Finite Group Databasebeta

← C3×#10589615 #13152410

Subgroups — C3×#10589615 #13152410

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×#63491C212yesmaximal
3C3×C4×#611C211yesmaximal, characteristic
4C3×#63488C212yesmaximal, complemented
5C3×#63486C211yesmaximal, characteristic, complemented
6C3×#63460C212yesmaximal, complemented
7C3×#63459C211yesmaximal, characteristic, complemented
8C3×#63454C211yesmaximal, characteristic, complemented
9C3×#63452C211yesmaximal, characteristic, complemented
10C3×#63451C212yesmaximal, complemented
11C3×#63449C211yesmaximal, characteristic, complemented
12C3×#5513—31nomaximal
13#10589615C311yesmaximal, characteristic, direct product factor
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3×C4×C9C2212yes—
16C2×C3×#611C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20C3×C9×C4⋊C4C2212yes—
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×#3676C2212yes—
23C3×#3676C2211yescharacteristic
24C3×(C2×D9)⋊C4C2211yescharacteristic
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2212yes—
29C3×C4⋊Dic9C2212yes—
30C3×Dic9⋊C4C2212yes—
31C3×Dic9⋊C4C2211yescharacteristic
32C3×Dic9⋊C4C2212yes—
33C3×Dic9⋊C4C2212yes—
34C3×C4×Dic9C2212yes—
35C3×C4×Dic9C2212yes—
36C3×C4×Dic9C2211yescharacteristic
37C2×C32×C4⋊C4D311yescharacteristic
38C3×#929—32no—
39C3×C4×#67—31no—
40C3×#926—32no—
41C3×#924—31no—
42C3×#898—32no—
43C3×#897—31no—
44C3×#892—31no—
45C3×#890—31no—
46C3×#889—32no—
47C3×#887—31no—
48C2×C9×C4⋊C4—21no—
49C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
50#63491C2×C312yes—
51C4×#611C2×C311yescharacteristic
52#63488C2×C312yescomplemented
53#63486C2×C311yescharacteristic, complemented
54#63460C2×C312yescomplemented
55#63459C2×C311yescharacteristic, complemented
56#63454C2×C311yescharacteristic, complemented
57#63452C2×C311yescharacteristic, complemented
58#63451C2×C312yescomplemented
59#63449C2×C311yescharacteristic, complemented
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9—24no—
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C3×#611—22no—
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C3×C4×D9—22no—
75C3×Dic18—22no—
76C3×#552—91no—
77#5513—31no—
78C22×C32×C4C2×D311yescharacteristic
79C22×C32×C4C2×D312yes—
80C2×C3×#67—31no—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C2×D312yes—
85C32×C4⋊C4C2×D311yescharacteristic
86C3×#217—32no—
87C3×#217—31no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—31no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—32no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—31no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—32no—
101C22×C4×C9—21no—
102C22×C4×C9—22no—
103C22×C4×C9C22×C312yes—
104C22×C4×C9C22×C311yescharacteristic
105C2×#611C22×C311yescharacteristic
106C2×C4×D9C22×C311yescharacteristic
107C2×Dic18C22×C311yescharacteristic
108C9×C4⋊C4—21no—
109C9×C4⋊C4—22no—
110C9×C4⋊C4—21no—
111C9×C4⋊C4C22×C311yescharacteristic
112C9×C4⋊C4C22×C312yes—
113C9×C4⋊C4C22×C311yescharacteristic
114#3676C22×C311yescharacteristic
115#3676C22×C312yes—
116(C2×D9)⋊C4C22×C311yescharacteristic
117(C2×D9)⋊C4C22×C312yes—
118(C2×D9)⋊C4C22×C312yes—
119C4⋊Dic9C22×C312yes—
120C4⋊Dic9C22×C312yes—
121C4⋊Dic9C22×C311yescharacteristic
122Dic9⋊C4C22×C311yescharacteristic
123Dic9⋊C4C22×C312yes—
124Dic9⋊C4C22×C312yes—
125Dic9⋊C4C22×C312yes—
126C4×Dic9C22×C312yes—
127C4×Dic9C22×C312yes—
128C4×Dic9C22×C311yescharacteristic
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9#4112yes—
132C22×C3×C9C2411yescharacteristic
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C3×C4×C9—21no—
137C3×C4×C9—22no—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3×Dic9—22no—
141C3×Dic9—21no—
142C3×Dic9—21no—
143C3×Dic9—21no—
144C3×Dic9—22no—
145C3×#126—92no—
146C3×#125—92no—
147C3×#125—91no—
148C3×#123—92no—
149C3×C22⋊Q8—91no—
150C3×C22⋊Q8—91no—
151C3×C4×Q8—91no—
152C3×C4×D4—91no—
153C3×#117—92no—
154C3×#117—91no—
155C2×C3×C4⋊C4—21no—
156C2×C3×C4⋊C4D911yescharacteristic, complemented
157C2×C3×C4⋊C4C3×D311yescharacteristic
158#929—32no—
159C4×#67—31no—
160#926—32no—
161#924—31no—
162#898—32no—
163#897—31no—
164#892—31no—
165#890—31no—
166#889—32no—
167#887—31no—
168C23×C32C22×D311yescharacteristic
169C22×C3×D3—31no—
170C2×C32×C4—22no—
171C2×C32×C4—24no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C3×#67—62no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C3×C4×D3—62no—
183C3×Dic6—62no—
184C23×C9—21no—
185C23×C9C23×C311yescharacteristic
186C22×D9C23×C311yescharacteristic
187C2×C4×C9—24no—
188C2×C4×C9—24no—
189C2×C4×C9—22no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—22no—
193C2×C4×C9—22no—
194C2×C4×C9—44no—
195C2×C4×C9C23×C312yes—
196C2×C4×C9C23×C311yescharacteristic
197C2×C4×C9C23×C312yes—
198C2×C4×C9C23×C311yescharacteristic
199#611—22no—
200C2×Dic9C23×C311yescharacteristic
201C2×Dic9C23×C311yescharacteristic
202C2×Dic9C23×C311yescharacteristic
203C2×Dic9C23×C312yes—
204C2×Dic9C23×C312yes—
205C4×D9—22no—
206Dic18—22no—
207#552—91noSylow
208C2×C3×C9—22no—
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C92−1+411yescharacteristic
212C3×D9—41no—
213C2×C3×Q8—91no—
214C2×C3×D4—91no—
215C22×C3×C4—21no—
216C22×C3×C4—22no—
217C22×C3×C4—91no—
218C22×C3×C4C2×D912yes—
219C22×C3×C4C2×D911yescharacteristic
220C22×C3×C4C2×C3×D311yescharacteristic
221C22×C3×C4C2×C3×D312yes—
222C2×#67—31no—
223C2×C4×D3—31no—
224C2×Dic6—31no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4C2×D911yescharacteristic
236C3×C4⋊C4C2×D911yescharacteristic
237C3×C4⋊C4C2×D912yes—
238C3×C4⋊C4C2×C3×D311yescharacteristic
239C3×C4⋊C4C2×C3×D312yes—
240C3×C4⋊C4C2×C3×D311yescharacteristic
241C3×C22⋊C4—91no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—91no—
245C3×C22⋊C4—92no—
246C3×C42—92no—
247C3×C42—92no—
248C3×C42—91no—
249#217—32no—
250#217—31no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—31no—
254C4⋊Dic3—32no—
255C4⋊Dic3—31no—
256C4⋊Dic3—32no—
257Dic3⋊C4—32no—
258Dic3⋊C4—31no—
259Dic3⋊C4—32no—
260Dic3⋊C4—32no—
261C4×Dic3—31no—
262C4×Dic3—32no—
263C4×Dic3—32no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32#23712yes—
267C22×C32C23×D311yescharacteristic
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C32×C4—21no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4#23512yes—
275C3×Dic3—62no—
276C3×Dic3—61no—
277C3×Dic3—61no—
278C3×Dic3—61no—
279C3×Dic3—62no—
280C22×C9—22no—
281C22×C9—22no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—21no—
285C22×C9—22no—
286C22×C9C3×#4112yes—
287C22×C9C24×C311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—21no—
291C4×C9—22no—
292C4×C9—21no—
293C4×C9—22no—
294C4×C9—22no—
295C4×C9—22no—
296C4×C9—42no—
297C4×C9—42no—
298C4×C9C3×#4112yes—
299Dic9—22no—
300Dic9—21no—
301Dic9—21no—
302Dic9—21no—
303Dic9—22no—
304#126—92no—
305#125—92no—
306#125—91no—
307#123—92no—
308C22⋊Q8—91no—
309C22⋊Q8—91no—
310C4×Q8—91no—
311C4×D4—91no—
312#117—92no—
313#117—91no—
314C2×C4⋊C4C3×D911yescharacteristic, complemented
315C3×C9#55211yescharacteristic, complemented, Sylow
316C23×C3—21no—
317C23×C3—91no—
318C23×C3C22×D911yescharacteristic
319C23×C3C22×C3×D311yescharacteristic
320C22×D3—31no—
321C3×Q8—182no—
322C3×D4—182no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—44no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—92no—
336C2×C3×C4—92no—
337C2×C3×C4—91no—
338C2×C3×C4—182no—
339C2×C3×C4C22×D912yes—
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C22×D912yes—
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×C3×D312yes—
344C2×C3×C4C22×C3×D311yescharacteristic
345C2×C3×C4C22×C3×D312yes—
346C2×C3×C4C22×C3×D311yescharacteristic
347#67—62no—
348C2×Dic3—31no—
349C2×Dic3—31no—
350C2×Dic3—31no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C4×D3—62no—
354Dic6—62no—
355C2×C32—22no—
356C2×C32C2×#23511yescharacteristic
357C2×C32C2×#23711yescharacteristic
358C2×C32#100711yescharacteristic
359C3×D3—121no—
360C2×C9—24no—
361C2×C9—22no—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9—21no—
365C2×C9C2×C3×#4111yescharacteristic
366C2×C9C2×C3×#4111yescharacteristic
367C2×C9C3×2−1+411yescharacteristic
368D9—41no—
369C2×Q8—91no—
370C2×D4—91no—
371C22×C4—91no—
372C22×C4C2×C3×D912yes—
373C22×C4C2×C3×D911yescharacteristic
374C4⋊C4—92no—
375C4⋊C4—91no—
376C4⋊C4—92no—
377C4⋊C4—91no—
378C4⋊C4—92no—
379C4⋊C4—92no—
380C4⋊C4—92no—
381C4⋊C4C2×C3×D911yescharacteristic
382C4⋊C4C2×C3×D911yescharacteristic
383C4⋊C4C2×C3×D912yes—
384C22⋊C4—91no—
385C22⋊C4—92no—
386C22⋊C4—92no—
387C22⋊C4—91no—
388C22⋊C4—92no—
389C42—92no—
390C42—92no—
391C42—91no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—22no—
400C22×C3—181no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3#369912yes—
404C22×C3C23×D911yescharacteristic
405C22×C3C3×#23712yes—
406C22×C3C23×C3×D311yescharacteristic
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C3×C4—22no—
411C3×C4—21no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—21no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—181no—
421C3×C4—182no—
422C3×C4—181no—
423C3×C4—181no—
424C3×C4—182no—
425C3×C4#369712yes—
426C3×C4C3×#23512yes—
427Dic3—62no—
428Dic3—61no—
429Dic3—61no—
430Dic3—61no—
431Dic3—62no—
432C32#551311yescharacteristic
433C9—21no—
434C9C3×#55211yescharacteristic, complemented
435C23—91no—
436C23C22×C3×D911yescharacteristic
437Q8—182no—
438D4—182no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—92no—
444C2×C4—92no—
445C2×C4—91no—
446C2×C4—182no—
447C2×C4C22×C3×D912yes—
448C2×C4C22×C3×D911yescharacteristic
449C2×C4C22×C3×D912yes—
450C2×C4C22×C3×D911yescharacteristic
451C2×C3—24no—
452C2×C3—22no—
453C2×C3—22no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—361no—
458C2×C3C2×#369711yescharacteristic
459C2×C3C2×#369911yescharacteristic
460C2×C3#6371311yescharacteristic
461C2×C3C2×C3×#23511yescharacteristic
462C2×C3C2×C3×#23711yescharacteristic
463C2×C3C3×#100711yescharacteristic
464D3—121no—
465C22—21no—
466C22—21no—
467C22—181no—
468C22—181no—
469C22—181no—
470C22C3×#369912yes—
471C22C23×C3×D911yescharacteristic
472C4—21no—
473C4—22no—
474C4—22no—
475C4—181no—
476C4—182no—
477C4—181no—
478C4—181no—
479C4—182no—
480C4C3×#369712yes—
481C3—21no—
482C3#1058961511yescharacteristic, direct product factor
483C3C3×#551311yescharacteristic
484C2—22no—
485C2—361no—
486C2C2×C3×#369711yescharacteristic
487C2C2×C3×#369911yescharacteristic
488C2C3×#6371311yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#552
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×C4⋊C4C2semidirect
4. C3×#63488C2semidirect
5. C3×#63486C2semidirect
6. C3×#63460C2semidirect
7. C3×#63459C2semidirect
8. C3×#63454C2semidirect
9. C3×#63452C2semidirect
10. C3×#63451C2semidirect
11. C3×#63449C2semidirect
13. #10589615C3direct (N × H)
49. C2×C9×C4⋊C4C2×C3semidirect
52. #63488C2×C3semidirect
53. #63486C2×C3semidirect
54. #63460C2×C3semidirect
55. #63459C2×C3semidirect
56. #63454C2×C3semidirect
57. #63452C2×C3semidirect
58. #63451C2×C3semidirect
59. #63449C2×C3semidirect
156. C2×C3×C4⋊C4D9semidirect
314. C2×C4⋊C4C3×D9semidirect
315. C3×C9#552semidirect
434. C9C3×#552semidirect
482. C3#10589615direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×C4⋊C42
C3×#634912
C3×C4×#6112
C3×#634882
C3×#634862
C3×#634602
C3×#634592
C3×#634542
C3×#634522
C3×#634512
C3×#634492
C3×#55133
#105896153