Finite Group Databasebeta

← C3×#81308 #11870702

Subgroups — C3×#81308 #11870702

Order 1152.

913 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×#5596C211yesmaximal, characteristic
2C2×C3×#887C211yesmaximal, characteristic, complemented
3C32×#481C211yesmaximal, characteristic, complemented
4C3×#5040C211yesmaximal, characteristic, complemented
5C3×#4968C211yesmaximal, characteristic, complemented
6C3×#4854C212yesmaximal, complemented
7C3×#4852C212yesmaximal, complemented
8C3×#4762C212yesmaximal, complemented
9C3×#4758C212yesmaximal, complemented
10C3×#4738C211yesmaximal, characteristic, complemented
11C3×#4718C211yesmaximal, characteristic, complemented
12C3×#3103—31nomaximal
13#81308C311yesmaximal, characteristic, direct product factor
14C2×C3×#237—21no—
15C22×C3×Dic6C2211yescharacteristic
16C32×#121C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C3×#940C2211yescharacteristic
19C3×#939—21no—
20C3×D4×Dic3C2211yescharacteristic
21C2×C3×#214C2212yes—
22C3×#930C2211yescharacteristic
23C2×C3×Dic3⋊C4—21no—
24C2×C3×C4×Dic3—21no—
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×C3×#204C2212yes—
27C3×#893C2211yescharacteristic
28C3×#888C2211yescharacteristic
29C3×#887—21no—
30C3×#887—21no—
31C3×#887C2212yes—
32C3×#887C2212yes—
33C3×#885—21no—
34C3×#67⋊C4—21no—
35C32×#106C2212yes—
36C32×#102C2212yes—
37C32×C22⋊C8C2211yescharacteristic
38C3×#830C2211yescharacteristic
39C3×#816C2212yes—
40C3×Dic3⋊C8C2212yes—
41C3×C8×Dic3C2212yes—
42C3×#810C2212yes—
43C3×#809C2212yes—
44C3×#808C2212yes—
45C3×#532—31no—
46C2×C3×C4×Q8—31no—
47#5596C2×C311yescharacteristic
48C2×#887C2×C311yescharacteristic, complemented
49C3×#481—21no—
50C3×#481—31no—
51C3×#481D311yescharacteristic, complemented
52C3×#481C2×C311yescharacteristic, complemented
53C3×#475—32no—
54C3×#460—32no—
55C3×#448—31no—
56C3×#448—31no—
57C3×C4×SD16—32no—
58C3×C4×SD16—32no—
59C3×#431—31no—
60#5040C2×C311yescharacteristic, complemented
61#4968C2×C311yescharacteristic, complemented
62#4854C2×C312yescomplemented
63#4852C2×C312yescomplemented
64#4762C2×C312yescomplemented
65#4758C2×C312yescomplemented
66#4738C2×C311yescharacteristic, complemented
67#4718C2×C311yescharacteristic, complemented
68C2×C32×D4—21no—
69C2×C32×D4C2311yescharacteristic
70C22×C32×C4—21no—
71C22×C32×C4C2311yescharacteristic
72C2×C3×#67—21no—
73C2×C3×#67—21no—
74C2×C3×#67—21no—
75C22×C3×Dic3—21no—
76C22×C3×Dic3—21no—
77C22×C3×Dic3D411yescharacteristic
78C22×C3×Dic3D411yescharacteristic
79C3×#237—44no—
80C2×C3×D12C2311yescharacteristic
81C2×C3×C4×D3—21no—
82C2×C3×Dic6—21no—
83C2×C3×Dic6—24no—
84C2×C3×Dic6—21no—
85C2×C3×Dic6C2312yes—
86C2×C32×C8C2312yes—
87C32×C4⋊C4—21no—
88C32×C4⋊C4C2312yes—
89C32×C4⋊C4C2311yescharacteristic
90C32×C22⋊C4—21no—
91C3×#217—21no—
92C3×#214—24no—
93C3×(C2×D3)⋊C4—21no—
94C3×(C2×D3)⋊C4—21no—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×Dic3⋊C4—21no—
97C3×Dic3⋊C4—22no—
98C3×Dic3⋊C4—22no—
99C3×C4×Dic3—22no—
100C3×C4×Dic3—22no—
101C3×C4×Dic3D411yescharacteristic
102C3×C4×Dic3D411yescharacteristic
103C3×C4×Dic3C2311yescharacteristic
104C3×C4×Dic3C2311yescharacteristic
105C2×C3×C3⋊C8C2312yes—
106C3×#204—24no—
107#3103—31noSylow
108C2×C3×#41—61no—
109C22×C3×Q8—31no—
110C2×#237—21no—
111C22×Dic6C22×C311yescharacteristic
112C2×C3×SD16—32no—
113C2×C3×SD16—32no—
114C3×C4⋊Q8—31no—
115C3×#127—31no—
116C3×#124—61no—
117C3×#121—21no—
118C3×#121—31no—
119C3×#121—61no—
120C3×#121C2×D311yescharacteristic
121C3×#121C22×C311yescharacteristic
122C3×C4×Q8—32no—
123C3×C4×Q8—32no—
124C3×C4×Q8—61no—
125C3×C4×Q8—61no—
126C3×C4×D4—31no—
127C3×C4×D4—31no—
128C3×C4×D4—61no—
129C2×C3×C4⋊C4—21no—
130C2×C3×C4⋊C4—61no—
131C2×C3×C4⋊C4C2×D311yescharacteristic
132C2×C3×C4⋊C4C22×C311yescharacteristic
133C2×C3×C42—31no—
134C2×C3×C42—61no—
135#940C22×C311yescharacteristic
136#939—21no—
137D4×Dic3C22×C311yescharacteristic
138C2×#214C22×C312yes—
139#930C22×C311yescharacteristic
140C2×Dic3⋊C4—21no—
141C2×C4×Dic3—21no—
142C2×C4×Dic3C22×C311yescharacteristic
143C2×#204C22×C312yes—
144#893C22×C311yescharacteristic
145#888C22×C311yescharacteristic
146#887—21no—
147#887—21no—
148#887C22×C312yes—
149#887C22×C312yes—
150#885—21no—
151#67⋊C4—21no—
152C3×#106—22no—
153C3×#106—32no—
154C3×#106C2×D312yes—
155C3×#106C22×C312yes—
156C3×C4⋊C8—32no—
157C3×#103—32no—
158C3×#103—32no—
159C3×#102—22no—
160C3×#102—32no—
161C3×#102C2×D312yes—
162C3×#102C22×C312yes—
163C3×C22⋊C8—21no—
164C3×C22⋊C8—31no—
165C3×C22⋊C8C2×D311yescharacteristic
166C3×C22⋊C8C22×C311yescharacteristic
167C3×C4×C8—32no—
168#830C22×C311yescharacteristic
169#816C22×C312yes—
170Dic3⋊C8C22×C312yes—
171C8×Dic3C22×C312yes—
172#810C22×C312yes—
173#809C22×C312yes—
174#808C22×C312yes—
175C23×C32—21no—
176C23×C32C2×D411yescharacteristic
177C22×C3×D3—21no—
178C32×D4—22no—
179C32×D4—42no—
180C2×C32×C4—22no—
181C2×C32×C4—22no—
182C2×C32×C4—21no—
183C2×C32×C4—42no—
184C2×C32×C4C2×D411yescharacteristic
185C2×C32×C4C2411yescharacteristic
186C3×#67—42no—
187C3×#67—42no—
188C3×#67—42no—
189C2×C3×Dic3—24no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—41no—
193C2×C3×Dic3—42no—
194C2×C3×Dic3SD1614yescomplemented
195C2×C3×Dic3C2×D411yescharacteristic
196C2×C3×Dic3C2×D411yescharacteristic
197C2×C3×Dic3C2×D411yescharacteristic
198C2×C3×Dic3C2×D411yescharacteristic
199C3×D12—22no—
200C3×C4×D3—42no—
201C3×Dic6—24no—
202C3×Dic6—22no—
203C3×Dic6—42no—
204C3×Dic6#4114yescomplemented
205C32×C8—22no—
206C3×C3⋊C8—22no—
207#532—31no—
208C2×C4×Q8—31no—
209#481—31no—
210#481C3×D311yescharacteristic, complemented
211#475—32no—
212#460—32no—
213#448—31no—
214#448—31no—
215C4×SD16—32no—
216C4×SD16—32no—
217#431—31no—
218C3×#41—124no—
219C2×C3×Q8—32no—
220C2×C3×Q8—61no—
221C2×C3×Q8—64no—
222C2×C3×Q8—61no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—31no—
227C2×C3×D4—41no—
228C2×C3×D4—61no—
229C2×C3×D4—61no—
230C2×C3×D4—61no—
231C2×C3×D4C22×D311yescharacteristic
232C2×C3×D4C23×C311yescharacteristic
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—31no—
237C22×C3×C4—31no—
238C22×C3×C4—41no—
239C22×C3×C4—61no—
240C22×C3×C4—61no—
241C22×C3×C4—61no—
242C22×C3×C4C22×D311yescharacteristic
243C22×C3×C4C23×C311yescharacteristic
244C2×#67—21no—
245C2×#67—21no—
246C2×#67—21no—
247C22×Dic3—21no—
248C22×Dic3—21no—
249C22×Dic3C3×D411yescharacteristic
250C22×Dic3C3×D411yescharacteristic
251#237—44no—
252C2×D12C23×C311yescharacteristic
253C2×C4×D3—21no—
254C2×Dic6—21no—
255C2×Dic6—24no—
256C2×Dic6—21no—
257C2×Dic6C23×C312yes—
258C3×SD16—64no—
259C3×SD16—64no—
260C2×C3×C8—22no—
261C2×C3×C8—32no—
262C2×C3×C8C22×D312yes—
263C2×C3×C8C23×C312yes—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—22no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—31no—
269C3×C4⋊C4—41no—
270C3×C4⋊C4—62no—
271C3×C4⋊C4—62no—
272C3×C4⋊C4—61no—
273C3×C4⋊C4C22×D312yes—
274C3×C4⋊C4C22×D311yescharacteristic
275C3×C4⋊C4C23×C312yes—
276C3×C4⋊C4C23×C311yescharacteristic
277C3×C22⋊C4—21no—
278C3×C22⋊C4—21no—
279C3×C22⋊C4—41no—
280C3×C22⋊C4—61no—
281C3×C22⋊C4—61no—
282C3×C22⋊C4—61no—
283C3×C42—31no—
284C3×C42—31no—
285C3×C42—31no—
286C3×C42—31no—
287C3×C42—62no—
288C3×C42—62no—
289#217—21no—
290#214—24no—
291(C2×D3)⋊C4—21no—
292(C2×D3)⋊C4—21no—
293C4⋊Dic3C23×C311yescharacteristic
294Dic3⋊C4—21no—
295Dic3⋊C4—22no—
296Dic3⋊C4—22no—
297C4×Dic3—22no—
298C4×Dic3—22no—
299C4×Dic3C3×D411yescharacteristic
300C4×Dic3C3×D411yescharacteristic
301C4×Dic3C23×C311yescharacteristic
302C4×Dic3C23×C311yescharacteristic
303C2×C3⋊C8C23×C312yes—
304#204—24no—
305C22×C32—21no—
306C22×C32—21no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32C2×SD1612yes—
311C22×C32C22×D411yescharacteristic
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4—42no—
318C32×C4C2×#4112yes—
319C3×Dic3—24no—
320C3×Dic3—21no—
321C3×Dic3—21no—
322C3×Dic3—41no—
323C3×Dic3C2×SD1612yes—
324C3×Dic3C2×SD1612yes—
325C2×#41—61no—
326C22×Q8—31no—
327C2×SD16—32no—
328C2×SD16—32no—
329C4⋊Q8—31no—
330#127—31no—
331#124—61no—
332#121—31no—
333#121—61no—
334#121C2×C3×D311yescharacteristic
335C4×Q8—32no—
336C4×Q8—32no—
337C4×Q8—61no—
338C4×Q8—61no—
339C4×D4—31no—
340C4×D4—31no—
341C4×D4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4C2×C3×D311yescharacteristic
344C2×C42—31no—
345C2×C42—61no—
346#106—32no—
347#106C2×C3×D312yes—
348C4⋊C8—32no—
349#103—32no—
350#103—32no—
351#102—32no—
352#102C2×C3×D312yes—
353C22⋊C8—31no—
354C22⋊C8C2×C3×D311yescharacteristic
355C4×C8—32no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—41no—
360C23×C3—61no—
361C23×C3D3×D411yescharacteristic
362C23×C3C2×C3×D411yescharacteristic
363C22×D3—21no—
364C3×Q8—34no—
365C3×Q8—64no—
366C3×Q8—62no—
367C3×Q8—122no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—42no—
371C3×D4—42no—
372C3×D4—42no—
373C3×D4—62no—
374C3×D4—82no—
375C3×D4—122no—
376C3×D4—122no—
377C3×D4—122no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—34no—
387C2×C3×C4—31no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—42no—
394C2×C3×C4—42no—
395C2×C3×C4—42no—
396C2×C3×C4—64no—
397C2×C3×C4—62no—
398C2×C3×C4—61no—
399C2×C3×C4—82no—
400C2×C3×C4—122no—
401C2×C3×C4—122no—
402C2×C3×C4—121no—
403C2×C3×C4D3×D411yescharacteristic
404C2×C3×C4C2×C3×D411yescharacteristic
405C2×C3×C4C23×D311yescharacteristic
406C2×C3×C4C24×C311yescharacteristic
407#67—42no—
408#67—42no—
409#67—42no—
410C2×Dic3—24no—
411C2×Dic3—22no—
412C2×Dic3—21no—
413C2×Dic3—41no—
414C2×Dic3—42no—
415C2×Dic3C3×SD1614yescomplemented
416C2×Dic3C2×C3×D411yescharacteristic
417C2×Dic3C2×C3×D411yescharacteristic
418C2×Dic3C2×C3×D411yescharacteristic
419C2×Dic3C2×C3×D411yescharacteristic
420D12—22no—
421C4×D3—42no—
422Dic6—24no—
423Dic6—22no—
424Dic6—42no—
425Dic6C3×#4114yescomplemented
426C3×C8—22no—
427C3×C8—22no—
428C3×C8—42no—
429C3×C8—62no—
430C3⋊C8—22no—
431C2×C32—22no—
432C2×C32—81no—
433C2×C32#52511yescharacteristic
434C2×C32C22×SD1611yescharacteristic
435C2×C32#57511yescharacteristic
436C3×D3—81no—
437#41—124no—
438C2×Q8—32no—
439C2×Q8—61no—
440C2×Q8—64no—
441C2×Q8—61no—
442C2×D4—21no—
443C2×D4—31no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4C22×C3×D311yescharacteristic
448C22×C4—21no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4C22×C3×D311yescharacteristic
455SD16—64no—
456SD16—64no—
457C2×C8—32no—
458C2×C8C22×C3×D312yes—
459C4⋊C4—21no—
460C4⋊C4—31no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C4⋊C4—61no—
464C4⋊C4C22×C3×D312yes—
465C4⋊C4C22×C3×D311yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—61no—
468C22⋊C4—61no—
469C22⋊C4—61no—
470C42—31no—
471C42—31no—
472C42—31no—
473C42—31no—
474C42—62no—
475C42—62no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—22no—
481C22×C3—21no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—121no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3D3×SD1612yes—
497C22×C3C2×C3×SD1612yes—
498C22×C3C2×D3×D411yescharacteristic
499C22×C3C22×C3×D411yescharacteristic
500C2×D3—41no—
501C2×D3—41no—
502C2×D3—41no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—22no—
506C3×C4—32no—
507C3×C4—32no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4—41no—
512C3×C4—42no—
513C3×C4—64no—
514C3×C4—61no—
515C3×C4—61no—
516C3×C4—82no—
517C3×C4—81no—
518C3×C4—121no—
519C3×C4D3×#4112yes—
520C3×C4C2×C3×#4112yes—
521Dic3—21no—
522Dic3—21no—
523Dic3—24no—
524Dic3—41no—
525Dic3C2×C3×SD1612yes—
526Dic3C2×C3×SD1612yes—
527C32#310311yescharacteristic, complemented, Sylow
528C23—21no—
529C23—61no—
530C23C3×D3×D411yescharacteristic
531Q8—34no—
532Q8—64no—
533Q8—62no—
534Q8—122no—
535D4—22no—
536D4—42no—
537D4—62no—
538D4—122no—
539D4—122no—
540D4—122no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—34no—
545C2×C4—31no—
546C2×C4—31no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—42no—
550C2×C4—64no—
551C2×C4—62no—
552C2×C4—61no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—121no—
556C2×C4C3×D3×D411yescharacteristic
557C2×C4C23×C3×D311yescharacteristic
558C8—22no—
559C8—62no—
560C2×C3—22no—
561C2×C3—22no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—42no—
566C2×C3—81no—
567C2×C3—81no—
568C2×C3—161no—
569C2×C3—241no—
570C2×C3#565611yescharacteristic
571C2×C3C2×D3×SD1611yescharacteristic
572C2×C3#577311yescharacteristic
573C2×C3C3×#52511yescharacteristic
574C2×C3C22×C3×SD1611yescharacteristic
575C2×C3C3×#57511yescharacteristic
576D3—81no—
577C22—21no—
578C22—21no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—121no—
583C22—121no—
584C22—121no—
585C22C3×D3×SD1612yes—
586C22C2×C3×D3×D411yescharacteristic
587C4—21no—
588C4—32no—
589C4—32no—
590C4—41no—
591C4—42no—
592C4—61no—
593C4—61no—
594C4—64no—
595C4—121no—
596C4C3×D3×#4112yes—
597C3—21no—
598C3#8130811yescharacteristic, direct product factor
599C3C3×#310311yescharacteristic, complemented
600C2—22no—
601C2—81no—
602C2—241no—
603C2C3×#565611yescharacteristic
604C2C2×C3×D3×SD1611yescharacteristic
605C2C3×#577311yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#481C2
socleC22×C32C22×D4
prime radicalC3×#5596C2
hypercenterC3×#481D3
hypocenterC3C3×#3103
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
2. C2×C3×#887C2semidirect
3. C32×#481C2semidirect
4. C3×#5040C2semidirect
5. C3×#4968C2semidirect
6. C3×#4854C2semidirect
7. C3×#4852C2semidirect
8. C3×#4762C2semidirect
9. C3×#4758C2semidirect
10. C3×#4738C2semidirect
11. C3×#4718C2semidirect
13. #81308C3direct (N × H)
48. C2×#887C2×C3semidirect
51. C3×#481D3semidirect
52. C3×#481C2×C3semidirect
60. #5040C2×C3semidirect
61. #4968C2×C3semidirect
62. #4854C2×C3semidirect
63. #4852C2×C3semidirect
64. #4762C2×C3semidirect
65. #4758C2×C3semidirect
66. #4738C2×C3semidirect
67. #4718C2×C3semidirect
194. C2×C3×Dic3SD16semidirect
204. C3×Dic6#41semidirect
210. #481C3×D3semidirect
415. C2×Dic3C3×SD16semidirect
425. Dic6C3×#41semidirect
527. C32#3103semidirect
598. C3#81308direct (N × H)
599. C3C3×#3103semidirect

Maximal subgroups

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

subgroupindex
C3×#55962
C2×C3×#8872
C32×#4812
C3×#50402
C3×#49682
C3×#48542
C3×#48522
C3×#47622
C3×#47582
C3×#47382
C3×#47182
C3×#31033
#813083