Finite Group Databasebeta

← C3×#19072 #10637679

Subgroups — C3×#19072 #10637679

Order 768.

992 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3545C211yesmaximal, characteristic, complemented
2C3×#3180C212yesmaximal, complemented
3C3×#3153C211yesmaximal, characteristic, complemented
4C3×#1661C212yesmaximal, complemented
5C3×#1660C211yesmaximal, characteristic, complemented
6C3×#1659C211yesmaximal, characteristic, complemented
7C3×#1657C211yesmaximal, characteristic, complemented
8C3×#1656C212yesmaximal, complemented
9C3×#1655C211yesmaximal, characteristic, complemented
10C3×#1524C212yesmaximal, complemented
11C2×C3×#329C211yesmaximal, characteristic, complemented
12#19072C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137—21no—
14C2×C3×AGL(1,ℤ/8)—21no—
15C3×#563C2211yescharacteristic
16C3×#562C2212yes—
17C3×H(3,𝔽4)C2211yescharacteristic
18C3×#555—21no—
19C3×#551—21no—
20C3×#551—22no—
21C3×#546—22no—
22C3×#542—22no—
23C3×#538—22no—
24C3×#536—21no—
25C3×#534—21no—
26C3×#532C2211yescharacteristic, complemented
27C3×#529C2212yes—
28C2×C3×#124C2211yescharacteristic
29C3×#501C2212yescomplemented
30C3×#497C2211yescharacteristic
31C3×#497C2211yescharacteristic, complemented
32C3×#496C2211yescharacteristic
33C3×#496C2211yescharacteristic, complemented
34C3×#496C2212yescomplemented
35C3×#495—21no—
36C3×#495—21no—
37C3×#490C2212yescomplemented
38C3×#489C2212yescomplemented
39C3×#466—22no—
40C3×#465—22no—
41C3×#442—22no—
42C3×#440—22no—
43C3×C4×M16C2211yescharacteristic, complemented
44C2×C3×#97C2212yescomplemented
45C3×#330C2212yescomplemented
46C3×#330C2212yescomplemented
47C3×#329C2211yescharacteristic
48C3×#329C2211yescharacteristic
49C3×#329C2212yescomplemented
50C3×#329C2212yescomplemented
51C3×#329C2212yescomplemented
52#3545C2×C311yescharacteristic, complemented
53#3180C2×C312yescomplemented
54#3153C2×C311yescharacteristic, complemented
55#1661C2×C312yescomplemented
56#1660C2×C311yescharacteristic, complemented
57#1659C2×C311yescharacteristic, complemented
58#1657C2×C311yescharacteristic, complemented
59#1656C2×C312yescomplemented
60#1655C2×C311yescharacteristic, complemented
61#1524C2×C312yescomplemented
62C2×#329C2×C311yescharacteristic, complemented
63C2×C3×#41—21no—
64C2×C3×#41—21no—
65C22×C3×Q8D411yescharacteristic
66C22×C3×D4—21no—
67C22×C3×D4D411yescharacteristic
68C3×#137—44no—
69C3×AGL(1,ℤ/8)—44no—
70C2×C3×Q16—21no—
71C2×C3×Q16—21no—
72C2×C3×SD16—21no—
73C2×C3×SD16—21no—
74C2×C3×SD16—22no—
75C2×C3×SD16—21no—
76C2×C3×SD16—21no—
77C2×C3×D8—21no—
78C2×C3×D8—21no—
79C2×C3×D8—22no—
80C2×C3×M16—21no—
81C22×C3×C8—22no—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8C2311yescharacteristic
85C3×#127C2311yescharacteristic
86C3×#126—22no—
87C3×#126—22no—
88C3×#126—22no—
89C3×#125C2312yes—
90C3×#124—21no—
91C3×#124—22no—
92C3×#124—21no—
93C3×#124—22no—
94C3×#124—22no—
95C3×#124—21no—
96C3×#124D411yescharacteristic
97C3×#124D411yescharacteristic
98C3×#124C2312yes—
99C3×#124C2311yescharacteristic
100C3×#124C2311yescharacteristic
101C3×#123—21no—
102C3×#123—22no—
103C3×#123—22no—
104C3×#123—21no—
105C3×#123—22no—
106C3×#123—22no—
107C3×#123—22no—
108C3×C22⋊Q8—22no—
109C3×C22⋊Q8—22no—
110C3×C22⋊Q8—22no—
111C3×C22⋊Q8—22no—
112C3×C22⋊Q8D412yes—
113C3×#121—22no—
114C3×#121—22no—
115C3×#121—22no—
116C3×#121—22no—
117C3×#121—22no—
118C3×#121D412yes—
119C3×#120—21no—
120C3×#120—22no—
121C3×#120—21no—
122C3×#120—22no—
123C3×C4×Q8D412yes—
124C3×C4×D4—22no—
125C3×C4×D4—22no—
126C3×C4×D4—22no—
127C3×C4×D4—22no—
128C3×C4×D4D412yes—
129C3×#117—21no—
130C3×#117—22no—
131C3×#117—22no—
132C2×C3×C22⋊C4—21no—
133C2×C3×C22⋊C4—22no—
134C2×C3×C22⋊C4—22no—
135C2×C3×C42C2311yescharacteristic
136C3×#107—22no—
137C3×#106—22no—
138C3×#103—22no—
139C3×#102—22no—
140C3×#102—22no—
141C3×#102—22no—
142C3×#97—22no—
143C3×#97C2312yes—
144C3×#97C2312yes—
145C3×#97C2311yescharacteristic
146C3×#97C2311yescharacteristic
147C3×C4×C8—21no—
148C2×#137—21no—
149C2×AGL(1,ℤ/8)—21no—
150#563C22×C311yescharacteristic
151#562C22×C312yes—
152H(3,𝔽4)C22×C311yescharacteristic
153#555—21no—
154#551—21no—
155#551—22no—
156#546—22no—
157#542—22no—
158#538—22no—
159#536—21no—
160#534—21no—
161#532C22×C311yescharacteristic, complemented
162#529C22×C312yes—
163C2×#124C22×C311yescharacteristic
164#501C22×C312yescomplemented
165#497C22×C311yescharacteristic
166#497C22×C311yescharacteristic, complemented
167#496C22×C311yescharacteristic
168#496C22×C311yescharacteristic, complemented
169#496C22×C312yescomplemented
170#495—21no—
171#495—21no—
172#490C22×C312yescomplemented
173#489C22×C312yescomplemented
174#466—22no—
175#465—22no—
176#442—22no—
177#440—22no—
178C4×M16C22×C311yescharacteristic, complemented
179C2×#97C22×C312yescomplemented
180#330C22×C312yescomplemented
181#330C22×C312yescomplemented
182#329C22×C311yescharacteristic
183#329C22×C311yescharacteristic
184#329C22×C312yescomplemented
185#329C22×C312yescomplemented
186#329C22×C312yescomplemented
187C24×C3—21no—
188C24×C3—21no—
189C3×#41—42no—
190C3×#41—42no—
191C3×#41—82no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—42no—
196C2×C3×Q8C2×D412yes—
197C2×C3×Q8C2×D411yescharacteristic
198C2×C3×Q8C2×D411yescharacteristic
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—21no—
203C2×C3×D4—22no—
204C2×C3×D4—21no—
205C2×C3×D4—22no—
206C2×C3×D4—21no—
207C2×C3×D4—42no—
208C2×C3×D4—41no—
209C2×C3×D4—82no—
210C2×C3×D4C2×D412yes—
211C2×C3×D4C2×D411yescharacteristic
212C2×C3×D4C2×D411yescharacteristic
213C22×C3×C4—21no—
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—21no—
217C22×C3×C4—22no—
218C22×C3×C4—41no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C3×Q16—42no—
222C3×Q16—42no—
223C3×SD16—44no—
224C3×SD16—42no—
225C3×SD16—42no—
226C3×SD16—42no—
227C3×SD16—42no—
228C3×D8—44no—
229C3×D8—42no—
230C3×D8—42no—
231C3×M16—24no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—44no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4C2×D412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—22no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—22no—
262C3×C22⋊C4—42no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—22no—
266C3×C42C2×D411yescharacteristic
267C3×C42C2×D412yes—
268C3×C42C2411yescharacteristic
269C2×#41—21no—
270C2×#41—21no—
271C22×Q8C3×D411yescharacteristic
272C22×D4—21no—
273C22×D4C3×D411yescharacteristic
274#137—44no—
275AGL(1,ℤ/8)—44no—
276C2×Q16—21no—
277C2×Q16—21no—
278C2×SD16—21no—
279C2×SD16—21no—
280C2×SD16—22no—
281C2×SD16—21no—
282C2×SD16—21no—
283C2×D8—21no—
284C2×D8—21no—
285C2×D8—22no—
286C2×M16—21no—
287C22×C8—22no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8C23×C311yescharacteristic
291#127C23×C311yescharacteristic
292#126—22no—
293#126—22no—
294#126—22no—
295#125C23×C312yes—
296#124—21no—
297#124—22no—
298#124—21no—
299#124—22no—
300#124—22no—
301#124—21no—
302#124C3×D411yescharacteristic
303#124C3×D411yescharacteristic
304#124C23×C311yescharacteristic
305#124C23×C311yescharacteristic
306#124C23×C312yes—
307#123—21no—
308#123—22no—
309#123—22no—
310#123—21no—
311#123—22no—
312#123—22no—
313#123—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8C3×D412yes—
319#121—22no—
320#121—22no—
321#121—22no—
322#121—22no—
323#121—22no—
324#121C3×D412yes—
325#120—21no—
326#120—22no—
327#120—21no—
328#120—22no—
329C4×Q8C3×D412yes—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4C3×D412yes—
335#117—21no—
336#117—22no—
337#117—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C42C23×C311yescharacteristic
342#107—22no—
343#106—22no—
344#103—22no—
345#102—22no—
346#102—22no—
347#102—22no—
348#97—22no—
349#97C23×C312yes—
350#97C23×C312yes—
351#97C23×C311yescharacteristic
352#97C23×C311yescharacteristic
353C4×C8—21no—
354C23×C3—21no—
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—42no—
358C23×C3—41no—
359C23×C3—82no—
360C23×C3—81no—
361C23×C3—82no—
362C23×C3#12011yescharacteristic
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—42no—
366C3×Q8—41no—
367C3×Q8—41no—
368C3×Q8—81no—
369C3×Q8—81no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—42no—
373C3×D4—41no—
374C3×D4—42no—
375C3×D4—41no—
376C3×D4—42no—
377C3×D4—44no—
378C3×D4—81no—
379C3×D4—82no—
380C3×D4—82no—
381C3×D4—82no—
382C3×D4—81no—
383C3×D4—82no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—82no—
394C2×C3×C4—82no—
395C2×C3×C4—82no—
396C2×C3×C4—81no—
397C2×C3×C4—82no—
398C2×C3×C4#12012yes—
399C2×C3×C4#12011yescharacteristic
400C2×C3×C4C22×D412yes—
401C2×C3×C4C22×D411yescharacteristic
402C3×C8—24no—
403C3×C8—22no—
404C3×C8—22no—
405C3×C8—42no—
406C24—21no—
407C24—21no—
408#41—42no—
409#41—42no—
410#41—82no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—42no—
415C2×Q8C2×C3×D412yes—
416C2×Q8C2×C3×D411yescharacteristic
417C2×Q8C2×C3×D411yescharacteristic
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—41no—
428C2×D4—82no—
429C2×D4C2×C3×D412yes—
430C2×D4C2×C3×D411yescharacteristic
431C2×D4C2×C3×D411yescharacteristic
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4C2×C3×D411yescharacteristic
439C22×C4C2×C3×D412yes—
440Q16—42no—
441Q16—42no—
442SD16—42no—
443SD16—42no—
444SD16—42no—
445SD16—42no—
446SD16—44no—
447D8—42no—
448D8—42no—
449D8—44no—
450M16—24no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—44no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4C2×C3×D412yes—
465C4⋊C4C2×C3×D412yes—
466C22⋊C4—22no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—42no—
482C42—21no—
483C42—21no—
484C42—22no—
485C42C2×C3×D411yescharacteristic
486C42C2×C3×D412yes—
487C42C24×C311yescharacteristic
488C22×C3—22no—
489C22×C3—42no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—81no—
497C22×C3—82no—
498C22×C3—162no—
499C22×C3—162no—
500C22×C3#45312yes—
501C22×C3C2×#12011yescharacteristic
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—22no—
505C3×C4—22no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—81no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—42no—
516C23—41no—
517C23—82no—
518C23—81no—
519C23—82no—
520C23C3×#12011yescharacteristic
521Q8—41no—
522Q8—41no—
523Q8—42no—
524Q8—41no—
525Q8—41no—
526Q8—81no—
527Q8—81no—
528D4—41no—
529D4—41no—
530D4—42no—
531D4—41no—
532D4—42no—
533D4—41no—
534D4—42no—
535D4—44no—
536D4—81no—
537D4—82no—
538D4—81no—
539D4—82no—
540D4—82no—
541D4—82no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4C3×#12012yes—
557C2×C4C3×#12011yescharacteristic
558C2×C4C22×C3×D412yes—
559C2×C4C22×C3×D411yescharacteristic
560C8—24no—
561C8—22no—
562C8—22no—
563C8—42no—
564C2×C3—22no—
565C2×C3—82no—
566C2×C3—81no—
567C2×C3—81no—
568C2×C3—161no—
569C2×C3C2×#45311yescharacteristic
570C2×C3#301812yes—
571C22—22no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—82no—
581C22—162no—
582C22—162no—
583C22C3×#45312yes—
584C22C2×C3×#12011yescharacteristic
585C4—21no—
586C4—21no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—81no—
595C3#1907211yescharacteristic, direct product factor, Sylow
596C2—22no—
597C2—81no—
598C2—82no—
599C2—81no—
600C2—161no—
601C2C2×C3×#45311yescharacteristic
602C2C3×#301812yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3545C2semidirect
2. C3×#3180C2semidirect
3. C3×#3153C2semidirect
4. C3×#1661C2semidirect
5. C3×#1660C2semidirect
6. C3×#1659C2semidirect
7. C3×#1657C2semidirect
8. C3×#1656C2semidirect
9. C3×#1655C2semidirect
10. C3×#1524C2semidirect
11. C2×C3×#329C2semidirect
12. #19072C3direct (N × H)
26. C3×#532C22semidirect
29. C3×#501C22semidirect
31. C3×#497C22semidirect
33. C3×#496C22semidirect
34. C3×#496C22semidirect
37. C3×#490C22semidirect
38. C3×#489C22semidirect
43. C3×C4×M16C22semidirect
44. C2×C3×#97C22semidirect
45. C3×#330C22semidirect
46. C3×#330C22semidirect
49. C3×#329C22semidirect
50. C3×#329C22semidirect
51. C3×#329C22semidirect
52. #3545C2×C3semidirect
53. #3180C2×C3semidirect
54. #3153C2×C3semidirect
55. #1661C2×C3semidirect
56. #1660C2×C3semidirect
57. #1659C2×C3semidirect
58. #1657C2×C3semidirect
59. #1656C2×C3semidirect
60. #1655C2×C3semidirect
61. #1524C2×C3semidirect
62. C2×#329C2×C3semidirect
161. #532C22×C3semidirect
164. #501C22×C3semidirect
166. #497C22×C3semidirect
168. #496C22×C3semidirect
169. #496C22×C3semidirect
172. #490C22×C3semidirect
173. #489C22×C3semidirect
178. C4×M16C22×C3semidirect
179. C2×#97C22×C3semidirect
180. #330C22×C3semidirect
181. #330C22×C3semidirect
184. #329C22×C3semidirect
185. #329C22×C3semidirect
186. #329C22×C3semidirect
595. C3#19072direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35452
C3×#31802
C3×#31532
C3×#16612
C3×#16602
C3×#16592
C3×#16572
C3×#16562
C3×#16552
C3×#15242
C2×C3×#3292
#190723