
248#117—92no—
249C2×C4⋊C4—92no—
250C2×C4⋊C4C3×D911yescharacteristic, complemented
251C3×C9#52011yescharacteristic, complemented, Sylow
252C23×C3—21no—
253C23×C3—92no—
254C23×C3C22×D911yescharacteristic
255C23×C3C22×C3×D311yescharacteristic
256C22×D3—32no—
257C3×Q8—94no—
258C3×D4—98no—
259C3×D4—94no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—44no—
267C2×C3×C4—92no—
268C2×C3×C4—98no—
269C2×C3×C4—92no—
270C2×C3×C4—92no—
271C2×C3×C4—364no—
272C2×C3×C4C4×D914yescomplemented
273C2×C3×C4C22×D914yes—
274C2×C3×C4C22×D911yescharacteristic
275C2×C3×C4C22×D911yescharacteristic
276C2×C3×C4C3×C4×D314yes—
277C2×C3×C4C22×C3×D314yes—
278C2×C3×C4C22×C3×D311yescharacteristic
279C2×C3×C4C22×C3×D311yescharacteristic
280#67—38no—
281C2×Dic3—32no—
282C2×Dic3—32no—
283C2×Dic3—32no—
284D12—34no—
285C4×D3—38no—
286C4×D3—124no—
287Dic6—34no—
288C2×C32—22no—
289C2×C32C22×C4×D311yescharacteristic
290C2×C32#100411yescharacteristic
291C2×C32#100711yescharacteristic
292C3×D3—64no—
293C2×C9—24no—
294C2×C9—22no—
295C2×C9—21no—
296C2×C9—21no—
297C2×C9—21no—
298C2×C9C23×C3×C411yescharacteristic
299C2×C9C3×H(5,𝔽2)11yescharacteristic
300C2×C9C3×2−1+411yescharacteristic
301D9—24no—
302#41—98no—
303C2×Q8—91no—
304C2×D4—92no—
305C2×D4—91no—
306C22×C4—92no—
307C22×C4—94no—
308C22×C4C2×C3×D912yes—
309C22×C4C2×C3×D911yescharacteristic
310C4⋊C4—91no—
311C4⋊C4—94no—
312C4⋊C4—91no—
313C4⋊C4C2×C3×D912yes—
314C4⋊C4C2×C3×D912yes—
315C22⋊C4—92no—
316C22⋊C4—94no—
317C42—94no—
318C42—92no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—94no—
328C22×C3—182no—
329C22×C3—182no—
330C22×C3C2×C4×D912yes—
331C22×C3C23×D911yescharacteristic
332C22×C3C2×C3×C4×D312yes—
333C22×C3C23×C3×D311yescharacteristic
334C2×D3—34no—
335C2×D3—62no—
336C2×D3—62no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—44no—
342C3×C4—94no—
343C3×C4—182no—
344C3×C4—182no—
345C3×C4C2×C4×D912yes—
346C3×C4C2×C4×D912yes—
347C3×C4C2×C3×C4×D312yes—
348C3×C4C2×C3×C4×D312yes—
349Dic3—34no—
350Dic3—62no—
351Dic3—62no—
352C32#550411yescharacteristic
353C9—21no—
354C9C3×#52011yescharacteristic, complemented
355C23—92no—
356C23C22×C3×D911yescharacteristic
357Q8—94no—
358D4—98no—
359D4—94no—
360C2×C4—42no—
361C2×C4—92no—
362C2×C4—98no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4—364no—
366C2×C4C3×C4×D914yescomplemented
367C2×C4C22×C3×D914yes—
368C2×C4C22×C3×D911yescharacteristic
369C2×C4C22×C3×D911yescharacteristic
370C2×C3—24no—
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—184no—
377C2×C3C22×C4×D911yescharacteristic
378C2×C3#6371011yescharacteristic
379C2×C3#6371311yescharacteristic
380C2×C3C22×C3×C4×D311yescharacteristic
381C2×C3C3×#100411yescharacteristic
382C2×C3C3×#100711yescharacteristic
383D3—64no—
384C22—21no—
385C22—21no—
386C22—94no—
387C22—182no—
388C22—182no—
389C22C2×C3×C4×D912yes—
390C22C23×C3×D911yescharacteristic
391C4—24no—
392C4—94no—
393C4—182no—
394C4—182no—
395C4C2×C3×C4×D912yes—
396C4C2×C3×C4×D912yes—
397C3—21no—
398C3#1058960611yescharacteristic, direct product factor
399C3C3×#550411yescharacteristic
400C2—22no—
401C2—184no—
402C2C22×C3×C4×D911yescharacteristic
403C2C3×#6371011yescharacteristic
404C2C3×#6371311yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutatorC2×C9C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC22×C32C23×D3
prime radicalC2×C3×#3697C2
cosocleC22×C9C24×C3
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#520
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C9×C4⋊C4C2semidirect
3. C3×C4×#611C2semidirect
4. C3×#63488C2semidirect
5. C3×#63455C2semidirect
6. C3×#63454C2semidirect
7. C3×C4⋊C4×D9C2semidirect
8. C3×#63449C2semidirect
10. #10589606C3direct (N × H)
14. C3×#3697C4semidirect
38. C2×C9×C4⋊C4C2×C3semidirect
39. C4×#611C2×C3semidirect
40. #63488C2×C3semidirect
41. #63455C2×C3semidirect
42. #63454C2×C3semidirect
43. C4⋊C4×D9C2×C3semidirect
44. #63449C2×C3semidirect
48. C2×C3×C4×C9C2×C4semidirect
52. C3×#611C2×C4semidirect
56. C3×D36C2×C4semidirect
58. C3×C4×D9C2×C4semidirect
59. C3×Dic18C2×C4semidirect
84. #3697C3×C4semidirect
122. C2×C3×C4⋊C4D9semidirect
154. C2×C4×C9C2×C3×C4semidirect
158. #611C2×C3×C4semidirect
162. D36C2×C3×C4semidirect
164. C4×D9C2×C3×C4semidirect
165. Dic18C2×C3×C4semidirect
250. C2×C4⋊C4C3×D9semidirect
251. C3×C9#520semidirect
272. C2×C3×C4C4×D9semidirect
354. C9C3×#520semidirect
366. C2×C4C3×C4×D9semidirect
398. C3#10589606direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×C3×C9×C4⋊C42
C3×C4×#6112
C3×#634882
C3×#634552
C3×#634542
C3×C4⋊C4×D92
C3×#634492
C3×#55043
#105896063