Finite Group Databasebeta

← C3×#10589793 #13153153

Subgroups — C3×#10589793 #13153153

Order 1728.

758 conjugacy classes of subgroups.

The subgroup lattice has 492 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×#63457C211yesmaximal, characteristic
5C3×#63456C212yesmaximal, complemented
6C3×#63455C212yesmaximal, complemented
7C3×#63455C211yesmaximal, characteristic, complemented
8C3×#63454C212yesmaximal, complemented
9C3×#63449C211yesmaximal, characteristic, complemented
10C3×#63437C211yesmaximal, characteristic, complemented
11C3×C42×D9C211yesmaximal, characteristic, complemented
12C3×#5691—31nomaximal
13#10589793C311yesmaximal, characteristic, direct product factor
14C2×C3×D36C2211yescharacteristic
15C2×C3×D36C2212yes—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C3×C4×D9C2212yes—
20C2×C3×Dic18C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C4⋊C4C2212yes—
23C3×C9×C4⋊C4C2211yescharacteristic
24C3×C9×C4⋊C4C2212yes—
25C3×C42×C9C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C3×C4⋊Dic9C2212yes—
32C3×Dic9⋊C4C2212yes—
33C3×C4×Dic9C2211yescharacteristic
34C3×C4×Dic9C2211yescharacteristic
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2212yes—
37C32×#125D311yescharacteristic
38C3×#898—32no—
39C3×#897—31no—
40C3×#895—31no—
41C3×#894—32no—
42C3×#893—32no—
43C3×#893—31no—
44C3×#892—32no—
45C3×#887—31no—
46C3×#875—31no—
47C3×C42×D3—31no—
48C9×#125—21no—
49C9×#125C2×C311yescharacteristic, complemented
50#63460C2×C312yescomplemented
51#63459C2×C311yescharacteristic, complemented
52#63457C2×C311yescharacteristic
53#63456C2×C312yescomplemented
54#63455C2×C311yescharacteristic, complemented
55#63455C2×C312yescomplemented
56#63454C2×C312yescomplemented
57#63449C2×C311yescharacteristic, complemented
58#63437C2×C311yescharacteristic, complemented
59C42×D9C2×C311yescharacteristic, complemented
60C22×C3×D9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×Dic9C2312yes—
68C2×C3×Dic9C2311yescharacteristic
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C3×D36—24no—
72C3×D36—22no—
73C3×C4×D9—22no—
74C3×C4×D9—22no—
75C3×C4×D9—22no—
76C3×C4×D9—24no—
77C3×Dic18—22no—
78C3×#529—91no—
79#5691—31no—
80C2×C3×D12—32no—
81C2×C3×D12—31no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×Dic6—31no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D312yes—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C2×D312yes—
91C32×C42C2×D311yescharacteristic
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—32no—
96C3×(C2×D3)⋊C4—32no—
97C3×C4⋊Dic3—32no—
98C3×Dic3⋊C4—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C2×D36C22×C311yescharacteristic
104C2×D36C22×C312yes—
105C2×C4×D9C22×C311yescharacteristic
106C2×C4×D9C22×C311yescharacteristic
107C2×C4×D9C22×C311yescharacteristic
108C2×C4×D9C22×C312yes—
109C2×Dic18C22×C311yescharacteristic
110C9×C4⋊C4—21no—
111C9×C4⋊C4—21no—
112C9×C4⋊C4—22no—
113C9×C4⋊C4—22no—
114C9×C4⋊C4C22×C311yescharacteristic
115C9×C4⋊C4C22×C312yes—
116C9×C4⋊C4C22×C311yescharacteristic
117C9×C4⋊C4C22×C312yes—
118C42×C9—21no—
119C42×C9C22×C311yescharacteristic
120(C2×D9)⋊C4C22×C312yes—
121(C2×D9)⋊C4C22×C312yes—
122(C2×D9)⋊C4C22×C312yes—
123(C2×D9)⋊C4C22×C312yes—
124(C2×D9)⋊C4C22×C312yes—
125C4⋊Dic9C22×C312yes—
126Dic9⋊C4C22×C312yes—
127C4×Dic9C22×C311yescharacteristic
128C4×Dic9C22×C311yescharacteristic
129C4×Dic9C22×C311yescharacteristic
130C4×Dic9C22×C312yes—
131C22×C3×C9C2411yescharacteristic
132C2×C3×D9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—22no—
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9#4112yes—
138C3×C4×C9—21no—
139C3×C4×C9—21no—
140C3×C4×C9—22no—
141C3×C4×C9—22no—
142C3×C4×C9#4112yes—
143C3×Dic9—22no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×Dic9#4112yes—
147C3×#126—92no—
148C3×#125—21no—
149C3×#125D911yescharacteristic, complemented
150C3×#125C3×D311yescharacteristic
151C3×#124—91no—
152C3×#123—92no—
153C3×C22⋊Q8—91no—
154C3×#121—91no—
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C3×C4×D4—92no—
158C3×#117—92no—
159C2×C3×C42—91no—
160#898—32no—
161#897—31no—
162#895—31no—
163#894—32no—
164#893—31no—
165#893—32no—
166#892—32no—
167#887—31no—
168#875—31no—
169C42×D3—31no—
170C22×C3×D3—31no—
171C22×C3×D3—32no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D311yescharacteristic
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C3×D12—64no—
182C3×D12—62no—
183C3×C4×D3—62no—
184C3×C4×D3—64no—
185C3×C4×D3—62no—
186C3×C4×D3—62no—
187C3×Dic6—62no—
188C22×D9C23×C312yes—
189C22×D9C23×C311yescharacteristic
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—22no—
194C2×C4×C9—22no—
195C2×C4×C9C23×C312yes—
196C2×C4×C9C23×C311yescharacteristic
197C2×C4×C9C23×C311yescharacteristic
198C2×C4×C9C23×C312yes—
199C2×C4×C9C23×C311yescharacteristic
200C2×Dic9C23×C311yescharacteristic
201C2×Dic9C23×C311yescharacteristic
202C2×Dic9C23×C312yes—
203C2×Dic9C23×C311yescharacteristic
204D36—24no—
205D36—22no—
206C4×D9—22no—
207C4×D9—22no—
208C4×D9—22no—
209C4×D9—24no—
210Dic18—22no—
211#529—91noSylow
212C2×C3×C9C2×#4111yescharacteristic
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9C2×#4111yescharacteristic
215C3×D9—22no—
216C3×D9—42no—
217C2×C3×Q8—91no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C22×C3×C4—92no—
224C2×D12—31no—
225C2×D12—32no—
226C2×C4×D3—31no—
227C2×C4×D3—31no—
228C2×C4×D3—31no—
229C2×C4×D3—32no—
230C2×Dic6—31no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4C2×D912yes—
238C3×C4⋊C4C2×D911yescharacteristic
239C3×C4⋊C4C2×D912yes—
240C3×C4⋊C4C2×D911yescharacteristic
241C3×C4⋊C4C2×C3×D311yescharacteristic
242C3×C4⋊C4C2×C3×D312yes—
243C3×C4⋊C4C2×C3×D311yescharacteristic
244C3×C4⋊C4C2×C3×D312yes—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C42—21no—
251C3×C42—91no—
252C3×C42—91no—
253C3×C42—91no—
254C3×C42—92no—
255C3×C42C2×D911yescharacteristic
256C3×C42C2×C3×D311yescharacteristic
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—32no—
262C4⋊Dic3—32no—
263Dic3⋊C4—32no—
264C4×Dic3—31no—
265C4×Dic3—31no—
266C4×Dic3—31no—
267C4×Dic3—32no—
268C22×C32C23×D311yescharacteristic
269C2×C3×D3—32no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C32×C4—21no—
276C32×C4—22no—
277C32×C4—21no—
278C32×C4—22no—
279C32×C4#23912yes—
280C3×Dic3—32no—
281C3×Dic3—62no—
282C3×Dic3—61no—
283C3×Dic3—61no—
284C22×C9—21no—
285C22×C9C24×C311yescharacteristic
286C2×D9—21no—
287C2×D9—21no—
288C2×D9—22no—
289C2×D9—22no—
290C2×D9—22no—
291C2×D9C3×#4112yes—
292C4×C9—21no—
293C4×C9—21no—
294C4×C9—22no—
295C4×C9—22no—
296C4×C9—22no—
297C4×C9—41no—
298C4×C9—41no—
299C4×C9—42no—
300C4×C9—42no—
301C4×C9C3×#4112yes—
302Dic9—22no—
303Dic9—21no—
304Dic9—21no—
305Dic9C3×#4112yes—
306#126—92no—
307#125C3×D911yescharacteristic, complemented
308#124—91no—
309#123—92no—
310C22⋊Q8—91no—
311#121—91no—
312C4×Q8—91no—
313C4×D4—91no—
314C4×D4—92no—
315#117—92no—
316C2×C42—91no—
317C3×C9#52911yescharacteristic, complemented, Sylow
318C23×C3—91no—
319C23×C3—92no—
320C22×D3—31no—
321C22×D3—32no—
322C3×Q8—182no—
323C3×D4—184no—
324C3×D4—182no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—92no—
334C2×C3×C4—182no—
335C2×C3×C4—184no—
336C2×C3×C4—182no—
337C2×C3×C4—182no—
338C2×C3×C4C22×D911yescharacteristic
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C22×D912yes—
341C2×C3×C4C22×D912yes—
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×C3×D312yes—
344C2×C3×C4C22×C3×D311yescharacteristic
345C2×C3×C4C22×C3×D311yescharacteristic
346C2×C3×C4C22×C3×D312yes—
347C2×C3×C4C22×C3×D311yescharacteristic
348C2×Dic3—31no—
349C2×Dic3—31no—
350C2×Dic3—31no—
351C2×Dic3—32no—
352D12—64no—
353D12—62no—
354C4×D3—62no—
355C4×D3—64no—
356C4×D3—62no—
357C4×D3—62no—
358Dic6—62no—
359C2×C32C2×#23911yescharacteristic
360C2×C32D3×#4111yescharacteristic
361C2×C32D3×#4111yescharacteristic
362C3×D3—62no—
363C3×D3—122no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9C2×C3×#4111yescharacteristic
368C2×C9C2×C3×#4111yescharacteristic
369C2×C9C2×C3×#4111yescharacteristic
370D9—22no—
371D9—42no—
372C2×Q8—91no—
373C2×D4—92no—
374C2×D4—91no—
375C22×C4—91no—
376C22×C4—91no—
377C22×C4—91no—
378C22×C4—92no—
379C4⋊C4—92no—
380C4⋊C4—92no—
381C4⋊C4C2×C3×D912yes—
382C4⋊C4C2×C3×D911yescharacteristic
383C4⋊C4C2×C3×D912yes—
384C4⋊C4C2×C3×D911yescharacteristic
385C22⋊C4—92no—
386C22⋊C4—92no—
387C22⋊C4—92no—
388C22⋊C4—92no—
389C22⋊C4—92no—
390C42—91no—
391C42—91no—
392C42—91no—
393C42—92no—
394C42C2×C3×D911yescharacteristic
395C22×C3—21no—
396C22×C3—92no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3—182no—
400C22×C3—182no—
401C22×C3—182no—
402C22×C3C23×D911yescharacteristic
403C22×C3C23×C3×D311yescharacteristic
404C2×D3—32no—
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—62no—
410C3×C4—21no—
411C3×C4—22no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—21no—
415C3×C4—22no—
416C3×C4—21no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—42no—
422C3×C4—42no—
423C3×C4—92no—
424C3×C4—181no—
425C3×C4—182no—
426C3×C4—181no—
427C3×C4#370112yes—
428C3×C4C3×#23912yes—
429Dic3—32no—
430Dic3—62no—
431Dic3—61no—
432Dic3—61no—
433C32#569111yescharacteristic
434C9—21no—
435C9C3×#52911yescharacteristic, complemented
436C23—91no—
437C23—92no—
438Q8—182no—
439D4—184no—
440D4—182no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—92no—
445C2×C4—182no—
446C2×C4—184no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4C22×C3×D911yescharacteristic
450C2×C4C22×C3×D911yescharacteristic
451C2×C4C22×C3×D912yes—
452C2×C4C22×C3×D912yes—
453C2×C4C22×C3×D911yescharacteristic
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—182no—
458C2×C3—362no—
459C2×C3C2×#370111yescharacteristic
460C2×C3#41×D911yescharacteristic
461C2×C3#41×D911yescharacteristic
462C2×C3C2×C3×#23911yescharacteristic
463C2×C3C3×D3×#4111yescharacteristic
464C2×C3C3×D3×#4111yescharacteristic
465D3—62no—
466D3—122no—
467C22—92no—
468C22—181no—
469C22—181no—
470C22—182no—
471C22—182no—
472C22—182no—
473C22C23×C3×D911yescharacteristic
474C4—21no—
475C4—22no—
476C4—21no—
477C4—22no—
478C4—92no—
479C4—181no—
480C4—182no—
481C4—181no—
482C4C3×#370112yes—
483C3—21no—
484C3#1058979311yescharacteristic, direct product factor
485C3C3×#569111yescharacteristic
486C2—182no—
487C2—362no—
488C2C2×C3×#370111yescharacteristic
489C2C3×#41×D911yescharacteristic
490C2C3×#41×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#125C2
socleC22×C32C23×D3
prime radicalC3×#63457C2
hypercenterC3×#125D9
hypocenterC9C3×#529
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
5. C3×#63456C2semidirect
6. C3×#63455C2semidirect
7. C3×#63455C2semidirect
8. C3×#63454C2semidirect
9. C3×#63449C2semidirect
10. C3×#63437C2semidirect
11. C3×C42×D9C2semidirect
13. #10589793C3direct (N × H)
49. C9×#125C2×C3semidirect
50. #63460C2×C3semidirect
51. #63459C2×C3semidirect
53. #63456C2×C3semidirect
54. #63455C2×C3semidirect
55. #63455C2×C3semidirect
56. #63454C2×C3semidirect
57. #63449C2×C3semidirect
58. #63437C2×C3semidirect
59. C42×D9C2×C3semidirect
149. C3×#125D9semidirect
307. #125C3×D9semidirect
317. C3×C9#529semidirect
435. C9C3×#529semidirect
484. C3#10589793direct (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×#634572
C3×#634562
C3×#634552
C3×#634542
C3×#634492
C3×#634372
C3×C42×D92
C3×#56913
#105897933