Finite Group Databasebeta

← C3×#10589881 #13153510

Subgroups — C3×#10589881 #13153510

Order 1728.

804 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63716C211yesmaximal, characteristic, complemented
2C3×#63713C211yesmaximal, characteristic, complemented
3C3×C9×#137C211yesmaximal, characteristic, complemented
4C3×#63513C211yesmaximal, characteristic, complemented
5C2×C3×#3674C211yesmaximal, characteristic, complemented
6C3×#63480C212yesmaximal, complemented
7C3×Q16×D9C212yesmaximal, complemented
8C3×#63478C212yesmaximal, complemented
9C3×#63477C212yesmaximal, complemented
10C3×#63471C211yesmaximal, characteristic, complemented
11C3×#63469C211yesmaximal, characteristic, complemented
12C3×#5779—31nomaximal
13#10589881C311yesmaximal, characteristic, direct product factor
14C3×C9×#41C2211yescharacteristic
15C2×C3×Q8×C9C2211yescharacteristic
16C3×#3701—21no—
17C3×#3701C2212yes—
18C3×Q8×D9—21no—
19C3×Q8×D9C2212yes—
20C3×Q8×D9C2212yes—
21C3×#3699—22no—
22C3×#3699C2212yes—
23C3×#3697—22no—
24C3×#3697—21no—
25C3×#3697C2211yescharacteristic
26C3×#3697C2211yescharacteristic
27C2×C3×Dic18—21no—
28C2×C3×Dic18C2211yescharacteristic
29C3×C9×Q16C2212yes—
30C3×C9×SD16C2212yes—
31C3×C9×M16C2211yescharacteristic
32C3×#3675C2211yescharacteristic
33C3×#3674C2212yes—
34C3×#3674C2212yes—
35C3×#3674C2211yescharacteristic
36C3×#3673C2211yescharacteristic
37C3×#3672C2211yescharacteristic
38C2×C3×C9⋊C8C2211yescharacteristic
39C3×#3664C2212yes—
40C3×#3663C2212yes—
41C3×C8×D9C2212yes—
42C3×Dic36C2212yes—
43C3×#1010—31no—
44C3×#1007—31no—
45C32×#137D311yescharacteristic
46C3×#950—31no—
47C2×C3×#216—31no—
48C3×#918—32no—
49C3×D3×Q16—32no—
50C3×#916—32no—
51C3×#915—32no—
52C3×#909—31no—
53C3×#907—31no—
54#63716C2×C311yescharacteristic, complemented
55#63713C2×C311yescharacteristic, complemented
56C9×#137—21no—
57C9×#137C2×C311yescharacteristic, complemented
58#63513C2×C311yescharacteristic, complemented
59C2×#3674C2×C311yescharacteristic, complemented
60#63480C2×C312yescomplemented
61Q16×D9C2×C312yescomplemented
62#63478C2×C312yescomplemented
63#63477C2×C312yescomplemented
64#63471C2×C311yescharacteristic, complemented
65#63469C2×C311yescharacteristic, complemented
66C3×Q8×C9—21no—
67C3×Q8×C9C2311yescharacteristic
68C3×Q8×C9C2312yes—
69C3×D4×C9—21no—
70C3×D4×C9C2311yescharacteristic
71C2×C3×C4×C9—21no—
72C2×C3×C4×C9—21no—
73C2×C3×C4×C9C2311yescharacteristic
74C3×#611—21no—
75C3×#611—22no—
76C3×#611D412yes—
77C2×C3×Dic9—21no—
78C2×C3×Dic9—22no—
79C3×D36—22no—
80C3×D36D411yescharacteristic
81C3×D36C2311yescharacteristic
82C3×C4×D9—22no—
83C3×C4×D9—22no—
84C3×C4×D9—21no—
85C3×C4×D9—22no—
86C3×C4×D9C2312yes—
87C3×Dic18—21no—
88C3×Dic18—22no—
89C3×Dic18—22no—
90C3×Dic18D411yescharacteristic
91C3×Dic18C2312yes—
92C3×Dic18C2311yescharacteristic
93C3×C8×C9C2312yes—
94C3×C9⋊C8C2311yescharacteristic
95C3×C9⋊C8C2311yescharacteristic
96C3×#578—91no—
97#5779—31no—
98C32×#41C2×D311yescharacteristic
99C2×C32×Q8C2×D311yescharacteristic
100C3×#239—32no—
101C3×#239—61no—
102C3×D3×Q8—32no—
103C3×D3×Q8—32no—
104C3×D3×Q8—61no—
105C3×#237—32no—
106C3×#237—62no—
107C3×#235—31no—
108C3×#235—31no—
109C3×#235—61no—
110C3×#235—62no—
111C2×C3×Dic6—31no—
112C2×C3×Dic6—61no—
113C32×Q16C2×D312yes—
114C32×SD16C2×D312yes—
115C32×M16C2×D311yescharacteristic
116C3×#216—32no—
117C3×#216—32no—
118C3×#216—31no—
119C3×#215—31no—
120C3×#214—31no—
121C3×#213—31no—
122C2×C3×C3⋊C8—31no—
123C3×Dic12—32no—
124C3×#204—32no—
125C3×ΓL(1,𝔽25)—32no—
126C3×D3×C8—32no—
127C9×#41—21no—
128C9×#41C22×C311yescharacteristic
129C2×Q8×C9—21no—
130C2×Q8×C9C22×C311yescharacteristic
131#3701—21no—
132#3701C22×C312yes—
133Q8×D9—21no—
134Q8×D9C22×C312yes—
135Q8×D9C22×C312yes—
136#3699—22no—
137#3699C22×C312yes—
138#3697—22no—
139#3697—21no—
140#3697C22×C311yescharacteristic
141#3697C22×C311yescharacteristic
142C2×Dic18—21no—
143C2×Dic18C22×C311yescharacteristic
144C9×Q16—22no—
145C9×Q16C22×C312yes—
146C9×SD16—22no—
147C9×SD16C22×C312yes—
148C9×M16—21no—
149C9×M16C22×C311yescharacteristic
150#3675C22×C311yescharacteristic
151#3674C22×C312yes—
152#3674C22×C312yes—
153#3674C22×C311yescharacteristic
154#3673C22×C311yescharacteristic
155#3672C22×C311yescharacteristic
156C2×C9⋊C8C22×C311yescharacteristic
157#3664C22×C312yes—
158#3663C22×C312yes—
159C8×D9C22×C312yes—
160Dic36C22×C312yes—
161C22×C3×C9—21no—
162C22×C3×C9C2×D411yescharacteristic
163C2×C3×D9—21no—
164C2×C3×D9C2×D412yes—
165C3×C4×C9—21no—
166C3×C4×C9—22no—
167C3×C4×C9C2×D411yescharacteristic
168C3×C4×C9C2411yescharacteristic
169C3×Dic9—21no—
170C3×Dic9—22no—
171C3×Dic9C2×D412yes—
172C3×2−1+4—91no—
173C3×2−1+4—91no—
174#1010—31no—
175#1007—31no—
176C3×#137—21no—
177C3×#137—92no—
178C3×#137—92no—
179C3×#137—91no—
180C3×#137D911yescharacteristic, complemented
181C3×#137C3×D311yescharacteristic
182C3×#135—91no—
183C3×#135—92no—
184C2×C3×Q16—91no—
185C2×C3×Q16—92no—
186C3×#131—91no—
187#950—31no—
188C2×#216—31no—
189#918—32no—
190D3×Q16—32no—
191#916—32no—
192#915—32no—
193#909—31no—
194#907—31no—
195C32×Q8—21no—
196C32×Q8C22×D311yescharacteristic
197C32×Q8C22×D312yes—
198C32×D4—21no—
199C32×D4C22×D311yescharacteristic
200C2×C32×C4—21no—
201C2×C32×C4—21no—
202C2×C32×C4C22×D311yescharacteristic
203C3×#67—32no—
204C3×#67—62no—
205C3×#67—61no—
206C2×C3×Dic3—62no—
207C2×C3×Dic3—61no—
208C3×D12—31no—
209C3×D12—31no—
210C3×D12—62no—
211C3×C4×D3—32no—
212C3×C4×D3—62no—
213C3×C4×D3—62no—
214C3×C4×D3—62no—
215C3×C4×D3—61no—
216C3×Dic6—31no—
217C3×Dic6—32no—
218C3×Dic6—31no—
219C3×Dic6—62no—
220C3×Dic6—62no—
221C3×Dic6—61no—
222C32×C8C22×D312yes—
223C3×C3⋊C8—31no—
224C3×C3⋊C8—31no—
225Q8×C9—21no—
226Q8×C9—22no—
227Q8×C9—21no—
228Q8×C9—41no—
229Q8×C9C23×C312yes—
230Q8×C9C23×C311yescharacteristic
231D4×C9—21no—
232D4×C9—21no—
233D4×C9—41no—
234D4×C9C23×C311yescharacteristic
235C2×C4×C9—21no—
236C2×C4×C9—21no—
237C2×C4×C9—21no—
238C2×C4×C9—41no—
239C2×C4×C9—41no—
240C2×C4×C9C23×C311yescharacteristic
241#611—21no—
242#611—22no—
243#611C3×D412yes—
244C2×Dic9—21no—
245C2×Dic9—22no—
246D36—22no—
247D36C3×D411yescharacteristic
248D36C23×C311yescharacteristic
249C4×D9—22no—
250C4×D9—22no—
251C4×D9—21no—
252C4×D9—22no—
253C4×D9C23×C312yes—
254Dic18—21no—
255Dic18—22no—
256Dic18—22no—
257Dic18C3×D411yescharacteristic
258Dic18C23×C311yescharacteristic
259Dic18C23×C312yes—
260C8×C9—22no—
261C8×C9C23×C312yes—
262C9⋊C8C23×C311yescharacteristic
263C9⋊C8C23×C311yescharacteristic
264#578—91noSylow
265C2×C3×C9—21no—
266C2×C3×C9—41no—
267C2×C3×C9C22×D411yescharacteristic
268C3×D9—22no—
269C3×D9—41no—
270C3×#41—21no—
271C3×#41—91no—
272C3×#41—92no—
273C3×#41—92no—
274C3×#41—91no—
275C3×#41—182no—
276C3×#41—181no—
277C3×#41—182no—
278C3×#41—181no—
279C3×#41C2×D911yescharacteristic
280C3×#41C2×C3×D311yescharacteristic
281C2×C3×Q8—21no—
282C2×C3×Q8—92no—
283C2×C3×Q8—92no—
284C2×C3×Q8—91no—
285C2×C3×Q8—181no—
286C2×C3×Q8—181no—
287C2×C3×Q8C2×D911yescharacteristic
288C2×C3×Q8C2×C3×D311yescharacteristic
289#239—32no—
290#239—61no—
291D3×Q8—32no—
292D3×Q8—32no—
293D3×Q8—61no—
294#237—32no—
295#237—62no—
296#235—31no—
297#235—31no—
298#235—61no—
299#235—62no—
300C2×Dic6—31no—
301C2×Dic6—61no—
302C3×Q16—22no—
303C3×Q16—91no—
304C3×Q16—92no—
305C3×Q16—92no—
306C3×Q16—92no—
307C3×Q16C2×D912yes—
308C3×Q16C2×C3×D312yes—
309C3×SD16—22no—
310C3×SD16—91no—
311C3×SD16—91no—
312C3×SD16—92no—
313C3×SD16C2×D912yes—
314C3×SD16C2×C3×D312yes—
315C3×D8—91no—
316C3×M16—21no—
317C3×M16—92no—
318C3×M16C2×D911yescharacteristic
319C3×M16C2×C3×D311yescharacteristic
320C2×C3×C8—91no—
321C2×C3×C8—92no—
322#216—32no—
323#216—32no—
324#216—31no—
325#215—31no—
326#214—31no—
327#213—31no—
328C2×C3⋊C8—31no—
329Dic12—32no—
330#204—32no—
331ΓL(1,𝔽25)—32no—
332D3×C8—32no—
333C22×C32—21no—
334C22×C32D3×D411yescharacteristic
335C2×C3×D3—32no—
336C2×C3×D3—61no—
337C32×C4—21no—
338C32×C4—22no—
339C32×C4D3×D411yescharacteristic
340C32×C4C23×D311yescharacteristic
341C3×Dic3—32no—
342C3×Dic3—61no—
343C3×Dic3—62no—
344C22×C9—21no—
345C22×C9—21no—
346C22×C9—41no—
347C22×C9C2×C3×D411yescharacteristic
348C2×D9—21no—
349C2×D9C2×C3×D412yes—
350C4×C9—21no—
351C4×C9—22no—
352C4×C9—21no—
353C4×C9—21no—
354C4×C9—41no—
355C4×C9—42no—
356C4×C9C2×C3×D411yescharacteristic
357C4×C9C24×C311yescharacteristic
358Dic9—21no—
359Dic9—22no—
360Dic9C2×C3×D412yes—
3612−1+4—91no—
3622−1+4—91no—
363#137—92no—
364#137—92no—
365#137—91no—
366#137C3×D911yescharacteristic, complemented
367#135—91no—
368#135—92no—
369C2×Q16—91no—
370C2×Q16—92no—
371#131—91no—
372C3×C9#57811yescharacteristic, complemented, Sylow
373C3×Q8—21no—
374C3×Q8—21no—
375C3×Q8—22no—
376C3×Q8—21no—
377C3×Q8—41no—
378C3×Q8—91no—
379C3×Q8—92no—
380C3×Q8—91no—
381C3×Q8—182no—
382C3×Q8—182no—
383C3×Q8—181no—
384C3×Q8C22×D911yescharacteristic
385C3×Q8C22×D912yes—
386C3×Q8C22×C3×D311yescharacteristic
387C3×Q8C22×C3×D312yes—
388C3×D4—21no—
389C3×D4—21no—
390C3×D4—21no—
391C3×D4—41no—
392C3×D4—92no—
393C3×D4—91no—
394C3×D4—91no—
395C3×D4—182no—
396C3×D4—182no—
397C3×D4—181no—
398C3×D4C22×D911yescharacteristic
399C3×D4C22×C3×D311yescharacteristic
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—92no—
408C2×C3×C4—182no—
409C2×C3×C4—182no—
410C2×C3×C4—182no—
411C2×C3×C4—182no—
412C2×C3×C4—181no—
413C2×C3×C4—181no—
414C2×C3×C4C22×D911yescharacteristic
415C2×C3×C4C22×C3×D311yescharacteristic
416#67—32no—
417#67—62no—
418#67—61no—
419C2×Dic3—62no—
420C2×Dic3—61no—
421D12—31no—
422D12—31no—
423D12—62no—
424C4×D3—32no—
425C4×D3—62no—
426C4×D3—62no—
427C4×D3—62no—
428C4×D3—61no—
429Dic6—31no—
430Dic6—32no—
431Dic6—31no—
432Dic6—62no—
433Dic6—62no—
434Dic6—61no—
435C3×C8—22no—
436C3×C8—91no—
437C3×C8—91no—
438C3×C8C22×D912yes—
439C3×C8C22×C3×D312yes—
440C3⋊C8—31no—
441C3⋊C8—31no—
442C2×C32—21no—
443C2×C32—41no—
444C2×C32C2×D3×D411yescharacteristic
445C3×D3—62no—
446C3×D3—121no—
447C2×C9—21no—
448C2×C9—21no—
449C2×C9—41no—
450C2×C9—41no—
451C2×C9—81no—
452C2×C9C22×C3×D411yescharacteristic
453D9—22no—
454D9—41no—
455#41—91no—
456#41—92no—
457#41—92no—
458#41—91no—
459#41—182no—
460#41—181no—
461#41—182no—
462#41—181no—
463#41C2×C3×D911yescharacteristic
464C2×Q8—92no—
465C2×Q8—92no—
466C2×Q8—91no—
467C2×Q8—181no—
468C2×Q8—181no—
469C2×Q8C2×C3×D911yescharacteristic
470Q16—91no—
471Q16—92no—
472Q16—92no—
473Q16—92no—
474Q16C2×C3×D912yes—
475SD16—91no—
476SD16—91no—
477SD16—92no—
478SD16C2×C3×D912yes—
479D8—91no—
480M16—92no—
481M16C2×C3×D911yescharacteristic
482C2×C8—91no—
483C2×C8—92no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—41no—
488C22×C3—92no—
489C22×C3—181no—
490C22×C3D4×D911yescharacteristic
491C22×C3C3×D3×D411yescharacteristic
492C2×D3—32no—
493C2×D3—61no—
494C3×C4—21no—
495C3×C4—22no—
496C3×C4—21no—
497C3×C4—22no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—92no—
503C3×C4—181no—
504C3×C4—182no—
505C3×C4D4×D911yescharacteristic
506C3×C4C23×D911yescharacteristic
507C3×C4C3×D3×D411yescharacteristic
508C3×C4C23×C3×D311yescharacteristic
509Dic3—32no—
510Dic3—61no—
511Dic3—62no—
512C32#577911yescharacteristic
513C9—21no—
514C9C3×#57811yescharacteristic, complemented
515Q8—21no—
516Q8—91no—
517Q8—91no—
518Q8—92no—
519Q8—182no—
520Q8—182no—
521Q8—181no—
522Q8C22×C3×D911yescharacteristic
523Q8C22×C3×D912yes—
524D4—21no—
525D4—92no—
526D4—91no—
527D4—91no—
528D4—182no—
529D4—182no—
530D4—181no—
531D4C22×C3×D911yescharacteristic
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—92no—
535C2×C4—182no—
536C2×C4—182no—
537C2×C4—182no—
538C2×C4—182no—
539C2×C4—181no—
540C2×C4—181no—
541C2×C4C22×C3×D911yescharacteristic
542C8—91no—
543C8—91no—
544C8C22×C3×D912yes—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—41no—
549C2×C3—41no—
550C2×C3—41no—
551C2×C3—81no—
552C2×C3—182no—
553C2×C3—361no—
554C2×C3C2×D4×D911yescharacteristic
555C2×C3C2×C3×D3×D411yescharacteristic
556D3—62no—
557D3—121no—
558C22—21no—
559C22—92no—
560C22—181no—
561C22C3×D4×D911yescharacteristic
562C4—21no—
563C4—22no—
564C4—92no—
565C4—181no—
566C4—182no—
567C4C3×D4×D911yescharacteristic
568C4C23×C3×D911yescharacteristic
569C3—21no—
570C3#1058988111yescharacteristic, direct product factor
571C3C3×#577911yescharacteristic
572C2—21no—
573C2—41no—
574C2—182no—
575C2—361no—
576C2C2×C3×D4×D911yescharacteristic
subgroupG/H
centerC2×C3C2×D4×D9
commutator, cosocleC4×C9C24×C3
FrattiniC3×C4C23×C3×D3
FittingC3×C9×#137C2
socleC2×C32C2×D3×D4
hypercenterC3×#137D9
hypocenterC9C3×#578
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#137

Complements

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

NH ≅ G/Ntype
1. C3×#63716C2semidirect
2. C3×#63713C2semidirect
3. C3×C9×#137C2semidirect
4. C3×#63513C2semidirect
5. C2×C3×#3674C2semidirect
6. C3×#63480C2semidirect
7. C3×Q16×D9C2semidirect
8. C3×#63478C2semidirect
9. C3×#63477C2semidirect
10. C3×#63471C2semidirect
11. C3×#63469C2semidirect
13. #10589881C3direct (N × H)
54. #63716C2×C3semidirect
55. #63713C2×C3semidirect
57. C9×#137C2×C3semidirect
58. #63513C2×C3semidirect
59. C2×#3674C2×C3semidirect
60. #63480C2×C3semidirect
61. Q16×D9C2×C3semidirect
62. #63478C2×C3semidirect
63. #63477C2×C3semidirect
64. #63471C2×C3semidirect
65. #63469C2×C3semidirect
180. C3×#137D9semidirect
366. #137C3×D9semidirect
372. C3×C9#578semidirect
514. C9C3×#578semidirect
570. C3#10589881direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637162
C3×#637132
C3×C9×#1372
C3×#635132
C2×C3×#36742
C3×#634802
C3×Q16×D92
C3×#634782
C3×#634772
C3×#634712
C3×#634692
C3×#57793
#105898813