Finite Group Databasebeta

← C3×#10591533 #13158477

Subgroups — C3×#10591533 #13158477

Order 1728.

990 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1004C211yesmaximal, characteristic
2C32×#935C211yesmaximal, characteristic, complemented
3C3×#63961C212yesmaximal, complemented
4C3×#63661C211yesmaximal, characteristic, complemented
5C3×#63580C212yesmaximal, complemented
6C3×#5243—31nomaximal
7C3×#5053—31nomaximal
8#10591533C311yesmaximal, characteristic, direct product factor
9C2×C33×D4C2211yescharacteristic
10C2×C32×#67—21no—
11C2×C32×#67C2212yes—
12C22×C32×Dic3C2211yescharacteristic, complemented
13C32×#237—22no—
14C32×D3×D4—22no—
15C32×#235—21no—
16C3×#3757C2212yescomplemented
17C32×#217—21no—
18C32×#217C2211yescharacteristic, complemented
19C32×Dic3⋊C4—21no—
20C3×#3724—22no—
21C3×(C2×D3)⋊Dic3—22no—
22C32×H(5,𝔽2)—31no—
23C3×#1004—21no—
24C3×#1004D311yescharacteristic
25C3×#1004C2×C311yescharacteristic
26C32×#123—31no—
27C3×#935—21no—
28C3×#935—32no—
29C3×#935D311yescharacteristic, complemented
30C3×#935C2×C311yescharacteristic, complemented
31C3×#929—32no—
32#63961C2×C312yescomplemented
33#63661C2×C311yescharacteristic, complemented
34C3×#834—31no—
35C3×#834—31no—
36C3×#834—32no—
37C3×#834—61no—
38C3×#806—32no—
39#63580C2×C312yescomplemented
40C23×C33D411yescharacteristic
41C23×C33C2311yescharacteristic
42C22×C32×D3—21no—
43C22×C32×D3D412yes—
44C33×D4—22no—
45C2×C33×C4D411yescharacteristic
46C2×C3×C3⋊Dic3—22no—
47C32×#67—22no—
48C32×#67—22no—
49C32×#67—22no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—22no—
54C2×C32×Dic3D412yes—
55C32×D12—21no—
56C32×C4×D3—22no—
57C32×Dic6—21no—
58C3×#458—91no—
59#5243—31no—
60#5053—31no—
61C32×#41—62no—
62C32×#41—61no—
63C2×C32×D4—21no—
64C2×C32×D4—21no—
65C2×C32×D4—21no—
66C2×C32×D4—32no—
67C2×C32×D4—41no—
68C2×C32×D4—62no—
69C2×C32×D4—61no—
70C2×C32×D4C2×D311yescharacteristic
71C2×C32×D4C2×D311yescharacteristic
72C2×C32×D4C22×C311yescharacteristic
73C22×C32×C4—31no—
74C2×C3×#67—22no—
75C2×C3×#67—22no—
76C2×C3×#67—21no—
77C2×C3×#67—21no—
78C2×C3×#67C2×D312yes—
79C2×C3×#67C22×C312yes—
80C22×C3×Dic3—21no—
81C22×C3×Dic3C2×D311yescharacteristic, complemented
82C22×C3×Dic3C22×C311yescharacteristic, complemented
83C3×#237—24no—
84C3×#237—22no—
85C3×#237—22no—
86C3×D3×D4—24no—
87C3×D3×D4—22no—
88C3×D3×D4—22no—
89C3×#235—22no—
90C3×#235—21no—
91C3×#235—21no—
92C32×C4⋊C4—61no—
93C32×C22⋊C4—31no—
94C32×C22⋊C4—61no—
95#3757C22×C312yescomplemented
96C3×#217—22no—
97C3×#217—21no—
98C3×#217—21no—
99C3×#217—21no—
100C3×#217—32no—
101C3×#217—32no—
102C3×#217—62no—
103C3×#217—62no—
104C3×#217C2×D311yescharacteristic, complemented
105C3×#217C22×C311yescharacteristic, complemented
106C3×(C2×D3)⋊C4—62no—
107C3×Dic3⋊C4—22no—
108C3×Dic3⋊C4—21no—
109C3×Dic3⋊C4—21no—
110C3×Dic3⋊C4—62no—
111C3×Dic3⋊C4—62no—
112#3724—22no—
113(C2×D3)⋊Dic3—22no—
114C22×C33—21no—
115C22×C33—41no—
116C22×C33—41no—
117C22×C33C2×D412yes—
118C22×C33C2×D411yescharacteristic
119C2×C32×D3—22no—
120C2×C32×D3—42no—
121C2×C32×D3—81no—
122C33×C4—21no—
123C3×C3⋊Dic3—42no—
124C32×Dic3—22no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C3×H(5,𝔽2)—31no—
128C3×H(5,𝔽2)—31no—
129C3×H(5,𝔽2)—61no—
130#1004C3×D311yescharacteristic
131C3×#123—31no—
132C3×#123—31no—
133C3×#123—61no—
134C3×#123—92no—
135#935—32no—
136#935C3×D311yescharacteristic, complemented
137#929—32no—
138C3×#99—92no—
139C3×#99—91no—
140#834—31no—
141#834—31no—
142#834—32no—
143#834—61no—
144#806—32no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32—32no—
152C23×C32—41no—
153C23×C32—41no—
154C23×C32—61no—
155C23×C32#6711yescharacteristic
156C23×C32#6711yescharacteristic
157C23×C32C3×D411yescharacteristic
158C23×C32C22×D311yescharacteristic
159C23×C32C22×D311yescharacteristic
160C23×C32C23×C311yescharacteristic
161C22×C3×D3—22no—
162C22×C3×D3—21no—
163C22×C3×D3—21no—
164C22×C3×D3—22no—
165C22×C3×D3#6712yes—
166C22×C3×D3C3×D412yes—
167C32×Q8—61no—
168C32×D4—22no—
169C32×D4—24no—
170C32×D4—22no—
171C32×D4—22no—
172C32×D4—42no—
173C32×D4—42no—
174C32×D4—44no—
175C32×D4—61no—
176C32×D4—62no—
177C32×D4—62no—
178C32×D4—62no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—32no—
183C2×C32×C4—41no—
184C2×C32×C4—62no—
185C2×C32×C4—61no—
186C2×C32×C4—61no—
187C2×C32×C4—61no—
188C2×C32×C4—62no—
189C2×C32×C4#6711yescharacteristic
190C2×C32×C4#6711yescharacteristic
191C2×C32×C4C3×D411yescharacteristic
192C2×C3⋊Dic3—22no—
193C3×#67—24no—
194C3×#67—24no—
195C3×#67—22no—
196C3×#67—22no—
197C3×#67—24no—
198C3×#67—22no—
199C3×#67—22no—
200C3×#67—22no—
201C3×#67—22no—
202C2×C3×Dic3—22no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—21no—
205C2×C3×Dic3—22no—
206C2×C3×Dic3—22no—
207C2×C3×Dic3—22no—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—21no—
210C2×C3×Dic3—22no—
211C2×C3×Dic3—21no—
212C2×C3×Dic3—21no—
213C2×C3×Dic3—22no—
214C2×C3×Dic3—42no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—62no—
217C2×C3×Dic3—122no—
218C2×C3×Dic3#6712yes—
219C2×C3×Dic3C3×D412yes—
220C3×D12—22no—
221C3×D12—21no—
222C3×D12—21no—
223C3×C4×D3—24no—
224C3×C4×D3—22no—
225C3×C4×D3—22no—
226C3×Dic6—22no—
227C3×Dic6—21no—
228C3×Dic6—21no—
229#458—91noSylow
230C2×C33—21no—
231C2×C33—22no—
232C2×C33—41no—
233C2×C33#12011yescharacteristic
234C32×D3—42no—
235C3×#41—62no—
236C3×#41—62no—
237C3×#41—62no—
238C3×#41—64no—
239C3×#41—61no—
240C3×#41—61no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4—32no—
245C2×C3×D4—32no—
246C2×C3×D4—41no—
247C2×C3×D4—62no—
248C2×C3×D4—62no—
249C2×C3×D4—62no—
250C2×C3×D4—64no—
251C2×C3×D4—62no—
252C2×C3×D4—61no—
253C2×C3×D4—61no—
254C2×C3×D4D3211yescharacteristic
255C2×C3×D4C2×C3×D311yescharacteristic
256C2×C3×D4C2×C3×D311yescharacteristic
257C22×C3×C4—31no—
258C22×C3×C4—31no—
259C22×C3×C4—61no—
260C2×#67—21no—
261C2×#67C2×C3×D312yes—
262C22×Dic3C2×C3×D311yescharacteristic, complemented
263#237—22no—
264D3×D4—22no—
265#235—21no—
266C3×C4⋊C4—62no—
267C3×C4⋊C4—61no—
268C3×C4⋊C4—61no—
269C3×C4⋊C4—182no—
270C3×C22⋊C4—31no—
271C3×C22⋊C4—31no—
272C3×C22⋊C4—61no—
273C3×C22⋊C4—62no—
274C3×C22⋊C4—61no—
275C3×C22⋊C4—61no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4—182no—
278#217—21no—
279#217—32no—
280#217—32no—
281#217—62no—
282#217—62no—
283#217C2×C3×D311yescharacteristic, complemented
284(C2×D3)⋊C4—62no—
285Dic3⋊C4—21no—
286Dic3⋊C4—62no—
287Dic3⋊C4—62no—
288C22×C32—21no—
289C22×C32—22no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—22no—
294C22×C32—22no—
295C22×C32—21no—
296C22×C32—22no—
297C22×C32—21no—
298C22×C32—41no—
299C22×C32—42no—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—42no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—42no—
310C22×C32—42no—
311C22×C32—62no—
312C22×C32—81no—
313C22×C32—81no—
314C22×C32—81no—
315C22×C32—81no—
316C22×C32—82no—
317C22×C32—82no—
318C22×C32—122no—
319C22×C32—241no—
320C22×C32D3×D412yes—
321C22×C32C2×#6711yescharacteristic
322C22×C32C2×#6711yescharacteristic
323C22×C32C2×#6712yes—
324C22×C32C2×C3×D412yes—
325C22×C32C2×C3×D411yescharacteristic
326C2×C3×D3—24no—
327C2×C3×D3—22no—
328C2×C3×D3—22no—
329C2×C3×D3—42no—
330C2×C3×D3—42no—
331C2×C3×D3—82no—
332C2×C3×D3—82no—
333C2×C3×D3—81no—
334C2×C3×D3—81no—
335C32×C4—21no—
336C32×C4—22no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—41no—
340C32×C4—41no—
341C32×C4—42no—
342C32×C4—62no—
343C32×C4—121no—
344C32×C4—121no—
345C3⋊Dic3—42no—
346C3×Dic3—24no—
347C3×Dic3—22no—
348C3×Dic3—22no—
349C3×Dic3—42no—
350C3×Dic3—41no—
351C3×Dic3—41no—
352C3×Dic3—42no—
353C3×Dic3—41no—
354C3×Dic3—41no—
355C3×Dic3—122no—
356C3×Dic3—122no—
357C3×Dic3—242no—
358H(5,𝔽2)—31no—
359#123—31no—
360#123—92no—
361#99—92no—
362#99—91no—
363C33#45811yescharacteristic, complemented, Sylow
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—32no—
371C23×C3—32no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—62no—
375C23×C3—62no—
376C23×C3—61no—
377C23×C3—61no—
378C23×C3#62511yescharacteristic
379C23×C3C3×#6711yescharacteristic
380C23×C3C3×#6711yescharacteristic
381C23×C3C2×D3211yescharacteristic
382C23×C3C22×C3×D311yescharacteristic
383C23×C3C22×C3×D311yescharacteristic
384C22×D3—21no—
385C22×D3C3×#6712yes—
386C3×Q8—62no—
387C3×Q8—61no—
388C3×Q8—61no—
389C3×D4—24no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—22no—
393C3×D4—42no—
394C3×D4—42no—
395C3×D4—44no—
396C3×D4—62no—
397C3×D4—61no—
398C3×D4—61no—
399C3×D4—64no—
400C3×D4—62no—
401C3×D4—62no—
402C3×D4—64no—
403C3×D4—64no—
404C3×D4—62no—
405C3×D4—62no—
406C3×D4—62no—
407C3×D4—62no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—32no—
412C2×C3×C4—32no—
413C2×C3×C4—41no—
414C2×C3×C4—62no—
415C2×C3×C4—64no—
416C2×C3×C4—62no—
417C2×C3×C4—62no—
418C2×C3×C4—62no—
419C2×C3×C4—61no—
420C2×C3×C4—61no—
421C2×C3×C4—62no—
422C2×C3×C4—62no—
423C2×C3×C4—61no—
424C2×C3×C4—61no—
425C2×C3×C4—62no—
426C2×C3×C4—62no—
427C2×C3×C4—61no—
428C2×C3×C4—61no—
429C2×C3×C4—122no—
430C2×C3×C4—182no—
431C2×C3×C4#62511yescharacteristic
432C2×C3×C4C3×#6711yescharacteristic
433C2×C3×C4C3×#6711yescharacteristic
434#67—22no—
435#67—22no—
436#67—22no—
437C2×Dic3—21no—
438C2×Dic3—22no—
439C2×Dic3—21no—
440C2×Dic3—21no—
441C2×Dic3—62no—
442C2×Dic3—62no—
443C2×Dic3—122no—
444C2×Dic3C3×#6712yes—
445D12—21no—
446C4×D3—22no—
447Dic6—21no—
448C2×C32—21no—
449C2×C32—22no—
450C2×C32—22no—
451C2×C32—21no—
452C2×C32—22no—
453C2×C32—21no—
454C2×C32—22no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—41no—
459C2×C32—41no—
460C2×C32—41no—
461C2×C32—42no—
462C2×C32—42no—
463C2×C32—42no—
464C2×C32—42no—
465C2×C32—41no—
466C2×C32—42no—
467C2×C32—41no—
468C2×C32—41no—
469C2×C32—82no—
470C2×C32—81no—
471C2×C32—81no—
472C2×C32—82no—
473C2×C32—122no—
474C2×C32#93711yescharacteristic
475C2×C32#95311yescharacteristic
476C2×C32C3×#12011yescharacteristic
477C3×D3—44no—
478C3×D3—42no—
479C3×D3—42no—
480#41—62no—
481#41—61no—
482C2×D4—32no—
483C2×D4—62no—
484C2×D4—61no—
485C2×D4C3×D3211yescharacteristic
486C22×C4—31no—
487C4⋊C4—61no—
488C4⋊C4—182no—
489C22⋊C4—31no—
490C22⋊C4—61no—
491C22⋊C4—92no—
492C22⋊C4—182no—
493C22×C3—22no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—42no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—41no—
515C22×C3—42no—
516C22×C3—64no—
517C22×C3—62no—
518C22×C3—62no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—82no—
524C22×C3—82no—
525C22×C3—122no—
526C22×C3—122no—
527C22×C3—242no—
528C22×C3—242no—
529C22×C3—241no—
530C22×C3—241no—
531C22×C3C2×#62511yescharacteristic
532C22×C3D3×#6712yes—
533C22×C3C3×D3×D412yes—
534C22×C3C2×C3×#6711yescharacteristic
535C22×C3C2×C3×#6711yescharacteristic
536C22×C3C2×C3×#6712yes—
537C2×D3—22no—
538C2×D3—42no—
539C2×D3—81no—
540C3×C4—22no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—42no—
547C3×C4—64no—
548C3×C4—62no—
549C3×C4—62no—
550C3×C4—122no—
551C3×C4—122no—
552C3×C4—121no—
553C3×C4—121no—
554C3×C4—121no—
555C3×C4—121no—
556C3×C4—362no—
557Dic3—22no—
558Dic3—41no—
559Dic3—41no—
560Dic3—122no—
561Dic3—122no—
562Dic3—242no—
563C32—21no—
564C32—21no—
565C32—21no—
566C32—41no—
567C32#505311yescharacteristic, complemented
568C32#524311yescharacteristic, complemented
569C32C3×#45811yescharacteristic, complemented
570C23—32no—
571C23—61no—
572C23C3×#62511yescharacteristic
573C23C2×C3×D3211yescharacteristic
574Q8—61no—
575D4—22no—
576D4—61no—
577D4—62no—
578D4—62no—
579D4—62no—
580C2×C4—32no—
581C2×C4—62no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—62no—
586C2×C4—182no—
587C2×C4C3×#62511yescharacteristic
588C2×C3—22no—
589C2×C3—21no—
590C2×C3—22no—
591C2×C3—22no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—22no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—42no—
599C2×C3—42no—
600C2×C3—41no—
601C2×C3—41no—
602C2×C3—42no—
603C2×C3—42no—
604C2×C3—42no—
605C2×C3—41no—
606C2×C3—41no—
607C2×C3—41no—
608C2×C3—41no—
609C2×C3—82no—
610C2×C3—81no—
611C2×C3—81no—
612C2×C3—82no—
613C2×C3—124no—
614C2×C3—122no—
615C2×C3—122no—
616C2×C3#6397611yescharacteristic
617C2×C3C3×#93711yescharacteristic
618C2×C3C3×#95311yescharacteristic
619D3—42no—
620C22—21no—
621C22—41no—
622C22—41no—
623C22—62no—
624C22—122no—
625C22—241no—
626C22C2×C3×#62511yescharacteristic
627C22C3×D3×#6712yes—
628C4—21no—
629C4—62no—
630C4—121no—
631C4—121no—
632C4—362no—
633C3—21no—
634C3—21no—
635C3—21no—
636C3—41no—
637C3#1059153311yescharacteristic, direct product factor
638C3C3×#505311yescharacteristic, complemented
639C3C3×#524311yescharacteristic, complemented
640C2—21no—
641C2—22no—
642C2—41no—
643C2—122no—
644C2C3×#6397611yescharacteristic
subgroupG/H
centerC2×C3#63976
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33#120
prime radicalC32×#1004C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#458
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C32×#935C2semidirect
3. C3×#63961C2semidirect
4. C3×#63661C2semidirect
5. C3×#63580C2semidirect
8. #10591533C3direct (N × H)
12. C22×C32×Dic3C22semidirect
16. C3×#3757C22semidirect
18. C32×#217C22semidirect
29. C3×#935D3semidirect
30. C3×#935C2×C3semidirect
32. #63961C2×C3semidirect
33. #63661C2×C3semidirect
39. #63580C2×C3semidirect
81. C22×C3×Dic3C2×D3semidirect
82. C22×C3×Dic3C22×C3semidirect
95. #3757C22×C3semidirect
104. C3×#217C2×D3semidirect
105. C3×#217C22×C3semidirect
136. #935C3×D3semidirect
262. C22×Dic3C2×C3×D3semidirect
283. #217C2×C3×D3semidirect
363. C33#458semidirect
567. C32#5053semidirect
568. C32#5243semidirect
569. C32C3×#458semidirect
637. C3#10591533direct (N × H)
638. C3C3×#5053semidirect
639. C3C3×#5243semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
C32×#9352
C3×#639612
C3×#636612
C3×#635802
C3×#52433
C3×#50533
#105915333