Finite Group Databasebeta

← C3×#10590400 #13157417

Subgroups — C3×#10590400 #13157417

Order 1728.

778 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64297C211yesmaximal, characteristic, complemented
2C32×#875C211yesmaximal, characteristic, complemented
3C3×#63830C211yesmaximal, characteristic, complemented
4C3×#63823C211yesmaximal, characteristic, complemented
5C3×#63632C211yesmaximal, characteristic, complemented
6C32×#805C211yesmaximal, characteristic, complemented
7C3×#63560C211yesmaximal, characteristic, complemented
8C3×#5062—31nomaximal
9C3×#4827—31nomaximal
10#10590400C311yesmaximal, characteristic, direct product factor
11C3×#3828C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3804—22no—
16C3×#3799—21no—
17C3×#3797—21no—
18C3×#3796—21no—
19C3×D3×Dic6—21no—
20C33×C42C2211yescharacteristic
21C3×#3748C2211yescharacteristic
22C32×(C2×D3)⋊C4—22no—
23C32×#208C2211yescharacteristic
24C3×#3716—21no—
25C3×#3715—21no—
26C3×#3714—21no—
27C3×#3713—21no—
28C3×#1010—31no—
29C3×#1007—31no—
30#64297C2×C311yescharacteristic, complemented
31C32×#124—31no—
32C3×#949—31no—
33C3×#932—31no—
34C3×#932—31no—
35C3×#908—31no—
36C3×#875—21no—
37C3×#875D311yescharacteristic, complemented
38C3×#875C2×C311yescharacteristic, complemented
39#63830C2×C311yescharacteristic, complemented
40#63823C2×C311yescharacteristic, complemented
41C32×#104—31no—
42#63632C2×C311yescharacteristic, complemented
43C3×#836—31no—
44C3×#824—31no—
45C3×#805—21no—
46C3×#805—31no—
47C3×#805—31no—
48C3×#805—61no—
49C3×#805D311yescharacteristic, complemented
50C3×#805C2×C311yescharacteristic, complemented
51#63560C2×C311yescharacteristic, complemented
52C22×C32×D3—21no—
53C2×C33×C4—21no—
54C2×C33×C4C2311yescharacteristic
55C3×#638—21no—
56C3×C4⋊(C3⋊D3)D411yescharacteristic
57C3×C4×C3⋊D3—21no—
58C3×#634D411yescharacteristic
59C32×#67—21no—
60C2×C32×Dic3D412yes—
61C32×D12—21no—
62C32×D12—21no—
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#5062—31no—
76#4827—31no—
77C32×#41—31no—
78C2×C32×Q8—31no—
79C2×C32×D4—31no—
80#3828C22×C311yescharacteristic
81C3×#239—61no—
82C3×#239—61no—
83C3×D3×Q8—61no—
84C3×D3×Q8—61no—
85C3×D3×Q8—61no—
86C3×#237—61no—
87C3×#237—62no—
88C3×#235—21no—
89C3×#235—31no—
90C3×#235—31no—
91C3×#235—61no—
92C3×#235—62no—
93C3×#235—61no—
94C3×#235C2×D311yescharacteristic
95C3×#235C22×C311yescharacteristic
96C2×C3×D12—21no—
97C2×C3×D12C2×D311yescharacteristic
98C2×C3×D12C22×C311yescharacteristic
99C2×C3×Dic6—21no—
100C2×C3×Dic6—61no—
101C2×C3×Dic6C2×D311yescharacteristic
102C2×C3×Dic6C22×C311yescharacteristic
103#3804—22no—
104#3799—21no—
105#3797—21no—
106#3796—21no—
107D3×Dic6—21no—
108C32×M16—31no—
109C32×C22⋊C4—62no—
110C32×C42—21no—
111C32×C42—21no—
112C32×C42—21no—
113C32×C42—41no—
114C32×C42C2×D311yescharacteristic
115C32×C42C2×D311yescharacteristic
116C32×C42C22×C311yescharacteristic
117#3748C22×C311yescharacteristic
118C3×#215—61no—
119C3×#214—61no—
120C3×#214—61no—
121C3×#213—61no—
122C3×#213—61no—
123C3×#213—61no—
124C3×(C2×D3)⋊C4—24no—
125C3×(C2×D3)⋊C4—22no—
126C3×(C2×D3)⋊C4—22no—
127C3×#208—21no—
128C3×#208—31no—
129C3×#208—31no—
130C3×#208—61no—
131C3×#208C2×D311yescharacteristic
132C3×#208C22×C311yescharacteristic
133C3×D24—61no—
134C3×#204—61no—
135#3716—21no—
136#3715—21no—
137#3714—21no—
138#3713—21no—
139C22×C33C2×D411yescharacteristic
140C2×C3×C3⋊D3—21no—
141C2×C32×D3—21no—
142C2×C32×D3—42no—
143C2×C32×D3—41no—
144C33×C4—41no—
145C33×C4C2×D411yescharacteristic
146C33×C4C2×D411yescharacteristic
147C3×C3⋊Dic3—21no—
148C32×Dic3—22no—
149C32×Dic3—21no—
150C3×2−1+4—91no—
151#1010—31no—
152#1007—31no—
153C3×AGL(1,ℤ/8)—91no—
154C3×AGL(1,ℤ/8)—91no—
155C3×#124—31no—
156C3×#124—31no—
157C3×#124—61no—
158#949—31no—
159#932—31no—
160#932—31no—
161#908—31no—
162#875C3×D311yescharacteristic, complemented
163C3×#104—31no—
164C3×#104—31no—
165C3×#104—61no—
166C3×#104—91no—
167C3×#101—91no—
168#836—31no—
169#824—31no—
170#805—31no—
171#805—31no—
172#805—61no—
173#805C3×D311yescharacteristic, complemented
174C23×C32—61no—
175C22×C3×D3—22no—
176C22×C3×D3—21no—
177C22×C3×D3—21no—
178C32×Q8—31no—
179C32×Q8—61no—
180C32×Q8—61no—
181C32×D4—31no—
182C32×D4—61no—
183C32×D4—61no—
184C32×D4—61no—
185C2×C32×C4—21no—
186C2×C32×C4—22no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—32no—
193C2×C32×C4—41no—
194C2×C32×C4—41no—
195C2×C32×C4—41no—
196C2×C32×C4—42no—
197C2×C32×C4—61no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C23×C311yescharacteristic
201#638—21no—
202C4⋊(C3⋊D3)C3×D411yescharacteristic
203C4×C3⋊D3—21no—
204#634C3×D411yescharacteristic
205C3×#67—21no—
206C3×#67—21no—
207C3×#67—41no—
208C3×#67—62no—
209C3×#67—61no—
210C3×#67—61no—
211C3×#67—121no—
212C2×C3×Dic3—22no—
213C2×C3×Dic3—62no—
214C2×C3×Dic3#6712yes—
215C2×C3×Dic3C3×D412yes—
216C3×D12—21no—
217C3×D12—21no—
218C3×D12—21no—
219C3×D12—21no—
220C3×D12—21no—
221C3×D12—31no—
222C3×D12—31no—
223C3×D12—41no—
224C3×D12—41no—
225C3×D12—61no—
226C3×D12—62no—
227C3×D12D1211yescharacteristic
228C3×D12C3×D411yescharacteristic
229C3×C4×D3—21no—
230C3×C4×D3—21no—
231C3×C4×D3—41no—
232C3×C4×D3—62no—
233C3×C4×D3—62no—
234C3×C4×D3—62no—
235C3×C4×D3—61no—
236C3×C4×D3—61no—
237C3×C4×D3—121no—
238C3×Dic6—21no—
239C3×Dic6—21no—
240C3×Dic6—21no—
241C3×Dic6—21no—
242C3×Dic6—21no—
243C3×Dic6—31no—
244C3×Dic6—31no—
245C3×Dic6—41no—
246C3×Dic6—41no—
247C3×Dic6—61no—
248C3×Dic6—62no—
249C3×Dic6—62no—
250C3×Dic6D1211yescharacteristic
251C3×Dic6C3×D411yescharacteristic
252#627—22no—
253#626—22no—
254#624—22no—
255D3×Dic3—22no—
256C32×C8—61no—
257C3⋊(C3⋊C8)—21no—
258C3×C3⋊C8—21no—
259C3×C3⋊C8—21no—
260C3×C3⋊C8—41no—
261C3×C3⋊C8—61no—
262C3×C3⋊C8—61no—
263C3×C3⋊C8—121no—
264#454—91noSylow
265C2×C33—21no—
266C2×C33#12011yescharacteristic
267C3×C3⋊D3—41no—
268C32×D3—41no—
269C32×D3—81no—
270C3×#41—31no—
271C3×#41—31no—
272C3×#41—61no—
273C3×#41—91no—
274C3×#41—182no—
275C3×#41—181no—
276C3×#41—181no—
277C2×C3×Q8—31no—
278C2×C3×Q8—31no—
279C2×C3×Q8—61no—
280C2×C3×Q8—181no—
281C2×C3×Q8—181no—
282C2×C3×D4—31no—
283C2×C3×D4—31no—
284C2×C3×D4—61no—
285#239—61no—
286#239—61no—
287D3×Q8—61no—
288D3×Q8—61no—
289D3×Q8—61no—
290#237—61no—
291#237—62no—
292#235—31no—
293#235—31no—
294#235—61no—
295#235—61no—
296#235—62no—
297#235C2×C3×D311yescharacteristic
298C2×D12C2×C3×D311yescharacteristic
299C2×Dic6—61no—
300C2×Dic6C2×C3×D311yescharacteristic
301C3×SD16—181no—
302C3×SD16—181no—
303C3×D8—181no—
304C3×D8—181no—
305C3×M16—31no—
306C3×M16—31no—
307C3×M16—61no—
308C3×M16—91no—
309C3×C22⋊C4—62no—
310C3×C22⋊C4—62no—
311C3×C22⋊C4—64no—
312C3×C42—21no—
313C3×C42—21no—
314C3×C42—21no—
315C3×C42—41no—
316C3×C42D3211yescharacteristic
317C3×C42C2×C3×D311yescharacteristic
318C3×C42C2×C3×D311yescharacteristic
319#215—61no—
320#214—61no—
321#214—61no—
322#213—61no—
323#213—61no—
324#213—61no—
325(C2×D3)⋊C4—22no—
326#208—31no—
327#208—31no—
328#208—61no—
329#208C2×C3×D311yescharacteristic
330D24—61no—
331#204—61no—
332C22×C32—21no—
333C22×C32—21no—
334C22×C32—21no—
335C22×C32—41no—
336C22×C32—61no—
337C22×C32—122no—
338C22×C32—121no—
339C22×C32D3×D411yescharacteristic
340C22×C32C2×#6711yescharacteristic
341C22×C32C2×C3×D411yescharacteristic
342C2×C3⋊D3—21no—
343C2×C3×D3—21no—
344C2×C3×D3—21no—
345C2×C3×D3—41no—
346C2×C3×D3—42no—
347C2×C3×D3—44no—
348C2×C3×D3—42no—
349C2×C3×D3—42no—
350C2×C3×D3—41no—
351C2×C3×D3—41no—
352C2×C3×D3—61no—
353C2×C3×D3—61no—
354C2×C3×D3—121no—
355C32×C4—21no—
356C32×C4—21no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—41no—
362C32×C4—41no—
363C32×C4—41no—
364C32×C4—42no—
365C32×C4—41no—
366C32×C4—41no—
367C32×C4—42no—
368C32×C4—42no—
369C32×C4—44no—
370C32×C4—62no—
371C32×C4—61no—
372C32×C4C2×D1211yescharacteristic
373C32×C4D3×D411yescharacteristic
374C32×C4D3×D411yescharacteristic
375C32×C4D3×D411yescharacteristic
376C32×C4C2×C3×D411yescharacteristic
377C32×C4C2×C3×D411yescharacteristic
378C3⋊Dic3—21no—
379C3×Dic3—21no—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—21no—
383C3×Dic3—41no—
384C3×Dic3—42no—
385C3×Dic3—61no—
386C3×Dic3—61no—
387C3×Dic3—121no—
3882−1+4—91no—
389AGL(1,ℤ/8)—91no—
390AGL(1,ℤ/8)—91no—
391#124—31no—
392#104—31no—
393#104—91no—
394#101—91no—
395C33#45411yescharacteristic, complemented, Sylow
396C23×C3—62no—
397C23×C3—61no—
398C23×C3—61no—
399C22×D3—21no—
400C3×Q8—31no—
401C3×Q8—31no—
402C3×Q8—61no—
403C3×Q8—61no—
404C3×Q8—61no—
405C3×Q8—61no—
406C3×Q8—61no—
407C3×Q8—91no—
408C3×Q8—121no—
409C3×Q8—121no—
410C3×Q8—182no—
411C3×D4—31no—
412C3×D4—31no—
413C3×D4—61no—
414C3×D4—61no—
415C3×D4—61no—
416C3×D4—61no—
417C3×D4—61no—
418C3×D4—61no—
419C3×D4—61no—
420C3×D4—91no—
421C3×D4—121no—
422C3×D4—121no—
423C3×D4—121no—
424C3×D4—182no—
425C3×D4—181no—
426C2×C3×C4—22no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—21no—
430C2×C3×C4—21no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—32no—
434C2×C3×C4—32no—
435C2×C3×C4—41no—
436C2×C3×C4—41no—
437C2×C3×C4—42no—
438C2×C3×C4—41no—
439C2×C3×C4—62no—
440C2×C3×C4—61no—
441C2×C3×C4—61no—
442C2×C3×C4—121no—
443C2×C3×C4—182no—
444C2×C3×C4—182no—
445C2×C3×C4—181no—
446C2×C3×C4C2×D3211yescharacteristic
447C2×C3×C4C22×C3×D311yescharacteristic
448C2×C3×C4C22×C3×D311yescharacteristic
449#67—21no—
450#67—62no—
451#67—61no—
452#67—61no—
453#67—121no—
454C2×Dic3—62no—
455C2×Dic3C3×#6712yes—
456D12—21no—
457D12—21no—
458D12—31no—
459D12—31no—
460D12—61no—
461D12—62no—
462D12C3×D1211yescharacteristic
463C4×D3—21no—
464C4×D3—62no—
465C4×D3—62no—
466C4×D3—62no—
467C4×D3—61no—
468C4×D3—61no—
469C4×D3—121no—
470Dic6—21no—
471Dic6—21no—
472Dic6—31no—
473Dic6—31no—
474Dic6—61no—
475Dic6—62no—
476Dic6—62no—
477Dic6C3×D1211yescharacteristic
478C3×C8—61no—
479C3×C8—61no—
480C3×C8—121no—
481C3×C8—181no—
482C3⋊C8—21no—
483C3⋊C8—61no—
484C3⋊C8—61no—
485C3⋊C8—121no—
486C2×C32—21no—
487C2×C32—22no—
488C2×C32—21no—
489C2×C32—21no—
490C2×C32—21no—
491C2×C32—21no—
492C2×C32—21no—
493C2×C32—41no—
494C2×C32—41no—
495C2×C32—41no—
496C2×C32—42no—
497C2×C32—121no—
498C2×C32—241no—
499C2×C32#88211yescharacteristic
500C2×C32#93711yescharacteristic
501C2×C32C3×#12011yescharacteristic
502C3⋊D3—41no—
503C3×D3—41no—
504C3×D3—41no—
505C3×D3—81no—
506C3×D3—82no—
507C3×D3—81no—
508C3×D3—81no—
509C3×D3—121no—
510C3×D3—121no—
511C3×D3—241no—
512#41—31no—
513#41—91no—
514#41—182no—
515#41—181no—
516#41—181no—
517C2×Q8—31no—
518C2×Q8—181no—
519C2×Q8—181no—
520C2×D4—31no—
521SD16—181no—
522SD16—181no—
523D8—181no—
524D8—181no—
525M16—31no—
526M16—91no—
527C22⋊C4—62no—
528C42C3×D3211yescharacteristic
529C22×C3—21no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—41no—
533C22×C3—61no—
534C22×C3—61no—
535C22×C3—121no—
536C22×C3—122no—
537C22×C3—124no—
538C22×C3—122no—
539C22×C3—122no—
540C22×C3—121no—
541C22×C3—121no—
542C22×C3—181no—
543C22×C3D3×#6711yescharacteristic
544C22×C3C3×D3×D411yescharacteristic
545C22×C3C2×C3×#6711yescharacteristic
546C2×D3—21no—
547C2×D3—42no—
548C2×D3—41no—
549C2×D3—61no—
550C2×D3—61no—
551C2×D3—121no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—42no—
559C3×C4—41no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4—42no—
563C3×C4—42no—
564C3×C4—44no—
565C3×C4—41no—
566C3×C4—41no—
567C3×C4—62no—
568C3×C4—61no—
569C3×C4—62no—
570C3×C4—61no—
571C3×C4—122no—
572C3×C4—121no—
573C3×C4—181no—
574C3×C4D3×D1211yescharacteristic
575C3×C4#380211yescharacteristic
576C3×C4C2×C3×D1211yescharacteristic
577C3×C4C3×D3×D411yescharacteristic
578C3×C4C3×D3×D411yescharacteristic
579C3×C4C3×D3×D411yescharacteristic
580Dic3—21no—
581Dic3—22no—
582Dic3—61no—
583Dic3—61no—
584Dic3—121no—
585C32—21no—
586C32—21no—
587C32—21no—
588C32—41no—
589C32#482711yescharacteristic, complemented
590C32#506211yescharacteristic, complemented
591C32C3×#45411yescharacteristic, complemented
592C23—61no—
593Q8—31no—
594Q8—61no—
595Q8—61no—
596Q8—91no—
597Q8—182no—
598D4—31no—
599D4—61no—
600D4—61no—
601D4—61no—
602D4—91no—
603D4—182no—
604D4—181no—
605C2×C4—21no—
606C2×C4—32no—
607C2×C4—61no—
608C2×C4—182no—
609C2×C4—182no—
610C2×C4—181no—
611C2×C4C2×C3×D3211yescharacteristic
612C8—61no—
613C8—181no—
614C2×C3—22no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—41no—
622C2×C3—41no—
623C2×C3—42no—
624C2×C3—41no—
625C2×C3—121no—
626C2×C3—121no—
627C2×C3—241no—
628C2×C3—242no—
629C2×C3—241no—
630C2×C3—241no—
631C2×C3—361no—
632C2×C3#6392211yescharacteristic
633C2×C3C3×#88211yescharacteristic
634C2×C3C3×#93711yescharacteristic
635D3—41no—
636D3—81no—
637D3—121no—
638D3—121no—
639D3—241no—
640C22—61no—
641C22—122no—
642C22—121no—
643C22—181no—
644C22C3×D3×#6711yescharacteristic
645C4—41no—
646C4—62no—
647C4—61no—
648C4—181no—
649C4C3×D3×D1211yescharacteristic
650C4C3×#380211yescharacteristic
651C3—21no—
652C3—21no—
653C3—21no—
654C3—41no—
655C3#1059040011yescharacteristic, direct product factor
656C3C3×#482711yescharacteristic, complemented
657C3C3×#506211yescharacteristic, complemented
658C2—21no—
659C2—121no—
660C2—241no—
661C2—361no—
662C2C3×#6392211yescharacteristic
subgroupG/H
centerC2×C3#63922
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC2×C33#120
hypercenterC3×C42D32
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×#64297C2semidirect
2. C32×#875C2semidirect
3. C3×#63830C2semidirect
4. C3×#63823C2semidirect
5. C3×#63632C2semidirect
6. C32×#805C2semidirect
7. C3×#63560C2semidirect
10. #10590400C3direct (N × H)
30. #64297C2×C3semidirect
37. C3×#875D3semidirect
38. C3×#875C2×C3semidirect
39. #63830C2×C3semidirect
40. #63823C2×C3semidirect
42. #63632C2×C3semidirect
49. C3×#805D3semidirect
50. C3×#805C2×C3semidirect
51. #63560C2×C3semidirect
162. #875C3×D3semidirect
173. #805C3×D3semidirect
395. C33#454semidirect
589. C32#4827semidirect
590. C32#5062semidirect
591. C32C3×#454semidirect
655. C3#10590400direct (N × H)
656. C3C3×#4827semidirect
657. C3C3×#5062semidirect

Maximal subgroups

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

subgroupindex
C3×#642972
C32×#8752
C3×#638302
C3×#638232
C3×#636322
C32×#8052
C3×#635602
C3×#50623
C3×#48273
#105904003