Finite Group Databasebeta

← C3×#10595084 #13174681

Subgroups — C3×#10595084 #13174681

Order 1728.

1758 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3811C212yesmaximal, complemented
2C2×C3×#3752C211yesmaximal, characteristic, complemented
3C32×#930C212yesmaximal, complemented
4C3×#63980C212yesmaximal, complemented
5C3×#63922C214yesmaximal, complemented
6C3×#63854C214yesmaximal, complemented
7C3×#5605—32nomaximal
8#10595084C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3C2211yescharacteristic
10C22×C3×D32C2212yescomplemented
11C22×C33×C4C2211yescharacteristic, complemented
12C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
13C2×C32×#67C2214yescomplemented
14C2×C32×D12C2212yescomplemented
15C3×#3811—28no—
16C2×C3×#626C2214yes—
17C2×C3×#625C2212yescomplemented
18C2×C3×#624C2212yescomplemented
19C3×#3752C2214yescomplemented
20C32×(C2×D3)⋊C4C2214yescomplemented
21C32×C4⋊Dic3C2212yescomplemented
22C3×#3724C2212yescomplemented
23C3×(C2×D3)⋊Dic3C2214yescomplemented
24C2×C3×D3×D4—34no—
25C2×#3811C2×C312yescomplemented
26C32×#121—32no—
27C2×#3752C2×C311yescharacteristic, complemented
28C3×#939—34no—
29C3×#937—34no—
30C3×#930—22no—
31C3×#930D312yescomplemented
32C3×#930C2×C312yescomplemented
33C2×C3×(C2×D3)⋊C4—32no—
34C2×C3×(C2×D3)⋊C4—61no—
35C3×#894—34no—
36C3×#883—34no—
37C3×#882—34no—
38#63980C2×C312yescomplemented
39#63922C2×C314yescomplemented
40#63854C2×C314yescomplemented
41C23×C33C2311yescharacteristic
42C22×C3×C3⋊D3—41no—
43C22×C3×C3⋊D3—41no—
44C22×C3×C3⋊D3D414yescomplemented
45C22×C3×C3⋊D3C2312yes—
46C22×C32×D3C2314yes—
47C2×C3×D32—24no—
48C2×C3×D32—42no—
49C2×C3×D32—42no—
50C2×C33×C4—41no—
51C2×C33×C4C2312yes—
52C2×C3×C3⋊Dic3—41no—
53C2×C3×C3⋊Dic3C2312yes—
54C32×#67—28no—
55C2×C32×Dic3C2314yes—
56C32×D12—42no—
57C3×#626—28no—
58C3×#625—42no—
59C3×#624—24no—
60C3×#534—91no—
61#5605—32no—
62C23×C3⋊D3C22×C311yescharacteristic
63C23×C3×D3—32no—
64C23×C3×D3—34no—
65C23×C3×D3—61no—
66C22×D32C22×C312yescomplemented
67C2×C32×D4—34no—
68C2×C32×D4—32no—
69C22×C32×C4—21no—
70C22×C32×C4—22no—
71C22×C32×C4—41no—
72C22×C32×C4C2×D312yescomplemented
73C22×C32×C4C22×C311yescharacteristic, complemented
74C22×C3⋊Dic3C22×C311yescharacteristic, complemented
75C2×C3×#67—24no—
76C2×C3×#67—34no—
77C2×C3×#67—34no—
78C2×C3×#67C2×D314yescomplemented
79C2×C3×#67C22×C314yescomplemented
80C22×C3×Dic3—32no—
81C22×C3×Dic3—61no—
82C3×D3×D4—616no—
83C2×C3×D12—22no—
84C2×C3×D12—34no—
85C2×C3×D12C2×D312yescomplemented
86C2×C3×D12C22×C312yescomplemented
87C2×C3×C4×D3—34no—
88#3811—28no—
89C2×#626C22×C314yes—
90C2×#625C22×C312yescomplemented
91C2×#624C22×C312yescomplemented
92C32×C4⋊C4—32no—
93C32×C22⋊C4—34no—
94#3752C22×C314yescomplemented
95C3×#217—34no—
96C3×(C2×D3)⋊C4—24no—
97C3×(C2×D3)⋊C4—34no—
98C3×(C2×D3)⋊C4—34no—
99C3×(C2×D3)⋊C4—34no—
100C3×(C2×D3)⋊C4—64no—
101C3×(C2×D3)⋊C4C2×D314yescomplemented
102C3×(C2×D3)⋊C4C22×C314yescomplemented
103C3×C4⋊Dic3—22no—
104C3×C4⋊Dic3C2×D312yescomplemented
105C3×C4⋊Dic3C22×C312yescomplemented
106C3×Dic3⋊C4—34no—
107#3724C22×C312yescomplemented
108(C2×D3)⋊Dic3C22×C314yescomplemented
109C22×C33—21no—
110C22×C33—21no—
111C22×C33C2×D412yes—
112C22×C33C2411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—44no—
116C2×C3×C3⋊D3C2×D414yes—
117C2×C32×D3—24no—
118C2×C32×D3—24no—
119C2×C32×D3—24no—
120C3×D32—48no—
121C33×C4—22no—
122C3×C3⋊Dic3—22no—
123C32×Dic3—24no—
124C22×C3×D4—92no—
125C2×D3×D4—34no—
126C3×#123—94no—
127C3×#121—32no—
128C3×#121—32no—
129C3×#121—62no—
130C3×#121—92no—
131C3×#120—94no—
132C2×C3×C22⋊C4—91no—
133#939—34no—
134#937—34no—
135#930C3×D312yescomplemented
136C2×(C2×D3)⋊C4—32no—
137C2×(C2×D3)⋊C4—61no—
138#894—34no—
139#883—34no—
140#882—34no—
141C23×C32—21no—
142C23×C32—22no—
143C23×C32—34no—
144C23×C32—41no—
145C23×C32C22×D312yes—
146C23×C32C23×C311yescharacteristic
147C22×C3⋊D3—41no—
148C22×C3⋊D3—41no—
149C22×C3⋊D3C3×D414yescomplemented
150C22×C3⋊D3C23×C312yes—
151C22×C3×D3—24no—
152C22×C3×D3—34no—
153C22×C3×D3—38no—
154C22×C3×D3—62no—
155C22×C3×D3—64no—
156C22×C3×D3—68no—
157C22×C3×D3—124no—
158C22×C3×D3—124no—
159C22×C3×D3—122no—
160C22×C3×D3—122no—
161C22×C3×D3—122no—
162C22×C3×D3—122no—
163C22×C3×D3C22×D314yes—
164C22×C3×D3C23×C314yes—
165C2×D32—24no—
166C2×D32—42no—
167C2×D32—42no—
168C32×D4—68no—
169C32×D4—122no—
170C2×C32×C4—22no—
171C2×C32×C4—22no—
172C2×C32×C4—22no—
173C2×C32×C4—34no—
174C2×C32×C4—42no—
175C2×C32×C4—42no—
176C2×C32×C4—41no—
177C2×C32×C4—44no—
178C2×C32×C4—42no—
179C2×C32×C4—44no—
180C2×C32×C4C22×D312yes—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C23×C312yes—
183C2×C3⋊Dic3—41no—
184C2×C3⋊Dic3C23×C312yes—
185C3×#67—28no—
186C3×#67—28no—
187C3×#67—48no—
188C3×#67—68no—
189C3×#67—124no—
190C2×C3×Dic3—24no—
191C2×C3×Dic3—34no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3—122no—
194C2×C3×Dic3—122no—
195C2×C3×Dic3C22×D314yes—
196C2×C3×Dic3C23×C314yes—
197C3×D12—42no—
198C3×D12—42no—
199C3×D12—44no—
200C3×D12—68no—
201C3×C4×D3—68no—
202#626—28no—
203#625—42no—
204#624—24no—
205#534—91noSylow
206C2×C33—22no—
207C2×C33C22×D411yescharacteristic
208C2×C33H(5,𝔽2)11yescharacteristic
209C2×C33H(5,𝔽2)11yescharacteristic
210C3×C3⋊D3—24no—
211C32×D3—44no—
212C24×C3—92no—
213C24×C3—91no—
214C23×D3—32no—
215C23×D3—34no—
216C23×D3—61no—
217C2×C3×D4—34no—
218C2×C3×D4—34no—
219C2×C3×D4—32no—
220C2×C3×D4—32no—
221C2×C3×D4—62no—
222C2×C3×D4—64no—
223C2×C3×D4—94no—
224C2×C3×D4—92no—
225C2×C3×D4—188no—
226C22×C3×C4—21no—
227C22×C3×C4—22no—
228C22×C3×C4—41no—
229C22×C3×C4—92no—
230C22×C3×C4—91no—
231C22×C3×C4D3211yescharacteristic, complemented
232C22×C3×C4C2×C3×D312yescomplemented
233C2×#67—34no—
234C2×#67—34no—
235C2×#67C2×C3×D314yescomplemented
236C22×Dic3—32no—
237C22×Dic3—61no—
238D3×D4—616no—
239C2×D12—34no—
240C2×D12C2×C3×D312yescomplemented
241C2×C4×D3—34no—
242C3×C4⋊C4—32no—
243C3×C4⋊C4—32no—
244C3×C4⋊C4—62no—
245C3×C4⋊C4—92no—
246C3×C22⋊C4—34no—
247C3×C22⋊C4—34no—
248C3×C22⋊C4—64no—
249C3×C22⋊C4—94no—
250C3×C22⋊C4—94no—
251#217—34no—
252(C2×D3)⋊C4—34no—
253(C2×D3)⋊C4—34no—
254(C2×D3)⋊C4—34no—
255(C2×D3)⋊C4—64no—
256(C2×D3)⋊C4C2×C3×D314yescomplemented
257C4⋊Dic3C2×C3×D312yescomplemented
258Dic3⋊C4—34no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—21no—
266C22×C32—22no—
267C22×C32—22no—
268C22×C32—24no—
269C22×C32—42no—
270C22×C32—41no—
271C22×C32—42no—
272C22×C32—42no—
273C22×C32—42no—
274C22×C32—42no—
275C22×C32—64no—
276C22×C32—64no—
277C22×C32—64no—
278C22×C32D3×D414yes—
279C22×C32C2×C3×D412yes—
280C22×C32C23×D312yes—
281C22×C32C24×C311yescharacteristic
282C2×C3⋊D3—22no—
283C2×C3⋊D3—22no—
284C2×C3⋊D3—44no—
285C2×C3⋊D3C2×C3×D414yes—
286C2×C3×D3—24no—
287C2×C3×D3—24no—
288C2×C3×D3—24no—
289C2×C3×D3—24no—
290C2×C3×D3—24no—
291C2×C3×D3—24no—
292C2×C3×D3—38no—
293C2×C3×D3—44no—
294C2×C3×D3—44no—
295C2×C3×D3—44no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—122no—
300C2×C3×D3—122no—
301C2×C3×D3—1216no—
302C2×C3×D3—128no—
303C2×C3×D3—128no—
304D32—48no—
305C32×C4—22no—
306C32×C4—24no—
307C32×C4—22no—
308C32×C4—22no—
309C32×C4—42no—
310C32×C4—42no—
311C32×C4—44no—
312C32×C4—64no—
313C3⋊Dic3—22no—
314C3×Dic3—24no—
315C3×Dic3—24no—
316C3×Dic3—44no—
317C3×Dic3—64no—
318C3×Dic3—122no—
319C22×D4—92no—
320#123—94no—
321#121—32no—
322#121—92no—
323#120—94no—
324C2×C22⋊C4—91no—
325C33#53411yescharacteristic, complemented, Sylow
326C23×C3—21no—
327C23×C3—22no—
328C23×C3—34no—
329C23×C3—34no—
330C23×C3—41no—
331C23×C3—64no—
332C23×C3—92no—
333C23×C3—94no—
334C23×C3—184no—
335C23×C3—361no—
336C23×C3—361no—
337C23×C3—362no—
338C23×C3—362no—
339C23×C3C2×D3211yescharacteristic
340C23×C3C22×C3×D312yes—
341C22×D3—34no—
342C22×D3—38no—
343C22×D3—64no—
344C22×D3—62no—
345C22×D3—68no—
346C22×D3—124no—
347C22×D3—124no—
348C22×D3—122no—
349C22×D3—122no—
350C22×D3—122no—
351C22×D3—122no—
352C22×D3C22×C3×D314yes—
353C3×D4—68no—
354C3×D4—68no—
355C3×D4—128no—
356C3×D4—122no—
357C3×D4—122no—
358C3×D4—124no—
359C3×D4—188no—
360C3×D4—362no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—34no—
365C2×C3×C4—34no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—44no—
369C2×C3×C4—42no—
370C2×C3×C4—44no—
371C2×C3×C4—42no—
372C2×C3×C4—64no—
373C2×C3×C4—92no—
374C2×C3×C4—184no—
375C2×C3×C4—361no—
376C2×C3×C4C2×D3212yes—
377C2×C3×C4C22×C3×D312yes—
378C2×C3×C4C22×C3×D312yes—
379#67—28no—
380#67—68no—
381#67—124no—
382C2×Dic3—34no—
383C2×Dic3—62no—
384C2×Dic3—122no—
385C2×Dic3—122no—
386C2×Dic3C22×C3×D314yes—
387D12—42no—
388D12—68no—
389C4×D3—68no—
390C2×C32—22no—
391C2×C32—24no—
392C2×C32—24no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—22no—
397C2×C32—22no—
398C2×C32—22no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—44no—
403C2×C32—44no—
404C2×C32—124no—
405C2×C32C2×D3×D412yes—
406C2×C32#100412yes—
407C2×C32#100912yes—
408C2×C32C22×C3×D411yescharacteristic
409C2×C32C3×H(5,𝔽2)11yescharacteristic
410C2×C32C3×H(5,𝔽2)11yescharacteristic
411C3⋊D3—24no—
412C3×D3—44no—
413C3×D3—44no—
414C3×D3—68no—
415C3×D3—84no—
416C3×D3—124no—
417C24—92no—
418C24—91no—
419C2×D4—34no—
420C2×D4—32no—
421C2×D4—94no—
422C2×D4—92no—
423C2×D4—188no—
424C22×C4—92no—
425C22×C4—91no—
426C22×C4C3×D3211yescharacteristic, complemented
427C4⋊C4—32no—
428C4⋊C4—92no—
429C22⋊C4—34no—
430C22⋊C4—94no—
431C22⋊C4—94no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—21no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—24no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—41no—
447C22×C3—42no—
448C22×C3—64no—
449C22×C3—64no—
450C22×C3—64no—
451C22×C3—64no—
452C22×C3—64no—
453C22×C3—64no—
454C22×C3—94no—
455C22×C3—124no—
456C22×C3—124no—
457C22×C3—124no—
458C22×C3—182no—
459C22×C3—182no—
460C22×C3—364no—
461C22×C3—368no—
462C22×C3#380212yes—
463C22×C3C3×D3×D414yes—
464C22×C3C22×D3211yescharacteristic
465C22×C3C23×C3×D312yes—
466C2×D3—24no—
467C2×D3—24no—
468C2×D3—24no—
469C2×D3—38no—
470C2×D3—64no—
471C2×D3—64no—
472C2×D3—64no—
473C2×D3—122no—
474C2×D3—122no—
475C2×D3—1216no—
476C2×D3—128no—
477C2×D3—128no—
478C3×C4—24no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—44no—
485C3×C4—64no—
486C3×C4—64no—
487C3×C4—124no—
488C3×C4—182no—
489Dic3—24no—
490Dic3—64no—
491Dic3—122no—
492C32—21no—
493C32—22no—
494C32—41no—
495C32#560512yescomplemented
496C32C3×#53411yescharacteristic, complemented
497C23—34no—
498C23—92no—
499C23—94no—
500C23—184no—
501C23—361no—
502C23—361no—
503C23—362no—
504C23—362no—
505C23C2×C3×D3211yescharacteristic
506D4—68no—
507D4—122no—
508D4—188no—
509D4—362no—
510C2×C4—34no—
511C2×C4—41no—
512C2×C4—92no—
513C2×C4—184no—
514C2×C4—361no—
515C2×C4C2×C3×D3212yes—
516C2×C3—24no—
517C2×C3—22no—
518C2×C3—24no—
519C2×C3—22no—
520C2×C3—21no—
521C2×C3—22no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—22no—
525C2×C3—44no—
526C2×C3—44no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—124no—
531C2×C3—124no—
532C2×C3—184no—
533C2×C3—244no—
534C2×C3C2×#380211yescharacteristic
535C2×C3#6430811yescharacteristic
536C2×C3#6433011yescharacteristic
537C2×C3C2×C3×D3×D412yes—
538C2×C3C3×#100412yes—
539C2×C3C3×#100912yes—
540D3—44no—
541D3—68no—
542D3—124no—
543C22—21no—
544C22—21no—
545C22—64no—
546C22—64no—
547C22—64no—
548C22—94no—
549C22—182no—
550C22—182no—
551C22—364no—
552C22—368no—
553C22C3×#380212yes—
554C22C22×C3×D3211yescharacteristic
555C4—22no—
556C4—64no—
557C4—182no—
558C3—21no—
559C3—22no—
560C3—41no—
561C3#1059508411yescharacteristic, direct product factor
562C3C3×#560512yescomplemented
563C2—22no—
564C2—124no—
565C2—184no—
566C2C2×C3×#380211yescharacteristic
567C2C3×#6430811yescharacteristic
568C2C3×#6433011yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
hypercenterC22×C3×C4D32
hypocenterC32C3×#534
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3811C2semidirect
2. C2×C3×#3752C2semidirect
3. C32×#930C2semidirect
4. C3×#63980C2semidirect
5. C3×#63922C2semidirect
6. C3×#63854C2semidirect
8. #10595084C3direct (N × H)
10. C22×C3×D32C22semidirect
11. C22×C33×C4C22semidirect
12. C22×C3×C3⋊Dic3C22semidirect
13. C2×C32×#67C22semidirect
14. C2×C32×D12C22semidirect
17. C2×C3×#625C22semidirect
18. C2×C3×#624C22semidirect
19. C3×#3752C22semidirect
20. C32×(C2×D3)⋊C4C22semidirect
21. C32×C4⋊Dic3C22semidirect
22. C3×#3724C22semidirect
23. C3×(C2×D3)⋊Dic3C22semidirect
25. C2×#3811C2×C3semidirect
27. C2×#3752C2×C3semidirect
31. C3×#930D3semidirect
32. C3×#930C2×C3semidirect
38. #63980C2×C3semidirect
39. #63922C2×C3semidirect
40. #63854C2×C3semidirect
44. C22×C3×C3⋊D3D4semidirect
66. C22×D32C22×C3semidirect
72. C22×C32×C4C2×D3semidirect
73. C22×C32×C4C22×C3semidirect
74. C22×C3⋊Dic3C22×C3semidirect
78. C2×C3×#67C2×D3semidirect
79. C2×C3×#67C22×C3semidirect
85. C2×C3×D12C2×D3semidirect
86. C2×C3×D12C22×C3semidirect
90. C2×#625C22×C3semidirect
91. C2×#624C22×C3semidirect
94. #3752C22×C3semidirect
101. C3×(C2×D3)⋊C4C2×D3semidirect
102. C3×(C2×D3)⋊C4C22×C3semidirect
104. C3×C4⋊Dic3C2×D3semidirect
105. C3×C4⋊Dic3C22×C3semidirect
107. #3724C22×C3semidirect
108. (C2×D3)⋊Dic3C22×C3semidirect
135. #930C3×D3semidirect
149. C22×C3⋊D3C3×D4semidirect
231. C22×C3×C4D32semidirect
232. C22×C3×C4C2×C3×D3semidirect
235. C2×#67C2×C3×D3semidirect
240. C2×D12C2×C3×D3semidirect
256. (C2×D3)⋊C4C2×C3×D3semidirect
257. C4⋊Dic3C2×C3×D3semidirect
325. C33#534semidirect
426. C22×C4C3×D32semidirect
495. C32#5605semidirect
496. C32C3×#534semidirect
561. C3#10595084direct (N × H)
562. C3C3×#5605semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38112
C2×C3×#37522
C32×#9302
C3×#639802
C3×#639222
C3×#638542
C3×#56053
#105950843