Finite Group Databasebeta

← C3×#10590887 #13157849

Subgroups — C3×#10590887 #13157849

Order 1728.

782 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×D24C211yesmaximal, characteristic, complemented
2C2×C32×#204C211yesmaximal, characteristic, complemented
3C3×#63914C211yesmaximal, characteristic, complemented
4C3×#63863C211yesmaximal, characteristic, complemented
5C2×C3×#3715C211yesmaximal, characteristic, complemented
6C2×C3×#3713C211yesmaximal, characteristic, complemented
7C3×#63642C211yesmaximal, characteristic, complemented
8C3×#5176—31nomaximal
9C3×#5154—31nomaximal
10#10590887C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#625—22no—
16C2×C33×C8C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×D24—22no—
20C32×#204—22no—
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3715—22no—
23C3×#3713—22no—
24C2×C32×SD16—31no—
25C2×C32×D8—31no—
26C3×#947—31no—
27C2×C3×#215—31no—
28C3×#940—31no—
29C3×#940—31no—
30C3×#936—31no—
31C2×C3×#214—31no—
32C2×C3×#213—31no—
33C2×C3×#213—31no—
34C2×C3×D24—21no—
35C2×C3×D24D311yescharacteristic, complemented
36C2×C3×D24C2×C311yescharacteristic, complemented
37C2×C3×#204—21no—
38C2×C3×#204D311yescharacteristic, complemented
39C2×C3×#204C2×C311yescharacteristic, complemented
40#63914C2×C311yescharacteristic, complemented
41#63863C2×C311yescharacteristic, complemented
42C2×#3715C2×C311yescharacteristic, complemented
43C2×#3713C2×C311yescharacteristic, complemented
44#63642C2×C311yescharacteristic, complemented
45C3×#815—31no—
46C3×#815—31no—
47C3×#815—61no—
48C22×C32×D3—21no—
49C22×C32×D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3D412yes—
53C2×C32×Dic3—21no—
54C32×D12—21no—
55C32×D12—21no—
56C32×D12—21no—
57C32×D12—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×#625—44no—
63C33×C8D412yescomplemented
64C3×C3⋊(C3⋊C8)D412yescomplemented
65C3×#496—91no—
66#5176—31no—
67#5154—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—31no—
70C2×C32×D4—31no—
71C2×C32×D4—31no—
72C2×C3×#67—62no—
73C2×C3×#67—62no—
74C2×C3×D12—21no—
75C2×C3×D12—21no—
76C2×C3×D12—21no—
77C2×C3×D12C2×D311yescharacteristic
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C22×C311yescharacteristic
81C2×C3×D12C22×C311yescharacteristic
82C2×C3×D12C22×C311yescharacteristic
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C2×#625—22no—
87C32×SD16—62no—
88C32×D8—62no—
89C2×C32×C8—21no—
90C2×C32×C8—21no—
91C2×C32×C8—21no—
92C2×C32×C8—41no—
93C2×C32×C8C2×D311yescharacteristic
94C2×C32×C8C2×D311yescharacteristic
95C2×C32×C8C22×C311yescharacteristic
96C4×C3⋊Dic3C22×C311yescharacteristic
97C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
98C3×#217—62no—
99C3×#215—62no—
100C3×#214—62no—
101C3×#213—62no—
102C3×#213—62no—
103C3×(C2×D3)⋊C4—62no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—61no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—61no—
110C3×D24—24no—
111C3×D24—22no—
112C3×D24—22no—
113C3×#204—24no—
114C3×#204—22no—
115C3×#204—22no—
116(C2×D3)⋊Dic3—22no—
117#3715—22no—
118#3713—22no—
119C22×C33C2×D411yescharacteristic
120C2×C32×D3—41no—
121C2×C32×D3—41no—
122C2×C32×D3—41no—
123C2×C32×D3—42no—
124C2×C32×D3—42no—
125C2×C32×D3—42no—
126C33×C4C2×D411yescharacteristic
127C33×C4C2×D411yescharacteristic
128C3×C3⋊Dic3—22no—
129C32×Dic3—41no—
130C2×C3×SD16—31no—
131C2×C3×SD16—31no—
132C2×C3×SD16—61no—
133C2×C3×SD16—91no—
134C2×C3×D8—31no—
135C2×C3×D8—31no—
136C2×C3×D8—61no—
137C2×C3×D8—91no—
138C3×#127—91no—
139C3×#124—91no—
140#947—31no—
141C2×#215—31no—
142#940—31no—
143#940—31no—
144#936—31no—
145C2×#214—31no—
146C2×#213—31no—
147C2×#213—31no—
148C2×D24C3×D311yescharacteristic, complemented
149C2×#204C3×D311yescharacteristic, complemented
150C3×#97—91no—
151#815—31no—
152#815—31no—
153#815—61no—
154C23×C32—61no—
155C23×C32—61no—
156C23×C32—61no—
157C22×C3×D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—41no—
164C22×C3×D3—41no—
165C22×C3×D3—41no—
166C32×Q8—61no—
167C32×Q8—61no—
168C32×D4—61no—
169C32×D4—61no—
170C32×D4—61no—
171C32×D4—61no—
172C32×D4—61no—
173C32×D4—61no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—41no—
178C2×C32×C4—61no—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C23×C311yescharacteristic
182C2×C3⋊Dic3C3×D412yes—
183C3×#67—124no—
184C3×#67—124no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—41no—
190C2×C3×Dic3—62no—
191C3×D12—22no—
192C3×D12—22no—
193C3×D12—22no—
194C3×D12—21no—
195C3×D12—21no—
196C3×D12—21no—
197C3×D12—21no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—21no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—41no—
207C3×D12—41no—
208C3×D12—41no—
209C3×Dic6—22no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—41no—
215#625—44no—
216C32×C8—22no—
217C32×C8—22no—
218C32×C8—22no—
219C32×C8—42no—
220C32×C8#6712yescomplemented
221C32×C8#6712yescomplemented
222C32×C8C3×D412yescomplemented
223C3⋊(C3⋊C8)C3×D412yescomplemented
224C3×C3⋊C8—32no—
225C3×C3⋊C8—32no—
226C3×C3⋊C8—62no—
227#496—91noSylow
228C2×C33#12711yescharacteristic
229C2×C33AGL(1,ℤ/8)12yes—
230C32×D3—81no—
231C32×D3—81no—
232C32×D3—81no—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—61no—
236C2×C3×D4—31no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4—31no—
242C2×C3×D4—61no—
243C2×C3×D4—61no—
244C2×C3×D4—61no—
245C2×C3×D4—182no—
246C2×#67—62no—
247C2×#67—62no—
248C2×D12C2×C3×D311yescharacteristic
249C2×D12C2×C3×D311yescharacteristic
250C2×D12C2×C3×D311yescharacteristic
251C2×Dic6C2×C3×D311yescharacteristic
252C3×SD16—64no—
253C3×SD16—62no—
254C3×SD16—62no—
255C3×SD16—182no—
256C3×D8—64no—
257C3×D8—62no—
258C3×D8—62no—
259C3×D8—182no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8—41no—
264C2×C3×C8—91no—
265C2×C3×C8D3211yescharacteristic
266C2×C3×C8C2×C3×D311yescharacteristic
267C2×C3×C8C2×C3×D311yescharacteristic
268C3×C22⋊C4—182no—
269C3×C42—91no—
270#217—62no—
271#215—62no—
272#214—62no—
273#213—62no—
274#213—62no—
275(C2×D3)⋊C4—62no—
276C4×Dic3—31no—
277C4×Dic3—31no—
278C4×Dic3—61no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—61no—
282D24—22no—
283#204—22no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—41no—
288C22×C32—121no—
289C22×C32—121no—
290C22×C32—121no—
291C22×C32—122no—
292C22×C32—122no—
293C22×C32—122no—
294C22×C32D3×D411yescharacteristic
295C22×C32D3×D411yescharacteristic
296C22×C32C2×C3×D411yescharacteristic
297C2×C3×D3—42no—
298C2×C3×D3—42no—
299C2×C3×D3—42no—
300C2×C3×D3—42no—
301C2×C3×D3—42no—
302C2×C3×D3—42no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3—41no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—42no—
310C2×C3×D3—42no—
311C2×C3×D3—42no—
312C2×C3×D3—82no—
313C2×C3×D3—82no—
314C2×C3×D3—82no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—121no—
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4C2×#6711yescharacteristic
327C32×C4C2×#6711yescharacteristic
328C32×C4C2×C3×D411yescharacteristic
329C32×C4C2×C3×D411yescharacteristic
330C3⋊Dic3—22no—
331C3×Dic3—42no—
332C3×Dic3—41no—
333C3×Dic3—41no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—122no—
337C2×SD16—31no—
338C2×SD16—91no—
339C2×D8—31no—
340C2×D8—91no—
341#127—91no—
342#124—91no—
343#97—91no—
344C33#49611yescharacteristic, complemented, Sylow
345C23×C3—61no—
346C23×C3—61no—
347C23×C3—61no—
348C23×C3—61no—
349C23×C3—61no—
350C23×C3—61no—
351C23×C3—121no—
352C23×C3—121no—
353C23×C3—121no—
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C3×Q8—62no—
358C3×Q8—61no—
359C3×Q8—61no—
360C3×Q8—61no—
361C3×Q8—61no—
362C3×Q8—121no—
363C3×D4—62no—
364C3×D4—62no—
365C3×D4—62no—
366C3×D4—61no—
367C3×D4—61no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—61no—
372C3×D4—61no—
373C3×D4—61no—
374C3×D4—61no—
375C3×D4—61no—
376C3×D4—61no—
377C3×D4—61no—
378C3×D4—121no—
379C3×D4—121no—
380C3×D4—121no—
381C3×D4—364no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—92no—
389C2×C3×C4—121no—
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C22×C3×D311yescharacteristic
392C2×C3×C4C22×C3×D311yescharacteristic
393#67—124no—
394#67—124no—
395C2×Dic3—21no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—62no—
399D12—21no—
400D12—21no—
401D12—21no—
402D12—21no—
403D12—21no—
404D12—21no—
405Dic6—21no—
406Dic6—21no—
407C3×C8—22no—
408C3×C8—22no—
409C3×C8—22no—
410C3×C8—42no—
411C3×C8—92no—
412C3×C8#62512yescomplemented
413C3×C8C3×#6712yescomplemented
414C3×C8C3×#6712yescomplemented
415C3⋊C8—32no—
416C3⋊C8—32no—
417C3⋊C8—62no—
418C2×C32—22no—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—41no—
425C2×C32—42no—
426C2×C32—241no—
427C2×C32—241no—
428C2×C32—241no—
429C2×C32#91112yes—
430C2×C32#91412yes—
431C2×C32#94011yescharacteristic
432C2×C32#94011yescharacteristic
433C2×C32C3×#12711yescharacteristic
434C2×C32C3×AGL(1,ℤ/8)12yes—
435C3×D3—82no—
436C3×D3—82no—
437C3×D3—82no—
438C3×D3—81no—
439C3×D3—81no—
440C3×D3—81no—
441C3×D3—81no—
442C3×D3—81no—
443C3×D3—81no—
444C2×Q8—31no—
445C2×D4—31no—
446C2×D4—31no—
447C2×D4—31no—
448C2×D4—182no—
449SD16—62no—
450SD16—182no—
451D8—62no—
452D8—182no—
453C2×C8—91no—
454C2×C8C3×D3211yescharacteristic
455C22⋊C4—182no—
456C42—91no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—41no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3—122no—
464C22×C3—122no—
465C22×C3—122no—
466C22×C3—122no—
467C22×C3—122no—
468C22×C3—122no—
469C22×C3—122no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—242no—
477C22×C3—242no—
478C22×C3—242no—
479C22×C3#380211yescharacteristic
480C22×C3C3×D3×D411yescharacteristic
481C22×C3C3×D3×D411yescharacteristic
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—122no—
497C3×C4—121no—
498C3×C4—121no—
499C3×C4—182no—
500C3×C4#380211yescharacteristic
501C3×C4C2×#62511yescharacteristic
502C3×C4C3×D3×D411yescharacteristic
503C3×C4C3×D3×D411yescharacteristic
504C3×C4C2×C3×#6711yescharacteristic
505C3×C4C2×C3×#6711yescharacteristic
506Dic3—41no—
507Dic3—62no—
508Dic3—62no—
509Dic3—122no—
510C32—21no—
511C32—21no—
512C32—21no—
513C32—41no—
514C32#515411yescharacteristic, complemented
515C32#517611yescharacteristic, complemented
516C32C3×#49611yescharacteristic, complemented
517C23—61no—
518C23—61no—
519C23—61no—
520Q8—61no—
521Q8—61no—
522D4—61no—
523D4—61no—
524D4—61no—
525D4—61no—
526D4—61no—
527D4—61no—
528D4—364no—
529C2×C4—61no—
530C2×C4—92no—
531C2×C4C2×C3×D3211yescharacteristic
532C8—92no—
533C8C3×#62512yescomplemented
534C2×C3—22no—
535C2×C3—22no—
536C2×C3—22no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—42no—
541C2×C3—41no—
542C2×C3—242no—
543C2×C3—242no—
544C2×C3—242no—
545C2×C3—241no—
546C2×C3—241no—
547C2×C3—241no—
548C2×C3—241no—
549C2×C3—241no—
550C2×C3—241no—
551C2×C3#6379812yes—
552C2×C3#6391411yescharacteristic
553C2×C3C3×#91112yes—
554C2×C3C3×#91412yes—
555C2×C3C3×#94011yescharacteristic
556C2×C3C3×#94011yescharacteristic
557D3—81no—
558D3—81no—
559D3—81no—
560C22—122no—
561C22—122no—
562C22—122no—
563C22—121no—
564C22—121no—
565C22—121no—
566C22C3×#380211yescharacteristic
567C4—121no—
568C4—182no—
569C4C3×#380211yescharacteristic
570C4C2×C3×#62511yescharacteristic
571C3—21no—
572C3—21no—
573C3—21no—
574C3—41no—
575C3#1059088711yescharacteristic, direct product factor
576C3C3×#515411yescharacteristic, complemented
577C3C3×#517611yescharacteristic, complemented
578C2—241no—
579C2—241no—
580C2—241no—
581C2C3×#6379812yes—
582C2C3×#6391411yescharacteristic
subgroupG/H
centerC22×C3#3802
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#496
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C2×C32×D24C2semidirect
2. C2×C32×#204C2semidirect
3. C3×#63914C2semidirect
4. C3×#63863C2semidirect
5. C2×C3×#3715C2semidirect
6. C2×C3×#3713C2semidirect
7. C3×#63642C2semidirect
10. #10590887C3direct (N × H)
35. C2×C3×D24D3semidirect
36. C2×C3×D24C2×C3semidirect
38. C2×C3×#204D3semidirect
39. C2×C3×#204C2×C3semidirect
40. #63914C2×C3semidirect
41. #63863C2×C3semidirect
42. C2×#3715C2×C3semidirect
43. C2×#3713C2×C3semidirect
44. #63642C2×C3semidirect
63. C33×C8D4semidirect
64. C3×C3⋊(C3⋊C8)D4semidirect
148. C2×D24C3×D3semidirect
149. C2×#204C3×D3semidirect
220. C32×C8#67semidirect
221. C32×C8#67semidirect
222. C32×C8C3×D4semidirect
223. C3⋊(C3⋊C8)C3×D4semidirect
344. C33#496semidirect
412. C3×C8#625semidirect
413. C3×C8C3×#67semidirect
414. C3×C8C3×#67semidirect
514. C32#5154semidirect
515. C32#5176semidirect
516. C32C3×#496semidirect
533. C8C3×#625semidirect
575. C3#10590887direct (N × H)
576. C3C3×#5154semidirect
577. C3C3×#5176semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D242
C2×C32×#2042
C3×#639142
C3×#638632
C2×C3×#37152
C2×C3×#37132
C3×#636422
C3×#51763
C3×#51543
#105908873