Finite Group Databasebeta

← C3×#10595219 #13174764

Subgroups — C3×#10595219 #13174764

Order 1728.

1582 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C32×D4×Dic3C212yesmaximal, complemented
3C2×C3×#3724C212yesmaximal, complemented
4C3×#63957C212yesmaximal, complemented
5C3×#63952C214yesmaximal, complemented
6C3×#63900C211yesmaximal, characteristic, complemented
7C3×#63874C211yesmaximal, characteristic, complemented
8C3×#63844C212yesmaximal, complemented
9C3×#5546—32nomaximal
10#10595219C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638C2212yes—
13C22×C3×C3⋊Dic3C2212yes—
14C3×#3830C418yescomplemented
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C22×C32×Dic3C2214yes—
18C2×C3×#624C2212yes—
19C32×#217C2214yes—
20C32×C4⋊Dic3C2212yes—
21C32×C4×Dic3C2212yes—
22C3×#3724C2214yes—
23C3×#3724C2211yescharacteristic
24C3×#3724C2212yes—
25C3×#3724C2211yescharacteristic
26C3×#3722C2212yes—
27C3×Dic32C2214yes—
28C2×#3830C2×C311yescharacteristic
29C2×C3×#237—32no—
30C2×C3×#237—61no—
31C32×C4×D4—32no—
32C3×D4×Dic3—22no—
33C3×D4×Dic3D312yescomplemented
34C3×D4×Dic3C2×C312yescomplemented
35C3×C4×#67—34no—
36C2×C3×Dic3⋊C4—34no—
37C3×D3×C4⋊C4—32no—
38C3×#887—32no—
39C3×#67⋊C4—34no—
40C3×#877—34no—
41C3×#872—32no—
42C3×C4×Dic6—32no—
43C2×#3724C2×C312yescomplemented
44#63957C2×C312yescomplemented
45#63952C2×C314yescomplemented
46#63900C2×C311yescharacteristic, complemented
47#63874C2×C311yescharacteristic, complemented
48#63844C2×C312yescomplemented
49C23×C33C2312yes—
50C22×C3×C3⋊D3C2311yescharacteristic
51C33×D4C2×C414yescomplemented
52C2×C33×C4C2311yescharacteristic
53C3×#638C2×C418yescomplemented
54C2×C3×C3⋊Dic3C2×C418yescomplemented
55C2×C3×C3⋊Dic3C2312yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4×C3⋊D3C2×C414yescomplemented
58C3×#634C2×C414yescomplemented
59C2×C32×Dic3—44no—
60C2×C32×Dic3C2312yes—
61C2×C32×Dic3C2314yes—
62C2×C32×Dic3C2312yes—
63C3×#624—42no—
64C3×#520—91no—
65#5546—32no—
66C2×C32×D4—21no—
67C2×C32×D4—22no—
68C2×C32×D4—41no—
69C2×C32×D4C2×D312yes—
70C2×C32×D4C22×C311yescharacteristic
71C22×C32×C4—34no—
72C2×#638C22×C312yes—
73C22×C3⋊Dic3C22×C312yes—
74#3830C3×C418yescomplemented
75C2×C4×C3⋊D3C22×C311yescharacteristic
76C2×#634C22×C311yescharacteristic
77C2×C3×#67—34no—
78C2×C3×#67—62no—
79C22×C3×Dic3—24no—
80C22×C3×Dic3—34no—
81C22×C3×Dic3—62no—
82C22×C3×Dic3C2×D314yes—
83C22×C3×Dic3C22×C314yes—
84C3×#237—316no—
85C3×#237—68no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—34no—
88C2×C3×C4×D3—61no—
89C2×C3×Dic6—32no—
90C2×C3×Dic6—61no—
91C2×#624C22×C312yes—
92C32×C4⋊C4—32no—
93C32×C22⋊C4—34no—
94C32×C42—32no—
95C3×#217—24no—
96C3×#217C2×D314yes—
97C3×#217C22×C314yes—
98C3×(C2×D3)⋊C4—32no—
99C3×(C2×D3)⋊C4—32no—
100C3×C4⋊Dic3—22no—
101C3×C4⋊Dic3C2×D312yes—
102C3×C4⋊Dic3C22×C312yes—
103C3×Dic3⋊C4—32no—
104C3×Dic3⋊C4—34no—
105C3×Dic3⋊C4—34no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—34no—
108C3×C4×Dic3—22no—
109C3×C4×Dic3—34no—
110C3×C4×Dic3—34no—
111C3×C4×Dic3C2×D312yes—
112C3×C4×Dic3C22×C312yes—
113#3724C22×C314yes—
114#3724C22×C311yescharacteristic
115#3724C22×C311yescharacteristic
116#3724C22×C312yes—
117#3722C22×C312yes—
118Dic32C22×C314yes—
119C22×C33—22no—
120C22×C33—22no—
121C22×C33C22×C414yes—
122C22×C33C2411yescharacteristic
123C2×C3×C3⋊D3—21no—
124C2×C3×C3⋊D3—21no—
125C2×C3×C3⋊D3C22×C412yes—
126C33×C4C22×C412yes—
127C3×C3⋊Dic3C22×C414yes—
128C3×C3⋊Dic3C22×C412yes—
129C32×Dic3—22no—
130C32×Dic3—24no—
131C32×Dic3—22no—
132C2×C3×#41—91no—
133C2×#237—32no—
134C2×#237—61no—
135C3×C4×Q8—92no—
136C3×C4×D4—32no—
137C3×C4×D4—32no—
138C3×C4×D4—62no—
139C3×C4×D4—94no—
140C3×#117—91no—
141C3×#117—92no—
142C2×C3×C4⋊C4—91no—
143C2×C3×C4⋊C4—92no—
144D4×Dic3C3×D312yescomplemented
145C4×#67—34no—
146C2×Dic3⋊C4—34no—
147D3×C4⋊C4—32no—
148#887—32no—
149#67⋊C4—34no—
150#877—34no—
151#872—32no—
152C4×Dic6—32no—
153C23×C32—22no—
154C23×C32—24no—
155C23×C32—42no—
156C23×C32C22×D314yes—
157C23×C32C23×C312yes—
158C22×C3⋊D3C23×C311yescharacteristic
159C22×C3×D3—32no—
160C22×C3×D3—61no—
161C32×D4—24no—
162C32×D4—28no—
163C32×D4—44no—
164C32×D4C4×D318yescomplemented
165C32×D4C2×C3×C414yescomplemented
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4—32no—
169C2×C32×C4—34no—
170C2×C32×C4—32no—
171C2×C32×C4—41no—
172C2×C32×C4—124no—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C23×C311yescharacteristic
175#638C2×C3×C418yescomplemented
176C2×C3⋊Dic3C2×C3×C418yescomplemented
177C2×C3⋊Dic3C23×C312yes—
178C2×C3⋊Dic3C23×C311yescharacteristic
179C4×C3⋊D3C2×C3×C414yescomplemented
180#634C2×C3×C414yescomplemented
181C3×#67—316no—
182C3×#67—68no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—316no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—44no—
190C2×C3×Dic3—44no—
191C2×C3×Dic3—48no—
192C2×C3×Dic3—68no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3C22×D312yes—
196C2×C3×Dic3C22×D314yes—
197C2×C3×Dic3C22×D312yes—
198C2×C3×Dic3C23×C312yes—
199C2×C3×Dic3C23×C314yes—
200C2×C3×Dic3C23×C312yes—
201C3×C4×D3—38no—
202C3×C4×D3—64no—
203C3×C4×D3—124no—
204C3×Dic6—38no—
205C3×Dic6—64no—
206#624—42no—
207#520—91noSylow
208C2×C33—24no—
209C2×C33C23×C411yescharacteristic
210C2×C33H(5,𝔽2)11yescharacteristic
211C2×C332−1+411yescharacteristic
212C3×C3⋊D3—22no—
213C3×#41—98no—
214C2×C3×Q8—91no—
215C2×C3×D4—22no—
216C2×C3×D4—21no—
217C2×C3×D4—41no—
218C2×C3×D4—92no—
219C2×C3×D4D3211yescharacteristic
220C2×C3×D4C2×C3×D312yes—
221C22×C3×C4—34no—
222C22×C3×C4—34no—
223C22×C3×C4—64no—
224C22×C3×C4—92no—
225C22×C3×C4—92no—
226C22×C3×C4—91no—
227C2×#67—34no—
228C2×#67—62no—
229C22×Dic3—34no—
230C22×Dic3—62no—
231C22×Dic3C2×C3×D314yes—
232#237—316no—
233#237—68no—
234C2×C4×D3—32no—
235C2×C4×D3—34no—
236C2×C4×D3—61no—
237C2×Dic6—32no—
238C2×Dic6—61no—
239C3×C4⋊C4—32no—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—62no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—94no—
245C3×C4⋊C4—91no—
246C3×C22⋊C4—34no—
247C3×C22⋊C4—34no—
248C3×C22⋊C4—64no—
249C3×C22⋊C4—92no—
250C3×C42—32no—
251C3×C42—32no—
252C3×C42—62no—
253C3×C42—94no—
254#217C2×C3×D314yes—
255(C2×D3)⋊C4—32no—
256(C2×D3)⋊C4—32no—
257C4⋊Dic3C2×C3×D312yes—
258Dic3⋊C4—34no—
259Dic3⋊C4—32no—
260Dic3⋊C4—32no—
261Dic3⋊C4—34no—
262Dic3⋊C4—34no—
263C4×Dic3—34no—
264C4×Dic3—34no—
265C4×Dic3C2×C3×D312yes—
266C22×C32—24no—
267C22×C32—24no—
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—21no—
271C22×C32—22no—
272C22×C32—24no—
273C22×C32—28no—
274C22×C32—41no—
275C22×C32—44no—
276C22×C32—44no—
277C22×C32—42no—
278C22×C32—42no—
279C22×C32—44no—
280C22×C32—82no—
281C22×C32—82no—
282C22×C32C2×C4×D318yes—
283C22×C32C22×C3×C414yes—
284C22×C32C23×D312yes—
285C22×C32C24×C311yescharacteristic
286C2×C3⋊D3—21no—
287C2×C3⋊D3—21no—
288C2×C3⋊D3C22×C3×C412yes—
289C2×C3×D3—34no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—121no—
294C2×C3×D3—121no—
295C32×C4—22no—
296C32×C4—24no—
297C32×C4—42no—
298C32×C4—62no—
299C32×C4—64no—
300C32×C4—62no—
301C32×C4C2×C4×D314yes—
302C32×C4C22×C3×C412yes—
303C3⋊Dic3C22×C3×C414yes—
304C3⋊Dic3C22×C3×C412yes—
305C3×Dic3—24no—
306C3×Dic3—24no—
307C3×Dic3—22no—
308C3×Dic3—22no—
309C3×Dic3—24no—
310C3×Dic3—22no—
311C3×Dic3—22no—
312C3×Dic3—38no—
313C3×Dic3—34no—
314C3×Dic3—42no—
315C3×Dic3—44no—
316C3×Dic3—64no—
317C3×Dic3—62no—
318C2×#41—91no—
319C4×Q8—92no—
320C4×D4—32no—
321C4×D4—94no—
322#117—91no—
323#117—92no—
324C2×C4⋊C4—91no—
325C2×C4⋊C4—92no—
326C33#52011yescharacteristic, complemented, Sylow
327C23×C3—24no—
328C23×C3—22no—
329C23×C3—42no—
330C23×C3—91no—
331C23×C3C2×D3212yes—
332C23×C3C22×C3×D314yes—
333C22×D3—32no—
334C22×D3—61no—
335C3×Q8—94no—
336C3×D4—28no—
337C3×D4—24no—
338C3×D4—44no—
339C3×D4—98no—
340C3×D4#62414yescomplemented
341C3×D4C3×C4×D318yescomplemented
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—32no—
345C2×C3×C4—32no—
346C2×C3×C4—34no—
347C2×C3×C4—32no—
348C2×C3×C4—34no—
349C2×C3×C4—32no—
350C2×C3×C4—41no—
351C2×C3×C4—62no—
352C2×C3×C4—62no—
353C2×C3×C4—64no—
354C2×C3×C4—92no—
355C2×C3×C4—91no—
356C2×C3×C4—98no—
357C2×C3×C4—94no—
358C2×C3×C4—124no—
359C2×C3×C4—124no—
360C2×C3×C4—128no—
361C2×C3×C4—362no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C22×C3×D312yes—
364#67—316no—
365#67—68no—
366C2×Dic3—34no—
367C2×Dic3—32no—
368C2×Dic3—316no—
369C2×Dic3—44no—
370C2×Dic3—68no—
371C2×Dic3—62no—
372C2×Dic3—61no—
373C2×Dic3C22×C3×D312yes—
374C2×Dic3C22×C3×D314yes—
375C2×Dic3C22×C3×D312yes—
376C4×D3—38no—
377C4×D3—64no—
378C4×D3—124no—
379Dic6—38no—
380Dic6—64no—
381C2×C32—28no—
382C2×C32—24no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—22no—
387C2×C32—22no—
388C2×C32—22no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—48no—
393C2×C32—44no—
394C2×C32—84no—
395C2×C32C22×C4×D312yes—
396C2×C32#100412yes—
397C2×C32#101012yes—
398C2×C32C23×C3×C411yescharacteristic
399C2×C32C3×H(5,𝔽2)11yescharacteristic
400C2×C32C3×2−1+411yescharacteristic
401C3⋊D3—22no—
402C3×D3—64no—
403C3×D3—122no—
404#41—98no—
405C2×Q8—91no—
406C2×D4—92no—
407C2×D4C3×D3211yescharacteristic
408C22×C4—34no—
409C22×C4—92no—
410C22×C4—92no—
411C22×C4—91no—
412C4⋊C4—32no—
413C4⋊C4—91no—
414C4⋊C4—92no—
415C4⋊C4—94no—
416C4⋊C4—91no—
417C22⋊C4—34no—
418C22⋊C4—92no—
419C42—32no—
420C42—94no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—21no—
426C22×C3—22no—
427C22×C3—24no—
428C22×C3—28no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—44no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3—92no—
438C22×C3—181no—
439C22×C3—181no—
440C22×C3C2×#62414yes—
441C22×C3C2×C3×C4×D318yes—
442C22×C3C22×D3211yescharacteristic
443C22×C3C23×C3×D312yes—
444C2×D3—34no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—62no—
448C2×D3—121no—
449C2×D3—121no—
450C3×C4—22no—
451C3×C4—24no—
452C3×C4—42no—
453C3×C4—64no—
454C3×C4—62no—
455C3×C4—62no—
456C3×C4—64no—
457C3×C4—64no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—92no—
461C3×C4—94no—
462C3×C4—124no—
463C3×C4—122no—
464C3×C4C2×#62412yes—
465C3×C4C2×C3×C4×D314yes—
466Dic3—24no—
467Dic3—22no—
468Dic3—22no—
469Dic3—38no—
470Dic3—34no—
471Dic3—64no—
472Dic3—62no—
473C32—22no—
474C32—21no—
475C32—41no—
476C32#554612yescomplemented
477C32C3×#52011yescharacteristic, complemented
478C23—91no—
479C23C2×C3×D3212yes—
480Q8—94no—
481D4—98no—
482D4C3×#62414yescomplemented
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—34no—
486C2×C4—92no—
487C2×C4—91no—
488C2×C4—98no—
489C2×C4—94no—
490C2×C4—124no—
491C2×C4—362no—
492C2×C4C2×C3×D3211yescharacteristic
493C2×C3—28no—
494C2×C3—24no—
495C2×C3—21no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—21no—
501C2×C3—44no—
502C2×C3—48no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—84no—
507C2×C3—182no—
508C2×C3#6430911yescharacteristic
509C2×C3#6431311yescharacteristic
510C2×C3C22×#62411yescharacteristic
511C2×C3C22×C3×C4×D312yes—
512C2×C3C3×#100412yes—
513C2×C3C3×#101012yes—
514D3—64no—
515D3—122no—
516C22—22no—
517C22—22no—
518C22—92no—
519C22—181no—
520C22—181no—
521C22C2×C3×#62414yes—
522C22C22×C3×D3211yescharacteristic
523C4—62no—
524C4—62no—
525C4—64no—
526C4—92no—
527C4—94no—
528C4C2×C3×#62412yes—
529C3—21no—
530C3—22no—
531C3—41no—
532C3#1059521911yescharacteristic, direct product factor
533C3C3×#554612yescomplemented
534C2—24no—
535C2—182no—
536C2C3×#6430911yescharacteristic
537C2C3×#6431311yescharacteristic
538C2C22×C3×#62411yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
prime radicalC2×C3×#3830C2
cosocleC22×C32C24×C3
hypercenterC2×C3×D4D32
hypocenterC32C3×#520
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C32×D4×Dic3C2semidirect
3. C2×C3×#3724C2semidirect
4. C3×#63957C2semidirect
5. C3×#63952C2semidirect
6. C3×#63900C2semidirect
7. C3×#63874C2semidirect
8. C3×#63844C2semidirect
10. #10595219C3direct (N × H)
14. C3×#3830C4semidirect
33. C3×D4×Dic3D3semidirect
34. C3×D4×Dic3C2×C3semidirect
43. C2×#3724C2×C3semidirect
44. #63957C2×C3semidirect
45. #63952C2×C3semidirect
46. #63900C2×C3semidirect
47. #63874C2×C3semidirect
48. #63844C2×C3semidirect
51. C33×D4C2×C4semidirect
53. C3×#638C2×C4semidirect
54. C2×C3×C3⋊Dic3C2×C4semidirect
57. C3×C4×C3⋊D3C2×C4semidirect
58. C3×#634C2×C4semidirect
74. #3830C3×C4semidirect
144. D4×Dic3C3×D3semidirect
164. C32×D4C4×D3semidirect
165. C32×D4C2×C3×C4semidirect
175. #638C2×C3×C4semidirect
176. C2×C3⋊Dic3C2×C3×C4semidirect
179. C4×C3⋊D3C2×C3×C4semidirect
180. #634C2×C3×C4semidirect
326. C33#520semidirect
340. C3×D4#624semidirect
341. C3×D4C3×C4×D3semidirect
476. C32#5546semidirect
477. C32C3×#520semidirect
482. D4C3×#624semidirect
532. C3#10595219direct (N × H)
533. C3C3×#5546semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C32×D4×Dic32
C2×C3×#37242
C3×#639572
C3×#639522
C3×#639002
C3×#638742
C3×#638442
C3×#55463
#105952193