Finite Group Databasebeta

← C3×#10589631 #13152490

Subgroups — C3×#10589631 #13152490

Order 1728.

718 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#117C211yesmaximal, characteristic, complemented
2C3×C4×#611C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic, complemented
4C3×#63459C211yesmaximal, characteristic, complemented
5C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
6C3×#63452C211yesmaximal, characteristic, complemented
7C3×#63449C211yesmaximal, characteristic, complemented
8C3×#63445C212yesmaximal
9C3×#63441C212yesmaximal, complemented
10C3×#63434C212yesmaximal, complemented
11C3×#63432C212yesmaximal, complemented
12C3×#5529—31nomaximal
13#10589631C311yesmaximal, characteristic, direct product factor
14C22×C3×C4×C9C2211yescharacteristic
15C2×C3×#611C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×C3×Dic18C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C9×C22⋊C4C2212yes—
22C3×C42×C9C2212yes—
23C3×#3676C2212yes—
24C3×#3676C2211yescharacteristic
25C3×(C2×D9)⋊C4C2211yescharacteristic
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×Dic9⋊C4C2212yes—
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2212yes—
37C32×#117D311yescharacteristic
38C3×C4×#67—31no—
39C3×#924—31no—
40C3×#897—31no—
41C3×D3×C4⋊C4—31no—
42C3×#890—31no—
43C3×#887—31no—
44C3×#883—32no—
45C3×#879—32no—
46C3×#872—32no—
47C3×#870—32no—
48C9×#117—21no—
49C9×#117C2×C311yescharacteristic, complemented
50C4×#611C2×C311yescharacteristic
51#63486C2×C311yescharacteristic, complemented
52#63459C2×C311yescharacteristic, complemented
53C4⋊C4×D9C2×C311yescharacteristic, complemented
54#63452C2×C311yescharacteristic, complemented
55#63449C2×C311yescharacteristic, complemented
56#63445C2×C312yes—
57#63441C2×C312yescomplemented
58#63434C2×C312yescomplemented
59#63432C2×C312yescomplemented
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C3×#611—22no—
68C2×C3×Dic9C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9—24no—
74C3×C4×D9—22no—
75C3×Dic18—22no—
76C3×#552—91no—
77#5529—31no—
78C22×C32×C4C2×D311yescharacteristic
79C2×C3×#67—31no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C22⋊C4C2×D312yes—
86C32×C42C2×D312yes—
87C3×#217—31no—
88C3×#217—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—31no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—32no—
93C3×C4⋊Dic3—31no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—31no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—31no—
101C22×C4×C9—21no—
102C22×C4×C9C22×C311yescharacteristic
103C2×#611C22×C311yescharacteristic
104C2×C4×D9C22×C311yescharacteristic
105C2×C4×D9C22×C312yes—
106C2×Dic18C22×C311yescharacteristic
107C9×C4⋊C4—21no—
108C9×C4⋊C4—21no—
109C9×C4⋊C4C22×C311yescharacteristic
110C9×C4⋊C4C22×C311yescharacteristic
111C9×C22⋊C4—22no—
112C9×C22⋊C4C22×C312yes—
113C42×C9—22no—
114C42×C9C22×C312yes—
115#3676C22×C311yescharacteristic
116#3676C22×C312yes—
117(C2×D9)⋊C4C22×C311yescharacteristic
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—
126Dic9⋊C4C22×C312yes—
127C4×Dic9C22×C311yescharacteristic
128C4×Dic9C22×C312yes—
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9C2411yescharacteristic
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9#4112yes—
136C3×C4×C9—21no—
137C3×C4×C9—22no—
138C3×C4×C9—22no—
139C3×C4×C9#4112yes—
140C3×Dic9—22no—
141C3×Dic9—21no—
142C3×Dic9—22no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×#126—92no—
146C3×#125—91no—
147C3×#125—92no—
148C3×#123—92no—
149C3×C22⋊Q8—91no—
150C3×C22⋊Q8—91no—
151C3×C4×Q8—91no—
152C3×C4×D4—91no—
153C3×#117—21no—
154C3×#117—92no—
155C3×#117D911yescharacteristic, complemented
156C3×#117C3×D311yescharacteristic
157C2×C3×C4⋊C4—91no—
158C4×#67—31no—
159#924—31no—
160#897—31no—
161D3×C4⋊C4—31no—
162#890—31no—
163#887—31no—
164#883—32no—
165#879—32no—
166#872—32no—
167#870—32no—
168C23×C32C22×D311yescharacteristic
169C22×C3×D3—31no—
170C2×C32×C4—22no—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C3×#67—62no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—32no—
181C3×C4×D3—64no—
182C3×C4×D3—62no—
183C3×Dic6—62no—
184C23×C9—21no—
185C23×C9C23×C311yescharacteristic
186C22×D9C23×C311yescharacteristic
187C2×C4×C9—24no—
188C2×C4×C9—22no—
189C2×C4×C9—21no—
190C2×C4×C9—21no—
191C2×C4×C9—22no—
192C2×C4×C9—22no—
193C2×C4×C9C23×C312yes—
194C2×C4×C9C23×C311yescharacteristic
195C2×C4×C9C23×C312yes—
196C2×C4×C9C23×C311yescharacteristic
197#611—22no—
198C2×Dic9C23×C311yescharacteristic
199C2×Dic9C23×C311yescharacteristic
200C2×Dic9C23×C312yes—
201C2×Dic9C23×C311yescharacteristic
202C2×Dic9C23×C312yes—
203C4×D9—24no—
204C4×D9—22no—
205Dic18—22no—
206#552—91noSylow
207C2×C3×C9—41no—
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C92−1+411yescharacteristic
211C3×D9—22no—
212C2×C3×Q8—91no—
213C2×C3×D4—91no—
214C22×C3×C4—21no—
215C22×C3×C4—92no—
216C22×C3×C4—91no—
217C22×C3×C4C2×D911yescharacteristic
218C22×C3×C4C2×C3×D311yescharacteristic
219C2×#67—31no—
220C2×C4×D3—32no—
221C2×C4×D3—31no—
222C2×Dic6—31no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4—91no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4—92no—
233C3×C4⋊C4C2×D911yescharacteristic
234C3×C4⋊C4C2×D911yescharacteristic
235C3×C4⋊C4C2×C3×D311yescharacteristic
236C3×C4⋊C4C2×C3×D311yescharacteristic
237C3×C22⋊C4—22no—
238C3×C22⋊C4—91no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4C2×D912yes—
243C3×C22⋊C4C2×C3×D312yes—
244C3×C42—22no—
245C3×C42—92no—
246C3×C42—91no—
247C3×C42C2×D912yes—
248C3×C42C2×C3×D312yes—
249#217—31no—
250#217—32no—
251(C2×D3)⋊C4—31no—
252(C2×D3)⋊C4—32no—
253C4⋊Dic3—32no—
254C4⋊Dic3—32no—
255C4⋊Dic3—31no—
256Dic3⋊C4—31no—
257Dic3⋊C4—32no—
258Dic3⋊C4—32no—
259Dic3⋊C4—32no—
260Dic3⋊C4—32no—
261C4×Dic3—32no—
262C4×Dic3—31no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32C23×D311yescharacteristic
267C2×C3×D3—32no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C32×C4—21no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4#23512yes—
274C3×Dic3—62no—
275C3×Dic3—61no—
276C3×Dic3—62no—
277C3×Dic3—61no—
278C3×Dic3—61no—
279C22×C9—22no—
280C22×C9—21no—
281C22×C9—21no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—41no—
285C22×C9—41no—
286C22×C9C24×C311yescharacteristic
287C2×D9—21no—
288C2×D9—21no—
289C2×D9C3×#4112yes—
290C4×C9—22no—
291C4×C9—22no—
292C4×C9—21no—
293C4×C9—22no—
294C4×C9—22no—
295C4×C9—42no—
296C4×C9—42no—
297C4×C9C3×#4112yes—
298Dic9—22no—
299Dic9—21no—
300Dic9—22no—
301Dic9—21no—
302Dic9—21no—
303#126—92no—
304#125—91no—
305#125—92no—
306#123—92no—
307C22⋊Q8—91no—
308C22⋊Q8—91no—
309C4×Q8—91no—
310C4×D4—91no—
311#117—92no—
312#117C3×D911yescharacteristic, complemented
313C2×C4⋊C4—91no—
314C3×C9#55211yescharacteristic, complemented, Sylow
315C23×C3—21no—
316C23×C3—91no—
317C23×C3C22×D911yescharacteristic
318C23×C3C22×C3×D311yescharacteristic
319C22×D3—31no—
320C3×Q8—182no—
321C3×D4—182no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—91no—
330C2×C3×C4—92no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—92no—
334C2×C3×C4—184no—
335C2×C3×C4—182no—
336C2×C3×C4C22×D912yes—
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C22×D912yes—
339C2×C3×C4C22×D911yescharacteristic
340C2×C3×C4C22×C3×D312yes—
341C2×C3×C4C22×C3×D311yescharacteristic
342C2×C3×C4C22×C3×D312yes—
343C2×C3×C4C22×C3×D311yescharacteristic
344#67—62no—
345C2×Dic3—31no—
346C2×Dic3—31no—
347C2×Dic3—32no—
348C2×Dic3—31no—
349C2×Dic3—32no—
350C4×D3—64no—
351C4×D3—62no—
352Dic6—62no—
353C2×C32—41no—
354C2×C32C2×#23511yescharacteristic
355C2×C32D3×#4111yescharacteristic
356C2×C32#101011yescharacteristic
357C3×D3—62no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—42no—
362C2×C9—41no—
363C2×C9C2×C3×#4111yescharacteristic
364C2×C9C2×C3×#4111yescharacteristic
365C2×C9C3×2−1+411yescharacteristic
366D9—22no—
367C2×Q8—91no—
368C2×D4—91no—
369C22×C4—92no—
370C22×C4—91no—
371C22×C4C2×C3×D911yescharacteristic
372C4⋊C4—91no—
373C4⋊C4—92no—
374C4⋊C4—92no—
375C4⋊C4—92no—
376C4⋊C4—91no—
377C4⋊C4—92no—
378C4⋊C4—92no—
379C4⋊C4—92no—
380C4⋊C4C2×C3×D911yescharacteristic
381C4⋊C4C2×C3×D911yescharacteristic
382C22⋊C4—91no—
383C22⋊C4—92no—
384C22⋊C4—91no—
385C22⋊C4—92no—
386C22⋊C4C2×C3×D912yes—
387C42—92no—
388C42—91no—
389C42C2×C3×D912yes—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—92no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3C23×D911yescharacteristic
404C22×C3C23×C3×D311yescharacteristic
405C2×D3—32no—
406C2×D3—61no—
407C2×D3—61no—
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—181no—
419C3×C4—182no—
420C3×C4—182no—
421C3×C4—181no—
422C3×C4—181no—
423C3×C4#369712yes—
424C3×C4C3×#23512yes—
425Dic3—62no—
426Dic3—61no—
427Dic3—62no—
428Dic3—61no—
429Dic3—61no—
430C32#552911yescharacteristic
431C9—21no—
432C9C3×#55211yescharacteristic, complemented
433C23—91no—
434C23C22×C3×D911yescharacteristic
435Q8—182no—
436D4—182no—
437C2×C4—22no—
438C2×C4—91no—
439C2×C4—92no—
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—92no—
443C2×C4—184no—
444C2×C4—182no—
445C2×C4C22×C3×D912yes—
446C2×C4C22×C3×D911yescharacteristic
447C2×C4C22×C3×D912yes—
448C2×C4C22×C3×D911yescharacteristic
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—42no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—182no—
456C2×C3C2×#369711yescharacteristic
457C2×C3#41×D911yescharacteristic
458C2×C3#6371611yescharacteristic
459C2×C3C2×C3×#23511yescharacteristic
460C2×C3C3×D3×#4111yescharacteristic
461C2×C3C3×#101011yescharacteristic
462D3—62no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—92no—
467C22—181no—
468C22—181no—
469C22C23×C3×D911yescharacteristic
470C4—21no—
471C4—22no—
472C4—22no—
473C4—181no—
474C4—182no—
475C4—182no—
476C4—181no—
477C4—181no—
478C4C3×#369712yes—
479C3—21no—
480C3#1058963111yescharacteristic, direct product factor
481C3C3×#552911yescharacteristic
482C2—41no—
483C2—182no—
484C2C2×C3×#369711yescharacteristic
485C2C3×#41×D911yescharacteristic
486C2C3×#6371611yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#117C2
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC3×#117D9
hypocenterC9C3×#552
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
1. C3×C9×#117C2semidirect
3. C3×#63486C2semidirect
4. C3×#63459C2semidirect
5. C3×C4⋊C4×D9C2semidirect
6. C3×#63452C2semidirect
7. C3×#63449C2semidirect
9. C3×#63441C2semidirect
10. C3×#63434C2semidirect
11. C3×#63432C2semidirect
13. #10589631C3direct (N × H)
49. C9×#117C2×C3semidirect
51. #63486C2×C3semidirect
52. #63459C2×C3semidirect
53. C4⋊C4×D9C2×C3semidirect
54. #63452C2×C3semidirect
55. #63449C2×C3semidirect
57. #63441C2×C3semidirect
58. #63434C2×C3semidirect
59. #63432C2×C3semidirect
155. C3×#117D9semidirect
312. #117C3×D9semidirect
314. C3×C9#552semidirect
432. C9C3×#552semidirect
480. C3#10589631direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1172
C3×C4×#6112
C3×#634862
C3×#634592
C3×C4⋊C4×D92
C3×#634522
C3×#634492
C3×#634452
C3×#634412
C3×#634342
C3×#634322
C3×#55293
#105896313