Finite Group Databasebeta

← C3×#80360 #11870708

Subgroups — C3×#80360 #11870708

Order 1152.

920 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5716C211yesmaximal, characteristic
2C3×C4×D3×Q8C211yesmaximal, characteristic, complemented
3C32×#460C211yesmaximal, characteristic, complemented
4C3×#5170C212yesmaximal, complemented
5C3×Dic3×SD16C212yesmaximal, complemented
6C3×#5075C211yesmaximal, characteristic, complemented
7C3×#5020C211yesmaximal, characteristic, complemented
8C3×#4851C211yesmaximal, characteristic, complemented
9C3×#4833C211yesmaximal, characteristic, complemented
10C3×#4777C212yesmaximal, complemented
11C3×#4758C212yesmaximal, complemented
12C3×#3103—31nomaximal
13#80360C311yesmaximal, characteristic, direct product factor
14C2×C3×D3×Q8C2211yescharacteristic
15C2×C3×#237—21no—
16C2×C32×SD16C2212yes—
17C32×#127C2211yescharacteristic
18C32×C4×Q8C2211yescharacteristic
19C3×Q8×Dic3C2211yescharacteristic
20C3×#938—21no—
21C3×#938C2212yes—
22C3×#936—21no—
23C3×D4×Dic3—21no—
24C3×D4×Dic3C2212yes—
25C2×C3×#214C2212yes—
26C3×D3×C4⋊C4—21no—
27C3×D3×C4⋊C4C2211yescharacteristic
28C3×#887—21no—
29C3×#887C2211yescharacteristic
30C3×C42×D3—21no—
31C3×C42×D3C2211yescharacteristic
32C3×#869C2211yescharacteristic
33C3×C4×Dic6—21no—
34C3×C4×Dic6C2211yescharacteristic
35C32×C4⋊C8C2211yescharacteristic
36C32×#102C2212yes—
37C3×#835C2212yes—
38C3×#832C2212yes—
39C3×(C2×D3)⋊C8C2212yes—
40C3×#817C2212yes—
41C3×#816C2212yes—
42C3×C8×Dic3C2212yes—
43C3×#808C2212yes—
44C3×C4⋊(C3⋊C8)C2211yescharacteristic
45C3×#532—31no—
46C2×C3×C4×Q8—31no—
47#5716C2×C311yescharacteristic
48C4×D3×Q8C2×C311yescharacteristic, complemented
49C3×#481—32no—
50C3×#475—31no—
51C3×#475—31no—
52C3×#460—21no—
53C3×#460—31no—
54C3×#460D311yescharacteristic, complemented
55C3×#460C2×C311yescharacteristic, complemented
56C3×#448—32no—
57C3×C4×SD16—32no—
58C3×C4×SD16—32no—
59C3×#431—31no—
60#5170C2×C312yescomplemented
61Dic3×SD16C2×C312yescomplemented
62#5075C2×C311yescharacteristic, complemented
63#5020C2×C311yescharacteristic, complemented
64#4851C2×C311yescharacteristic, complemented
65#4833C2×C311yescharacteristic, complemented
66#4777C2×C312yescomplemented
67#4758C2×C312yescomplemented
68C2×C32×Q8C2311yescharacteristic
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4C2312yes—
72C2×C3×#67—22no—
73C22×C3×Dic3—22no—
74C3×D3×Q8—22no—
75C3×D3×Q8—22no—
76C3×#237—44no—
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3—21no—
79C2×C3×C4×D3D411yescharacteristic
80C2×C3×C4×D3D411yescharacteristic
81C2×C3×C4×D3C2311yescharacteristic
82C2×C3×Dic6—21no—
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2311yescharacteristic
85C32×SD16—24no—
86C2×C32×C8C2312yes—
87C32×C4⋊C4—21no—
88C32×C4⋊C4C2311yescharacteristic
89C32×C42—21no—
90C32×C42C2311yescharacteristic
91C3×#217—22no—
92C3×#217—22no—
93C3×#214—24no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3C2311yescharacteristic
97C3×C4⋊Dic3C2312yes—
98C3×Dic3⋊C4—21no—
99C3×Dic3⋊C4—21no—
100C3×Dic3⋊C4—21no—
101C3×C4×Dic3—21no—
102C3×C4×Dic3—21no—
103C3×C4×Dic3—21no—
104C3×C4×Dic3D411yescharacteristic
105C3×C4×Dic3D411yescharacteristic
106C3×C4×Dic3C2311yescharacteristic
107C2×C3×C3⋊C8C2312yes—
108#3103—31noSylow
109C2×C3×#41—61no—
110C22×C3×Q8—31no—
111C2×D3×Q8C22×C311yescharacteristic
112C2×#237—21no—
113C2×C3×SD16—22no—
114C2×C3×SD16—32no—
115C2×C3×SD16C2×D312yes—
116C2×C3×SD16C22×C312yes—
117C3×C4⋊Q8—31no—
118C3×#127—21no—
119C3×#127C2×D311yescharacteristic
120C3×#127C22×C311yescharacteristic
121C3×#124—61no—
122C3×#121—32no—
123C3×#121—61no—
124C3×C4×Q8—21no—
125C3×C4×Q8—31no—
126C3×C4×Q8—31no—
127C3×C4×Q8—31no—
128C3×C4×Q8—61no—
129C3×C4×Q8—61no—
130C3×C4×Q8C2×D311yescharacteristic
131C3×C4×Q8C22×C311yescharacteristic
132C3×C4×D4—32no—
133C3×C4×D4—61no—
134C2×C3×C4⋊C4—31no—
135C2×C3×C4⋊C4—61no—
136C2×C3×C42—31no—
137C2×C3×C42—61no—
138Q8×Dic3C22×C311yescharacteristic
139#938—21no—
140#938C22×C312yes—
141#936—21no—
142D4×Dic3—21no—
143D4×Dic3C22×C312yes—
144C2×#214C22×C312yes—
145D3×C4⋊C4—21no—
146D3×C4⋊C4C22×C311yescharacteristic
147#887—21no—
148#887C22×C311yescharacteristic
149C42×D3—21no—
150C42×D3C22×C311yescharacteristic
151#869C22×C311yescharacteristic
152C4×Dic6—21no—
153C4×Dic6C22×C311yescharacteristic
154C3×#106—32no—
155C3×#106—32no—
156C3×C4⋊C8—21no—
157C3×C4⋊C8—31no—
158C3×C4⋊C8C2×D311yescharacteristic
159C3×C4⋊C8C22×C311yescharacteristic
160C3×#103—32no—
161C3×#103—32no—
162C3×#102—22no—
163C3×#102—32no—
164C3×#102C2×D312yes—
165C3×#102C22×C312yes—
166C3×C22⋊C8—32no—
167C3×C4×C8—32no—
168#835C22×C312yes—
169#832C22×C312yes—
170(C2×D3)⋊C8C22×C312yes—
171#817C22×C312yes—
172#816C22×C312yes—
173C8×Dic3C22×C312yes—
174#808C22×C312yes—
175C4⋊(C3⋊C8)C22×C311yescharacteristic
176C23×C32—22no—
177C22×C3×D3C2×D411yescharacteristic
178C32×Q8—21no—
179C32×Q8#4112yescomplemented
180C32×D4—22no—
181C32×D4—22no—
182C32×D4—42no—
183C32×D4—42no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4C2×D411yescharacteristic
188C2×C32×C4C2411yescharacteristic
189C3×#67—44no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—44no—
194C2×C3×Dic3C2×D411yescharacteristic
195C2×C3×Dic3C2×D411yescharacteristic
196C2×C3×Dic3C2×D411yescharacteristic
197C3×C4×D3—24no—
198C3×C4×D3—21no—
199C3×C4×D3—21no—
200C3×C4×D3—41no—
201C3×C4×D3—42no—
202C3×C4×D3SD1614yescomplemented
203C3×Dic6—24no—
204C3×Dic6—21no—
205C3×Dic6—42no—
206C3×Dic6#4112yescomplemented
207C32×C8—22no—
208C3×C3⋊C8—22no—
209#532—31no—
210C2×C4×Q8—31no—
211#481—32no—
212#475—31no—
213#475—31no—
214#460—31no—
215#460C3×D311yescharacteristic, complemented
216#448—32no—
217C4×SD16—32no—
218C4×SD16—32no—
219#431—31no—
220C3×#41—124no—
221C2×C3×Q8—21no—
222C2×C3×Q8—31no—
223C2×C3×Q8—61no—
224C2×C3×Q8—62no—
225C2×C3×Q8—62no—
226C2×C3×Q8—61no—
227C2×C3×Q8C22×D311yescharacteristic
228C2×C3×Q8C23×C311yescharacteristic
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—22no—
234C2×C3×D4—41no—
235C2×C3×D4—41no—
236C2×C3×D4—62no—
237C2×C3×D4C22×D312yes—
238C2×C3×D4C23×C312yes—
239C22×C3×C4—31no—
240C22×C3×C4—31no—
241C22×C3×C4—31no—
242C22×C3×C4—61no—
243C22×C3×C4—61no—
244C22×C3×C4—62no—
245C2×#67—22no—
246C22×Dic3—22no—
247D3×Q8—22no—
248D3×Q8—22no—
249#237—44no—
250C2×C4×D3—21no—
251C2×C4×D3—21no—
252C2×C4×D3C3×D411yescharacteristic
253C2×C4×D3C3×D411yescharacteristic
254C2×C4×D3C23×C311yescharacteristic
255C2×Dic6—21no—
256C2×Dic6—21no—
257C2×Dic6C23×C311yescharacteristic
258C3×SD16—24no—
259C3×SD16—24no—
260C3×SD16—44no—
261C3×SD16—64no—
262C2×C3×C8—22no—
263C2×C3×C8—32no—
264C2×C3×C8C22×D312yes—
265C2×C3×C8C23×C312yes—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—21no—
269C3×C4⋊C4—31no—
270C3×C4⋊C4—32no—
271C3×C4⋊C4—41no—
272C3×C4⋊C4—61no—
273C3×C4⋊C4—61no—
274C3×C4⋊C4—61no—
275C3×C4⋊C4—61no—
276C3×C4⋊C4—61no—
277C3×C4⋊C4C22×D311yescharacteristic
278C3×C4⋊C4C23×C311yescharacteristic
279C3×C22⋊C4—62no—
280C3×C22⋊C4—62no—
281C3×C42—21no—
282C3×C42—21no—
283C3×C42—21no—
284C3×C42—31no—
285C3×C42—31no—
286C3×C42—31no—
287C3×C42—41no—
288C3×C42—61no—
289C3×C42—61no—
290C3×C42—61no—
291C3×C42C22×D311yescharacteristic
292C3×C42C23×C311yescharacteristic
293#217—22no—
294#217—22no—
295#214—24no—
296C4⋊Dic3—21no—
297C4⋊Dic3—21no—
298C4⋊Dic3C23×C312yes—
299C4⋊Dic3C23×C311yescharacteristic
300Dic3⋊C4—21no—
301Dic3⋊C4—21no—
302Dic3⋊C4—21no—
303C4×Dic3—21no—
304C4×Dic3—21no—
305C4×Dic3—21no—
306C4×Dic3C3×D411yescharacteristic
307C4×Dic3C3×D411yescharacteristic
308C4×Dic3C23×C311yescharacteristic
309C2×C3⋊C8C23×C312yes—
310C22×C32—42no—
311C22×C32—42no—
312C22×C32—42no—
313C22×C32C22×D411yescharacteristic
314C2×C3×D3—21no—
315C2×C3×D3—21no—
316C2×C3×D3C2×SD1612yes—
317C32×C4—22no—
318C32×C4—21no—
319C32×C4—41no—
320C32×C4C2×SD1612yes—
321C32×C4C2×#4111yescharacteristic
322C32×C4C2×#4111yescharacteristic
323C3×Dic3—21no—
324C3×Dic3—22no—
325C3×Dic3—21no—
326C3×Dic3—42no—
327C3×Dic3—41no—
328C3×Dic3C2×SD1612yes—
329C2×#41—61no—
330C22×Q8—31no—
331C2×SD16—32no—
332C2×SD16C2×C3×D312yes—
333C4⋊Q8—31no—
334#127C2×C3×D311yescharacteristic
335#124—61no—
336#121—32no—
337#121—61no—
338C4×Q8—31no—
339C4×Q8—31no—
340C4×Q8—31no—
341C4×Q8—61no—
342C4×Q8—61no—
343C4×Q8C2×C3×D311yescharacteristic
344C4×D4—32no—
345C4×D4—61no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—61no—
348C2×C42—31no—
349C2×C42—61no—
350#106—32no—
351#106—32no—
352C4⋊C8—31no—
353C4⋊C8C2×C3×D311yescharacteristic
354#103—32no—
355#103—32no—
356#102—32no—
357#102C2×C3×D312yes—
358C22⋊C8—32no—
359C4×C8—32no—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3—31no—
363C23×C3—42no—
364C22×D3C2×C3×D411yescharacteristic
365C3×Q8—21no—
366C3×Q8—21no—
367C3×Q8—22no—
368C3×Q8—32no—
369C3×Q8—41no—
370C3×Q8—64no—
371C3×Q8—61no—
372C3×Q8—122no—
373C3×Q8#23712yescomplemented
374C3×Q8C3×#4112yescomplemented
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—42no—
380C3×D4—42no—
381C3×D4—42no—
382C3×D4—42no—
383C3×D4—42no—
384C3×D4—42no—
385C3×D4—82no—
386C3×D4—82no—
387C3×D4—124no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—31no—
396C2×C3×C4—34no—
397C2×C3×C4—31no—
398C2×C3×C4—31no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—64no—
402C2×C3×C4—61no—
403C2×C3×C4—61no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—62no—
407C2×C3×C4—121no—
408C2×C3×C4—124no—
409C2×C3×C4—122no—
410C2×C3×C4D3×D411yescharacteristic
411C2×C3×C4D3×D411yescharacteristic
412C2×C3×C4C2×C3×D411yescharacteristic
413C2×C3×C4C2×C3×D411yescharacteristic
414C2×C3×C4C23×D311yescharacteristic
415C2×C3×C4C24×C311yescharacteristic
416#67—44no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—22no—
420C2×Dic3—44no—
421C2×Dic3C2×C3×D411yescharacteristic
422C2×Dic3C2×C3×D411yescharacteristic
423C2×Dic3C2×C3×D411yescharacteristic
424C4×D3—24no—
425C4×D3—21no—
426C4×D3—21no—
427C4×D3—41no—
428C4×D3—42no—
429C4×D3C3×SD1614yescomplemented
430Dic6—24no—
431Dic6—21no—
432Dic6—42no—
433Dic6C3×#4112yescomplemented
434C3×C8—22no—
435C3×C8—22no—
436C3×C8—42no—
437C3×C8—62no—
438C3⋊C8—22no—
439C2×C32—82no—
440C2×C32#52511yescharacteristic
441C2×C32C22×SD1611yescharacteristic
442C2×C32#57511yescharacteristic
443C3×D3—22no—
444#41—124no—
445C2×Q8—31no—
446C2×Q8—61no—
447C2×Q8—62no—
448C2×Q8—62no—
449C2×Q8—61no—
450C2×Q8C22×C3×D311yescharacteristic
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—62no—
454C2×D4C22×C3×D312yes—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—62no—
461SD16—24no—
462SD16—64no—
463C2×C8—32no—
464C2×C8C22×C3×D312yes—
465C4⋊C4—21no—
466C4⋊C4—31no—
467C4⋊C4—32no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4C22×C3×D311yescharacteristic
474C22⋊C4—62no—
475C22⋊C4—62no—
476C42—21no—
477C42—31no—
478C42—31no—
479C42—31no—
480C42—61no—
481C42—61no—
482C42—61no—
483C42C22×C3×D311yescharacteristic
484C22×C3—21no—
485C22×C3—32no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3—42no—
492C22×C3—61no—
493C22×C3—61no—
494C22×C3—82no—
495C22×C3—82no—
496C22×C3—82no—
497C22×C3C2×D3×D411yescharacteristic
498C22×C3C22×C3×D411yescharacteristic
499C2×D3—21no—
500C2×D3—21no—
501C2×D3C2×C3×SD1612yes—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—22no—
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—21no—
508C3×C4—21no—
509C3×C4—32no—
510C3×C4—41no—
511C3×C4—42no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4—61no—
515C3×C4—61no—
516C3×C4—62no—
517C3×C4—81no—
518C3×C4—122no—
519C3×C4—121no—
520C3×C4D3×SD1612yes—
521C3×C4C2×C3×SD1612yes—
522C3×C4C2×#23711yescharacteristic
523C3×C4D3×#4111yescharacteristic
524C3×C4C2×C3×#4111yescharacteristic
525C3×C4C2×C3×#4111yescharacteristic
526Dic3—21no—
527Dic3—21no—
528Dic3—22no—
529Dic3—42no—
530Dic3—41no—
531Dic3C2×C3×SD1612yes—
532C32#310311yescharacteristic, complemented, Sylow
533C23—22no—
534C23—31no—
535Q8—21no—
536Q8—32no—
537Q8—64no—
538Q8—61no—
539Q8—122no—
540Q8C3×#23712yescomplemented
541D4—22no—
542D4—22no—
543D4—42no—
544D4—42no—
545D4—124no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—31no—
549C2×C4—34no—
550C2×C4—31no—
551C2×C4—31no—
552C2×C4—64no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—121no—
559C2×C4—124no—
560C2×C4—122no—
561C2×C4C3×D3×D411yescharacteristic
562C2×C4C3×D3×D411yescharacteristic
563C2×C4C23×C3×D311yescharacteristic
564C8—22no—
565C8—62no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—62no—
570C2×C3—82no—
571C2×C3—82no—
572C2×C3—162no—
573C2×C3#560611yescharacteristic
574C2×C3C2×D3×SD1611yescharacteristic
575C2×C3#577311yescharacteristic
576C2×C3C3×#52511yescharacteristic
577C2×C3C22×C3×SD1611yescharacteristic
578C2×C3C3×#57511yescharacteristic
579D3—22no—
580C22—32no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—61no—
585C22—61no—
586C22C2×C3×D3×D411yescharacteristic
587C4—21no—
588C4—22no—
589C4—32no—
590C4—41no—
591C4—61no—
592C4—61no—
593C4—62no—
594C4—122no—
595C4—121no—
596C4C3×D3×SD1612yes—
597C4C2×C3×#23711yescharacteristic
598C4C3×D3×#4111yescharacteristic
599C3—21no—
600C3#8036011yescharacteristic, direct product factor
601C3C3×#310311yescharacteristic, complemented
602C2—62no—
603C2—82no—
604C2C3×#560611yescharacteristic
605C2C2×C3×D3×SD1611yescharacteristic
606C2C3×#577311yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#460C2
socleC22×C32C22×D4
prime radicalC3×#5716C2
hypercenterC3×#460D3
hypocenterC3C3×#3103
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#460

Complements

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

NH ≅ G/Ntype
2. C3×C4×D3×Q8C2semidirect
3. C32×#460C2semidirect
4. C3×#5170C2semidirect
5. C3×Dic3×SD16C2semidirect
6. C3×#5075C2semidirect
7. C3×#5020C2semidirect
8. C3×#4851C2semidirect
9. C3×#4833C2semidirect
10. C3×#4777C2semidirect
11. C3×#4758C2semidirect
13. #80360C3direct (N × H)
48. C4×D3×Q8C2×C3semidirect
54. C3×#460D3semidirect
55. C3×#460C2×C3semidirect
60. #5170C2×C3semidirect
61. Dic3×SD16C2×C3semidirect
62. #5075C2×C3semidirect
63. #5020C2×C3semidirect
64. #4851C2×C3semidirect
65. #4833C2×C3semidirect
66. #4777C2×C3semidirect
67. #4758C2×C3semidirect
179. C32×Q8#41semidirect
202. C3×C4×D3SD16semidirect
206. C3×Dic6#41semidirect
215. #460C3×D3semidirect
373. C3×Q8#237semidirect
374. C3×Q8C3×#41semidirect
429. C4×D3C3×SD16semidirect
433. Dic6C3×#41semidirect
532. C32#3103semidirect
540. Q8C3×#237semidirect
600. C3#80360direct (N × H)
601. C3C3×#3103semidirect

Maximal subgroups

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

subgroupindex
C3×#57162
C3×C4×D3×Q82
C32×#4602
C3×#51702
C3×Dic3×SD162
C3×#50752
C3×#50202
C3×#48512
C3×#48332
C3×#47772
C3×#47582
C3×#31033
#803603