Finite Group Databasebeta

← C3×#10591525 #13158552

Subgroups — C3×#10591525 #13158552

Order 1728.

850 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64297C211yesmaximal, characteristic, complemented
2C3×#64141C211yesmaximal, characteristic, complemented
3C32×#936C211yesmaximal, characteristic, complemented
4C32×#932C211yesmaximal, characteristic, complemented
5C3×#63570C211yesmaximal, characteristic, complemented
6C3×#63569C211yesmaximal, characteristic, complemented
7C3×#63560C211yesmaximal, characteristic, complemented
8C3×#5201—31nomaximal
9C3×#5062—31nomaximal
10#10591525C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C3×#3828C2211yescharacteristic
13C32×#235C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3804—22no—
16C3×#3799—21no—
17C3×#3797—21no—
18C3×#3796—21no—
19C3×D3×Dic6—21no—
20C3×#3754—21no—
21C3×#3753—21no—
22C3×#3748C2211yescharacteristic
23C32×#217—22no—
24C32×#214—21no—
25C32×#213—21no—
26C32×C4×Dic3C2211yescharacteristic
27C32×#208C2211yescharacteristic
28C3×#1010—31no—
29C3×#1007—31no—
30#64297C2×C311yescharacteristic, complemented
31C32×AGL(1,ℤ/8)—31no—
32C32×#124—31no—
33#64141C2×C311yescharacteristic, complemented
34C3×#936—21no—
35C3×#936D311yescharacteristic, complemented
36C3×#936C2×C311yescharacteristic, complemented
37C3×#932—21no—
38C3×#932—31no—
39C3×#932—31no—
40C3×#932—61no—
41C3×#932D311yescharacteristic, complemented
42C3×#932C2×C311yescharacteristic, complemented
43C3×#837—31no—
44C3×#836—31no—
45C3×#825—31no—
46C3×#824—31no—
47C3×#805—31no—
48C3×#805—31no—
49#63570C2×C311yescharacteristic, complemented
50#63569C2×C311yescharacteristic, complemented
51#63560C2×C311yescharacteristic, complemented
52C23×C33—21no—
53C33×D4—21no—
54C33×D4—21no—
55C2×C33×C4C2311yescharacteristic
56C3×#638—21no—
57C3×C4⋊(C3⋊D3)D411yescharacteristic
58C3×C4×C3⋊D3—21no—
59C3×#634D411yescharacteristic
60C32×#67—21no—
61C2×C32×Dic3—21no—
62C2×C32×Dic3D412yes—
63C32×D12D411yescharacteristic
64C32×C4×D3—21no—
65C32×Dic6—21no—
66C32×Dic6—21no—
67C32×Dic6D411yescharacteristic
68C3×#627—22no—
69C3×#626—22no—
70C3×#624—22no—
71C3×D3×Dic3—22no—
72C3×C3⋊(C3⋊C8)—21no—
73C32×C3⋊C8—21no—
74C3×#454—91no—
75#5201—31no—
76#5062—31no—
77C32×#41—31no—
78C2×C32×Q8—31no—
79C2×C32×D4—21no—
80C2×C32×D4—21no—
81C2×C32×D4—21no—
82C2×C32×D4—41no—
83C2×C32×D4C2×D311yescharacteristic
84C2×C32×D4C2×D311yescharacteristic
85C2×C32×D4C22×C311yescharacteristic
86#3828C22×C311yescharacteristic
87C3×#239—61no—
88C3×#239—61no—
89C3×D3×Q8—61no—
90C3×D3×Q8—61no—
91C3×D3×Q8—61no—
92C3×#237—61no—
93C3×#237—62no—
94C3×#235—21no—
95C3×#235—31no—
96C3×#235—31no—
97C3×#235—61no—
98C3×#235—62no—
99C3×#235—61no—
100C3×#235C2×D311yescharacteristic
101C3×#235C22×C311yescharacteristic
102C2×C3×Dic6—21no—
103C2×C3×Dic6—61no—
104C2×C3×Dic6C2×D311yescharacteristic
105C2×C3×Dic6C22×C311yescharacteristic
106#3804—22no—
107#3799—21no—
108#3797—21no—
109#3796—21no—
110D3×Dic6—21no—
111C32×SD16—61no—
112C32×D8—61no—
113C32×M16—31no—
114C32×C22⋊C4—62no—
115C32×C42—31no—
116#3754—21no—
117#3753—21no—
118#3748C22×C311yescharacteristic
119C3×#217—24no—
120C3×#217—22no—
121C3×#217—22no—
122C3×#214—22no—
123C3×#214—21no—
124C3×#214—21no—
125C3×#214—61no—
126C3×#214—61no—
127C3×#214—62no—
128C3×#213—22no—
129C3×#213—21no—
130C3×#213—21no—
131C3×#213—61no—
132C3×#213—61no—
133C3×#213—62no—
134C3×C4×Dic3—21no—
135C3×C4×Dic3C2×D311yescharacteristic
136C3×C4×Dic3C22×C311yescharacteristic
137C3×#208—21no—
138C3×#208—31no—
139C3×#208—31no—
140C3×#208—61no—
141C3×#208C2×D311yescharacteristic
142C3×#208C22×C311yescharacteristic
143C22×C33—42no—
144C22×C33—41no—
145C22×C33C2×D411yescharacteristic
146C2×C3×C3⋊D3—21no—
147C2×C32×D3—21no—
148C33×C4C2×D411yescharacteristic
149C33×C4C2×D411yescharacteristic
150C3×C3⋊Dic3—21no—
151C32×Dic3—22no—
152C32×Dic3—21no—
153C32×Dic3—41no—
154C3×2−1+4—91no—
155#1010—31no—
156#1007—31no—
157C3×AGL(1,ℤ/8)—31no—
158C3×AGL(1,ℤ/8)—31no—
159C3×AGL(1,ℤ/8)—61no—
160C3×AGL(1,ℤ/8)—91no—
161C3×#124—31no—
162C3×#124—31no—
163C3×#124—61no—
164#936C3×D311yescharacteristic, complemented
165#932—31no—
166#932—31no—
167#932—61no—
168#932C3×D311yescharacteristic, complemented
169C3×#104—91no—
170C3×#104—91no—
171C3×#101—91no—
172#837—31no—
173#836—31no—
174#825—31no—
175#824—31no—
176#805—31no—
177#805—31no—
178C23×C32—21no—
179C23×C32—22no—
180C23×C32—21no—
181C23×C32—21no—
182C23×C32—41no—
183C23×C32—41no—
184C23×C32—42no—
185C32×Q8—31no—
186C32×Q8—61no—
187C32×Q8—61no—
188C32×D4—21no—
189C32×D4—21no—
190C32×D4—22no—
191C32×D4—22no—
192C32×D4—21no—
193C32×D4—21no—
194C32×D4—21no—
195C32×D4—21no—
196C32×D4—31no—
197C32×D4—41no—
198C32×D4—41no—
199C32×D4—41no—
200C32×D4—41no—
201C32×D4—42no—
202C32×D4—42no—
203C32×D4—61no—
204C2×C32×C4—21no—
205C2×C32×C4—21no—
206C2×C32×C4—21no—
207C2×C32×C4—32no—
208C2×C32×C4—41no—
209C2×C32×C4—61no—
210C2×C32×C4—61no—
211C2×C32×C4C22×D311yescharacteristic
212C2×C32×C4C22×D311yescharacteristic
213C2×C32×C4C23×C311yescharacteristic
214#638—21no—
215C4⋊(C3⋊D3)C3×D411yescharacteristic
216C4×C3⋊D3—21no—
217#634C3×D411yescharacteristic
218C3×#67—21no—
219C3×#67—21no—
220C3×#67—41no—
221C3×#67—62no—
222C3×#67—61no—
223C3×#67—61no—
224C3×#67—121no—
225C2×C3×Dic3—22no—
226C2×C3×Dic3—21no—
227C2×C3×Dic3—21no—
228C2×C3×Dic3—22no—
229C2×C3×Dic3—62no—
230C2×C3×Dic3#6712yes—
231C2×C3×Dic3C3×D412yes—
232C3×D12—21no—
233C3×D12—31no—
234C3×D12—31no—
235C3×D12—61no—
236C3×D12—62no—
237C3×D12#6711yescharacteristic
238C3×D12C3×D411yescharacteristic
239C3×C4×D3—21no—
240C3×C4×D3—21no—
241C3×C4×D3—41no—
242C3×C4×D3—62no—
243C3×C4×D3—62no—
244C3×C4×D3—62no—
245C3×C4×D3—61no—
246C3×C4×D3—61no—
247C3×C4×D3—121no—
248C3×Dic6—21no—
249C3×Dic6—21no—
250C3×Dic6—21no—
251C3×Dic6—21no—
252C3×Dic6—21no—
253C3×Dic6—31no—
254C3×Dic6—31no—
255C3×Dic6—41no—
256C3×Dic6—41no—
257C3×Dic6—61no—
258C3×Dic6—62no—
259C3×Dic6—62no—
260C3×Dic6#6711yescharacteristic
261C3×Dic6C3×D411yescharacteristic
262#627—22no—
263#626—22no—
264#624—22no—
265D3×Dic3—22no—
266C32×C8—61no—
267C3⋊(C3⋊C8)—21no—
268C3×C3⋊C8—22no—
269C3×C3⋊C8—21no—
270C3×C3⋊C8—21no—
271C3×C3⋊C8—62no—
272C3×C3⋊C8—61no—
273C3×C3⋊C8—61no—
274#454—91noSylow
275C2×C33—21no—
276C2×C33—81no—
277C2×C33#12011yescharacteristic
278C3×C3⋊D3—41no—
279C32×D3—41no—
280C3×#41—31no—
281C3×#41—31no—
282C3×#41—61no—
283C3×#41—91no—
284C3×#41—182no—
285C3×#41—181no—
286C3×#41—181no—
287C2×C3×Q8—31no—
288C2×C3×Q8—31no—
289C2×C3×Q8—61no—
290C2×C3×Q8—181no—
291C2×C3×Q8—181no—
292C2×C3×D4—21no—
293C2×C3×D4—21no—
294C2×C3×D4—21no—
295C2×C3×D4—41no—
296C2×C3×D4D3211yescharacteristic
297C2×C3×D4C2×C3×D311yescharacteristic
298C2×C3×D4C2×C3×D311yescharacteristic
299#239—61no—
300#239—61no—
301D3×Q8—61no—
302D3×Q8—61no—
303D3×Q8—61no—
304#237—61no—
305#237—62no—
306#235—31no—
307#235—31no—
308#235—61no—
309#235—61no—
310#235—62no—
311#235C2×C3×D311yescharacteristic
312C2×Dic6—61no—
313C2×Dic6C2×C3×D311yescharacteristic
314C3×SD16—61no—
315C3×SD16—61no—
316C3×SD16—62no—
317C3×SD16—181no—
318C3×D8—61no—
319C3×D8—61no—
320C3×D8—62no—
321C3×D8—181no—
322C3×M16—31no—
323C3×M16—31no—
324C3×M16—61no—
325C3×M16—91no—
326C3×C22⋊C4—62no—
327C3×C22⋊C4—62no—
328C3×C22⋊C4—64no—
329C3×C42—31no—
330C3×C42—31no—
331C3×C42—61no—
332#217—22no—
333#214—21no—
334#214—61no—
335#214—61no—
336#214—62no—
337#213—21no—
338#213—61no—
339#213—61no—
340#213—62no—
341C4×Dic3C2×C3×D311yescharacteristic
342#208—31no—
343#208—31no—
344#208—61no—
345#208C2×C3×D311yescharacteristic
346C22×C32—21no—
347C22×C32—21no—
348C22×C32—21no—
349C22×C32—41no—
350C22×C32—42no—
351C22×C32—42no—
352C22×C32—42no—
353C22×C32—42no—
354C22×C32—44no—
355C22×C32—41no—
356C22×C32—41no—
357C22×C32—41no—
358C22×C32—44no—
359C22×C32—42no—
360C22×C32—42no—
361C22×C32—61no—
362C22×C32—82no—
363C22×C32—82no—
364C22×C32—84no—
365C22×C32D3×D411yescharacteristic
366C22×C32C2×#6711yescharacteristic
367C22×C32C2×C3×D411yescharacteristic
368C2×C3⋊D3—21no—
369C2×C3×D3—21no—
370C2×C3×D3—21no—
371C2×C3×D3—41no—
372C2×C3×D3—61no—
373C2×C3×D3—61no—
374C2×C3×D3—121no—
375C32×C4—21no—
376C32×C4—21no—
377C32×C4—21no—
378C32×C4—21no—
379C32×C4—21no—
380C32×C4—21no—
381C32×C4—41no—
382C32×C4—41no—
383C32×C4—61no—
384C32×C4—62no—
385C32×C4—121no—
386C32×C4D3×D411yescharacteristic
387C32×C4D3×D411yescharacteristic
388C32×C4D3×D411yescharacteristic
389C32×C4C2×#6711yescharacteristic
390C32×C4C2×C3×D411yescharacteristic
391C32×C4C2×C3×D411yescharacteristic
392C3⋊Dic3—21no—
393C3×Dic3—22no—
394C3×Dic3—21no—
395C3×Dic3—22no—
396C3×Dic3—21no—
397C3×Dic3—41no—
398C3×Dic3—42no—
399C3×Dic3—42no—
400C3×Dic3—41no—
401C3×Dic3—41no—
402C3×Dic3—61no—
403C3×Dic3—61no—
404C3×Dic3—121no—
4052−1+4—91no—
406AGL(1,ℤ/8)—31no—
407AGL(1,ℤ/8)—91no—
408#124—31no—
409#104—91no—
410#104—91no—
411#101—91no—
412C33#45411yescharacteristic, complemented, Sylow
413C23×C3—22no—
414C23×C3—21no—
415C23×C3—21no—
416C23×C3—21no—
417C23×C3—41no—
418C23×C3—41no—
419C23×C3—42no—
420C3×Q8—31no—
421C3×Q8—31no—
422C3×Q8—61no—
423C3×Q8—61no—
424C3×Q8—61no—
425C3×Q8—61no—
426C3×Q8—61no—
427C3×Q8—91no—
428C3×Q8—121no—
429C3×Q8—121no—
430C3×Q8—182no—
431C3×D4—22no—
432C3×D4—22no—
433C3×D4—21no—
434C3×D4—21no—
435C3×D4—21no—
436C3×D4—21no—
437C3×D4—21no—
438C3×D4—21no—
439C3×D4—31no—
440C3×D4—31no—
441C3×D4—41no—
442C3×D4—41no—
443C3×D4—41no—
444C3×D4—41no—
445C3×D4—42no—
446C3×D4—42no—
447C3×D4—61no—
448C3×D4—61no—
449C3×D4—61no—
450C3×D4—91no—
451C3×D4—121no—
452C3×D4—182no—
453C3×D4—181no—
454C2×C3×C4—21no—
455C2×C3×C4—21no—
456C2×C3×C4—21no—
457C2×C3×C4—32no—
458C2×C3×C4—32no—
459C2×C3×C4—41no—
460C2×C3×C4—62no—
461C2×C3×C4—61no—
462C2×C3×C4—61no—
463C2×C3×C4—62no—
464C2×C3×C4—61no—
465C2×C3×C4—61no—
466C2×C3×C4—121no—
467C2×C3×C4—182no—
468C2×C3×C4—182no—
469C2×C3×C4—181no—
470C2×C3×C4C2×D3211yescharacteristic
471C2×C3×C4C22×C3×D311yescharacteristic
472C2×C3×C4C22×C3×D311yescharacteristic
473#67—21no—
474#67—62no—
475#67—61no—
476#67—61no—
477#67—121no—
478C2×Dic3—21no—
479C2×Dic3—62no—
480C2×Dic3C3×#6712yes—
481D12—31no—
482D12—31no—
483D12—61no—
484D12—62no—
485D12C3×#6711yescharacteristic
486C4×D3—21no—
487C4×D3—62no—
488C4×D3—62no—
489C4×D3—62no—
490C4×D3—61no—
491C4×D3—61no—
492C4×D3—121no—
493Dic6—21no—
494Dic6—21no—
495Dic6—31no—
496Dic6—31no—
497Dic6—61no—
498Dic6—62no—
499Dic6—62no—
500Dic6C3×#6711yescharacteristic
501C3×C8—62no—
502C3×C8—61no—
503C3×C8—61no—
504C3×C8—181no—
505C3⋊C8—21no—
506C3⋊C8—62no—
507C3⋊C8—61no—
508C3⋊C8—61no—
509C2×C32—21no—
510C2×C32—22no—
511C2×C32—21no—
512C2×C32—21no—
513C2×C32—21no—
514C2×C32—21no—
515C2×C32—21no—
516C2×C32—41no—
517C2×C32—41no—
518C2×C32—41no—
519C2×C32—42no—
520C2×C32—82no—
521C2×C32—82no—
522C2×C32—82no—
523C2×C32—84no—
524C2×C32—81no—
525C2×C32—81no—
526C2×C32—81no—
527C2×C32—121no—
528C2×C32#93711yescharacteristic
529C2×C32#93711yescharacteristic
530C2×C32C3×#12011yescharacteristic
531C3⋊D3—41no—
532C3×D3—41no—
533C3×D3—41no—
534C3×D3—81no—
535C3×D3—121no—
536C3×D3—121no—
537C3×D3—241no—
538#41—31no—
539#41—91no—
540#41—182no—
541#41—181no—
542#41—181no—
543C2×Q8—31no—
544C2×Q8—181no—
545C2×Q8—181no—
546C2×D4C3×D3211yescharacteristic
547SD16—61no—
548SD16—181no—
549D8—61no—
550D8—181no—
551M16—31no—
552M16—91no—
553C22⋊C4—62no—
554C42—31no—
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—42no—
561C22×C3—42no—
562C22×C3—44no—
563C22×C3—42no—
564C22×C3—42no—
565C22×C3—44no—
566C22×C3—41no—
567C22×C3—41no—
568C22×C3—41no—
569C22×C3—41no—
570C22×C3—61no—
571C22×C3—61no—
572C22×C3—82no—
573C22×C3—82no—
574C22×C3—84no—
575C22×C3—121no—
576C22×C3—181no—
577C22×C3D3×#6711yescharacteristic
578C22×C3C3×D3×D411yescharacteristic
579C22×C3C2×C3×#6711yescharacteristic
580C2×D3—21no—
581C2×D3—61no—
582C2×D3—61no—
583C2×D3—121no—
584C3×C4—21no—
585C3×C4—21no—
586C3×C4—21no—
587C3×C4—21no—
588C3×C4—21no—
589C3×C4—21no—
590C3×C4—41no—
591C3×C4—41no—
592C3×C4—61no—
593C3×C4—62no—
594C3×C4—62no—
595C3×C4—61no—
596C3×C4—122no—
597C3×C4—121no—
598C3×C4—122no—
599C3×C4—121no—
600C3×C4—121no—
601C3×C4—181no—
602C3×C4D3×#6711yescharacteristic
603C3×C4#381111yescharacteristic
604C3×C4C3×D3×D411yescharacteristic
605C3×C4C3×D3×D411yescharacteristic
606C3×C4C3×D3×D411yescharacteristic
607C3×C4C2×C3×#6711yescharacteristic
608Dic3—22no—
609Dic3—21no—
610Dic3—41no—
611Dic3—61no—
612Dic3—61no—
613Dic3—121no—
614C32—21no—
615C32—21no—
616C32—21no—
617C32—41no—
618C32#506211yescharacteristic, complemented
619C32#520111yescharacteristic, complemented
620C32C3×#45411yescharacteristic, complemented
621C23—21no—
622Q8—31no—
623Q8—61no—
624Q8—61no—
625Q8—91no—
626Q8—182no—
627D4—21no—
628D4—21no—
629D4—31no—
630D4—61no—
631D4—91no—
632D4—182no—
633D4—181no—
634C2×C4—32no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—182no—
638C2×C4—182no—
639C2×C4—181no—
640C2×C4C2×C3×D3211yescharacteristic
641C8—61no—
642C8—181no—
643C2×C3—22no—
644C2×C3—21no—
645C2×C3—21no—
646C2×C3—21no—
647C2×C3—21no—
648C2×C3—21no—
649C2×C3—21no—
650C2×C3—41no—
651C2×C3—41no—
652C2×C3—42no—
653C2×C3—41no—
654C2×C3—82no—
655C2×C3—82no—
656C2×C3—82no—
657C2×C3—84no—
658C2×C3—81no—
659C2×C3—81no—
660C2×C3—81no—
661C2×C3—121no—
662C2×C3—121no—
663C2×C3—241no—
664C2×C3—361no—
665C2×C3#6398311yescharacteristic
666C2×C3C3×#93711yescharacteristic
667C2×C3C3×#93711yescharacteristic
668D3—41no—
669D3—121no—
670D3—121no—
671D3—241no—
672C22—42no—
673C22—41no—
674C22—61no—
675C22—181no—
676C22C3×D3×#6711yescharacteristic
677C4—62no—
678C4—61no—
679C4—121no—
680C4—181no—
681C4C3×D3×#6711yescharacteristic
682C4C3×#381111yescharacteristic
683C3—21no—
684C3—21no—
685C3—21no—
686C3—41no—
687C3#1059152511yescharacteristic, direct product factor
688C3C3×#506211yescharacteristic, complemented
689C3C3×#520111yescharacteristic, complemented
690C2—21no—
691C2—81no—
692C2—121no—
693C2—361no—
694C2C3×#6398311yescharacteristic
subgroupG/H
centerC2×C3#63983
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33#120
hypercenterC2×C3×D4D32
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64297C2semidirect
2. C3×#64141C2semidirect
3. C32×#936C2semidirect
4. C32×#932C2semidirect
5. C3×#63570C2semidirect
6. C3×#63569C2semidirect
7. C3×#63560C2semidirect
10. #10591525C3direct (N × H)
30. #64297C2×C3semidirect
33. #64141C2×C3semidirect
35. C3×#936D3semidirect
36. C3×#936C2×C3semidirect
41. C3×#932D3semidirect
42. C3×#932C2×C3semidirect
49. #63570C2×C3semidirect
50. #63569C2×C3semidirect
51. #63560C2×C3semidirect
164. #936C3×D3semidirect
168. #932C3×D3semidirect
412. C33#454semidirect
618. C32#5062semidirect
619. C32#5201semidirect
620. C32C3×#454semidirect
687. C3#10591525direct (N × H)
688. C3C3×#5062semidirect
689. C3C3×#5201semidirect

Maximal subgroups

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

subgroupindex
C3×#642972
C3×#641412
C32×#9362
C32×#9322
C3×#635702
C3×#635692
C3×#635602
C3×#52013
C3×#50623
#105915253