Finite Group Databasebeta

← C3×#10595134 #13174776

Subgroups — C3×#10595134 #13174776

Order 1728.

1504 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3805C212yesmaximal, complemented
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C32×#938C212yesmaximal, complemented
4C3×#63982C212yesmaximal, complemented
5C3×#63969C214yesmaximal, complemented
6C3×#63919C211yesmaximal, characteristic, complemented
7C3×C4×#625C211yesmaximal, characteristic, complemented
8C3×D3×C4⋊Dic3C212yesmaximal, complemented
9C3×#5611—32nomaximal
10#10595134C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3C2212yes—
13C2×C32×#67C2214yes—
14C2×C32×C4×D3C2212yes—
15C2×C3×#627C2212yes—
16C2×C3×#625C2211yescharacteristic
17C3×#3805—28no—
18C2×C3×D3×Dic3C2214yes—
19C3×#3757C2212yes—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C32×#217C2214yes—
23C32×C4⋊Dic3C2212yes—
24C3×#3724C2212yes—
25C3×#3724C2211yescharacteristic
26C3×Dic3⋊Dic3C2214yes—
27C3×(C2×D3)⋊Dic3C2212yes—
28C2×C3×#237—34no—
29C2×#3805C2×C312yescomplemented
30C32×#121—32no—
31D4×C3⋊Dic3C2×C311yescharacteristic, complemented
32C3×#938—22no—
33C3×#938D312yescomplemented
34C3×#938C2×C312yescomplemented
35C3×#935—34no—
36C3×D4×Dic3—32no—
37C3×D4×Dic3—61no—
38C3×C4×#67—32no—
39C2×C3×C4⋊Dic3—32no—
40C3×D3×C4⋊C4—32no—
41C3×#888—32no—
42C3×#883—34no—
43C3×#878—34no—
44#63982C2×C312yescomplemented
45#63969C2×C314yescomplemented
46#63919C2×C311yescharacteristic, complemented
47C4×#625C2×C311yescharacteristic, complemented
48D3×C4⋊Dic3C2×C312yescomplemented
49C23×C33C2312yes—
50C22×C32×D3C2312yes—
51C33×D4—22no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—24no—
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C2×C3×C3⋊Dic3C2312yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C32×#67—28no—
58C2×C32×Dic3C2312yes—
59C2×C32×Dic3C2314yes—
60C32×C4×D3—24no—
61C3×#627—24no—
62C3×#625D414yescomplemented
63C3×D3×Dic3—28no—
64C3×#547—91no—
65#5611—32no—
66C2×C32×D4—21no—
67C2×C32×D4—22no—
68C2×C32×D4—34no—
69C2×C32×D4—41no—
70C2×C32×D4C2×D312yes—
71C2×C32×D4C22×C311yescharacteristic
72C22×C32×C4—32no—
73C22×C3⋊Dic3C22×C312yes—
74C2×C3×#67—24no—
75C2×C3×#67—32no—
76C2×C3×#67C2×D314yes—
77C2×C3×#67C22×C314yes—
78C22×C3×Dic3—34no—
79C22×C3×Dic3—34no—
80C22×C3×Dic3—62no—
81C3×#237—616no—
82C2×C3×C4×D3—22no—
83C2×C3×C4×D3—34no—
84C2×C3×C4×D3C2×D312yes—
85C2×C3×C4×D3C22×C312yes—
86C2×C3×Dic6—34no—
87C2×#627C22×C312yes—
88C2×#625C22×C311yescharacteristic
89#3805—28no—
90C2×D3×Dic3C22×C314yes—
91C32×C4⋊C4—32no—
92C32×C22⋊C4—34no—
93#3757C22×C312yes—
94C4⋊(C3⋊Dic3)C22×C311yescharacteristic
95C4×C3⋊Dic3C22×C311yescharacteristic
96C3×#217—24no—
97C3×#217—34no—
98C3×#217—32no—
99C3×#217—62no—
100C3×#217C2×D314yes—
101C3×#217C22×C314yes—
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3—22no—
104C3×C4⋊Dic3—32no—
105C3×C4⋊Dic3—34no—
106C3×C4⋊Dic3—61no—
107C3×C4⋊Dic3C2×D312yes—
108C3×C4⋊Dic3C22×C312yes—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—34no—
111C3×Dic3⋊C4—34no—
112C3×C4×Dic3—32no—
113C3×C4×Dic3—61no—
114#3724C22×C311yescharacteristic
115#3724C22×C312yes—
116Dic3⋊Dic3C22×C314yes—
117(C2×D3)⋊Dic3C22×C312yes—
118C22×C33—22no—
119C22×C33—22no—
120C22×C33—22no—
121C22×C33C2411yescharacteristic
122C2×C32×D3—22no—
123C2×C32×D3—22no—
124C2×C32×D3C2×D414yes—
125C33×C4#4112yescomplemented
126C3×C3⋊Dic3—22no—
127C3×C3⋊Dic3—21no—
128C3×C3⋊Dic3C2×D412yes—
129C32×Dic3—24no—
130C32×Dic3—22no—
131C2×C3×#41—92no—
132C2×#237—34no—
133C3×C4⋊Q8—91no—
134C3×#123—94no—
135C3×C22⋊Q8—92no—
136C3×#121—32no—
137C3×#121—32no—
138C3×#121—62no—
139C3×C4×D4—91no—
140C3×C4×D4—91no—
141C2×C3×C4⋊C4—92no—
142#938C3×D312yescomplemented
143#935—34no—
144D4×Dic3—32no—
145D4×Dic3—61no—
146C4×#67—32no—
147C2×C4⋊Dic3—32no—
148D3×C4⋊C4—32no—
149#888—32no—
150#883—34no—
151#878—34no—
152C23×C32—22no—
153C23×C32—24no—
154C23×C32—32no—
155C23×C32—42no—
156C23×C32C22×D314yes—
157C23×C32C23×C312yes—
158C22×C3×D3—22no—
159C22×C3×D3C22×D312yes—
160C22×C3×D3C23×C312yes—
161C32×D4—22no—
162C32×D4—24no—
163C32×D4—24no—
164C32×D4—44no—
165C32×D4—44no—
166C32×D4—68no—
167C2×C32×C4—21no—
168C2×C32×C4—22no—
169C2×C32×C4—34no—
170C2×C32×C4—32no—
171C2×C32×C4—41no—
172C2×C32×C4—64no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C23×C311yescharacteristic
175C2×C3⋊Dic3—24no—
176C2×C3⋊Dic3C23×C311yescharacteristic
177C2×C3⋊Dic3C23×C312yes—
178C2×C3⋊Dic3C23×C311yescharacteristic
179C3×#67—28no—
180C3×#67—28no—
181C3×#67—38no—
182C3×#67—48no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—68no—
192C2×C3×Dic3—68no—
193C2×C3×Dic3—124no—
194C2×C3×Dic3C22×D312yes—
195C2×C3×Dic3C22×D314yes—
196C2×C3×Dic3C23×C312yes—
197C2×C3×Dic3C23×C314yes—
198C3×C4×D3—24no—
199C3×C4×D3—24no—
200C3×C4×D3—44no—
201C3×C4×D3—68no—
202C3×Dic6—68no—
203#627—24no—
204#625C3×D414yescomplemented
205D3×Dic3—28no—
206#547—91noSylow
207C2×C33—42no—
208C2×C33C22×D411yescharacteristic
209C2×C33C2×#4111yescharacteristic
210C2×C332−1+411yescharacteristic
211C32×D3—24no—
212C3×#41—188no—
213C2×C3×Q8—92no—
214C2×C3×D4—21no—
215C2×C3×D4—22no—
216C2×C3×D4—34no—
217C2×C3×D4—34no—
218C2×C3×D4—41no—
219C2×C3×D4—64no—
220C2×C3×D4—91no—
221C2×C3×D4D3211yescharacteristic
222C2×C3×D4C2×C3×D312yes—
223C22×C3×C4—32no—
224C22×C3×C4—32no—
225C22×C3×C4—62no—
226C22×C3×C4—94no—
227C22×C3×C4—92no—
228C2×#67—32no—
229C2×#67C2×C3×D314yes—
230C22×Dic3—34no—
231C22×Dic3—34no—
232C22×Dic3—62no—
233#237—616no—
234C2×C4×D3—34no—
235C2×C4×D3C2×C3×D312yes—
236C2×Dic6—34no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—62no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—94no—
243C3×C4⋊C4—91no—
244C3×C22⋊C4—34no—
245C3×C22⋊C4—34no—
246C3×C22⋊C4—64no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C42—91no—
250#217—34no—
251#217—32no—
252#217—62no—
253#217C2×C3×D314yes—
254(C2×D3)⋊C4—32no—
255C4⋊Dic3—32no—
256C4⋊Dic3—34no—
257C4⋊Dic3—61no—
258C4⋊Dic3C2×C3×D312yes—
259Dic3⋊C4—32no—
260Dic3⋊C4—34no—
261Dic3⋊C4—34no—
262C4×Dic3—32no—
263C4×Dic3—61no—
264C22×C32—22no—
265C22×C32—24no—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—24no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—34no—
274C22×C32—41no—
275C22×C32—44no—
276C22×C32—44no—
277C22×C32—42no—
278C22×C32—42no—
279C22×C32—44no—
280C22×C32—44no—
281C22×C32—62no—
282C22×C32—62no—
283C22×C32—82no—
284C22×C32—82no—
285C22×C32C23×D312yes—
286C22×C32C24×C311yescharacteristic
287C2×C3×D3—22no—
288C2×C3×D3—22no—
289C2×C3×D3—22no—
290C2×C3×D3—22no—
291C2×C3×D3—24no—
292C2×C3×D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3D3×D414yes—
295C2×C3×D3C2×C3×D414yes—
296C32×C4—22no—
297C32×C4—24no—
298C32×C4—42no—
299C32×C4—64no—
300C32×C4—62no—
301C32×C4#23714yescomplemented
302C32×C4C3×#4112yescomplemented
303C3⋊Dic3—22no—
304C3⋊Dic3—21no—
305C3⋊Dic3C2×C3×D412yes—
306C3×Dic3—24no—
307C3×Dic3—22no—
308C3×Dic3—24no—
309C3×Dic3—22no—
310C3×Dic3—34no—
311C3×Dic3—42no—
312C3×Dic3—44no—
313C3×Dic3—62no—
314C3×Dic3—64no—
315C3×Dic3—62no—
316C3×Dic3—121no—
317C3×Dic3—122no—
318C2×#41—92no—
319C4⋊Q8—91no—
320#123—94no—
321C22⋊Q8—92no—
322#121—32no—
323C4×D4—91no—
324C4×D4—91no—
325C2×C4⋊C4—92no—
326C33#54711yescharacteristic, complemented, Sylow
327C23×C3—22no—
328C23×C3—24no—
329C23×C3—32no—
330C23×C3—32no—
331C23×C3—42no—
332C23×C3—62no—
333C23×C3C2×D3212yes—
334C23×C3C22×C3×D314yes—
335C22×D3C22×C3×D312yes—
336C3×Q8—184no—
337C3×D4—24no—
338C3×D4—22no—
339C3×D4—24no—
340C3×D4—44no—
341C3×D4—44no—
342C3×D4—68no—
343C3×D4—68no—
344C3×D4—94no—
345C3×D4—128no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—32no—
349C2×C3×C4—34no—
350C2×C3×C4—34no—
351C2×C3×C4—32no—
352C2×C3×C4—41no—
353C2×C3×C4—62no—
354C2×C3×C4—64no—
355C2×C3×C4—64no—
356C2×C3×C4—64no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—92no—
360C2×C3×C4—124no—
361C2×C3×C4—188no—
362C2×C3×C4—184no—
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C22×C3×D312yes—
365#67—28no—
366#67—38no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—34no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—62no—
373C2×Dic3—68no—
374C2×Dic3—68no—
375C2×Dic3—124no—
376C2×Dic3C22×C3×D312yes—
377C2×Dic3C22×C3×D314yes—
378C4×D3—24no—
379C4×D3—68no—
380Dic6—68no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—22no—
385C2×C32—22no—
386C2×C32—22no—
387C2×C32—41no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—42no—
391C2×C32—44no—
392C2×C32—44no—
393C2×C32—64no—
394C2×C32—84no—
395C2×C32—84no—
396C2×C32C2×D3×D412yes—
397C2×C32C2×#23712yes—
398C2×C32#101012yes—
399C2×C32C22×C3×D411yescharacteristic
400C2×C32C2×C3×#4111yescharacteristic
401C2×C32C3×2−1+411yescharacteristic
402C3×D3—24no—
403C3×D3—24no—
404C3×D3—44no—
405#41—188no—
406C2×Q8—92no—
407C2×D4—34no—
408C2×D4—91no—
409C2×D4C3×D3211yescharacteristic
410C22×C4—32no—
411C22×C4—94no—
412C22×C4—92no—
413C4⋊C4—32no—
414C4⋊C4—94no—
415C4⋊C4—91no—
416C4⋊C4—92no—
417C4⋊C4—91no—
418C22⋊C4—34no—
419C22⋊C4—92no—
420C22⋊C4—92no—
421C42—91no—
422C22×C3—24no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—24no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—21no—
430C22×C3—22no—
431C22×C3—34no—
432C22×C3—34no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3—44no—
439C22×C3—41no—
440C22×C3—64no—
441C22×C3—62no—
442C22×C3—62no—
443C22×C3—62no—
444C22×C3—62no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3—122no—
448C22×C3—122no—
449C22×C3C22×D3211yescharacteristic
450C22×C3C23×C3×D312yes—
451C2×D3—22no—
452C2×D3—22no—
453C2×D3C3×D3×D414yes—
454C3×C4—24no—
455C3×C4—22no—
456C3×C4—42no—
457C3×C4—64no—
458C3×C4—62no—
459C3×C4—64no—
460C3×C4—62no—
461C3×C4—92no—
462C3×C4—122no—
463C3×C4—124no—
464C3×C4—181no—
465C3×C4—182no—
466C3×C4#380512yescomplemented
467C3×C4C3×#23714yescomplemented
468Dic3—24no—
469Dic3—22no—
470Dic3—34no—
471Dic3—62no—
472Dic3—64no—
473Dic3—62no—
474Dic3—122no—
475Dic3—121no—
476C32—21no—
477C32—22no—
478C32—41no—
479C32#561112yescomplemented
480C32C3×#54711yescharacteristic, complemented
481C23—32no—
482C23C2×C3×D3212yes—
483Q8—184no—
484D4—22no—
485D4—68no—
486D4—94no—
487C2×C4—32no—
488C2×C4—34no—
489C2×C4—64no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—92no—
493C2×C4—188no—
494C2×C4—184no—
495C2×C4C2×C3×D3211yescharacteristic
496C2×C3—22no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—21no—
501C2×C3—22no—
502C2×C3—44no—
503C2×C3—42no—
504C2×C3—44no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—64no—
509C2×C3—64no—
510C2×C3—84no—
511C2×C3—84no—
512C2×C3—124no—
513C2×C3#6430911yescharacteristic
514C2×C3D32×D411yescharacteristic
515C2×C3C2×#380511yescharacteristic
516C2×C3C2×C3×D3×D412yes—
517C2×C3C2×C3×#23712yes—
518C2×C3C3×#101012yes—
519D3—24no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—34no—
524C22—62no—
525C22—62no—
526C22C22×C3×D3211yescharacteristic
527C4—62no—
528C4—64no—
529C4—92no—
530C4—181no—
531C4—182no—
532C4C3×#380512yescomplemented
533C3—21no—
534C3—22no—
535C3—41no—
536C3#1059513411yescharacteristic, direct product factor
537C3C3×#561112yescomplemented
538C2—42no—
539C2—64no—
540C2C3×#6430911yescharacteristic
541C2C3×D32×D411yescharacteristic
542C2C2×C3×#380511yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
hypercenterC2×C3×D4D32
hypocenterC32C3×#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3805C2semidirect
2. C3×D4×C3⋊Dic3C2semidirect
3. C32×#938C2semidirect
4. C3×#63982C2semidirect
5. C3×#63969C2semidirect
6. C3×#63919C2semidirect
7. C3×C4×#625C2semidirect
8. C3×D3×C4⋊Dic3C2semidirect
10. #10595134C3direct (N × H)
29. C2×#3805C2×C3semidirect
31. D4×C3⋊Dic3C2×C3semidirect
33. C3×#938D3semidirect
34. C3×#938C2×C3semidirect
44. #63982C2×C3semidirect
45. #63969C2×C3semidirect
46. #63919C2×C3semidirect
47. C4×#625C2×C3semidirect
48. D3×C4⋊Dic3C2×C3semidirect
62. C3×#625D4semidirect
125. C33×C4#41semidirect
142. #938C3×D3semidirect
204. #625C3×D4semidirect
301. C32×C4#237semidirect
302. C32×C4C3×#41semidirect
326. C33#547semidirect
466. C3×C4#3805semidirect
467. C3×C4C3×#237semidirect
479. C32#5611semidirect
480. C32C3×#547semidirect
532. C4C3×#3805semidirect
536. C3#10595134direct (N × H)
537. C3C3×#5611semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38052
C3×D4×C3⋊Dic32
C32×#9382
C3×#639822
C3×#639692
C3×#639192
C3×C4×#6252
C3×D3×C4⋊Dic32
C3×#56113
#105951343