Finite Group Databasebeta

← C3×#81281 #11869624

Subgroups — C3×#81281 #11869624

Order 1152.

897 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5606C211yesmaximal, characteristic
2C2×C3×#892C211yesmaximal, characteristic, complemented
3C32×#480C211yesmaximal, characteristic, complemented
4C3×#5034C211yesmaximal, characteristic, complemented
5C3×#4971C211yesmaximal, characteristic, complemented
6C3×#4888C212yesmaximal, complemented
7C3×#4880C212yesmaximal, complemented
8C3×#4784C212yesmaximal, complemented
9C3×#4772C212yesmaximal, complemented
10C3×D3×C22⋊C8C211yesmaximal, characteristic, complemented
11C3×#4722C211yesmaximal, characteristic, complemented
12C3×#3091—31nomaximal
13#81281C311yesmaximal, characteristic, direct product factor
14C2×C3×#237C2211yescharacteristic
15C22×C3×C4×D3C2211yescharacteristic
16C32×#121C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C3×#938C2211yescharacteristic
19C3×#937—21no—
20C3×D4×Dic3—21no—
21C2×C3×(C2×D3)⋊C4—21no—
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C3×#924C2211yescharacteristic
24C2×C3×C4×Dic3—21no—
25C2×C3×D3×C8C2212yes—
26C3×#897C2211yescharacteristic
27C3×#892—21no—
28C3×#892—21no—
29C3×#892C2212yes—
30C3×#892C2212yes—
31C3×#892C2211yescharacteristic
32C3×#883—21no—
33C3×#67⋊C4—21no—
34C32×#107C2212yescomplemented
35C32×#102C2212yescomplemented
36C32×C22⋊C8C2211yescharacteristic
37C3×#832C2212yescomplemented
38C3×#830C2211yescharacteristic
39C3×(C2×D3)⋊C8C2212yes—
40C3×#818C2212yescomplemented
41C3×#816C2212yescomplemented
42C3×#810C2212yescomplemented
43C3×#808C2212yescomplemented
44C3×#808C2212yescomplemented
45C3×#525—31no—
46C2×C3×#117—31no—
47#5606C2×C311yescharacteristic
48C2×#892C2×C311yescharacteristic, complemented
49C3×#485—32no—
50C3×#484—31no—
51C3×#483—31no—
52C3×#482—32no—
53C3×#481—31no—
54C3×#480—21no—
55C3×#480D311yescharacteristic, complemented
56C3×#480C2×C311yescharacteristic, complemented
57C3×#427—32no—
58C3×#416—32no—
59C2×C3×C22⋊C8—31no—
60#5034C2×C311yescharacteristic, complemented
61#4971C2×C311yescharacteristic, complemented
62#4888C2×C312yescomplemented
63#4880C2×C312yescomplemented
64#4784C2×C312yescomplemented
65#4772C2×C312yescomplemented
66D3×C22⋊C8C2×C311yescharacteristic, complemented
67#4722C2×C311yescharacteristic, complemented
68C23×C3×D3D411yescharacteristic
69C2×C32×D4—21no—
70C2×C32×D4C2311yescharacteristic
71C22×C32×C4—21no—
72C22×C32×C4C2311yescharacteristic
73C2×C3×#67—21no—
74C2×C3×#67—21no—
75C22×C3×Dic3—21no—
76C22×C3×Dic3—21no—
77C22×C3×Dic3D411yescharacteristic
78C3×#237—24no—
79C2×C3×C4×D3—24no—
80C2×C3×C4×D3D411yescharacteristic
81C2×C3×C4×D3D411yescharacteristic
82C2×C3×C4×D3C2311yescharacteristic
83C2×C3×C4×D3C2311yescharacteristic
84C2×C3×Dic6C2311yescharacteristic
85C2×C32×C8C2312yes—
86C32×C4⋊C4—21no—
87C32×C4⋊C4C2312yes—
88C32×C4⋊C4C2311yescharacteristic
89C32×C22⋊C4—21no—
90C3×#217—21no—
91C3×#217—21no—
92C3×(C2×D3)⋊C4—21no—
93C3×(C2×D3)⋊C4—21no—
94C3×(C2×D3)⋊C4—22no—
95C3×(C2×D3)⋊C4—22no—
96C3×C4⋊Dic3—21no—
97C3×C4⋊Dic3—21no—
98C3×C4⋊Dic3C2312yes—
99C3×C4⋊Dic3C2311yescharacteristic
100C3×Dic3⋊C4—21no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3—22no—
103C3×C4×Dic3—22no—
104C2×C3×C3⋊C8C2312yes—
105C3×D3×C8—24no—
106#3091—31noSylow
107C2×C3×#41—31no—
108C23×C3×C4—31no—
109C2×#237C22×C311yescharacteristic
110C22×C4×D3C22×C311yescharacteristic
111C22×C3×C8—32no—
112C3×#123—61no—
113C3×C22⋊Q8—31no—
114C3×C22⋊Q8—31no—
115C3×#121—21no—
116C3×#121—31no—
117C3×#121C2×D311yescharacteristic
118C3×#121C22×C311yescharacteristic
119C3×#120—61no—
120C3×C4×D4—61no—
121C3×C4×D4—61no—
122C3×#117—32no—
123C3×#117—31no—
124C3×#117—32no—
125C3×#117—61no—
126C3×#117—61no—
127C2×C3×C4⋊C4—21no—
128C2×C3×C4⋊C4—31no—
129C2×C3×C4⋊C4C2×D311yescharacteristic
130C2×C3×C4⋊C4C22×C311yescharacteristic
131C2×C3×C22⋊C4—61no—
132C2×C3×C42—61no—
133#938C22×C311yescharacteristic
134#937—21no—
135D4×Dic3—21no—
136C2×(C2×D3)⋊C4—21no—
137C2×C4⋊Dic3C22×C311yescharacteristic
138#924C22×C311yescharacteristic
139C2×C4×Dic3—21no—
140C2×D3×C8C22×C312yes—
141#897C22×C311yescharacteristic
142#892—21no—
143#892—21no—
144#892C22×C311yescharacteristic
145#892C22×C312yes—
146#892C22×C312yes—
147#883—21no—
148#67⋊C4—21no—
149C3×#107—22no—
150C3×#107—32no—
151C3×#107C2×D312yescomplemented
152C3×#107C22×C312yescomplemented
153C3×#106—32no—
154C3×#106—32no—
155C3×#103—32no—
156C3×#103—32no—
157C3×#102—22no—
158C3×#102—32no—
159C3×#102C2×D312yescomplemented
160C3×#102C22×C312yescomplemented
161C3×C22⋊C8—21no—
162C3×C22⋊C8—31no—
163C3×C22⋊C8—32no—
164C3×C22⋊C8C2×D311yescharacteristic
165C3×C22⋊C8C22×C311yescharacteristic
166#832C22×C312yescomplemented
167#830C22×C311yescharacteristic
168(C2×D3)⋊C8C22×C312yes—
169#818C22×C312yescomplemented
170#816C22×C312yescomplemented
171#810C22×C312yescomplemented
172#808C22×C312yescomplemented
173#808C22×C312yescomplemented
174C23×C32—21no—
175C23×C32C2×D411yescharacteristic
176C22×C3×D3—22no—
177C22×C3×D3—22no—
178C22×C3×D3—41no—
179C22×C3×D3C2×D411yescharacteristic
180C22×C3×D3C2×D411yescharacteristic
181C32×D4—22no—
182C32×D4—42no—
183C2×C32×C4—22no—
184C2×C32×C4—22no—
185C2×C32×C4—21no—
186C2×C32×C4—42no—
187C2×C32×C4C2×D411yescharacteristic
188C2×C32×C4C2411yescharacteristic
189C3×#67—42no—
190C3×#67—42no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—42no—
196C2×C3×Dic3—42no—
197C2×C3×Dic3C2×D411yescharacteristic
198C2×C3×Dic3C2×D411yescharacteristic
199C3×C4×D3—22no—
200C3×C4×D3—22no—
201C3×C4×D3—22no—
202C3×C4×D3#4114yes—
203C3×Dic6—22no—
204C32×C8—22no—
205C3×C3⋊C8—22no—
206#525—31no—
207C2×#117—31no—
208#485—32no—
209#484—31no—
210#483—31no—
211#482—32no—
212#481—31no—
213#480C3×D311yescharacteristic, complemented
214#427—32no—
215#416—32no—
216C2×C22⋊C8—31no—
217C24×C3—31no—
218C23×D3C3×D411yescharacteristic
219C3×#41—64no—
220C2×C3×Q8—31no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—41no—
225C2×C3×D4—61no—
226C2×C3×D4—61no—
227C2×C3×D4C22×D311yescharacteristic
228C2×C3×D4C23×C311yescharacteristic
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—31no—
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—41no—
238C22×C3×C4—64no—
239C22×C3×C4—61no—
240C22×C3×C4—61no—
241C22×C3×C4C22×D311yescharacteristic
242C22×C3×C4C23×C311yescharacteristic
243C2×#67—21no—
244C2×#67—21no—
245C22×Dic3—21no—
246C22×Dic3—21no—
247C22×Dic3C3×D411yescharacteristic
248#237—24no—
249C2×C4×D3—24no—
250C2×C4×D3C3×D411yescharacteristic
251C2×C4×D3C3×D411yescharacteristic
252C2×C4×D3C23×C311yescharacteristic
253C2×C4×D3C23×C311yescharacteristic
254C2×Dic6C23×C311yescharacteristic
255C2×C3×C8—22no—
256C2×C3×C8—32no—
257C2×C3×C8—64no—
258C2×C3×C8C22×D312yes—
259C2×C3×C8C23×C312yes—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—22no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—32no—
265C3×C4⋊C4—31no—
266C3×C4⋊C4—41no—
267C3×C4⋊C4—61no—
268C3×C4⋊C4—61no—
269C3×C4⋊C4—61no—
270C3×C4⋊C4C22×D312yes—
271C3×C4⋊C4C22×D311yescharacteristic
272C3×C4⋊C4C23×C311yescharacteristic
273C3×C4⋊C4C23×C312yes—
274C3×C22⋊C4—21no—
275C3×C22⋊C4—21no—
276C3×C22⋊C4—41no—
277C3×C22⋊C4—62no—
278C3×C22⋊C4—62no—
279C3×C22⋊C4—61no—
280C3×C22⋊C4—61no—
281C3×C22⋊C4—61no—
282C3×C22⋊C4—61no—
283C3×C42—62no—
284C3×C42—62no—
285C3×C42—61no—
286#217—21no—
287#217—21no—
288(C2×D3)⋊C4—21no—
289(C2×D3)⋊C4—21no—
290(C2×D3)⋊C4—22no—
291(C2×D3)⋊C4—22no—
292C4⋊Dic3—21no—
293C4⋊Dic3—21no—
294C4⋊Dic3C23×C311yescharacteristic
295C4⋊Dic3C23×C312yes—
296Dic3⋊C4—21no—
297C4×Dic3—21no—
298C4×Dic3—22no—
299C4×Dic3—22no—
300C2×C3⋊C8C23×C312yes—
301D3×C8—24no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32#13512yes—
308C22×C32C22×D411yescharacteristic
309C2×C3×D3—21no—
310C2×C3×D3—21no—
311C2×C3×D3—21no—
312C2×C3×D3—21no—
313C2×C3×D3—21no—
314C2×C3×D3—44no—
315C2×C3×D3#12312yes—
316C32×C4—21no—
317C32×C4—42no—
318C32×C4—41no—
319C32×C4C2×#4112yes—
320C3×Dic3—21no—
321C3×Dic3—41no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3#12312yes—
325C2×#41—31no—
326C23×C4—31no—
327C22×C8—32no—
328#123—61no—
329C22⋊Q8—31no—
330C22⋊Q8—31no—
331#121—31no—
332#121C2×C3×D311yescharacteristic
333#120—61no—
334C4×D4—61no—
335C4×D4—61no—
336#117—32no—
337#117—32no—
338#117—31no—
339#117—61no—
340#117—61no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4C2×C3×D311yescharacteristic
343C2×C22⋊C4—61no—
344C2×C42—61no—
345#107—32no—
346#107C2×C3×D312yescomplemented
347#106—32no—
348#106—32no—
349#103—32no—
350#103—32no—
351#102—32no—
352#102C2×C3×D312yescomplemented
353C22⋊C8—31no—
354C22⋊C8—32no—
355C22⋊C8C2×C3×D311yescharacteristic
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—31no—
360C23×C3—31no—
361C23×C3—41no—
362C23×C3—62no—
363C23×C3—62no—
364C23×C3—121no—
365C23×C3D3×D411yescharacteristic
366C23×C3C2×C3×D411yescharacteristic
367C22×D3—22no—
368C22×D3—22no—
369C22×D3—41no—
370C22×D3C2×C3×D411yescharacteristic
371C22×D3C2×C3×D411yescharacteristic
372C3×Q8—62no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—42no—
378C3×D4—82no—
379C3×D4—122no—
380C3×D4—122no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—31no—
390C2×C3×C4—34no—
391C2×C3×C4—31no—
392C2×C3×C4—42no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—42no—
396C2×C3×C4—42no—
397C2×C3×C4—62no—
398C2×C3×C4—62no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4—62no—
402C2×C3×C4—61no—
403C2×C3×C4—61no—
404C2×C3×C4—82no—
405C2×C3×C4—122no—
406C2×C3×C4—122no—
407C2×C3×C4D3×D411yescharacteristic
408C2×C3×C4C2×C3×D411yescharacteristic
409C2×C3×C4C23×D311yescharacteristic
410C2×C3×C4C24×C311yescharacteristic
411#67—42no—
412#67—42no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—42no—
418C2×Dic3—42no—
419C2×Dic3C2×C3×D411yescharacteristic
420C2×Dic3C2×C3×D411yescharacteristic
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—22no—
424C4×D3C3×#4114yes—
425Dic6—22no—
426C3×C8—22no—
427C3×C8—22no—
428C3×C8—42no—
429C3×C8—62no—
430C3⋊C8—22no—
431C2×C32—22no—
432C2×C32—81no—
433C2×C32C2×#12311yescharacteristic
434C2×C32C2×#13511yescharacteristic
435C2×C32#57511yescharacteristic
436C3×D3—22no—
437C3×D3—41no—
438C24—31no—
439#41—64no—
440C2×Q8—31no—
441C2×D4—21no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4C22×C3×D311yescharacteristic
445C22×C4—21no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—64no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4C22×C3×D311yescharacteristic
455C2×C8—32no—
456C2×C8—64no—
457C2×C8C22×C3×D312yes—
458C4⋊C4—21no—
459C4⋊C4—32no—
460C4⋊C4—31no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4C22×C3×D312yes—
465C4⋊C4C22×C3×D311yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—61no—
470C22⋊C4—61no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C42—62no—
474C42—62no—
475C42—61no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—32no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—61no—
492C22×C3—61no—
493C22×C3—61no—
494C22×C3—61no—
495C22×C3—61no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—124no—
500C22×C3#91212yes—
501C22×C3C3×#13512yes—
502C22×C3C2×D3×D411yescharacteristic
503C22×C3C22×C3×D411yescharacteristic
504C2×D3—21no—
505C2×D3—21no—
506C2×D3—21no—
507C2×D3—21no—
508C2×D3—21no—
509C2×D3—44no—
510C2×D3C3×#12312yes—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—22no—
514C3×C4—32no—
515C3×C4—41no—
516C3×C4—42no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—61no—
521C3×C4—82no—
522C3×C4—81no—
523C3×C4—121no—
524C3×C4—122no—
525C3×C4—121no—
526C3×C4D3×#4112yes—
527C3×C4C2×C3×#4112yes—
528Dic3—21no—
529Dic3—42no—
530Dic3—41no—
531Dic3—41no—
532Dic3C3×#12312yes—
533C32#309111yescharacteristic, complemented, Sylow
534C23—21no—
535C23—31no—
536C23—31no—
537C23—62no—
538C23—62no—
539C23—121no—
540C23C3×D3×D411yescharacteristic
541Q8—62no—
542D4—22no—
543D4—42no—
544D4—122no—
545D4—122no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—31no—
550C2×C4—34no—
551C2×C4—31no—
552C2×C4—42no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—122no—
561C2×C4—122no—
562C2×C4C3×D3×D411yescharacteristic
563C2×C4C23×C3×D311yescharacteristic
564C8—22no—
565C8—62no—
566C2×C3—22no—
567C2×C3—22no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—42no—
572C2×C3—62no—
573C2×C3—81no—
574C2×C3—81no—
575C2×C3—121no—
576C2×C3—161no—
577C2×C3D3×#12311yescharacteristic
578C2×C3C2×#91211yescharacteristic
579C2×C3#577711yescharacteristic
580C2×C3C2×C3×#12311yescharacteristic
581C2×C3C2×C3×#13511yescharacteristic
582C2×C3C3×#57511yescharacteristic
583D3—22no—
584D3—41no—
585C22—21no—
586C22—21no—
587C22—32no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—61no—
592C22—61no—
593C22—61no—
594C22—61no—
595C22—61no—
596C22—124no—
597C22C3×#91212yes—
598C22C2×C3×D3×D411yescharacteristic
599C4—21no—
600C4—32no—
601C4—42no—
602C4—41no—
603C4—61no—
604C4—122no—
605C4—121no—
606C4—121no—
607C4C3×D3×#4112yes—
608C3—21no—
609C3#8128111yescharacteristic, direct product factor
610C3C3×#309111yescharacteristic, complemented
611C2—22no—
612C2—62no—
613C2—81no—
614C2—121no—
615C2C3×D3×#12311yescharacteristic
616C2C2×C3×#91211yescharacteristic
617C2C3×#577711yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#480C2
socleC22×C32C22×D4
prime radicalC3×#5606C2
hypercenterC3×#480D3
hypocenterC3C3×#3091
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#480

Complements

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

NH ≅ G/Ntype
2. C2×C3×#892C2semidirect
3. C32×#480C2semidirect
4. C3×#5034C2semidirect
5. C3×#4971C2semidirect
6. C3×#4888C2semidirect
7. C3×#4880C2semidirect
8. C3×#4784C2semidirect
9. C3×#4772C2semidirect
10. C3×D3×C22⋊C8C2semidirect
11. C3×#4722C2semidirect
13. #81281C3direct (N × H)
34. C32×#107C22semidirect
35. C32×#102C22semidirect
37. C3×#832C22semidirect
40. C3×#818C22semidirect
41. C3×#816C22semidirect
42. C3×#810C22semidirect
43. C3×#808C22semidirect
44. C3×#808C22semidirect
48. C2×#892C2×C3semidirect
55. C3×#480D3semidirect
56. C3×#480C2×C3semidirect
60. #5034C2×C3semidirect
61. #4971C2×C3semidirect
62. #4888C2×C3semidirect
63. #4880C2×C3semidirect
64. #4784C2×C3semidirect
65. #4772C2×C3semidirect
66. D3×C22⋊C8C2×C3semidirect
67. #4722C2×C3semidirect
151. C3×#107C2×D3semidirect
152. C3×#107C22×C3semidirect
159. C3×#102C2×D3semidirect
160. C3×#102C22×C3semidirect
166. #832C22×C3semidirect
169. #818C22×C3semidirect
170. #816C22×C3semidirect
171. #810C22×C3semidirect
172. #808C22×C3semidirect
173. #808C22×C3semidirect
213. #480C3×D3semidirect
346. #107C2×C3×D3semidirect
352. #102C2×C3×D3semidirect
533. C32#3091semidirect
609. C3#81281direct (N × H)
610. C3C3×#3091semidirect

Maximal subgroups

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

subgroupindex
C3×#56062
C2×C3×#8922
C32×#4802
C3×#50342
C3×#49712
C3×#48882
C3×#48802
C3×#47842
C3×#47722
C3×D3×C22⋊C82
C3×#47222
C3×#30913
#812813