Finite Group Databasebeta

← C3×#78960 #11864414

Subgroups — C3×#78960 #11864414

Order 1152.

1539 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1007C211yesmaximal, characteristic, complemented
2C22×C3×#215C211yesmaximal, characteristic, complemented
3C2×C3×#908C211yesmaximal, characteristic, complemented
4C3×#5525C211yesmaximal, characteristic, complemented
5C32×#419C211yesmaximal, characteristic, complemented
6C3×#5129C211yesmaximal, characteristic, complemented
7C3×#5002C211yesmaximal, characteristic, complemented
8C3×#4810C214yesmaximal, complemented
9C3×#4806C214yesmaximal, complemented
10C3×#3010—31nomaximal
11#78960C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2211yescharacteristic
13C3×#1007—24no—
14C3×#1007—24no—
15C2×C3×#239—21no—
16C2×C3×#239C2212yescomplemented
17C2×C3×D3×Q8—21no—
18C2×C3×D3×Q8C2212yescomplemented
19C2×C3×#235—22no—
20C2×C3×#235C2211yescharacteristic
21C2×C3×#235C2211yescharacteristic
22C22×C3×D12C2211yescharacteristic
23C2×C32×M16C2211yescharacteristic
24C32×#117C2211yescharacteristic
25C2×C3×#215—24no—
26C2×C3×#215C2214yes—
27C22×C3×C3⋊C8C2211yescharacteristic
28C3×#908—24no—
29C2×C3×D24C2212yes—
30C2×C3×#204C2212yes—
31C3×#895C2214yes—
32C3×#882—22no—
33C3×#875—21no—
34C3×#874—21no—
35C32×#103C2214yescomplemented
36C3×(C2×D3)⋊C8C2214yescomplemented
37C3×#810C2212yescomplemented
38C3×#809C2212yescomplemented
39C2×C3×2−1+4—31no—
40C2×#1007C2×C311yescharacteristic, complemented
41C2×C3×AGL(1,ℤ/8)—31no—
42C22×C3×SD16—31no—
43C3×#535—31no—
44C22×#215C2×C311yescharacteristic, complemented
45C2×#908C2×C311yescharacteristic, complemented
46#5525C2×C311yescharacteristic, complemented
47C3×#449—34no—
48C3×#448—34no—
49C3×#419—21no—
50C3×#419D311yescharacteristic, complemented
51C3×#419C2×C311yescharacteristic, complemented
52C3×#416—31no—
53C3×#408—31no—
54#5129C2×C311yescharacteristic, complemented
55#5002C2×C311yescharacteristic, complemented
56#4810C2×C314yescomplemented
57#4806C2×C314yescomplemented
58C23×C3×D3—21no—
59C2×C32×Q8—21no—
60C2×C32×Q8—22no—
61C2×C32×Q8D414yescomplemented
62C2×C32×Q8C2312yes—
63C22×C32×C4—21no—
64C22×C32×C4C2311yescharacteristic
65C2×C3×#67—21no—
66C2×C3×#67D412yes—
67C3×#239—28no—
68C3×#239—24no—
69C3×#239—24no—
70C3×D3×Q8—28no—
71C3×D3×Q8—24no—
72C3×D3×Q8—24no—
73C3×#235—216no—
74C3×#235—22no—
75C3×#235—22no—
76C3×#235—22no—
77C3×#235D414yescomplemented
78C2×C3×D12—22no—
79C2×C3×D12—22no—
80C2×C3×D12—22no—
81C2×C3×D12—21no—
82C2×C3×D12—24no—
83C2×C3×D12D411yescharacteristic
84C2×C3×D12C2312yes—
85C2×C3×D12C2311yescharacteristic
86C2×C3×C4×D3—22no—
87C2×C3×C4×D3—21no—
88C2×C3×C4×D3—22no—
89C2×C3×C4×D3C2312yes—
90C2×C3×Dic6—22no—
91C2×C3×Dic6D411yescharacteristic
92C2×C3×Dic6C2311yescharacteristic
93C32×M16—22no—
94C2×C32×C8C2312yes—
95C32×C4⋊C4C2312yes—
96C32×C22⋊C4—21no—
97C32×C42—21no—
98C3×#215—28no—
99C3×(C2×D3)⋊C4—24no—
100C2×C3×C3⋊C8—22no—
101C2×C3×C3⋊C8C2312yes—
102C3×D24—24no—
103C3×#204—24no—
104#3010—31noSylow
105C3×2−1+4—64no—
106C3×2−1+4—64no—
107C2×C3×#41—31no—
108C2×C3×#41—32no—
109C2×C3×#41—31no—
110C2×C3×#41—62no—
111C2×C3×#41—61no—
112C22×C3×Q8—21no—
113C22×C3×Q8—32no—
114C22×C3×Q8—61no—
115C22×C3×Q8C2×D311yescharacteristic
116C22×C3×Q8C22×C311yescharacteristic
117C22×C3×D4—31no—
118#1007—24no—
119#1007—24no—
120C2×#239—21no—
121C2×#239C22×C312yescomplemented
122C2×D3×Q8—21no—
123C2×D3×Q8C22×C312yescomplemented
124C2×#235—22no—
125C2×#235C22×C311yescharacteristic
126C2×#235C22×C311yescharacteristic
127C22×D12C22×C311yescharacteristic
128C3×AGL(1,ℤ/8)—64no—
129C2×C3×SD16—34no—
130C2×C3×SD16—32no—
131C2×C3×SD16—64no—
132C2×C3×D8—32no—
133C2×C3×M16—21no—
134C2×C3×M16C2×D311yescharacteristic
135C2×C3×M16C22×C311yescharacteristic
136C22×C3×C8—31no—
137C3×#127—61no—
138C3×#124—61no—
139C3×#121—34no—
140C3×#120—62no—
141C3×#117—21no—
142C3×#117C2×D311yescharacteristic
143C3×#117C22×C311yescharacteristic
144C2×#215—24no—
145C2×#215C22×C314yes—
146C22×C3⋊C8C22×C311yescharacteristic
147#908—24no—
148C2×D24C22×C312yes—
149C2×#204C22×C312yes—
150#895C22×C314yes—
151#882—22no—
152#875—21no—
153#874—21no—
154C3×#103—24no—
155C3×#103—32no—
156C3×#103C2×D314yescomplemented
157C3×#103C22×C314yescomplemented
158C3×#102—32no—
159C3×C22⋊C8—34no—
160(C2×D3)⋊C8C22×C314yescomplemented
161#810C22×C312yescomplemented
162#809C22×C312yescomplemented
163C23×C32C2×D411yescharacteristic
164C22×C3×D3—21no—
165C22×C3×D3—22no—
166C22×C3×D3—42no—
167C22×C3×D3—82no—
168C22×C3×D3C2×D412yes—
169C32×Q8—24no—
170C32×Q8—22no—
171C32×Q8C2×D414yes—
172C2×C32×C4—24no—
173C2×C32×C4—22no—
174C2×C32×C4—22no—
175C2×C32×C4C2×D412yes—
176C2×C32×C4C2×D411yescharacteristic
177C2×C32×C4C2×D412yes—
178C2×C32×C4C2411yescharacteristic
179C3×#67—24no—
180C3×#67—24no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3C2×D412yes—
183C3×D12—28no—
184C3×D12—22no—
185C3×D12—21no—
186C3×D12—22no—
187C3×D12—22no—
188C3×D12—42no—
189C3×D12—84no—
190C3×D12C2×D412yes—
191C3×C4×D3—28no—
192C3×C4×D3—28no—
193C3×C4×D3—24no—
194C3×C4×D3—22no—
195C3×C4×D3—22no—
196C3×Dic6—28no—
197C3×Dic6—22no—
198C3×Dic6—21no—
199C3×Dic6C2×D412yes—
200C32×C8—22no—
201C3×C3⋊C8—22no—
202C2×2−1+4—31no—
203C2×AGL(1,ℤ/8)—31no—
204C22×SD16—31no—
205#535—31no—
206#449—34no—
207#448—34no—
208#419C3×D311yescharacteristic, complemented
209#416—31no—
210#408—31no—
211C24×C3—61no—
212C23×D3—21no—
213C3×#41—34no—
214C3×#41—616no—
215C3×#41—64no—
216C3×#41—68no—
217C3×#41—64no—
218C3×#41—62no—
219C3×#41—62no—
220C3×#41—62no—
221C2×C3×Q8—21no—
222C2×C3×Q8—22no—
223C2×C3×Q8—21no—
224C2×C3×Q8—22no—
225C2×C3×Q8—24no—
226C2×C3×Q8—22no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—41no—
230C2×C3×Q8—42no—
231C2×C3×Q8—62no—
232C2×C3×Q8—64no—
233C2×C3×Q8—68no—
234C2×C3×Q8—64no—
235C2×C3×Q8D1214yescomplemented
236C2×C3×Q8C3×D414yescomplemented
237C2×C3×Q8C22×D312yes—
238C2×C3×Q8C23×C312yes—
239C2×C3×D4—32no—
240C2×C3×D4—31no—
241C2×C3×D4—31no—
242C2×C3×D4—32no—
243C2×C3×D4—62no—
244C2×C3×D4—64no—
245C2×C3×D4—61no—
246C2×C3×D4—62no—
247C2×C3×D4—62no—
248C2×C3×D4—61no—
249C22×C3×C4—21no—
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4—32no—
253C22×C3×C4—41no—
254C22×C3×C4—62no—
255C22×C3×C4—62no—
256C22×C3×C4—61no—
257C22×C3×C4C22×D311yescharacteristic
258C22×C3×C4C23×C311yescharacteristic
259C2×#67—21no—
260C2×#67C3×D412yes—
261#239—28no—
262#239—24no—
263#239—24no—
264D3×Q8—28no—
265D3×Q8—24no—
266D3×Q8—24no—
267#235—216no—
268#235—22no—
269#235—22no—
270#235—22no—
271#235C3×D414yescomplemented
272C2×D12—22no—
273C2×D12—22no—
274C2×D12—22no—
275C2×D12—21no—
276C2×D12—24no—
277C2×D12C3×D411yescharacteristic
278C2×D12C23×C312yes—
279C2×D12C23×C311yescharacteristic
280C2×C4×D3—22no—
281C2×C4×D3—21no—
282C2×C4×D3—22no—
283C2×C4×D3C23×C312yes—
284C2×Dic6—22no—
285C2×Dic6C3×D411yescharacteristic
286C2×Dic6C23×C311yescharacteristic
287C3×SD16—64no—
288C3×SD16—68no—
289C3×D8—64no—
290C3×M16—22no—
291C3×M16—22no—
292C3×M16—42no—
293C2×C3×C8—22no—
294C2×C3×C8—32no—
295C2×C3×C8—62no—
296C2×C3×C8C22×D312yes—
297C2×C3×C8C23×C312yes—
298C3×C4⋊C4—22no—
299C3×C4⋊C4C22×D312yes—
300C3×C4⋊C4C23×C312yes—
301C3×C22⋊C4—21no—
302C3×C22⋊C4—21no—
303C3×C22⋊C4—41no—
304C3×C22⋊C4—64no—
305C3×C42—21no—
306C3×C42—21no—
307C3×C42—41no—
308#215—28no—
309(C2×D3)⋊C4—24no—
310C2×C3⋊C8—22no—
311C2×C3⋊C8C23×C312yes—
312D24—24no—
313#204—24no—
314C22×C32—22no—
315C22×C32#12012yes—
316C22×C32C22×D411yescharacteristic
317C2×C3×D3—22no—
318C2×C3×D3—22no—
319C2×C3×D3—24no—
320C2×C3×D3—42no—
321C2×C3×D3—42no—
322C2×C3×D3—44no—
323C2×C3×D3—84no—
324C32×C4—24no—
325C32×C4—42no—
326C32×C4#12012yes—
327C32×C4C22×D411yescharacteristic
328C32×C4C22×D411yescharacteristic
329C3×Dic3—22no—
330C3×Dic3—22no—
3312−1+4—64no—
3322−1+4—64no—
333C2×#41—31no—
334C2×#41—32no—
335C2×#41—31no—
336C2×#41—62no—
337C2×#41—61no—
338C22×Q8—32no—
339C22×Q8—61no—
340C22×Q8C2×C3×D311yescharacteristic
341C22×D4—31no—
342AGL(1,ℤ/8)—64no—
343C2×SD16—34no—
344C2×SD16—32no—
345C2×SD16—64no—
346C2×D8—32no—
347C2×M16C2×C3×D311yescharacteristic
348C22×C8—31no—
349#127—61no—
350#124—61no—
351#121—34no—
352#120—62no—
353#117C2×C3×D311yescharacteristic
354#103—32no—
355#103C2×C3×D314yescomplemented
356#102—32no—
357C22⋊C8—34no—
358C23×C3—21no—
359C23×C3—32no—
360C23×C3—61no—
361C23×C3—62no—
362C23×C3—122no—
363C23×C3—242no—
364C23×C3D3×D411yescharacteristic
365C23×C3C2×C3×D411yescharacteristic
366C22×D3—21no—
367C22×D3—22no—
368C22×D3—42no—
369C22×D3—82no—
370C22×D3C2×C3×D412yes—
371C3×Q8—24no—
372C3×Q8—24no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—24no—
376C3×Q8—32no—
377C3×Q8—44no—
378C3×Q8—42no—
379C3×Q8—68no—
380C3×Q8—62no—
381C3×Q8—61no—
382C3×Q8C2×D1214yes—
383C3×Q8C2×C3×D414yes—
384C3×D4—32no—
385C3×D4—68no—
386C3×D4—64no—
387C3×D4—64no—
388C3×D4—62no—
389C3×D4—61no—
390C3×D4—62no—
391C3×D4—62no—
392C3×D4—122no—
393C3×D4—244no—
394C2×C3×C4—24no—
395C2×C3×C4—24no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—21no—
403C2×C3×C4—22no—
404C2×C3×C4—32no—
405C2×C3×C4—44no—
406C2×C3×C4—42no—
407C2×C3×C4—42no—
408C2×C3×C4—68no—
409C2×C3×C4—68no—
410C2×C3×C4—61no—
411C2×C3×C4—64no—
412C2×C3×C4—62no—
413C2×C3×C4—62no—
414C2×C3×C4C2×D1212yes—
415C2×C3×C4D3×D411yescharacteristic
416C2×C3×C4D3×D412yes—
417C2×C3×C4C2×C3×D412yes—
418C2×C3×C4C2×C3×D411yescharacteristic
419C2×C3×C4C2×C3×D412yes—
420C2×C3×C4C23×D311yescharacteristic
421C2×C3×C4C24×C311yescharacteristic
422#67—24no—
423#67—24no—
424C2×Dic3—21no—
425C2×Dic3C2×C3×D412yes—
426D12—28no—
427D12—22no—
428D12—21no—
429D12—22no—
430D12—22no—
431D12—42no—
432D12—84no—
433D12C2×C3×D412yes—
434C4×D3—28no—
435C4×D3—28no—
436C4×D3—24no—
437C4×D3—22no—
438C4×D3—22no—
439Dic6—28no—
440Dic6—22no—
441Dic6—21no—
442Dic6C2×C3×D412yes—
443C3×C8—22no—
444C3×C8—22no—
445C3×C8—42no—
446C3×C8—62no—
447C3⋊C8—22no—
448C2×C32—22no—
449C2×C32C2×#12011yescharacteristic
450C2×C32#57712yes—
451C3×D3—42no—
452C3×D3—42no—
453C3×D3—82no—
454C24—61no—
455#41—34no—
456#41—616no—
457#41—64no—
458#41—68no—
459#41—64no—
460#41—62no—
461#41—62no—
462#41—62no—
463C2×Q8—21no—
464C2×Q8—22no—
465C2×Q8—31no—
466C2×Q8—31no—
467C2×Q8—62no—
468C2×Q8—64no—
469C2×Q8—68no—
470C2×Q8—64no—
471C2×Q8C3×D1214yescomplemented
472C2×Q8C22×C3×D312yes—
473C2×D4—32no—
474C2×D4—31no—
475C2×D4—32no—
476C2×D4—31no—
477C2×D4—62no—
478C2×D4—64no—
479C2×D4—61no—
480C2×D4—62no—
481C2×D4—62no—
482C2×D4—61no—
483C22×C4—21no—
484C22×C4—32no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—61no—
488C22×C4C22×C3×D311yescharacteristic
489SD16—64no—
490SD16—68no—
491D8—64no—
492M16—22no—
493C2×C8—32no—
494C2×C8—62no—
495C2×C8C22×C3×D312yes—
496C4⋊C4C22×C3×D312yes—
497C22⋊C4—21no—
498C22⋊C4—64no—
499C42—21no—
500C22×C3—22no—
501C22×C3—22no—
502C22×C3—22no—
503C22×C3—21no—
504C22×C3—42no—
505C22×C3—62no—
506C22×C3—62no—
507C22×C3—64no—
508C22×C3—122no—
509C22×C3—122no—
510C22×C3—124no—
511C22×C3—244no—
512C22×C3#88212yes—
513C22×C3C3×#12012yes—
514C22×C3C2×D3×D411yescharacteristic
515C22×C3C22×C3×D411yescharacteristic
516C2×D3—24no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—42no—
520C2×D3—44no—
521C2×D3—42no—
522C2×D3—84no—
523C3×C4—24no—
524C3×C4—24no—
525C3×C4—22no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—44no—
529C3×C4—42no—
530C3×C4—42no—
531C3×C4—62no—
532C3×C4—62no—
533C3×C4—82no—
534C3×C4#88212yes—
535C3×C4C3×#12012yes—
536C3×C4C22×D1211yescharacteristic
537C3×C4C2×D3×D411yescharacteristic
538C3×C4C22×C3×D411yescharacteristic
539C3×C4C22×C3×D411yescharacteristic
540Dic3—22no—
541Dic3—22no—
542C32#301011yescharacteristic, complemented, Sylow
543C23—32no—
544C23—61no—
545C23—62no—
546C23—122no—
547C23—242no—
548C23C3×D3×D411yescharacteristic
549Q8—24no—
550Q8—22no—
551Q8—32no—
552Q8—68no—
553Q8—62no—
554Q8—61no—
555Q8C2×C3×D1214yes—
556D4—32no—
557D4—68no—
558D4—64no—
559D4—64no—
560D4—62no—
561D4—61no—
562D4—62no—
563D4—62no—
564D4—122no—
565D4—244no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—32no—
570C2×C4—68no—
571C2×C4—68no—
572C2×C4—61no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4C2×C3×D1212yes—
577C2×C4C3×D3×D411yescharacteristic
578C2×C4C3×D3×D412yes—
579C2×C4C23×C3×D311yescharacteristic
580C8—22no—
581C8—62no—
582C2×C3—22no—
583C2×C3—22no—
584C2×C3—21no—
585C2×C3—22no—
586C2×C3—42no—
587C2×C3—122no—
588C2×C3—122no—
589C2×C3—242no—
590C2×C3C2×#88211yescharacteristic
591C2×C3#577812yes—
592C2×C3C2×C3×#12011yescharacteristic
593C2×C3C3×#57712yes—
594D3—42no—
595D3—42no—
596D3—82no—
597C22—22no—
598C22—62no—
599C22—62no—
600C22—64no—
601C22—122no—
602C22—122no—
603C22—124no—
604C22—244no—
605C22C3×#88212yes—
606C22C2×C3×D3×D411yescharacteristic
607C4—24no—
608C4—42no—
609C4—62no—
610C4—62no—
611C4C3×#88212yes—
612C4C22×C3×D1211yescharacteristic
613C4C2×C3×D3×D411yescharacteristic
614C3—21no—
615C3#7896011yescharacteristic, direct product factor
616C3C3×#301011yescharacteristic, complemented
617C2—22no—
618C2—122no—
619C2—122no—
620C2—242no—
621C2C2×C3×#88211yescharacteristic
622C2C3×#577812yes—
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#419C2
socleC22×C32C22×D4
hypercenterC3×#419D3
hypocenterC3C3×#3010
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#419

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1007C2semidirect
2. C22×C3×#215C2semidirect
3. C2×C3×#908C2semidirect
4. C3×#5525C2semidirect
5. C32×#419C2semidirect
6. C3×#5129C2semidirect
7. C3×#5002C2semidirect
8. C3×#4810C2semidirect
9. C3×#4806C2semidirect
11. #78960C3direct (N × H)
16. C2×C3×#239C22semidirect
18. C2×C3×D3×Q8C22semidirect
35. C32×#103C22semidirect
36. C3×(C2×D3)⋊C8C22semidirect
37. C3×#810C22semidirect
38. C3×#809C22semidirect
40. C2×#1007C2×C3semidirect
44. C22×#215C2×C3semidirect
45. C2×#908C2×C3semidirect
46. #5525C2×C3semidirect
50. C3×#419D3semidirect
51. C3×#419C2×C3semidirect
54. #5129C2×C3semidirect
55. #5002C2×C3semidirect
56. #4810C2×C3semidirect
57. #4806C2×C3semidirect
61. C2×C32×Q8D4semidirect
77. C3×#235D4semidirect
121. C2×#239C22×C3semidirect
123. C2×D3×Q8C22×C3semidirect
156. C3×#103C2×D3semidirect
157. C3×#103C22×C3semidirect
160. (C2×D3)⋊C8C22×C3semidirect
161. #810C22×C3semidirect
162. #809C22×C3semidirect
208. #419C3×D3semidirect
235. C2×C3×Q8D12semidirect
236. C2×C3×Q8C3×D4semidirect
271. #235C3×D4semidirect
355. #103C2×C3×D3semidirect
471. C2×Q8C3×D12semidirect
542. C32#3010semidirect
615. C3#78960direct (N × H)
616. C3C3×#3010semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10072
C22×C3×#2152
C2×C3×#9082
C3×#55252
C32×#4192
C3×#51292
C3×#50022
C3×#48102
C3×#48062
C3×#30103
#789603