Finite Group Databasebeta

← C3×#10589796 #13153156

Subgroups — C3×#10589796 #13153156

Order 1728.

698 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#125C211yesmaximal, characteristic, complemented
2C3×#63460C212yesmaximal, complemented
3C3×#63459C211yesmaximal, characteristic, complemented
4C3×#63459C211yesmaximal, characteristic, complemented
5C3×#63456C212yesmaximal
6C3×#63454C211yesmaximal, characteristic, complemented
7C3×#63454C212yesmaximal, complemented
8C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
9C3×#63452C212yesmaximal, complemented
10C3×C4×D36C211yesmaximal, characteristic
11C3×C4×Dic18C211yesmaximal, characteristic, complemented
12C3×#5694—31nomaximal
13#10589796C311yesmaximal, characteristic, direct product factor
14C2×C3×D36C2211yescharacteristic
15C2×C3×C4×D9C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×C3×Dic18C2211yescharacteristic
19C3×C9×C4⋊C4C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C4⋊C4C2212yes—
23C3×C42×C9C2211yescharacteristic
24C3×(C2×D9)⋊C4C2211yescharacteristic
25C3×(C2×D9)⋊C4C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×C4⋊Dic9C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×C4⋊Dic9C2212yes—
32C3×C4⋊Dic9C2212yes—
33C3×Dic9⋊C4C2212yes—
34C3×Dic9⋊C4C2212yes—
35C3×C4×Dic9C2212yes—
36C3×C4×Dic9C2212yes—
37C32×#125D311yescharacteristic
38C3×#898—32no—
39C3×#897—31no—
40C3×#897—31no—
41C3×#894—32no—
42C3×#892—31no—
43C3×#892—32no—
44C3×D3×C4⋊C4—31no—
45C3×#890—32no—
46C3×C4×D12—31no—
47C3×C4×Dic6—31no—
48C9×#125—21no—
49C9×#125C2×C311yescharacteristic, complemented
50#63460C2×C312yescomplemented
51#63459C2×C311yescharacteristic, complemented
52#63459C2×C311yescharacteristic, complemented
53#63456C2×C312yes—
54#63454C2×C312yescomplemented
55#63454C2×C311yescharacteristic, complemented
56C4⋊C4×D9C2×C311yescharacteristic, complemented
57#63452C2×C312yescomplemented
58C4×D36C2×C311yescharacteristic
59C4×Dic18C2×C311yescharacteristic, complemented
60C22×C3×D9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×Dic9C2312yes—
68C2×C3×Dic9C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C3×D36—22no—
72C3×C4×D9—24no—
73C3×C4×D9—22no—
74C3×C4×D9—22no—
75C3×Dic18—22no—
76C3×#552—91no—
77#5694—31no—
78C2×C3×D12—31no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6—31no—
83C32×C4⋊C4C2×D312yes—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C2×D312yes—
87C32×C42C2×D311yescharacteristic
88C3×(C2×D3)⋊C4—31no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—32no—
96C3×C4⋊Dic3—31no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—32no—
101C2×D36C22×C311yescharacteristic
102C2×C4×D9C22×C311yescharacteristic
103C2×C4×D9C22×C311yescharacteristic
104C2×C4×D9C22×C312yes—
105C2×Dic18C22×C311yescharacteristic
106C9×C4⋊C4—21no—
107C9×C4⋊C4—21no—
108C9×C4⋊C4—22no—
109C9×C4⋊C4—22no—
110C9×C4⋊C4C22×C312yes—
111C9×C4⋊C4C22×C311yescharacteristic
112C9×C4⋊C4C22×C311yescharacteristic
113C9×C4⋊C4C22×C312yes—
114C42×C9—21no—
115C42×C9C22×C311yescharacteristic
116(C2×D9)⋊C4C22×C311yescharacteristic
117(C2×D9)⋊C4C22×C311yescharacteristic
118(C2×D9)⋊C4C22×C312yes—
119(C2×D9)⋊C4C22×C312yes—
120(C2×D9)⋊C4C22×C312yes—
121C4⋊Dic9C22×C311yescharacteristic
122C4⋊Dic9C22×C311yescharacteristic
123C4⋊Dic9C22×C312yes—
124C4⋊Dic9C22×C312yes—
125Dic9⋊C4C22×C312yes—
126Dic9⋊C4C22×C312yes—
127C4×Dic9C22×C312yes—
128C4×Dic9C22×C312yes—
129C22×C3×C9C2411yescharacteristic
130C2×C3×D9—21no—
131C2×C3×D9—21no—
132C2×C3×D9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9#4112yes—
136C3×C4×C9—22no—
137C3×C4×C9—21no—
138C3×C4×C9—21no—
139C3×C4×C9—22no—
140C3×C4×C9#4112yes—
141C3×Dic9—22no—
142C3×Dic9—21no—
143C3×Dic9—21no—
144C3×Dic9—22no—
145C3×#126—92no—
146C3×#125—21no—
147C3×#125—92no—
148C3×#125D911yescharacteristic, complemented
149C3×#125C3×D311yescharacteristic
150C3×#123—92no—
151C3×C22⋊Q8—91no—
152C3×C22⋊Q8—91no—
153C3×C4×Q8—91no—
154C3×C4×D4—91no—
155C3×#117—91no—
156C3×#117—92no—
157C2×C3×C4⋊C4—91no—
158#898—32no—
159#897—31no—
160#897—31no—
161#894—32no—
162#892—32no—
163#892—31no—
164D3×C4⋊C4—31no—
165#890—32no—
166C4×D12—31no—
167C4×Dic6—31no—
168C22×C3×D3—31no—
169C22×C3×D3—31no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C3×Dic3—32no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—31no—
179C3×D12—62no—
180C3×C4×D3—64no—
181C3×C4×D3—62no—
182C3×C4×D3—62no—
183C3×Dic6—62no—
184C22×D9C23×C311yescharacteristic
185C22×D9C23×C311yescharacteristic
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—22no—
190C2×C4×C9—22no—
191C2×C4×C9C23×C311yescharacteristic
192C2×C4×C9C23×C312yes—
193C2×C4×C9C23×C311yescharacteristic
194C2×C4×C9C23×C312yes—
195C2×C4×C9C23×C311yescharacteristic
196C2×Dic9C23×C312yes—
197C2×Dic9C23×C311yescharacteristic
198C2×Dic9C23×C312yes—
199C2×Dic9C23×C311yescharacteristic
200D36—22no—
201C4×D9—24no—
202C4×D9—22no—
203C4×D9—22no—
204Dic18—22no—
205#552—91noSylow
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C92−1+411yescharacteristic
209C3×D9—22no—
210C3×D9—41no—
211C2×C3×Q8—91no—
212C2×C3×D4—91no—
213C22×C3×C4—92no—
214C22×C3×C4—91no—
215C22×C3×C4—91no—
216C2×D12—31no—
217C2×C4×D3—31no—
218C2×C4×D3—31no—
219C2×C4×D3—32no—
220C2×Dic6—31no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—92no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4C2×D911yescharacteristic
232C3×C4⋊C4C2×D911yescharacteristic
233C3×C4⋊C4C2×D912yes—
234C3×C4⋊C4C2×D912yes—
235C3×C4⋊C4C2×C3×D312yes—
236C3×C4⋊C4C2×C3×D311yescharacteristic
237C3×C4⋊C4C2×C3×D311yescharacteristic
238C3×C4⋊C4C2×C3×D312yes—
239C3×C22⋊C4—91no—
240C3×C22⋊C4—92no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—91no—
243C3×C22⋊C4—92no—
244C3×C42—21no—
245C3×C42—92no—
246C3×C42—92no—
247C3×C42C2×D911yescharacteristic
248C3×C42C2×C3×D311yescharacteristic
249(C2×D3)⋊C4—31no—
250(C2×D3)⋊C4—31no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—32no—
257C4⋊Dic3—32no—
258Dic3⋊C4—32no—
259Dic3⋊C4—32no—
260C4×Dic3—32no—
261C4×Dic3—32no—
262C22×C32C23×D311yescharacteristic
263C2×C3×D3—32no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C32×C4—22no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—22no—
273C32×C4#23912yes—
274C3×Dic3—62no—
275C3×Dic3—61no—
276C3×Dic3—61no—
277C3×Dic3—62no—
278C22×C9—21no—
279C22×C9C24×C311yescharacteristic
280C2×D9—21no—
281C2×D9—21no—
282C2×D9—21no—
283C2×D9—21no—
284C2×D9—21no—
285C2×D9C3×#4112yes—
286C4×C9—22no—
287C4×C9—21no—
288C4×C9—21no—
289C4×C9—22no—
290C4×C9—22no—
291C4×C9—42no—
292C4×C9—41no—
293C4×C9—41no—
294C4×C9—42no—
295C4×C9C3×#4112yes—
296Dic9—22no—
297Dic9—21no—
298Dic9—21no—
299Dic9—22no—
300#126—92no—
301#125—92no—
302#125C3×D911yescharacteristic, complemented
303#123—92no—
304C22⋊Q8—91no—
305C22⋊Q8—91no—
306C4×Q8—91no—
307C4×D4—91no—
308#117—91no—
309#117—92no—
310C2×C4⋊C4—91no—
311C3×C9#55211yescharacteristic, complemented, Sylow
312C23×C3—91no—
313C23×C3—91no—
314C22×D3—31no—
315C22×D3—31no—
316C3×Q8—182no—
317C3×D4—182no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—91no—
324C2×C3×C4—92no—
325C2×C3×C4—92no—
326C2×C3×C4—91no—
327C2×C3×C4—184no—
328C2×C3×C4—182no—
329C2×C3×C4—182no—
330C2×C3×C4C22×D911yescharacteristic
331C2×C3×C4C22×D912yes—
332C2×C3×C4C22×D911yescharacteristic
333C2×C3×C4C22×D912yes—
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C22×C3×D311yescharacteristic
336C2×C3×C4C22×C3×D312yes—
337C2×C3×C4C22×C3×D311yescharacteristic
338C2×C3×C4C22×C3×D312yes—
339C2×C3×C4C22×C3×D311yescharacteristic
340C2×Dic3—32no—
341C2×Dic3—31no—
342C2×Dic3—32no—
343C2×Dic3—31no—
344D12—62no—
345C4×D3—64no—
346C4×D3—62no—
347C4×D3—62no—
348Dic6—62no—
349C2×C32C2×#23911yescharacteristic
350C2×C32D3×#4111yescharacteristic
351C2×C32#101011yescharacteristic
352C3×D3—62no—
353C3×D3—121no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9C2×C3×#4111yescharacteristic
358C2×C9C2×C3×#4111yescharacteristic
359C2×C9C3×2−1+411yescharacteristic
360D9—22no—
361D9—41no—
362C2×Q8—91no—
363C2×D4—91no—
364C22×C4—92no—
365C22×C4—91no—
366C22×C4—91no—
367C4⋊C4—91no—
368C4⋊C4—92no—
369C4⋊C4—92no—
370C4⋊C4—91no—
371C4⋊C4—92no—
372C4⋊C4—92no—
373C4⋊C4C2×C3×D911yescharacteristic
374C4⋊C4C2×C3×D911yescharacteristic
375C4⋊C4C2×C3×D912yes—
376C4⋊C4C2×C3×D912yes—
377C22⋊C4—91no—
378C22⋊C4—92no—
379C22⋊C4—92no—
380C22⋊C4—91no—
381C22⋊C4—92no—
382C42—92no—
383C42—92no—
384C42C2×C3×D911yescharacteristic
385C22×C3—21no—
386C22×C3—92no—
387C22×C3—181no—
388C22×C3—181no—
389C22×C3—181no—
390C22×C3—181no—
391C22×C3—181no—
392C22×C3C23×D911yescharacteristic
393C22×C3C23×C3×D311yescharacteristic
394C2×D3—32no—
395C2×D3—61no—
396C2×D3—61no—
397C2×D3—61no—
398C2×D3—61no—
399C2×D3—61no—
400C3×C4—22no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—42no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—42no—
413C3×C4—181no—
414C3×C4—181no—
415C3×C4—182no—
416C3×C4—182no—
417C3×C4#370112yes—
418C3×C4C3×#23912yes—
419Dic3—62no—
420Dic3—61no—
421Dic3—61no—
422Dic3—62no—
423C32#569411yescharacteristic
424C9—21no—
425C9C3×#55211yescharacteristic, complemented
426C23—91no—
427C23—91no—
428Q8—182no—
429D4—182no—
430C2×C4—91no—
431C2×C4—92no—
432C2×C4—92no—
433C2×C4—91no—
434C2×C4—184no—
435C2×C4—182no—
436C2×C4—182no—
437C2×C4C22×C3×D911yescharacteristic
438C2×C4C22×C3×D912yes—
439C2×C4C22×C3×D911yescharacteristic
440C2×C4C22×C3×D912yes—
441C2×C4C22×C3×D911yescharacteristic
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—182no—
446C2×C3—361no—
447C2×C3C2×#370111yescharacteristic
448C2×C3#41×D911yescharacteristic
449C2×C3#6371611yescharacteristic
450C2×C3C2×C3×#23911yescharacteristic
451C2×C3C3×D3×#4111yescharacteristic
452C2×C3C3×#101011yescharacteristic
453D3—62no—
454D3—121no—
455C22—92no—
456C22—181no—
457C22—181no—
458C22—181no—
459C22—181no—
460C22—181no—
461C22C23×C3×D911yescharacteristic
462C4—22no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—181no—
467C4—181no—
468C4—182no—
469C4—182no—
470C4C3×#370112yes—
471C3—21no—
472C3#1058979611yescharacteristic, direct product factor
473C3C3×#569411yescharacteristic
474C2—182no—
475C2—361no—
476C2C2×C3×#370111yescharacteristic
477C2C3×#41×D911yescharacteristic
478C2C3×#6371611yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#125C2
socleC22×C32C23×D3
prime radicalC2×C3×D36C22
hypercenterC3×#125D9
hypocenterC9C3×#552
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. C3×C9×#125C2semidirect
2. C3×#63460C2semidirect
3. C3×#63459C2semidirect
4. C3×#63459C2semidirect
6. C3×#63454C2semidirect
7. C3×#63454C2semidirect
8. C3×C4⋊C4×D9C2semidirect
9. C3×#63452C2semidirect
11. C3×C4×Dic18C2semidirect
13. #10589796C3direct (N × H)
49. C9×#125C2×C3semidirect
50. #63460C2×C3semidirect
51. #63459C2×C3semidirect
52. #63459C2×C3semidirect
54. #63454C2×C3semidirect
55. #63454C2×C3semidirect
56. C4⋊C4×D9C2×C3semidirect
57. #63452C2×C3semidirect
59. C4×Dic18C2×C3semidirect
148. C3×#125D9semidirect
302. #125C3×D9semidirect
311. C3×C9#552semidirect
425. C9C3×#552semidirect
472. C3#10589796direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1252
C3×#634602
C3×#634592
C3×#634562
C3×#634542
C3×C4⋊C4×D92
C3×#634522
C3×C4×D362
C3×C4×Dic182
C3×#56943
#105897963