Finite Group Databasebeta

← C3×#18320 #10636927

Subgroups — C3×#18320 #10636927

Order 768.

1008 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#539C211yesmaximal, characteristic, complemented
2C3×#3193C211yesmaximal, characteristic, complemented
3C3×#3192C211yesmaximal, characteristic
4C3×#2591C211yesmaximal, characteristic, complemented
5C3×#1615C214yesmaximal, complemented
6C3×#1597C214yesmaximal, complemented
7C3×#1508C211yesmaximal, characteristic, complemented
8C3×#1507C211yesmaximal, characteristic, complemented
9C3×#1463C211yesmaximal, characteristic, complemented
10#18320C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#539—22no—
12C3×#539—22no—
13C3×#539—22no—
14C3×#539C2214yes—
15C2×C3×#126—21no—
16C2×C3×#125—21no—
17C2×C3×#123—21no—
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121C2211yescharacteristic
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C22×C3×C4⋊C4C2211yescharacteristic
25C3×#484C2212yescomplemented
26C3×#481—21no—
27C3×#481C2212yes—
28C3×#480—21no—
29C3×#469—22no—
30C3×#468—22no—
31C3×#450C2212yescomplemented
32C3×#448—21no—
33C3×#447—21no—
34C3×#447C2212yes—
35C3×#407C2211yescharacteristic
36C3×#407C2211yescharacteristic
37C3×#400C2212yescomplemented
38C3×#394C2212yescomplemented
39C3×#388—22no—
40C3×#386—21no—
41C3×#386—21no—
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#328C2214yescomplemented
44C3×#327C2214yes—
45C3×#324C2214yescomplemented
46C2×#539C2×C311yescharacteristic, complemented
47#3193C2×C311yescharacteristic, complemented
48#3192C2×C311yescharacteristic
49#2591C2×C311yescharacteristic, complemented
50#1615C2×C314yescomplemented
51#1597C2×C314yescomplemented
52#1508C2×C311yescharacteristic, complemented
53#1507C2×C311yescharacteristic, complemented
54#1463C2×C311yescharacteristic, complemented
55C22×C3×Q8D411yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C2×C3×SD16—22no—
65C2×C3×D8—22no—
66C2×C3×M16—21no—
67C2×C3×M16—21no—
68C3×#126—24no—
69C3×#126—24no—
70C3×#125—24no—
71C3×#125—24no—
72C3×#123—24no—
73C3×#123—24no—
74C3×#123—24no—
75C3×#123—24no—
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8—21no—
79C3×C22⋊Q8—21no—
80C3×C22⋊Q8D412yes—
81C3×C22⋊Q8C2312yes—
82C3×#121—21no—
83C3×#121—21no—
84C3×#121—21no—
85C3×#121—21no—
86C3×#121—21no—
87C3×#121—21no—
88C3×#121D412yes—
89C3×#121C2312yes—
90C3×C4×D4—24no—
91C3×C4×D4—24no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4D412yes—
101C2×C3×C4⋊C4D411yescharacteristic
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C42—21no—
115C2×C3×C42—21no—
116C3×#107—22no—
117C3×#106—22no—
118C3×#103—22no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#102—22no—
122C3×C22⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×C22⋊C8C2312yes—
125C3×C22⋊C8C2312yes—
126C3×#95—22no—
127C3×#95—22no—
128C3×#95C2312yes—
129#539—22no—
130#539—22no—
131#539—22no—
132#539C22×C314yes—
133C2×#126—21no—
134C2×#125—21no—
135C2×#123—21no—
136C2×#123—21no—
137C2×C22⋊Q8—21no—
138C2×C22⋊Q8C22×C311yescharacteristic
139C2×#121C22×C311yescharacteristic
140C2×#121C22×C311yescharacteristic
141C2×C4×D4—21no—
142C22×C4⋊C4C22×C311yescharacteristic
143#484C22×C312yescomplemented
144#481—21no—
145#481C22×C312yes—
146#480—21no—
147#469—22no—
148#468—22no—
149#450C22×C312yescomplemented
150#448—21no—
151#447—21no—
152#447C22×C312yes—
153#407C22×C311yescharacteristic
154#407C22×C311yescharacteristic
155#400C22×C312yescomplemented
156#394C22×C312yescomplemented
157#388—22no—
158#386—21no—
159#386—21no—
160C4×C22⋊C4C22×C311yescharacteristic
161#328C22×C314yescomplemented
162#327C22×C314yes—
163#324C22×C314yescomplemented
164C24×C3—21no—
165C24×C3—21no—
166C24×C3C2×D411yescharacteristic
167C2×C3×Q8—21no—
168C2×C3×Q8—81no—
169C2×C3×Q8C2×D412yes—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—44no—
176C2×C3×D4—44no—
177C2×C3×D4—41no—
178C2×C3×D4—41no—
179C2×C3×D4—81no—
180C2×C3×D4—81no—
181C2×C3×D4C2×D412yes—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—24no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—42no—
193C22×C3×C4—81no—
194C22×C3×C4—81no—
195C22×C3×C4—82no—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C3×SD16—82no—
205C3×D8—82no—
206C3×M16—81no—
207C3×M16—81no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
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—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—44no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—41no—
235C3×C22⋊C4—41no—
236C3×C22⋊C4—82no—
237C3×C42—24no—
238C3×C42—81no—
239C22×Q8C3×D411yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C3×D411yescharacteristic
243C22×D4C23×C311yescharacteristic
244C23×C4—21no—
245C23×C4C3×D411yescharacteristic
246C23×C4C3×D411yescharacteristic
247C23×C4C23×C311yescharacteristic
248C2×SD16—22no—
249C2×D8—22no—
250C2×M16—21no—
251C2×M16—21no—
252#126—24no—
253#126—24no—
254#125—24no—
255#125—24no—
256#123—24no—
257#123—24no—
258#123—24no—
259#123—24no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264C22⋊Q8C3×D412yes—
265C22⋊Q8C23×C312yes—
266#121—21no—
267#121—21no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121—21no—
272#121C3×D412yes—
273#121C23×C312yes—
274C4×D4—24no—
275C4×D4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4C3×D412yes—
285C2×C4⋊C4C3×D411yescharacteristic
286C2×C4⋊C4C3×D411yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C312yes—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C42—21no—
299C2×C42—21no—
300#107—22no—
301#106—22no—
302#103—22no—
303#102—22no—
304#102—22no—
305#102—22no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8C23×C312yes—
309C22⋊C8C23×C312yes—
310#95—22no—
311#95—22no—
312#95C23×C312yes—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—41no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—41no—
323C23×C3—42no—
324C23×C3#12011yescharacteristic
325C23×C3#12011yescharacteristic
326C23×C3C22×D411yescharacteristic
327C3×Q8—42no—
328C3×Q8—81no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—42no—
332C3×D4—44no—
333C3×D4—44no—
334C3×D4—81no—
335C3×D4—82no—
336C3×D4—82no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—81no—
363C2×C3×C4—84no—
364C2×C3×C4#12011yescharacteristic
365C2×C3×C4#12011yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C3×C8—42no—
369C3×C8—42no—
370C24—21no—
371C24—21no—
372C24C2×C3×D411yescharacteristic
373C2×Q8—21no—
374C2×Q8—81no—
375C2×Q8C2×C3×D412yes—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4C2×C3×D412yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—82no—
402C22×C4C2×C3×D412yes—
403C22×C4C2×C3×D412yes—
404C22×C4C2×C3×D411yescharacteristic
405C22×C4C2×C3×D411yescharacteristic
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C24×C311yescharacteristic
410SD16—82no—
411D8—82no—
412M16—81no—
413M16—81no—
414C2×C8—22no—
415C2×C8—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4C2×C3×D412yes—
430C4⋊C4C2×C3×D412yes—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—44no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—82no—
443C42—24no—
444C42—81no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—44no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—84no—
457C22×C3—84no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3#45412yes—
462C22×C3C2×#12011yescharacteristic
463C22×C3#57512yes—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—82no—
474C3×C4—81no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23C3×#12011yescharacteristic
487C23C3×#12011yescharacteristic
488C23C22×C3×D411yescharacteristic
489Q8—42no—
490Q8—81no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—44no—
495D4—44no—
496D4—81no—
497D4—82no—
498D4—82no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—81no—
525C2×C4—84no—
526C2×C4C3×#12011yescharacteristic
527C2×C4C3×#12011yescharacteristic
528C2×C4C22×C3×D411yescharacteristic
529C2×C4C22×C3×D411yescharacteristic
530C8—42no—
531C8—42no—
532C2×C3—22no—
533C2×C3—22no—
534C2×C3—41no—
535C2×C3—82no—
536C2×C3—161no—
537C2×C3#300511yescharacteristic
538C2×C3C2×#45411yescharacteristic
539C2×C3#302611yescharacteristic
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—84no—
552C22—84no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22C3×#45412yes—
557C22C2×C3×#12011yescharacteristic
558C22C3×#57512yes—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—82no—
569C4—81no—
570C3#1832011yescharacteristic, direct product factor, Sylow
571C2—22no—
572C2—22no—
573C2—41no—
574C2—82no—
575C2—161no—
576C2C3×#300511yescharacteristic
577C2C2×C3×#45411yescharacteristic
578C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3192C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#539C2semidirect
2. C3×#3193C2semidirect
4. C3×#2591C2semidirect
5. C3×#1615C2semidirect
6. C3×#1597C2semidirect
7. C3×#1508C2semidirect
8. C3×#1507C2semidirect
9. C3×#1463C2semidirect
10. #18320C3direct (N × H)
25. C3×#484C22semidirect
31. C3×#450C22semidirect
37. C3×#400C22semidirect
38. C3×#394C22semidirect
43. C3×#328C22semidirect
45. C3×#324C22semidirect
46. C2×#539C2×C3semidirect
47. #3193C2×C3semidirect
49. #2591C2×C3semidirect
50. #1615C2×C3semidirect
51. #1597C2×C3semidirect
52. #1508C2×C3semidirect
53. #1507C2×C3semidirect
54. #1463C2×C3semidirect
143. #484C22×C3semidirect
149. #450C22×C3semidirect
155. #400C22×C3semidirect
156. #394C22×C3semidirect
161. #328C22×C3semidirect
163. #324C22×C3semidirect
570. C3#18320direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5392
C3×#31932
C3×#31922
C3×#25912
C3×#16152
C3×#15972
C3×#15082
C3×#15072
C3×#14632
#183203