Finite Group Databasebeta

← C3×#10589700 #13152789

Subgroups — C3×#10589700 #13152789

Order 1728.

964 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic, complemented
2C22×C3×Dic18C211yesmaximal, characteristic, complemented
3C3×C9×#121C211yesmaximal, characteristic, complemented
4C3×#63501C212yesmaximal, complemented
5C3×#63498C211yesmaximal, characteristic, complemented
6C3×D4×Dic9C212yesmaximal, complemented
7C3×C4×#611C211yesmaximal, characteristic, complemented
8C3×#63459C211yesmaximal, characteristic, complemented
9C3×#63449C211yesmaximal, characteristic, complemented
10C3×#63447C212yesmaximal, complemented
11C3×#63440C212yesmaximal, complemented
12C3×#5598—31nomaximal
13#10589700C311yesmaximal, characteristic, direct product factor
14C2×C3×D4×C9C2211yescharacteristic
15C2×C3×D4×C9C2212yes—
16C22×C3×C4×C9C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C2×C3×#611C2212yes—
19C22×C3×Dic9C2212yes—
20C22×C3×Dic9C2212yes—
21C3×#3699—24no—
22C2×C3×C4×D9C2211yescharacteristic
23C2×C3×Dic18—24no—
24C2×C3×Dic18C2211yescharacteristic
25C2×C3×Dic18C2212yes—
26C2×C3×Dic18C2211yescharacteristic
27C3×C9×C4⋊C4C2211yescharacteristic
28C3×C9×C22⋊C4C2212yes—
29C3×#3676C2211yescharacteristic
30C3×#3676C2212yes—
31C3×#3676C2212yes—
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C3×(C2×D9)⋊C4C2212yes—
34C3×C4⋊Dic9C2212yes—
35C3×Dic9⋊C4C2211yescharacteristic
36C3×Dic9⋊C4C2212yes—
37C3×C4×Dic9C2211yescharacteristic
38C3×C4×Dic9C2212yes—
39C2×C3×#237—31no—
40C22×C3×Dic6—31no—
41C32×#121D311yescharacteristic
42C3×#939—32no—
43C3×#936—31no—
44C3×D4×Dic3—32no—
45C3×C4×#67—31no—
46C3×#897—31no—
47C3×#887—31no—
48C3×#885—32no—
49C3×#878—32no—
50C2×#3699C2×C311yescharacteristic, complemented
51C22×Dic18C2×C311yescharacteristic, complemented
52C9×#121—21no—
53C9×#121C2×C311yescharacteristic, complemented
54#63501C2×C312yescomplemented
55#63498C2×C311yescharacteristic, complemented
56D4×Dic9C2×C312yescomplemented
57C4×#611C2×C311yescharacteristic, complemented
58#63459C2×C311yescharacteristic, complemented
59#63449C2×C311yescharacteristic, complemented
60#63447C2×C312yescomplemented
61#63440C2×C312yescomplemented
62C23×C3×C9C2312yes—
63C23×C3×C9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C3×D4×C9—24no—
66C3×D4×C9—22no—
67C2×C3×C4×C9—22no—
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2312yes—
70C2×C3×C4×C9C2311yescharacteristic
71C3×#611—22no—
72C3×#611—24no—
73C2×C3×Dic9—24no—
74C2×C3×Dic9—24no—
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2312yes—
80C3×C4×D9—22no—
81C3×Dic18—22no—
82C3×Dic18—24no—
83C3×Dic18D414yescomplemented
84C3×#548—91no—
85#5598—31no—
86C2×C32×D4C2×D312yes—
87C2×C32×D4C2×D311yescharacteristic
88C22×C32×C4C2×D311yescharacteristic
89C2×C3×#67—32no—
90C2×C3×#67—31no—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—32no—
93C3×#237—64no—
94C2×C3×C4×D3—31no—
95C2×C3×Dic6—31no—
96C2×C3×Dic6—32no—
97C2×C3×Dic6—31no—
98C2×C3×Dic6—64no—
99C32×C4⋊C4C2×D311yescharacteristic
100C32×C22⋊C4C2×D312yes—
101C3×#217—32no—
102C3×#217—32no—
103C3×#217—31no—
104C3×(C2×D3)⋊C4—31no—
105C3×(C2×D3)⋊C4—32no—
106C3×C4⋊Dic3—32no—
107C3×Dic3⋊C4—31no—
108C3×Dic3⋊C4—32no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—31no—
111C2×D4×C9—21no—
112C2×D4×C9—22no—
113C2×D4×C9C22×C312yes—
114C2×D4×C9C22×C311yescharacteristic
115C22×C4×C9—21no—
116C22×C4×C9C22×C311yescharacteristic
117C2×#611C22×C311yescharacteristic
118C2×#611C22×C312yes—
119C22×Dic9C22×C312yes—
120C22×Dic9C22×C312yes—
121#3699—24no—
122C2×C4×D9C22×C311yescharacteristic
123C2×Dic18—24no—
124C2×Dic18C22×C311yescharacteristic
125C2×Dic18C22×C312yes—
126C2×Dic18C22×C311yescharacteristic
127C9×C4⋊C4—21no—
128C9×C4⋊C4C22×C311yescharacteristic
129C9×C22⋊C4—22no—
130C9×C22⋊C4C22×C312yes—
131#3676C22×C311yescharacteristic
132#3676C22×C312yes—
133#3676C22×C312yes—
134(C2×D9)⋊C4C22×C311yescharacteristic
135(C2×D9)⋊C4C22×C312yes—
136C4⋊Dic9C22×C312yes—
137Dic9⋊C4C22×C311yescharacteristic
138Dic9⋊C4C22×C312yes—
139C4×Dic9C22×C311yescharacteristic
140C4×Dic9C22×C312yes—
141C22×C3×C9—21no—
142C22×C3×C9—21no—
143C22×C3×C9—22no—
144C22×C3×C9—22no—
145C22×C3×C9—22no—
146C22×C3×C9#4112yescomplemented
147C22×C3×C9C2411yescharacteristic
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9—21no—
151C3×C4×C9—21no—
152C3×C4×C9—22no—
153C3×C4×C9C2×D412yes—
154C3×Dic9—22no—
155C3×Dic9—21no—
156C3×Dic9—21no—
157C3×Dic9—21no—
158C3×Dic9C2×D414yes—
159C2×C3×#41—91no—
160C22×C3×Q8—91no—
161C2×#237—31no—
162C22×Dic6—31no—
163C3×#124—91no—
164C3×#124—92no—
165C3×C22⋊Q8—91no—
166C3×C22⋊Q8—92no—
167C3×#121—21no—
168C3×#121—92no—
169C3×#121D911yescharacteristic, complemented
170C3×#121C3×D311yescharacteristic
171C3×C4×Q8—91no—
172C3×C4×D4—92no—
173C3×C4×D4—91no—
174#939—32no—
175#936—31no—
176D4×Dic3—32no—
177C4×#67—31no—
178#897—31no—
179#887—31no—
180#885—32no—
181#878—32no—
182C23×C32C22×D312yes—
183C23×C32C22×D311yescharacteristic
184C22×C3×D3—31no—
185C32×D4—24no—
186C32×D4—22no—
187C2×C32×C4—22no—
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C3×#67—64no—
192C3×#67—62no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—64no—
199C2×C3×Dic3—64no—
200C3×C4×D3—62no—
201C3×Dic6—34no—
202C3×Dic6—64no—
203C3×Dic6—62no—
204C23×C9—21no—
205C23×C9—22no—
206C23×C9C23×C312yes—
207C23×C9C23×C311yescharacteristic
208C22×D9C23×C311yescharacteristic
209D4×C9—24no—
210D4×C9—22no—
211D4×C9—42no—
212D4×C9—44no—
213C2×C4×C9—22no—
214C2×C4×C9—21no—
215C2×C4×C9—21no—
216C2×C4×C9—22no—
217C2×C4×C9—42no—
218C2×C4×C9C23×C312yes—
219C2×C4×C9C23×C311yescharacteristic
220C2×C4×C9C23×C311yescharacteristic
221#611—22no—
222#611—24no—
223C2×Dic9—24no—
224C2×Dic9—24no—
225C2×Dic9C23×C311yescharacteristic
226C2×Dic9C23×C311yescharacteristic
227C2×Dic9C23×C311yescharacteristic
228C2×Dic9C23×C312yes—
229C2×Dic9C23×C312yes—
230C4×D9—22no—
231Dic18—22no—
232Dic18—24no—
233Dic18C3×D414yescomplemented
234#548—91noSylow
235C2×C3×C9—22no—
236C2×C3×C9—42no—
237C2×C3×C9C22×D411yescharacteristic
238C2×C3×C9C2×#4111yescharacteristic
239C2×C3×C92−1+411yescharacteristic
240C3×D9—41no—
241C3×#41—184no—
242C2×C3×Q8—91no—
243C2×C3×Q8—92no—
244C2×C3×Q8—91no—
245C2×C3×Q8—184no—
246C2×C3×D4—22no—
247C2×C3×D4—21no—
248C2×C3×D4—91no—
249C2×C3×D4—92no—
250C2×C3×D4C2×D912yes—
251C2×C3×D4C2×D911yescharacteristic
252C2×C3×D4C2×C3×D312yes—
253C2×C3×D4C2×C3×D311yescharacteristic
254C22×C3×C4—21no—
255C22×C3×C4—92no—
256C22×C3×C4—92no—
257C22×C3×C4—91no—
258C22×C3×C4C2×D911yescharacteristic
259C22×C3×C4C2×C3×D311yescharacteristic
260C2×#67—32no—
261C2×#67—31no—
262C22×Dic3—32no—
263C22×Dic3—32no—
264#237—64no—
265C2×C4×D3—31no—
266C2×Dic6—31no—
267C2×Dic6—32no—
268C2×Dic6—31no—
269C2×Dic6—64no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—91no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4C2×D911yescharacteristic
275C3×C4⋊C4C2×C3×D311yescharacteristic
276C3×C22⋊C4—22no—
277C3×C22⋊C4—91no—
278C3×C22⋊C4—92no—
279C3×C22⋊C4—92no—
280C3×C22⋊C4—91no—
281C3×C22⋊C4—92no—
282C3×C22⋊C4C2×D912yes—
283C3×C22⋊C4C2×C3×D312yes—
284C3×C42—92no—
285C3×C42—91no—
286#217—32no—
287#217—32no—
288#217—31no—
289(C2×D3)⋊C4—31no—
290(C2×D3)⋊C4—32no—
291C4⋊Dic3—32no—
292Dic3⋊C4—31no—
293Dic3⋊C4—32no—
294C4×Dic3—32no—
295C4×Dic3—31no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—22no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32#23712yes—
302C22×C32C23×D311yescharacteristic
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C32×C4—21no—
307C32×C4—22no—
308C32×C4D3×D412yes—
309C3×Dic3—34no—
310C3×Dic3—62no—
311C3×Dic3—61no—
312C3×Dic3—61no—
313C3×Dic3—61no—
314C22×C9—21no—
315C22×C9—21no—
316C22×C9—22no—
317C22×C9—22no—
318C22×C9—22no—
319C22×C9—21no—
320C22×C9—22no—
321C22×C9—41no—
322C22×C9—41no—
323C22×C9—42no—
324C22×C9—42no—
325C22×C9—42no—
326C22×C9C3×#4112yescomplemented
327C22×C9C24×C311yescharacteristic
328C2×D9—21no—
329C2×D9—21no—
330C2×D9—21no—
331C4×C9—21no—
332C4×C9—22no—
333C4×C9—22no—
334C4×C9—41no—
335C4×C9—42no—
336C4×C9C2×C3×D412yes—
337Dic9—22no—
338Dic9—21no—
339Dic9—21no—
340Dic9—21no—
341Dic9C2×C3×D414yes—
342C2×#41—91no—
343C22×Q8—91no—
344#124—91no—
345#124—92no—
346C22⋊Q8—91no—
347C22⋊Q8—92no—
348#121—92no—
349#121C3×D911yescharacteristic, complemented
350C4×Q8—91no—
351C4×D4—92no—
352C4×D4—91no—
353C3×C9#54811yescharacteristic, complemented, Sylow
354C23×C3—21no—
355C23×C3—22no—
356C23×C3—91no—
357C23×C3C22×D912yes—
358C23×C3C22×D911yescharacteristic
359C23×C3C22×C3×D312yes—
360C23×C3C22×C3×D311yescharacteristic
361C22×D3—31no—
362C3×Q8—94no—
363C3×Q8—184no—
364C3×Q8—182no—
365C3×D4—24no—
366C3×D4—24no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—42no—
370C3×D4—44no—
371C3×D4—184no—
372C3×D4—182no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—42no—
379C2×C3×C4—92no—
380C2×C3×C4—91no—
381C2×C3×C4—91no—
382C2×C3×C4—92no—
383C2×C3×C4—91no—
384C2×C3×C4—184no—
385C2×C3×C4—184no—
386C2×C3×C4—182no—
387C2×C3×C4C22×D912yes—
388C2×C3×C4C22×D911yescharacteristic
389C2×C3×C4C22×D911yescharacteristic
390C2×C3×C4C22×C3×D312yes—
391C2×C3×C4C22×C3×D311yescharacteristic
392C2×C3×C4C22×C3×D311yescharacteristic
393#67—64no—
394#67—62no—
395C2×Dic3—32no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—31no—
399C2×Dic3—32no—
400C2×Dic3—64no—
401C2×Dic3—64no—
402C4×D3—62no—
403Dic6—34no—
404Dic6—64no—
405Dic6—62no—
406C2×C32—22no—
407C2×C32—42no—
408C2×C32C2×D3×D411yescharacteristic
409C2×C32C2×#23711yescharacteristic
410C2×C32#101011yescharacteristic
411C3×D3—121no—
412C2×C9—22no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—42no—
417C2×C9—42no—
418C2×C9—82no—
419C2×C9C22×C3×D411yescharacteristic
420C2×C9C2×C3×#4111yescharacteristic
421C2×C9C3×2−1+411yescharacteristic
422D9—41no—
423#41—184no—
424C2×Q8—91no—
425C2×Q8—92no—
426C2×Q8—91no—
427C2×Q8—184no—
428C2×D4—91no—
429C2×D4—92no—
430C2×D4C2×C3×D912yes—
431C2×D4C2×C3×D911yescharacteristic
432C22×C4—92no—
433C22×C4—92no—
434C22×C4—91no—
435C22×C4C2×C3×D911yescharacteristic
436C4⋊C4—91no—
437C4⋊C4—92no—
438C4⋊C4—92no—
439C4⋊C4C2×C3×D911yescharacteristic
440C22⋊C4—91no—
441C22⋊C4—92no—
442C22⋊C4—92no—
443C22⋊C4—91no—
444C22⋊C4—92no—
445C22⋊C4C2×C3×D912yes—
446C42—92no—
447C42—91no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—21no—
459C22×C3—22no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3—181no—
468C22×C3#369912yescomplemented
469C22×C3C23×D911yescharacteristic
470C22×C3C3×#23712yes—
471C22×C3C23×C3×D311yescharacteristic
472C2×D3—61no—
473C2×D3—61no—
474C2×D3—61no—
475C3×C4—21no—
476C3×C4—22no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—41no—
481C3×C4—42no—
482C3×C4—94no—
483C3×C4—181no—
484C3×C4—182no—
485C3×C4—181no—
486C3×C4—181no—
487C3×C4D4×D912yes—
488C3×C4C3×D3×D412yes—
489Dic3—34no—
490Dic3—62no—
491Dic3—61no—
492Dic3—61no—
493Dic3—61no—
494C32#559811yescharacteristic
495C9—21no—
496C9C3×#54811yescharacteristic, complemented
497C23—91no—
498C23C22×C3×D912yes—
499C23C22×C3×D911yescharacteristic
500Q8—94no—
501Q8—184no—
502Q8—182no—
503D4—24no—
504D4—22no—
505D4—184no—
506D4—182no—
507C2×C4—22no—
508C2×C4—92no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—92no—
512C2×C4—91no—
513C2×C4—184no—
514C2×C4—184no—
515C2×C4—182no—
516C2×C4C22×C3×D912yes—
517C2×C4C22×C3×D911yescharacteristic
518C2×C4C22×C3×D911yescharacteristic
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—42no—
526C2×C3—42no—
527C2×C3—82no—
528C2×C3—361no—
529C2×C3C2×D4×D911yescharacteristic
530C2×C3C2×#369911yescharacteristic
531C2×C3#6371611yescharacteristic
532C2×C3C2×C3×D3×D411yescharacteristic
533C2×C3C2×C3×#23711yescharacteristic
534C2×C3C3×#101011yescharacteristic
535D3—121no—
536C22—21no—
537C22—21no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—181no—
542C22—181no—
543C22—181no—
544C22C3×#369912yescomplemented
545C22C23×C3×D911yescharacteristic
546C4—21no—
547C4—22no—
548C4—94no—
549C4—181no—
550C4—182no—
551C4—181no—
552C4—181no—
553C4C3×D4×D912yes—
554C3—21no—
555C3#1058970011yescharacteristic, direct product factor
556C3C3×#559811yescharacteristic
557C2—22no—
558C2—42no—
559C2—361no—
560C2C2×C3×D4×D911yescharacteristic
561C2C2×C3×#369911yescharacteristic
562C2C3×#6371611yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
hypercenterC3×#121D9
hypocenterC9C3×#548
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3699C2semidirect
2. C22×C3×Dic18C2semidirect
3. C3×C9×#121C2semidirect
4. C3×#63501C2semidirect
5. C3×#63498C2semidirect
6. C3×D4×Dic9C2semidirect
7. C3×C4×#611C2semidirect
8. C3×#63459C2semidirect
9. C3×#63449C2semidirect
10. C3×#63447C2semidirect
11. C3×#63440C2semidirect
13. #10589700C3direct (N × H)
50. C2×#3699C2×C3semidirect
51. C22×Dic18C2×C3semidirect
53. C9×#121C2×C3semidirect
54. #63501C2×C3semidirect
55. #63498C2×C3semidirect
56. D4×Dic9C2×C3semidirect
57. C4×#611C2×C3semidirect
58. #63459C2×C3semidirect
59. #63449C2×C3semidirect
60. #63447C2×C3semidirect
61. #63440C2×C3semidirect
83. C3×Dic18D4semidirect
146. C22×C3×C9#41semidirect
169. C3×#121D9semidirect
233. Dic18C3×D4semidirect
326. C22×C9C3×#41semidirect
349. #121C3×D9semidirect
353. C3×C9#548semidirect
468. C22×C3#3699semidirect
496. C9C3×#548semidirect
544. C22C3×#3699semidirect
555. C3#10589700direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C22×C3×Dic182
C3×C9×#1212
C3×#635012
C3×#634982
C3×D4×Dic92
C3×C4×#6112
C3×#634592
C3×#634492
C3×#634472
C3×#634402
C3×#55983
#105897003