Finite Group Databasebeta

← C3×#23587 #10642194

Subgroups — C3×#23587 #10642194

Order 768.

1274 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic, complemented
2C3×C4×AGL(1,ℤ/8)C212yesmaximal, complemented
3C3×C4×#127C211yesmaximal, characteristic, complemented
4C3×#2060C211yesmaximal, characteristic, complemented
5C3×#2039C211yesmaximal, characteristic, complemented
6C3×#2037C212yesmaximal, complemented
7C3×#2027C211yesmaximal, characteristic, complemented
8C3×#2006C211yesmaximal, characteristic, complemented
9C3×#2002C212yesmaximal, complemented
10C3×C4×#104C212yesmaximal, complemented
11C3×C4×#100C211yesmaximal, characteristic, complemented
12#23587C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C3×#544C2211yescharacteristic
15C3×#542—22no—
16C3×#540—22no—
17C3×#538—22no—
18C3×#537C2211yescharacteristic
19C3×#535C2212yes—
20C3×#532—22no—
21C2×C3×#127C2211yescharacteristic
22C3×#529—22no—
23C3×#525—22no—
24C3×C4×#41C2212yes—
25C2×C3×C4×D4—22no—
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×C4×D4C2211yescharacteristic
28C3×#442—22no—
29C3×#440—22no—
30C3×C4×SD16—22no—
31C3×C4×D8—22no—
32C3×#428C2212yes—
33C2×C3×#104C2212yes—
34C2×C3×#104C2212yes—
35C3×#418C2212yes—
36C3×#417C2212yes—
37C2×C3×#100C2212yes—
38C3×C4×M16C2212yes—
39C3×#390—22no—
40C3×#390C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#384C2211yescharacteristic
43C3×C43C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3460C2×C311yescharacteristic, complemented
49C4×AGL(1,ℤ/8)C2×C312yescomplemented
50C4×#127C2×C311yescharacteristic, complemented
51#2060C2×C311yescharacteristic, complemented
52#2039C2×C311yescharacteristic, complemented
53#2037C2×C312yescomplemented
54#2027C2×C311yescharacteristic, complemented
55#2006C2×C311yescharacteristic, complemented
56#2002C2×C312yescomplemented
57C4×#104C2×C312yescomplemented
58C4×#100C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×D4—22no—
62C22×C3×D4C2311yescharacteristic
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4D412yes—
66C3×AGL(1,ℤ/8)—24no—
67C3×AGL(1,ℤ/8)—24no—
68C2×C3×SD16—22no—
69C2×C3×D8—22no—
70C2×C3×M16C2312yes—
71C2×C3×M16C2312yes—
72C3×C4⋊Q8—22no—
73C3×#127—22no—
74C3×#127—22no—
75C3×#127—22no—
76C3×#126—24no—
77C3×#125—22no—
78C3×#124—24no—
79C3×#124—22no—
80C3×#123—24no—
81C3×#123—24no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×#121—22no—
86C3×#121—24no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#120—24no—
92C3×C4×Q8D412yescomplemented
93C3×C4×Q8D412yescomplemented
94C3×C4×D4—24no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—22no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4—48no—
102C3×C4×D4D412yescomplemented
103C3×C4×D4D412yescomplemented
104C3×#117—22no—
105C3×#117—22no—
106C3×#117C2312yes—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4—22no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4D412yes—
114C2×C3×C42—21no—
115C2×C3×C42—22no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×#107—22no—
121C3×#106—22no—
122C3×#104—24no—
123C3×#104—24no—
124C3×#103—22no—
125C3×#102—22no—
126C3×#102—22no—
127C3×#102—22no—
128C3×#100—24no—
129C3×#97—22no—
130C3×C4×C8—22no—
131C2×AGL(1,ℤ/8)C22×C312yes—
132#544C22×C311yescharacteristic
133#542—22no—
134#540—22no—
135#538—22no—
136#537C22×C311yescharacteristic
137#535C22×C312yes—
138#532—22no—
139C2×#127C22×C311yescharacteristic
140#529—22no—
141#525—22no—
142C4×#41C22×C312yes—
143C2×C4×D4—22no—
144C2×C4×D4C22×C311yescharacteristic
145C2×C4×D4C22×C311yescharacteristic
146#442—22no—
147#440—22no—
148C4×SD16—22no—
149C4×D8—22no—
150#428C22×C312yes—
151C2×#104C22×C312yes—
152C2×#104C22×C312yes—
153#418C22×C312yes—
154#417C22×C312yes—
155C2×#100C22×C312yes—
156C4×M16C22×C312yes—
157#390—22no—
158#390C22×C311yescharacteristic
159#390C22×C311yescharacteristic
160#384C22×C311yescharacteristic
161C43C22×C311yescharacteristic
162#343C22×C311yescharacteristic
163#343C22×C311yescharacteristic
164#339C22×C312yes—
165#339C22×C312yes—
166C24×C3—21no—
167C24×C3C2×D412yes—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—22no—
172C3×#41—84no—
173C2×C3×Q8—22no—
174C2×C3×Q8C2×D412yes—
175C2×C3×D4—28no—
176C2×C3×D4—22no—
177C2×C3×D4—24no—
178C2×C3×D4—24no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—44no—
188C2×C3×D4—44no—
189C2×C3×D4C2×D412yes—
190C2×C3×D4#4112yescomplemented
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—21no—
199C22×C3×C4—44no—
200C22×C3×C4—44no—
201C22×C3×C4—41no—
202C22×C3×C4—41no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×SD16—28no—
208C3×D8—28no—
209C3×M16—22no—
210C3×M16—22no—
211C3×M16—22no—
212C3×M16#4114yescomplemented
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—44no—
223C3×C4⋊C4C2×D412yes—
224C3×C4⋊C4C2×D412yes—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—44no—
233C3×C22⋊C4—44no—
234C3×C22⋊C4—42no—
235C3×C42—24no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—22no—
239C3×C42—21no—
240C3×C42—22no—
241C3×C42—42no—
242C3×C42C2×D412yes—
243C3×C42C2×D412yes—
244C3×C42#4112yescomplemented
245C2×#41—22no—
246C2×#41C23×C312yes—
247C22×D4—22no—
248C22×D4C23×C311yescharacteristic
249C22×D4C23×C311yescharacteristic
250C23×C4—21no—
251C23×C4C3×D412yes—
252AGL(1,ℤ/8)—24no—
253AGL(1,ℤ/8)—24no—
254C2×SD16—22no—
255C2×D8—22no—
256C2×M16C23×C312yes—
257C2×M16C23×C312yes—
258C4⋊Q8—22no—
259#127—22no—
260#127—22no—
261#127—22no—
262#126—24no—
263#125—22no—
264#124—24no—
265#124—22no—
266#123—24no—
267#123—24no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271#121—22no—
272#121—24no—
273#121—22no—
274#121—22no—
275#121—22no—
276#121—22no—
277#120—24no—
278C4×Q8C3×D412yescomplemented
279C4×Q8C3×D412yescomplemented
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—48no—
288C4×D4C3×D412yescomplemented
289C4×D4C3×D412yescomplemented
290#117—22no—
291#117—22no—
292#117C23×C312yes—
293C2×C4⋊C4—22no—
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4C3×D412yes—
300C2×C42—21no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42C23×C311yescharacteristic
304C2×C42C23×C311yescharacteristic
305C2×C42C23×C311yescharacteristic
306#107—22no—
307#106—22no—
308#104—24no—
309#104—24no—
310#103—22no—
311#102—22no—
312#102—22no—
313#102—22no—
314#100—24no—
315#97—22no—
316C4×C8—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3C22×D411yescharacteristic
329C3×Q8—22no—
330C3×Q8—22no—
331C3×Q8—82no—
332C3×D4—22no—
333C3×D4—24no—
334C3×D4—22no—
335C3×D4—416no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—84no—
342C3×D4—82no—
343C3×D4—84no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—44no—
359C2×C3×C4—44no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—82no—
364C2×C3×C4—82no—
365C2×C3×C4—82no—
366C2×C3×C4—82no—
367C2×C3×C4#12012yes—
368C2×C3×C4#12011yescharacteristic
369C2×C3×C4#12011yescharacteristic
370C2×C3×C4C22×D412yes—
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4112yes—
373C2×C3×C4C2×#4111yescharacteristic
374C3×C8—24no—
375C3×C8—42no—
376C24—21no—
377C24C2×C3×D412yes—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41—84no—
383C2×Q8—22no—
384C2×Q8C2×C3×D412yes—
385C2×D4—28no—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4C2×C3×D412yes—
400C2×D4C3×#4112yescomplemented
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D412yes—
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C24×C311yescharacteristic
417SD16—28no—
418D8—28no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16C3×#4114yescomplemented
423C2×C8—22no—
424C2×C8—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—44no—
433C4⋊C4C2×C3×D412yes—
434C4⋊C4C2×C3×D412yes—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—24no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—21no—
450C42—22no—
451C42—42no—
452C42C2×C3×D412yes—
453C42C2×C3×D412yes—
454C42C3×#4112yescomplemented
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—82no—
467C22×C3—82no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3C2×#12011yescharacteristic
471C22×C3#52511yescharacteristic
472C22×C3#53211yescharacteristic
473C3×C4—24no—
474C3×C4—21no—
475C3×C4—22no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—81no—
482C3×C4#45312yes—
483C3×C4#52512yes—
484C3×C4#53212yes—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23C22×C3×D411yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—82no—
500D4—22no—
501D4—24no—
502D4—22no—
503D4—416no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—82no—
510D4—84no—
511D4—84no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4C3×#12012yes—
536C2×C4C3×#12011yescharacteristic
537C2×C4C3×#12011yescharacteristic
538C2×C4C22×C3×D412yes—
539C2×C4C2×C3×#4111yescharacteristic
540C2×C4C2×C3×#4112yes—
541C2×C4C2×C3×#4111yescharacteristic
542C8—24no—
543C8—42no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—82no—
547C2×C3—82no—
548C2×C3—82no—
549C2×C3#238811yescharacteristic
550C2×C3C2×#45311yescharacteristic
551C2×C3#301811yescharacteristic
552C22—21no—
553C22—21no—
554C22—44no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—82no—
564C22—82no—
565C22—81no—
566C22—81no—
567C22C2×C3×#12011yescharacteristic
568C22C3×#52511yescharacteristic
569C22C3×#53211yescharacteristic
570C4—24no—
571C4—21no—
572C4—22no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—81no—
579C4C3×#45312yes—
580C4C3×#52512yes—
581C4C3×#53212yes—
582C3#2358711yescharacteristic, direct product factor, Sylow
583C2—21no—
584C2—21no—
585C2—82no—
586C2—82no—
587C2—82no—
588C2C3×#238811yescharacteristic
589C2C2×C3×#45311yescharacteristic
590C2C3×#301811yescharacteristic
subgroupG/H
centerC2×C3×C4#120
commutator, Frattini, cosocleC22×C4C24×C3
socleC22×C3C2×#120
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3460C2semidirect
2. C3×C4×AGL(1,ℤ/8)C2semidirect
3. C3×C4×#127C2semidirect
4. C3×#2060C2semidirect
5. C3×#2039C2semidirect
6. C3×#2037C2semidirect
7. C3×#2027C2semidirect
8. C3×#2006C2semidirect
9. C3×#2002C2semidirect
10. C3×C4×#104C2semidirect
11. C3×C4×#100C2semidirect
12. #23587C3direct (N × H)
48. #3460C2×C3semidirect
49. C4×AGL(1,ℤ/8)C2×C3semidirect
50. C4×#127C2×C3semidirect
51. #2060C2×C3semidirect
52. #2039C2×C3semidirect
53. #2037C2×C3semidirect
54. #2027C2×C3semidirect
55. #2006C2×C3semidirect
56. #2002C2×C3semidirect
57. C4×#104C2×C3semidirect
58. C4×#100C2×C3semidirect
92. C3×C4×Q8D4semidirect
93. C3×C4×Q8D4semidirect
102. C3×C4×D4D4semidirect
103. C3×C4×D4D4semidirect
190. C2×C3×D4#41semidirect
212. C3×M16#41semidirect
244. C3×C42#41semidirect
278. C4×Q8C3×D4semidirect
279. C4×Q8C3×D4semidirect
288. C4×D4C3×D4semidirect
289. C4×D4C3×D4semidirect
400. C2×D4C3×#41semidirect
422. M16C3×#41semidirect
454. C42C3×#41semidirect
582. C3#23587direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34602
C3×C4×AGL(1,ℤ/8)2
C3×C4×#1272
C3×#20602
C3×#20392
C3×#20372
C3×#20272
C3×#20062
C3×#20022
C3×C4×#1042
C3×C4×#1002
#235873