Finite Group Databasebeta

← C3×#81309 #11870841

Subgroups — C3×#81309 #11870841

Order 1152.

965 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5600C211yesmaximal, characteristic, complemented
2C2×C3×#893C211yesmaximal, characteristic, complemented
3C32×#481C211yesmaximal, characteristic, complemented
4C3×#5036C211yesmaximal, characteristic
5C3×#4970C211yesmaximal, characteristic, complemented
6C3×#4871C212yesmaximal, complemented
7C3×#4852C212yesmaximal, complemented
8C3×#4787C212yesmaximal, complemented
9C3×#4756C212yesmaximal, complemented
10C3×#4731C211yesmaximal, characteristic, complemented
11C3×#4718C211yesmaximal, characteristic, complemented
12C3×#3105—31nomaximal
13#81309C311yesmaximal, characteristic, direct product factor
14C22×C3×D12C2211yescharacteristic
15C22×C3×C4×D3—21no—
16C32×#121C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C3×#936C2211yescharacteristic, complemented
19C3×#935—21no—
20C3×D4×Dic3—21no—
21C3×D4×Dic3C2211yescharacteristic, complemented
22C2×C3×#213C2212yes—
23C2×C3×(C2×D3)⋊C4—21no—
24C3×#924C2211yescharacteristic
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×C3×#204C2212yes—
27C3×#893—21no—
28C3×#893—21no—
29C3×#893C2212yes—
30C3×#893C2212yes—
31C3×#890C2211yescharacteristic, complemented
32C3×#887C2211yescharacteristic, complemented
33C3×#879—21no—
34C3×#877—21no—
35C32×#106C2212yescomplemented
36C32×#102C2212yescomplemented
37C32×C22⋊C8C2211yescharacteristic
38C3×#830C2211yescharacteristic
39C3×#821C2212yes—
40C3×Dic3⋊C8C2212yescomplemented
41C3×C8×Dic3C2212yescomplemented
42C3×#810C2212yescomplemented
43C3×#809C2212yes—
44C3×#807C2212yescomplemented
45C3×#529—31no—
46C2×C3×C4×D4—31no—
47#5600C2×C311yescharacteristic, complemented
48C2×#893C2×C311yescharacteristic, complemented
49C3×#484—31no—
50C3×#481—21no—
51C3×#481D311yescharacteristic, complemented
52C3×#481C2×C311yescharacteristic, complemented
53C3×#478—32no—
54C3×#463—32no—
55C3×#450—31no—
56C3×#447—31no—
57C3×C4×SD16—32no—
58C3×C4×D8—32no—
59C3×#431—31no—
60#5036C2×C311yescharacteristic
61#4970C2×C311yescharacteristic, complemented
62#4871C2×C312yescomplemented
63#4852C2×C312yescomplemented
64#4787C2×C312yescomplemented
65#4756C2×C312yescomplemented
66#4731C2×C311yescharacteristic, complemented
67#4718C2×C311yescharacteristic, complemented
68C23×C3×D3—21no—
69C2×C32×D4—21no—
70C2×C32×D4C2311yescharacteristic
71C22×C32×C4—21no—
72C22×C32×C4C2311yescharacteristic
73C22×C3×Dic3—21no—
74C22×C3×Dic3D411yescharacteristic
75C22×C3×Dic3D411yescharacteristic
76C2×C3×D12—21no—
77C2×C3×D12—24no—
78C2×C3×D12C2312yes—
79C2×C3×C4×D3—22no—
80C2×C3×C4×D3—22no—
81C2×C3×C4×D3—44no—
82C2×C3×Dic6C2311yescharacteristic
83C2×C32×C8C2312yes—
84C32×C4⋊C4—21no—
85C32×C4⋊C4C2311yescharacteristic
86C32×C4⋊C4C2312yes—
87C32×C22⋊C4—21no—
88C3×#217—21no—
89C3×#217—21no—
90C3×#217—21no—
91C3×#217—21no—
92C3×#213—24no—
93C3×(C2×D3)⋊C4—22no—
94C3×(C2×D3)⋊C4—22no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3C2311yescharacteristic
97C3×Dic3⋊C4—21no—
98C3×Dic3⋊C4—21no—
99C3×Dic3⋊C4—21no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3D411yescharacteristic
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3C2311yescharacteristic
104C3×C4×Dic3C2311yescharacteristic
105C2×C3×C3⋊C8C2312yes—
106C3×#204—24no—
107#3105—31noSylow
108C22×C3×D4—31no—
109C23×C3×C4—61no—
110C22×D12C22×C311yescharacteristic
111C22×C4×D3—21no—
112C2×C3×SD16—32no—
113C2×C3×D8—32no—
114C3×#126—61no—
115C3×#125—31no—
116C3×#124—31no—
117C3×#123—61no—
118C3×C22⋊Q8—31no—
119C3×#121—21no—
120C3×#121C2×D311yescharacteristic
121C3×#121C22×C311yescharacteristic
122C3×C4×Q8—31no—
123C3×C4×D4—31no—
124C3×C4×D4—32no—
125C3×C4×D4—32no—
126C3×C4×D4—61no—
127C3×C4×D4—61no—
128C3×C4×D4—61no—
129C3×#117—61no—
130C2×C3×C4⋊C4—21no—
131C2×C3×C4⋊C4C2×D311yescharacteristic
132C2×C3×C4⋊C4C22×C311yescharacteristic
133C2×C3×C22⋊C4—61no—
134C2×C3×C42—31no—
135#936C22×C311yescharacteristic, complemented
136#935—21no—
137D4×Dic3—21no—
138D4×Dic3C22×C311yescharacteristic, complemented
139C2×#213C22×C312yes—
140C2×(C2×D3)⋊C4—21no—
141#924C22×C311yescharacteristic
142C2×C4×Dic3C22×C311yescharacteristic
143C2×#204C22×C312yes—
144#893—21no—
145#893—21no—
146#893C22×C312yes—
147#893C22×C312yes—
148#890C22×C311yescharacteristic, complemented
149#887C22×C311yescharacteristic, complemented
150#879—21no—
151#877—21no—
152C3×#107—32no—
153C3×#106—22no—
154C3×#106C2×D312yescomplemented
155C3×#106C22×C312yescomplemented
156C3×C4⋊C8—32no—
157C3×#103—32no—
158C3×#102—22no—
159C3×#102—32no—
160C3×#102—32no—
161C3×#102C2×D312yescomplemented
162C3×#102C22×C312yescomplemented
163C3×C22⋊C8—21no—
164C3×C22⋊C8—31no—
165C3×C22⋊C8C2×D311yescharacteristic
166C3×C22⋊C8C22×C311yescharacteristic
167C3×C4×C8—32no—
168#830C22×C311yescharacteristic
169#821C22×C312yes—
170Dic3⋊C8C22×C312yescomplemented
171C8×Dic3C22×C312yescomplemented
172#810C22×C312yescomplemented
173#809C22×C312yes—
174#807C22×C312yescomplemented
175C23×C32—21no—
176C23×C32C2×D411yescharacteristic
177C22×C3×D3—24no—
178C22×C3×D3—22no—
179C22×C3×D3—42no—
180C22×C3×D3—42no—
181C32×D4—22no—
182C32×D4—42no—
183C2×C32×C4—22no—
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4—42no—
187C2×C32×C4C2×D411yescharacteristic
188C2×C32×C4C2411yescharacteristic
189C2×C3×Dic3—22no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—42no—
194C2×C3×Dic3C2×D411yescharacteristic
195C2×C3×Dic3C2×D411yescharacteristic
196C2×C3×Dic3C2×D411yescharacteristic
197C2×C3×Dic3C2×D411yescharacteristic
198C3×D12—24no—
199C3×D12—22no—
200C3×D12#4114yes—
201C3×C4×D3—44no—
202C3×C4×D3—44no—
203C3×Dic6—22no—
204C32×C8—22no—
205C3×C3⋊C8—22no—
206#529—31no—
207C2×C4×D4—31no—
208#484—31no—
209#481C3×D311yescharacteristic, complemented
210#478—32no—
211#463—32no—
212#450—31no—
213#447—31no—
214C4×SD16—32no—
215C4×D8—32no—
216#431—31no—
217C24×C3—61no—
218C23×D3—21no—
219C2×C3×Q8—31no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—32no—
224C2×C3×D4—41no—
225C2×C3×D4—64no—
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—41no—
235C22×C3×C4—62no—
236C22×C3×C4—62no—
237C22×C3×C4—61no—
238C22×C3×C4—124no—
239C22×C3×C4C22×D311yescharacteristic
240C22×C3×C4C23×C311yescharacteristic
241C22×Dic3—21no—
242C22×Dic3C3×D411yescharacteristic
243C22×Dic3C3×D411yescharacteristic
244C2×D12—21no—
245C2×D12—24no—
246C2×D12C23×C312yes—
247C2×C4×D3—22no—
248C2×C4×D3—22no—
249C2×C4×D3—44no—
250C2×Dic6C23×C311yescharacteristic
251C3×SD16—64no—
252C3×D8—64no—
253C2×C3×C8—22no—
254C2×C3×C8—32no—
255C2×C3×C8C22×D312yes—
256C2×C3×C8C23×C312yes—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—41no—
263C3×C4⋊C4—61no—
264C3×C4⋊C4—61no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—61no—
267C3×C4⋊C4C22×D311yescharacteristic
268C3×C4⋊C4C22×D312yes—
269C3×C4⋊C4C23×C312yes—
270C3×C4⋊C4C23×C311yescharacteristic
271C3×C22⋊C4—21no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—41no—
274C3×C22⋊C4—62no—
275C3×C22⋊C4—62no—
276C3×C22⋊C4—61no—
277C3×C22⋊C4—61no—
278C3×C22⋊C4—61no—
279C3×C22⋊C4—61no—
280C3×C42—31no—
281C3×C42—31no—
282C3×C42—31no—
283C3×C42—31no—
284C3×C42—61no—
285#217—21no—
286#217—21no—
287#217—21no—
288#217—21no—
289#213—24no—
290(C2×D3)⋊C4—22no—
291(C2×D3)⋊C4—22no—
292C4⋊Dic3—21no—
293C4⋊Dic3C23×C311yescharacteristic
294Dic3⋊C4—21no—
295Dic3⋊C4—21no—
296Dic3⋊C4—21no—
297C4×Dic3—21no—
298C4×Dic3C3×D411yescharacteristic
299C4×Dic3C3×D411yescharacteristic
300C4×Dic3C23×C311yescharacteristic
301C4×Dic3C23×C311yescharacteristic
302C2×C3⋊C8C23×C312yes—
303#204—24no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32#13512yes—
310C22×C32C22×D411yescharacteristic
311C2×C3×D3—24no—
312C2×C3×D3—48no—
313C2×C3×D3—42no—
314C2×C3×D3—42no—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4—42no—
318C32×C4C2×#4112yes—
319C3×Dic3—21no—
320C3×Dic3—21no—
321C3×Dic3—21no—
322C3×Dic3—41no—
323C3×Dic3—41no—
324C3×Dic3AGL(1,ℤ/8)12yes—
325C22×D4—31no—
326C23×C4—61no—
327C2×SD16—32no—
328C2×D8—32no—
329#126—61no—
330#125—31no—
331#124—31no—
332#123—61no—
333C22⋊Q8—31no—
334#121C2×C3×D311yescharacteristic
335C4×Q8—31no—
336C4×D4—31no—
337C4×D4—32no—
338C4×D4—32no—
339C4×D4—61no—
340C4×D4—61no—
341C4×D4—61no—
342#117—61no—
343C2×C4⋊C4C2×C3×D311yescharacteristic
344C2×C22⋊C4—61no—
345C2×C42—31no—
346#107—32no—
347#106C2×C3×D312yescomplemented
348C4⋊C8—32no—
349#103—32no—
350#102—32no—
351#102—32no—
352#102C2×C3×D312yescomplemented
353C22⋊C8—31no—
354C22⋊C8C2×C3×D311yescharacteristic
355C4×C8—32no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—41no—
360C23×C3—64no—
361C23×C3—62no—
362C23×C3—122no—
363C23×C3—122no—
364C23×C3D3×D411yescharacteristic
365C23×C3C2×C3×D411yescharacteristic
366C22×D3—24no—
367C22×D3—22no—
368C22×D3—42no—
369C22×D3—42no—
370C3×Q8—62no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—34no—
374C3×D4—42no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—64no—
378C3×D4—62no—
379C3×D4—82no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
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—61no—
400C2×C3×C4—61no—
401C2×C3×C4—82no—
402C2×C3×C4—124no—
403C2×C3×C4—122no—
404C2×C3×C4—124no—
405C2×C3×C4D3×D411yescharacteristic
406C2×C3×C4C2×C3×D411yescharacteristic
407C2×C3×C4C23×D311yescharacteristic
408C2×C3×C4C24×C311yescharacteristic
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—42no—
414C2×Dic3C2×C3×D411yescharacteristic
415C2×Dic3C2×C3×D411yescharacteristic
416C2×Dic3C2×C3×D411yescharacteristic
417C2×Dic3C2×C3×D411yescharacteristic
418D12—24no—
419D12—22no—
420D12C3×#4114yes—
421C4×D3—44no—
422C4×D3—44no—
423Dic6—22no—
424C3×C8—22no—
425C3×C8—22no—
426C3×C8—42no—
427C3×C8—62no—
428C3⋊C8—22no—
429C2×C32—22no—
430C2×C32—81no—
431C2×C32#52511yescharacteristic
432C2×C32C2×#13511yescharacteristic
433C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
434C3×D3—44no—
435C24—61no—
436C2×Q8—31no—
437C2×D4—21no—
438C2×D4—32no—
439C2×D4—64no—
440C2×D4—61no—
441C2×D4C22×C3×D311yescharacteristic
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—61no—
448C22×C4—124no—
449C22×C4C22×C3×D311yescharacteristic
450SD16—64no—
451D8—64no—
452C2×C8—32no—
453C2×C8C22×C3×D312yes—
454C4⋊C4—21no—
455C4⋊C4—31no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4C22×C3×D311yescharacteristic
461C4⋊C4C22×C3×D312yes—
462C22⋊C4—21no—
463C22⋊C4—62no—
464C22⋊C4—62no—
465C22⋊C4—61no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C22⋊C4—61no—
469C42—31no—
470C42—31no—
471C42—31no—
472C42—31no—
473C42—61no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—22no—
479C22×C3—21no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—64no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—128no—
493C22×C3—122no—
494C22×C3—122no—
495C22×C3#91612yes—
496C22×C3C3×#13512yes—
497C22×C3C2×D3×D411yescharacteristic
498C22×C3C22×C3×D411yescharacteristic
499C2×D3—24no—
500C2×D3—48no—
501C2×D3—42no—
502C2×D3—42no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—22no—
506C3×C4—32no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—42no—
510C3×C4—41no—
511C3×C4—42no—
512C3×C4—61no—
513C3×C4—61no—
514C3×C4—61no—
515C3×C4—82no—
516C3×C4—81no—
517C3×C4—121no—
518C3×C4—121no—
519C3×C4D3×#4112yes—
520C3×C4C2×C3×#4112yes—
521Dic3—21no—
522Dic3—21no—
523Dic3—21no—
524Dic3—41no—
525Dic3—41no—
526Dic3C3×AGL(1,ℤ/8)12yes—
527C32#310511yescharacteristic, complemented, Sylow
528C23—21no—
529C23—64no—
530C23—62no—
531C23—122no—
532C23—122no—
533C23C3×D3×D411yescharacteristic
534Q8—62no—
535D4—22no—
536D4—34no—
537D4—42no—
538D4—64no—
539D4—62no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—31no—
544C2×C4—31no—
545C2×C4—31no—
546C2×C4—31no—
547C2×C4—42no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—124no—
553C2×C4—122no—
554C2×C4—124no—
555C2×C4C3×D3×D411yescharacteristic
556C2×C4C23×C3×D311yescharacteristic
557C8—22no—
558C8—62no—
559C2×C3—22no—
560C2×C3—22no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—42no—
565C2×C3—81no—
566C2×C3—81no—
567C2×C3—124no—
568C2×C3—161no—
569C2×C3#565611yescharacteristic
570C2×C3C2×#91611yescharacteristic
571C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
572C2×C3C3×#52511yescharacteristic
573C2×C3C2×C3×#13511yescharacteristic
574C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
575D3—44no—
576C22—21no—
577C22—21no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—64no—
582C22—128no—
583C22—122no—
584C22—122no—
585C22C3×#91612yes—
586C22C2×C3×D3×D411yescharacteristic
587C4—21no—
588C4—32no—
589C4—41no—
590C4—42no—
591C4—61no—
592C4—61no—
593C4—61no—
594C4—121no—
595C4—121no—
596C4C3×D3×#4112yes—
597C3—21no—
598C3#8130911yescharacteristic, direct product factor
599C3C3×#310511yescharacteristic, complemented
600C2—22no—
601C2—81no—
602C2—124no—
603C2C3×#565611yescharacteristic
604C2C2×C3×#91611yescharacteristic
605C2C3×D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#481C2
socleC22×C32C22×D4
prime radicalC3×#5036C2
hypercenterC3×#481D3
hypocenterC3C3×#3105
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#481

Complements

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

NH ≅ G/Ntype
1. C3×#5600C2semidirect
2. C2×C3×#893C2semidirect
3. C32×#481C2semidirect
5. C3×#4970C2semidirect
6. C3×#4871C2semidirect
7. C3×#4852C2semidirect
8. C3×#4787C2semidirect
9. C3×#4756C2semidirect
10. C3×#4731C2semidirect
11. C3×#4718C2semidirect
13. #81309C3direct (N × H)
18. C3×#936C22semidirect
21. C3×D4×Dic3C22semidirect
31. C3×#890C22semidirect
32. C3×#887C22semidirect
35. C32×#106C22semidirect
36. C32×#102C22semidirect
40. C3×Dic3⋊C8C22semidirect
41. C3×C8×Dic3C22semidirect
42. C3×#810C22semidirect
44. C3×#807C22semidirect
47. #5600C2×C3semidirect
48. C2×#893C2×C3semidirect
51. C3×#481D3semidirect
52. C3×#481C2×C3semidirect
61. #4970C2×C3semidirect
62. #4871C2×C3semidirect
63. #4852C2×C3semidirect
64. #4787C2×C3semidirect
65. #4756C2×C3semidirect
66. #4731C2×C3semidirect
67. #4718C2×C3semidirect
135. #936C22×C3semidirect
138. D4×Dic3C22×C3semidirect
148. #890C22×C3semidirect
149. #887C22×C3semidirect
154. C3×#106C2×D3semidirect
155. C3×#106C22×C3semidirect
161. C3×#102C2×D3semidirect
162. C3×#102C22×C3semidirect
170. Dic3⋊C8C22×C3semidirect
171. C8×Dic3C22×C3semidirect
172. #810C22×C3semidirect
174. #807C22×C3semidirect
209. #481C3×D3semidirect
347. #106C2×C3×D3semidirect
352. #102C2×C3×D3semidirect
527. C32#3105semidirect
598. C3#81309direct (N × H)
599. C3C3×#3105semidirect

Maximal subgroups

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

subgroupindex
C3×#56002
C2×C3×#8932
C32×#4812
C3×#50362
C3×#49702
C3×#48712
C3×#48522
C3×#47872
C3×#47562
C3×#47312
C3×#47182
C3×#31053
#813093