Finite Group Databasebeta

← C3×#10591491 #13158504

Subgroups — C3×#10591491 #13158504

Order 1728.

762 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3754C211yesmaximal, characteristic
2C32×#936C211yesmaximal, characteristic
3C2×C32×#214C211yesmaximal, characteristic
4C3×#63866C211yesmaximal, characteristic, complemented
5C2×C3×#3719C211yesmaximal, characteristic, complemented
6C2×C3×#3718C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8C3×#5167—31nomaximal
9C3×#5060—31nomaximal
10#10591491C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3754—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—22no—
18C32×#214—22no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—22no—
22C3×#3719—22no—
23C3×#3718—22no—
24C2×C32×SD16—31no—
25C32×#124—31no—
26C2×#3754C2×C311yescharacteristic
27C3×#944—31no—
28C2×C3×#216—31no—
29C2×C3×#216—31no—
30C2×C3×#216—31no—
31C3×#936—21no—
32C3×#936D311yescharacteristic
33C3×#936C2×C311yescharacteristic
34C2×C3×#214—21no—
35C2×C3×#214—31no—
36C2×C3×#214—31no—
37C2×C3×#214—61no—
38C2×C3×#214D311yescharacteristic
39C2×C3×#214C2×C311yescharacteristic
40C2×C3×Dic12—31no—
41C3×#869—31no—
42#63866C2×C311yescharacteristic, complemented
43C2×#3719C2×C311yescharacteristic, complemented
44C2×#3718C2×C311yescharacteristic, complemented
45C3×#815—31no—
46C3×#803—31no—
47#63554C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C33×D4—21no—
50C33×D4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C32×C3⋊C8D412yes—
64C3×#497—91no—
65#5167—31no—
66#5060—31no—
67C2×C32×Q8—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C22×C311yescharacteristic
76C2×#634C22×C311yescharacteristic
77C2×C3×Dic6—21no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C2×D311yescharacteristic
84C2×C3×Dic6C22×C311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×SD16—62no—
87C2×C32×C8—31no—
88C32×C22⋊C4—62no—
89C32×C42—31no—
90#3754—22no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#217—24no—
93C3×#217—22no—
94C3×#217—22no—
95C3×#216—62no—
96C3×#216—62no—
97C3×#216—62no—
98C3×#214—24no—
99C3×#214—22no—
100C3×#214—22no—
101C3×#214—62no—
102C3×#214—62no—
103C3×#214—64no—
104C3×C4⋊Dic3—62no—
105C3×Dic3⋊C4—62no—
106C3×C4×Dic3—21no—
107C3×C4×Dic3C2×D311yescharacteristic
108C3×C4×Dic3C22×C311yescharacteristic
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—61no—
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C3×Dic12—62no—
116Dic3⋊Dic3—22no—
117#3719—22no—
118#3718—22no—
119C22×C33—41no—
120C22×C33—42no—
121C22×C33C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C33×C4C2×D411yescharacteristic
124C3×C3⋊Dic3—41no—
125C32×Dic3—22no—
126C32×Dic3—41no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×Q16—91no—
130C2×C3×SD16—31no—
131C2×C3×SD16—31no—
132C2×C3×SD16—61no—
133C2×C3×SD16—91no—
134C3×C4⋊Q8—91no—
135C3×#124—31no—
136C3×#124—31no—
137C3×#124—61no—
138#944—31no—
139C2×#216—31no—
140C2×#216—31no—
141C2×#216—31no—
142#936C3×D311yescharacteristic
143C2×#214—31no—
144C2×#214—31no—
145C2×#214—61no—
146C2×#214C3×D311yescharacteristic
147C2×Dic12—31no—
148#869—31no—
149C3×#97—91no—
150#815—31no—
151#803—31no—
152C23×C32—21no—
153C23×C32—22no—
154C23×C32—21no—
155C23×C32—21no—
156C23×C32—41no—
157C23×C32—41no—
158C23×C32—42no—
159C32×Q8—61no—
160C32×Q8—61no—
161C32×Q8—61no—
162C32×Q8—61no—
163C32×D4—21no—
164C32×D4—21no—
165C32×D4—22no—
166C32×D4—22no—
167C32×D4—21no—
168C32×D4—21no—
169C32×D4—21no—
170C32×D4—21no—
171C32×D4—41no—
172C32×D4—41no—
173C32×D4—41no—
174C32×D4—41no—
175C32×D4—42no—
176C32×D4—42no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—32no—
181C2×C32×C4—41no—
182C2×C32×C4—61no—
183C2×C32×C4—61no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C23×C311yescharacteristic
187C2×C3⋊Dic3—21no—
188#634—21no—
189#634—21no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—41no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—121no—
200C2×C3×Dic3#6712yes—
201C2×C3×Dic3C3×D412yes—
202C3×Dic6—22no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—41no—
212C3×Dic6—41no—
213C3×Dic6—41no—
214C3×Dic6—62no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—121no—
220C32×C8—32no—
221C3⋊(C3⋊C8)C3×D412yes—
222C3×C3⋊C8—22no—
223C3×C3⋊C8—32no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—62no—
226C3×C3⋊C8#6712yes—
227C3×C3⋊C8C3×D412yes—
228#497—91noSylow
229C2×C33—81no—
230C2×C33#12711yescharacteristic
231C2×C33#13712yes—
232C2×C3×Q8—31no—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×Q8—61no—
238C2×C3×Q8—91no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—41no—
243C2×C3×D4D3211yescharacteristic
244C2×C3×D4C2×C3×D311yescharacteristic
245C2×C3×D4C2×C3×D311yescharacteristic
246C2×Dic6—31no—
247C2×Dic6—31no—
248C2×Dic6—61no—
249C2×Dic6C2×C3×D311yescharacteristic
250C2×Dic6C2×C3×D311yescharacteristic
251C3×Q16—182no—
252C3×Q16—182no—
253C3×SD16—62no—
254C3×SD16—62no—
255C3×SD16—64no—
256C3×SD16—182no—
257C2×C3×C8—31no—
258C2×C3×C8—31no—
259C2×C3×C8—61no—
260C2×C3×C8—91no—
261C3×C4⋊C4—182no—
262C3×C22⋊C4—62no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—64no—
265C3×C42—31no—
266C3×C42—31no—
267C3×C42—61no—
268#217—22no—
269#216—62no—
270#216—62no—
271#216—62no—
272#214—22no—
273#214—62no—
274#214—62no—
275#214—64no—
276C4⋊Dic3—62no—
277Dic3⋊C4—62no—
278C4×Dic3C2×C3×D311yescharacteristic
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—61no—
282C2×C3⋊C8C2×C3×D311yescharacteristic
283Dic12—62no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—41no—
288C22×C32—42no—
289C22×C32—42no—
290C22×C32—42no—
291C22×C32—44no—
292C22×C32—42no—
293C22×C32—41no—
294C22×C32—41no—
295C22×C32—41no—
296C22×C32—44no—
297C22×C32—42no—
298C22×C32—42no—
299C22×C32—82no—
300C22×C32—82no—
301C22×C32—84no—
302C22×C32D3×D411yescharacteristic
303C22×C32C2×#6711yescharacteristic
304C22×C32C2×C3×D411yescharacteristic
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—62no—
314C32×C4—121no—
315C32×C4—121no—
316C32×C4D3×D411yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4C2×#6711yescharacteristic
320C32×C4C2×C3×D411yescharacteristic
321C32×C4C2×C3×D411yescharacteristic
322C3⋊Dic3—41no—
323C3×Dic3—22no—
324C3×Dic3—22no—
325C3×Dic3—42no—
326C3×Dic3—42no—
327C3×Dic3—42no—
328C3×Dic3—41no—
329C3×Dic3—41no—
330C3×Dic3—41no—
331C3×Dic3—41no—
332C3×Dic3—122no—
333C3×Dic3—121no—
334C3×Dic3—121no—
335C2×Q16—91no—
336C2×Q16—91no—
337C2×SD16—31no—
338C2×SD16—91no—
339C4⋊Q8—91no—
340#124—31no—
341#97—91no—
342C33#49711yescharacteristic, complemented, Sylow
343C23×C3—22no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—42no—
350C3×Q8—62no—
351C3×Q8—61no—
352C3×Q8—61no—
353C3×Q8—61no—
354C3×Q8—61no—
355C3×Q8—61no—
356C3×Q8—61no—
357C3×Q8—61no—
358C3×Q8—61no—
359C3×Q8—121no—
360C3×Q8—121no—
361C3×Q8—121no—
362C3×Q8—181no—
363C3×Q8—181no—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—21no—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—42no—
377C3×D4—42no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—32no—
382C2×C3×C4—32no—
383C2×C3×C4—41no—
384C2×C3×C4—62no—
385C2×C3×C4—62no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—121no—
391C2×C3×C4—181no—
392C2×C3×C4C2×D3211yescharacteristic
393C2×C3×C4C22×C3×D311yescharacteristic
394C2×C3×C4C22×C3×D311yescharacteristic
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—121no—
400C2×Dic3C3×#6712yes—
401Dic6—21no—
402Dic6—21no—
403Dic6—21no—
404Dic6—21no—
405Dic6—62no—
406Dic6—61no—
407Dic6—61no—
408Dic6—61no—
409Dic6—61no—
410Dic6—121no—
411C3×C8—32no—
412C3×C8—32no—
413C3×C8—62no—
414C3×C8—92no—
415C3⋊C8—32no—
416C3⋊C8—32no—
417C3⋊C8—62no—
418C3⋊C8C3×#6712yes—
419C2×C32—22no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—41no—
426C2×C32—42no—
427C2×C32—82no—
428C2×C32—82no—
429C2×C32—82no—
430C2×C32—84no—
431C2×C32—81no—
432C2×C32—81no—
433C2×C32—81no—
434C2×C32#91512yes—
435C2×C32#94011yescharacteristic
436C2×C32#94011yescharacteristic
437C2×C32#95112yes—
438C2×C32C3×#12711yescharacteristic
439C2×C32C3×#13712yes—
440C2×Q8—31no—
441C2×Q8—31no—
442C2×Q8—91no—
443C2×D4C3×D3211yescharacteristic
444Q16—182no—
445Q16—182no—
446SD16—62no—
447SD16—182no—
448C2×C8—31no—
449C2×C8—91no—
450C4⋊C4—182no—
451C22⋊C4—62no—
452C42—31no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—42no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—82no—
469C22×C3—82no—
470C22×C3—84no—
471C22×C3D3×#6711yescharacteristic
472C22×C3C3×D3×D411yescharacteristic
473C22×C3C2×C3×#6711yescharacteristic
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—62no—
483C3×C4—62no—
484C3×C4—122no—
485C3×C4—122no—
486C3×C4—122no—
487C3×C4—121no—
488C3×C4—121no—
489C3×C4—121no—
490C3×C4—121no—
491C3×C4—361no—
492C3×C4D3×#6711yescharacteristic
493C3×C4#381111yescharacteristic
494C3×C4C3×D3×D411yescharacteristic
495C3×C4C3×D3×D411yescharacteristic
496C3×C4C3×D3×D411yescharacteristic
497C3×C4C2×C3×#6711yescharacteristic
498Dic3—22no—
499Dic3—41no—
500Dic3—41no—
501Dic3—122no—
502Dic3—121no—
503Dic3—121no—
504C32—21no—
505C32—21no—
506C32—21no—
507C32—41no—
508C32#506011yescharacteristic, complemented
509C32#516711yescharacteristic, complemented
510C32C3×#49711yescharacteristic, complemented
511C23—21no—
512Q8—61no—
513Q8—61no—
514Q8—61no—
515Q8—61no—
516Q8—181no—
517Q8—181no—
518D4—21no—
519D4—21no—
520C2×C4—32no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—181no—
524C2×C4C2×C3×D3211yescharacteristic
525C8—32no—
526C8—92no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—22no—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—41no—
534C2×C3—42no—
535C2×C3—82no—
536C2×C3—82no—
537C2×C3—82no—
538C2×C3—84no—
539C2×C3—81no—
540C2×C3—81no—
541C2×C3—81no—
542C2×C3#6392912yes—
543C2×C3#6397911yescharacteristic
544C2×C3C3×#91512yes—
545C2×C3C3×#94011yescharacteristic
546C2×C3C3×#94011yescharacteristic
547C2×C3C3×#95112yes—
548C22—41no—
549C22—42no—
550C22C3×D3×#6711yescharacteristic
551C4—62no—
552C4—121no—
553C4—121no—
554C4—361no—
555C4C3×D3×#6711yescharacteristic
556C4C3×#381111yescharacteristic
557C3—21no—
558C3—21no—
559C3—21no—
560C3—41no—
561C3#1059149111yescharacteristic, direct product factor
562C3C3×#506011yescharacteristic, complemented
563C3C3×#516711yescharacteristic, complemented
564C2—81no—
565C2C3×#6392912yes—
566C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#497
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
4. C3×#63866C2semidirect
5. C2×C3×#3719C2semidirect
6. C2×C3×#3718C2semidirect
7. C3×#63554C2semidirect
10. #10591491C3direct (N × H)
42. #63866C2×C3semidirect
43. C2×#3719C2×C3semidirect
44. C2×#3718C2×C3semidirect
47. #63554C2×C3semidirect
342. C33#497semidirect
508. C32#5060semidirect
509. C32#5167semidirect
510. C32C3×#497semidirect
561. C3#10591491direct (N × H)
562. C3C3×#5060semidirect
563. C3C3×#5167semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C32×#9362
C2×C32×#2142
C3×#638662
C2×C3×#37192
C2×C3×#37182
C3×#635542
C3×#51673
C3×#50603
#105914913