Finite Group Databasebeta

← C3×#78959 #11864049

Subgroups — C3×#78959 #11864049

Order 1152.

1591 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
1C22×C3×#239C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic, complemented
3C2×C3×#908C211yesmaximal, characteristic, complemented
4C3×#5524C211yesmaximal, characteristic, complemented
5C32×#419C211yesmaximal, characteristic, complemented
6C3×#5126C211yesmaximal, characteristic, complemented
7C3×#5001C211yesmaximal, characteristic, complemented
8C3×#4810C214yesmaximal, complemented
9C3×#4808C214yesmaximal, complemented
10C3×#3005—31nomaximal
11#78959C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2211yescharacteristic
13C2×C3×#239—24no—
14C2×C3×#239—24no—
15C2×C3×#239—22no—
16C2×C3×#239C2214yes—
17C2×C3×#235C2211yescharacteristic
18C22×C3×D12—21no—
19C22×C3×D12C2211yescharacteristic
20C22×C3×C4×D3—21no—
21C22×C3×C4×D3C2211yescharacteristic
22C2×C32×M16C2211yescharacteristic
23C32×#117C2211yescharacteristic
24C2×C3×#216C2212yescomplemented
25C3×#942—24no—
26C2×C3×#215C2212yescomplemented
27C2×C3×#208C2211yescharacteristic
28C3×#908—24no—
29C2×C3×D24C2212yescomplemented
30C2×C3×#204C2212yescomplemented
31C3×#897C2212yescomplemented
32C3×#895C2212yescomplemented
33C3×#886—21no—
34C3×#882—21no—
35C3×C4×D12—22no—
36C32×#103C2214yescomplemented
37C3×(C2×D3)⋊C8C2214yescomplemented
38C3×#809C2214yescomplemented
39C22×C3×#41—31no—
40C22×#239C2×C311yescharacteristic, complemented
41C2×C3×#137—31no—
42C2×C3×AGL(1,ℤ/8)—31no—
43C3×#525—31no—
44C2×#942C2×C311yescharacteristic, complemented
45C2×#908C2×C311yescharacteristic, complemented
46#5524C2×C311yescharacteristic, complemented
47C3×#452—34no—
48C3×#449—34no—
49C3×#419—21no—
50C3×#419D311yescharacteristic, complemented
51C3×#419C2×C311yescharacteristic, complemented
52C3×#418—31no—
53C3×#407—31no—
54#5126C2×C311yescharacteristic, complemented
55#5001C2×C311yescharacteristic, complemented
56#4810C2×C314yescomplemented
57#4808C2×C314yescomplemented
58C23×C3×D3—21no—
59C23×C3×D3D411yescharacteristic
60C2×C32×Q8—22no—
61C2×C32×Q8—21no—
62C2×C32×Q8D414yescomplemented
63C2×C32×Q8C2312yes—
64C22×C32×C4—21no—
65C22×C32×C4C2311yescharacteristic
66C2×C3×#67—21no—
67C22×C3×Dic3D411yescharacteristic
68C3×#239—216no—
69C3×#239—28no—
70C3×#239—28no—
71C3×#235—22no—
72C3×#235—22no—
73C2×C3×D12—28no—
74C2×C3×D12—24no—
75C2×C3×D12—21no—
76C2×C3×D12—22no—
77C2×C3×D12—22no—
78C2×C3×D12D414yescomplemented
79C2×C3×D12C2311yescharacteristic
80C2×C3×D12C2312yes—
81C2×C3×C4×D3—28no—
82C2×C3×C4×D3—22no—
83C2×C3×C4×D3—22no—
84C2×C3×C4×D3—24no—
85C2×C3×C4×D3—21no—
86C2×C3×C4×D3D412yes—
87C2×C3×C4×D3C2312yes—
88C2×C3×Dic6C2311yescharacteristic
89C32×M16—22no—
90C2×C32×C8C2312yes—
91C32×C4⋊C4C2312yes—
92C32×C22⋊C4—21no—
93C32×C42—21no—
94C3×#216—24no—
95C3×#215—24no—
96C3×(C2×D3)⋊C4—22no—
97C3×(C2×D3)⋊C4—22no—
98C3×C4⋊Dic3—22no—
99C3×#208—22no—
100C2×C3×C3⋊C8C2312yes—
101C3×D24—24no—
102C3×#204—24no—
103#3005—31noSylow
104C2×C3×#41—34no—
105C2×C3×#41—31no—
106C2×C3×#41—62no—
107C2×C3×#41—64no—
108C2×C3×#41—64no—
109C22×C3×Q8—21no—
110C22×C3×Q8C2×D311yescharacteristic
111C22×C3×Q8C22×C311yescharacteristic
112C22×C3×D4—31no—
113C22×C3×D4—61no—
114C23×C3×C4—31no—
115C23×C3×C4—61no—
116C2×#239—24no—
117C2×#239—24no—
118C2×#239—22no—
119C2×#239C22×C314yes—
120C2×#235C22×C311yescharacteristic
121C22×D12—21no—
122C22×D12C22×C311yescharacteristic
123C22×C4×D3—21no—
124C22×C4×D3C22×C311yescharacteristic
125C3×#137—64no—
126C3×AGL(1,ℤ/8)—64no—
127C2×C3×Q16—32no—
128C2×C3×SD16—32no—
129C2×C3×SD16—32no—
130C2×C3×D8—32no—
131C2×C3×M16—21no—
132C2×C3×M16—31no—
133C2×C3×M16C2×D311yescharacteristic
134C2×C3×M16C22×C311yescharacteristic
135C3×#123—61no—
136C3×C22⋊Q8—32no—
137C3×#121—32no—
138C3×#120—61no—
139C3×C4×D4—62no—
140C3×#117—21no—
141C3×#117C2×D311yescharacteristic
142C3×#117C22×C311yescharacteristic
143C2×#216C22×C312yescomplemented
144#942—24no—
145C2×#215C22×C312yescomplemented
146C2×#208C22×C311yescharacteristic
147#908—24no—
148C2×D24C22×C312yescomplemented
149C2×#204C22×C312yescomplemented
150#897C22×C312yescomplemented
151#895C22×C312yescomplemented
152#886—21no—
153#882—21no—
154C4×D12—22no—
155C3×#103—24no—
156C3×#103C2×D314yescomplemented
157C3×#103C22×C314yescomplemented
158C3×#102—34no—
159C3×C22⋊C8—34no—
160(C2×D3)⋊C8C22×C314yescomplemented
161#809C22×C314yescomplemented
162C23×C32C2×D411yescharacteristic
163C22×C3×D3—24no—
164C22×C3×D3—22no—
165C22×C3×D3—22no—
166C22×C3×D3—21no—
167C22×C3×D3—42no—
168C22×C3×D3—44no—
169C22×C3×D3C2×D412yes—
170C32×Q8—24no—
171C32×Q8—22no—
172C32×Q8C2×D414yes—
173C2×C32×C4—24no—
174C2×C32×C4—22no—
175C2×C32×C4—22no—
176C2×C32×C4C2×D412yes—
177C2×C32×C4C2×D411yescharacteristic
178C2×C32×C4C2×D412yes—
179C2×C32×C4C2411yescharacteristic
180C3×#67—42no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3C2×D412yes—
184C3×D12—216no—
185C3×D12—24no—
186C3×D12—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—44no—
190C3×D12C2×D414yes—
191C3×C4×D3—216no—
192C3×C4×D3—24no—
193C3×C4×D3—22no—
194C3×C4×D3—22no—
195C3×C4×D3—42no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C32×C8—22no—
199C3×C3⋊C8—22no—
200C22×#41—31no—
201C2×#137—31no—
202C2×AGL(1,ℤ/8)—31no—
203#525—31no—
204#452—34no—
205#449—34no—
206#419C3×D311yescharacteristic, complemented
207#418—31no—
208#407—31no—
209C24×C3—31no—
210C24×C3—61no—
211C23×D3—21no—
212C23×D3C3×D411yescharacteristic
213C3×#41—68no—
214C3×#41—616no—
215C3×#41—68no—
216C3×#41—62no—
217C3×#41—62no—
218C2×C3×Q8—22no—
219C2×C3×Q8—21no—
220C2×C3×Q8—22no—
221C2×C3×Q8—21no—
222C2×C3×Q8—24no—
223C2×C3×Q8—22no—
224C2×C3×Q8—31no—
225C2×C3×Q8—41no—
226C2×C3×Q8—42no—
227C2×C3×Q8D1214yescomplemented
228C2×C3×Q8C3×D414yescomplemented
229C2×C3×Q8C22×D312yes—
230C2×C3×Q8C23×C312yes—
231C2×C3×D4—34no—
232C2×C3×D4—32no—
233C2×C3×D4—31no—
234C2×C3×D4—64no—
235C2×C3×D4—68no—
236C2×C3×D4—62no—
237C2×C3×D4—61no—
238C2×C3×D4—62no—
239C2×C3×D4—61no—
240C22×C3×C4—21no—
241C22×C3×C4—21no—
242C22×C3×C4—21no—
243C22×C3×C4—31no—
244C22×C3×C4—32no—
245C22×C3×C4—32no—
246C22×C3×C4—41no—
247C22×C3×C4—64no—
248C22×C3×C4—68no—
249C22×C3×C4—62no—
250C22×C3×C4—62no—
251C22×C3×C4—61no—
252C22×C3×C4C22×D311yescharacteristic
253C22×C3×C4C23×C311yescharacteristic
254C2×#67—21no—
255C22×Dic3C3×D411yescharacteristic
256#239—216no—
257#239—28no—
258#239—28no—
259#235—22no—
260#235—22no—
261C2×D12—28no—
262C2×D12—24no—
263C2×D12—22no—
264C2×D12—21no—
265C2×D12—22no—
266C2×D12C3×D414yescomplemented
267C2×D12C23×C311yescharacteristic
268C2×D12C23×C312yes—
269C2×C4×D3—28no—
270C2×C4×D3—22no—
271C2×C4×D3—22no—
272C2×C4×D3—24no—
273C2×C4×D3—21no—
274C2×C4×D3C3×D412yes—
275C2×C4×D3C23×C312yes—
276C2×Dic6C23×C311yescharacteristic
277C3×Q16—64no—
278C3×SD16—64no—
279C3×SD16—64no—
280C3×D8—64no—
281C3×M16—22no—
282C3×M16—22no—
283C3×M16—42no—
284C3×M16—62no—
285C2×C3×C8—22no—
286C2×C3×C8—32no—
287C2×C3×C8C22×D312yes—
288C2×C3×C8C23×C312yes—
289C3×C4⋊C4—22no—
290C3×C4⋊C4—62no—
291C3×C4⋊C4C22×D312yes—
292C3×C4⋊C4C23×C312yes—
293C3×C22⋊C4—21no—
294C3×C22⋊C4—21no—
295C3×C22⋊C4—41no—
296C3×C22⋊C4—62no—
297C3×C22⋊C4—62no—
298C3×C42—21no—
299C3×C42—21no—
300C3×C42—41no—
301#216—24no—
302#215—24no—
303(C2×D3)⋊C4—22no—
304(C2×D3)⋊C4—22no—
305C4⋊Dic3—22no—
306#208—22no—
307C2×C3⋊C8C23×C312yes—
308D24—24no—
309#204—24no—
310C22×C32—22no—
311C22×C32#12012yes—
312C22×C32C22×D411yescharacteristic
313C2×C3×D3—22no—
314C2×C3×D3—24no—
315C2×C3×D3—22no—
316C2×C3×D3—22no—
317C2×C3×D3—44no—
318C2×C3×D3—48no—
319C2×C3×D3—44no—
320C2×C3×D3—41no—
321C2×C3×D3—42no—
322C32×C4—24no—
323C32×C4—42no—
324C32×C4#12012yes—
325C32×C4C22×D411yescharacteristic
326C32×C4C22×D411yescharacteristic
327C3×Dic3—22no—
328C3×Dic3—41no—
329C2×#41—34no—
330C2×#41—31no—
331C2×#41—62no—
332C2×#41—64no—
333C2×#41—64no—
334C22×Q8C2×C3×D311yescharacteristic
335C22×D4—31no—
336C22×D4—61no—
337C23×C4—31no—
338C23×C4—61no—
339#137—64no—
340AGL(1,ℤ/8)—64no—
341C2×Q16—32no—
342C2×SD16—32no—
343C2×SD16—32no—
344C2×D8—32no—
345C2×M16—31no—
346C2×M16C2×C3×D311yescharacteristic
347#123—61no—
348C22⋊Q8—32no—
349#121—32no—
350#120—61no—
351C4×D4—62no—
352#117C2×C3×D311yescharacteristic
353#103C2×C3×D314yescomplemented
354#102—34no—
355C22⋊C8—34no—
356C23×C3—21no—
357C23×C3—32no—
358C23×C3—64no—
359C23×C3—62no—
360C23×C3—62no—
361C23×C3—61no—
362C23×C3—122no—
363C23×C3—124no—
364C23×C3D3×D411yescharacteristic
365C23×C3C2×C3×D411yescharacteristic
366C22×D3—22no—
367C22×D3—24no—
368C22×D3—22no—
369C22×D3—21no—
370C22×D3—42no—
371C22×D3—44no—
372C22×D3C2×C3×D412yes—
373C3×Q8—24no—
374C3×Q8—24no—
375C3×Q8—22no—
376C3×Q8—22no—
377C3×Q8—24no—
378C3×Q8—44no—
379C3×Q8—42no—
380C3×Q8—61no—
381C3×Q8—61no—
382C3×Q8C2×D1214yes—
383C3×Q8C2×C3×D414yes—
384C3×D4—34no—
385C3×D4—616no—
386C3×D4—64no—
387C3×D4—62no—
388C3×D4—61no—
389C3×D4—61no—
390C3×D4—124no—
391C3×D4—122no—
392C2×C3×C4—24no—
393C2×C3×C4—24no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—32no—
403C2×C3×C4—44no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—616no—
407C2×C3×C4—62no—
408C2×C3×C4—64no—
409C2×C3×C4—62no—
410C2×C3×C4—62no—
411C2×C3×C4—61no—
412C2×C3×C4—122no—
413C2×C3×C4C2×D1212yes—
414C2×C3×C4D3×D411yescharacteristic
415C2×C3×C4D3×D412yes—
416C2×C3×C4C2×C3×D412yes—
417C2×C3×C4C2×C3×D411yescharacteristic
418C2×C3×C4C2×C3×D412yes—
419C2×C3×C4C23×D311yescharacteristic
420C2×C3×C4C24×C311yescharacteristic
421#67—42no—
422C2×Dic3—22no—
423C2×Dic3—21no—
424C2×Dic3C2×C3×D412yes—
425D12—216no—
426D12—24no—
427D12—22no—
428D12—21no—
429D12—21no—
430D12—44no—
431D12C2×C3×D414yes—
432C4×D3—216no—
433C4×D3—24no—
434C4×D3—22no—
435C4×D3—22no—
436C4×D3—42no—
437Dic6—21no—
438Dic6—21no—
439C3×C8—22no—
440C3×C8—22no—
441C3×C8—42no—
442C3×C8—62no—
443C3⋊C8—22no—
444C2×C32—22no—
445C2×C32C2×#12011yescharacteristic
446C2×C32#57512yes—
447C3×D3—42no—
448C3×D3—44no—
449C3×D3—81no—
450C24—31no—
451C24—61no—
452#41—68no—
453#41—616no—
454#41—68no—
455#41—62no—
456#41—62no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—31no—
460C2×Q8C3×D1214yescomplemented
461C2×Q8C22×C3×D312yes—
462C2×D4—32no—
463C2×D4—31no—
464C2×D4—34no—
465C2×D4—64no—
466C2×D4—68no—
467C2×D4—62no—
468C2×D4—61no—
469C2×D4—62no—
470C2×D4—61no—
471C22×C4—21no—
472C22×C4—31no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—64no—
476C22×C4—68no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—61no—
480C22×C4C22×C3×D311yescharacteristic
481Q16—64no—
482SD16—64no—
483SD16—64no—
484D8—64no—
485M16—22no—
486M16—62no—
487C2×C8—32no—
488C2×C8C22×C3×D312yes—
489C4⋊C4—62no—
490C4⋊C4C22×C3×D312yes—
491C22⋊C4—21no—
492C22⋊C4—62no—
493C22⋊C4—62no—
494C42—21no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—21no—
499C22×C3—42no—
500C22×C3—64no—
501C22×C3—62no—
502C22×C3—62no—
503C22×C3—62no—
504C22×C3—124no—
505C22×C3—128no—
506C22×C3—124no—
507C22×C3—121no—
508C22×C3—122no—
509C22×C3#88212yes—
510C22×C3C3×#12012yes—
511C22×C3C2×D3×D411yescharacteristic
512C22×C3C22×C3×D411yescharacteristic
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—24no—
517C2×D3—44no—
518C2×D3—48no—
519C2×D3—44no—
520C2×D3—42no—
521C2×D3—41no—
522C3×C4—24no—
523C3×C4—24no—
524C3×C4—22no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—44no—
528C3×C4—42no—
529C3×C4—42no—
530C3×C4—62no—
531C3×C4—82no—
532C3×C4—121no—
533C3×C4#88212yes—
534C3×C4C3×#12012yes—
535C3×C4C22×D1211yescharacteristic
536C3×C4C2×D3×D411yescharacteristic
537C3×C4C22×C3×D411yescharacteristic
538C3×C4C22×C3×D411yescharacteristic
539Dic3—22no—
540Dic3—41no—
541C32#300511yescharacteristic, complemented, Sylow
542C23—32no—
543C23—64no—
544C23—62no—
545C23—62no—
546C23—61no—
547C23—122no—
548C23—124no—
549C23C3×D3×D411yescharacteristic
550Q8—24no—
551Q8—22no—
552Q8—61no—
553Q8—61no—
554Q8C2×C3×D1214yes—
555D4—34no—
556D4—616no—
557D4—64no—
558D4—62no—
559D4—61no—
560D4—61no—
561D4—124no—
562D4—122no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—32no—
567C2×C4—616no—
568C2×C4—62no—
569C2×C4—64no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—61no—
573C2×C4—122no—
574C2×C4C2×C3×D1212yes—
575C2×C4C3×D3×D411yescharacteristic
576C2×C4C3×D3×D412yes—
577C2×C4C23×C3×D311yescharacteristic
578C8—22no—
579C8—62no—
580C2×C3—22no—
581C2×C3—22no—
582C2×C3—21no—
583C2×C3—22no—
584C2×C3—42no—
585C2×C3—122no—
586C2×C3—124no—
587C2×C3—241no—
588C2×C3C2×#88211yescharacteristic
589C2×C3#577712yes—
590C2×C3C2×C3×#12011yescharacteristic
591C2×C3C3×#57512yes—
592D3—42no—
593D3—44no—
594D3—81no—
595C22—22no—
596C22—64no—
597C22—62no—
598C22—62no—
599C22—62no—
600C22—124no—
601C22—128no—
602C22—124no—
603C22—121no—
604C22—122no—
605C22C3×#88212yes—
606C22C2×C3×D3×D411yescharacteristic
607C4—24no—
608C4—42no—
609C4—62no—
610C4—121no—
611C4C3×#88212yes—
612C4C22×C3×D1211yescharacteristic
613C4C2×C3×D3×D411yescharacteristic
614C3—21no—
615C3#7895911yescharacteristic, direct product factor
616C3C3×#300511yescharacteristic, complemented
617C2—22no—
618C2—122no—
619C2—124no—
620C2—241no—
621C2C2×C3×#88211yescharacteristic
622C2C3×#577712yes—
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#419C2
socleC22×C32C22×D4
hypercenterC3×#419D3
hypocenterC3C3×#3005
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. C22×C3×#239C2semidirect
2. C2×C3×#942C2semidirect
3. C2×C3×#908C2semidirect
4. C3×#5524C2semidirect
5. C32×#419C2semidirect
6. C3×#5126C2semidirect
7. C3×#5001C2semidirect
8. C3×#4810C2semidirect
9. C3×#4808C2semidirect
11. #78959C3direct (N × H)
24. C2×C3×#216C22semidirect
26. C2×C3×#215C22semidirect
29. C2×C3×D24C22semidirect
30. C2×C3×#204C22semidirect
31. C3×#897C22semidirect
32. C3×#895C22semidirect
36. C32×#103C22semidirect
37. C3×(C2×D3)⋊C8C22semidirect
38. C3×#809C22semidirect
40. C22×#239C2×C3semidirect
44. C2×#942C2×C3semidirect
45. C2×#908C2×C3semidirect
46. #5524C2×C3semidirect
50. C3×#419D3semidirect
51. C3×#419C2×C3semidirect
54. #5126C2×C3semidirect
55. #5001C2×C3semidirect
56. #4810C2×C3semidirect
57. #4808C2×C3semidirect
62. C2×C32×Q8D4semidirect
78. C2×C3×D12D4semidirect
143. C2×#216C22×C3semidirect
145. C2×#215C22×C3semidirect
148. C2×D24C22×C3semidirect
149. C2×#204C22×C3semidirect
150. #897C22×C3semidirect
151. #895C22×C3semidirect
156. C3×#103C2×D3semidirect
157. C3×#103C22×C3semidirect
160. (C2×D3)⋊C8C22×C3semidirect
161. #809C22×C3semidirect
206. #419C3×D3semidirect
227. C2×C3×Q8D12semidirect
228. C2×C3×Q8C3×D4semidirect
266. C2×D12C3×D4semidirect
353. #103C2×C3×D3semidirect
460. C2×Q8C3×D12semidirect
541. C32#3005semidirect
615. C3#78959direct (N × H)
616. C3C3×#3005semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2392
C2×C3×#9422
C2×C3×#9082
C3×#55242
C32×#4192
C3×#51262
C3×#50012
C3×#48102
C3×#48082
C3×#30053
#789593