Finite Group Databasebeta

← C3×#18678 #10637285

Subgroups — C3×#18678 #10637285

Order 768.

1000 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×#3506C211yesmaximal, characteristic, complemented
2C3×#3246C211yesmaximal, characteristic, complemented
3C3×#3169C211yesmaximal, characteristic, complemented
4C3×#3036C211yesmaximal, characteristic, complemented
5C3×#1650C212yesmaximal, complemented
6C3×#1642C212yesmaximal, complemented
7C3×#1628C212yesmaximal, complemented
8C3×#1621C212yesmaximal, complemented
9C3×#1547C211yesmaximal, characteristic, complemented
10C3×#1493C211yesmaximal, characteristic, complemented
11C2×C3×D4⋊C8C211yesmaximal, characteristic, complemented
12#18678C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)—21no—
14C3×#559—22no—
15C3×#556C2212yes—
16C3×#555—21no—
17C3×#553—22no—
18C3×#550C2212yes—
19C3×#547—21no—
20C3×#538—22no—
21C3×#535—21no—
22C3×#533C2211yescharacteristic
23C3×#532C2211yescharacteristic
24C3×#532C2211yescharacteristic
25C3×#525—22no—
26C3×#520—21no—
27C2×C3×C4×D4C2211yescharacteristic
28C3×#500C2211yescharacteristic, complemented
29C3×#493C2211yescharacteristic, complemented
30C3×#486C2211yescharacteristic, complemented
31C3×#486C2211yescharacteristic, complemented
32C3×#476—21no—
33C3×#475C2212yescomplemented
34C3×#466—22no—
35C3×#465—22no—
36C3×#463—22no—
37C3×#461—21no—
38C3×#461C2212yes—
39C3×#460C2212yescomplemented
40C3×#459C2212yes—
41C3×C4×D8—22no—
42C3×#423C2211yescharacteristic
43C2×C3×C4⋊C8C2211yescharacteristic
44C3×#417—21no—
45C2×C3×C4×C8C2211yescharacteristic
46C3×#332C2212yescomplemented
47C3×#331C2212yescomplemented
48C3×Q8⋊C8C2212yescomplemented
49C3×Q8⋊C8C2212yescomplemented
50C3×D4⋊C8C2212yes—
51C3×D4⋊C8C2212yes—
52#3506C2×C311yescharacteristic, complemented
53#3246C2×C311yescharacteristic, complemented
54#3169C2×C311yescharacteristic, complemented
55#3036C2×C311yescharacteristic, complemented
56#1650C2×C312yescomplemented
57#1642C2×C312yescomplemented
58#1628C2×C312yescomplemented
59#1621C2×C312yescomplemented
60#1547C2×C311yescharacteristic, complemented
61#1493C2×C311yescharacteristic, complemented
62C2×D4⋊C8C2×C311yescharacteristic, complemented
63C2×C3×#41—22no—
64C2×C3×#41—21no—
65C2×C3×#41—21no—
66C22×C3×D4D411yescharacteristic
67C23×C3×C4—21no—
68C3×AGL(1,ℤ/8)—44no—
69C2×C3×SD16—22no—
70C2×C3×SD16—22no—
71C2×C3×D8—22no—
72C2×C3×D8—22no—
73C2×C3×M16—21no—
74C22×C3×C8—21no—
75C22×C3×C8—21no—
76C3×C4⋊Q8D411yescharacteristic
77C3×C4⋊Q8D411yescharacteristic
78C3×C4⋊Q8C2311yescharacteristic
79C3×C4⋊Q8C2311yescharacteristic
80C3×#127—22no—
81C3×#127C2311yescharacteristic
82C3×#127C2311yescharacteristic
83C3×#126—24no—
84C3×#125—21no—
85C3×#124—24no—
86C3×#124—22no—
87C3×#124—21no—
88C3×#124—21no—
89C3×#123—24no—
90C3×#123—24no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8D412yes—
94C3×#121—22no—
95C3×#121—22no—
96C3×#121—24no—
97C3×#121—21no—
98C3×#121—22no—
99C3×#121—21no—
100C3×#121—21no—
101C3×#121D412yes—
102C3×#120—24no—
103C3×C4×Q8—22no—
104C3×C4×Q8—21no—
105C3×C4×Q8C2312yes—
106C3×C4×D4—24no—
107C3×C4×D4—24no—
108C3×C4×D4—22no—
109C3×C4×D4—21no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×C4×D4D412yes—
115C3×C4×D4D412yes—
116C3×C4×D4C2312yes—
117C3×#117—22no—
118C3×#117—21no—
119C3×#117—22no—
120C2×C3×C4⋊C4—22no—
121C2×C3×C4⋊C4D411yescharacteristic
122C2×C3×C22⋊C4—21no—
123C2×C3×C42C2311yescharacteristic
124C3×#107—22no—
125C3×#106—22no—
126C3×C4⋊C8—21no—
127C3×C4⋊C8—21no—
128C3×C4⋊C8C2312yes—
129C3×C4⋊C8C2312yes—
130C3×#103—22no—
131C3×#103—22no—
132C3×#102—22no—
133C3×#102—22no—
134C3×#102—22no—
135C3×#102—22no—
136C3×C4×C8—21no—
137C3×C4×C8C2311yescharacteristic
138C3×C4×C8C2311yescharacteristic
139C2×AGL(1,ℤ/8)—21no—
140#559—22no—
141#556C22×C312yes—
142#555—21no—
143#553—22no—
144#550C22×C312yes—
145#547—21no—
146#538—22no—
147#535—21no—
148#533C22×C311yescharacteristic
149#532C22×C311yescharacteristic
150#532C22×C311yescharacteristic
151#525—22no—
152#520—21no—
153C2×C4×D4C22×C311yescharacteristic
154#500C22×C311yescharacteristic, complemented
155#493C22×C311yescharacteristic, complemented
156#486C22×C311yescharacteristic, complemented
157#486C22×C311yescharacteristic, complemented
158#476—21no—
159#475C22×C312yescomplemented
160#466—22no—
161#465—22no—
162#463—22no—
163#461—21no—
164#461C22×C312yes—
165#460C22×C312yescomplemented
166#459C22×C312yes—
167C4×D8—22no—
168#423C22×C311yescharacteristic
169C2×C4⋊C8C22×C311yescharacteristic
170#417—21no—
171C2×C4×C8C22×C311yescharacteristic
172#332C22×C312yescomplemented
173#331C22×C312yescomplemented
174Q8⋊C8C22×C312yescomplemented
175Q8⋊C8C22×C312yescomplemented
176D4⋊C8C22×C312yes—
177D4⋊C8C22×C312yes—
178C24×C3—21no—
179C3×#41—42no—
180C3×#41—42no—
181C3×#41—42no—
182C3×#41—42no—
183C3×#41—82no—
184C3×#41—82no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8C2×D412yes—
189C2×C3×D4—22no—
190C2×C3×D4—24no—
191C2×C3×D4—24no—
192C2×C3×D4—21no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—41no—
201C2×C3×D4C2×D412yes—
202C2×C3×D4C2×D412yes—
203C22×C3×C4—24no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—41no—
209C22×C3×C4—81no—
210C22×C3×C4—81no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C3×SD16—44no—
215C3×SD16—82no—
216C3×D8—44no—
217C3×D8—44no—
218C3×M16—42no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—42no—
224C2×C3×C8—81no—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4C2×D412yes—
236C3×C4⋊C4C2×D412yes—
237C3×C4⋊C4C2×D412yes—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248C3×C42—24no—
249C3×C42—22no—
250C3×C42C2×D411yescharacteristic
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42C2411yescharacteristic
254C2×#41—22no—
255C2×#41—21no—
256C2×#41—21no—
257C22×D4C3×D411yescharacteristic
258C23×C4—21no—
259AGL(1,ℤ/8)—44no—
260C2×SD16—22no—
261C2×SD16—22no—
262C2×D8—22no—
263C2×D8—22no—
264C2×M16—21no—
265C22×C8—21no—
266C22×C8—21no—
267C4⋊Q8C3×D411yescharacteristic
268C4⋊Q8C3×D411yescharacteristic
269C4⋊Q8C23×C311yescharacteristic
270C4⋊Q8C23×C311yescharacteristic
271#127—22no—
272#127C23×C311yescharacteristic
273#127C23×C311yescharacteristic
274#126—24no—
275#125—21no—
276#124—24no—
277#124—22no—
278#124—21no—
279#124—21no—
280#123—24no—
281#123—24no—
282C22⋊Q8—22no—
283C22⋊Q8—21no—
284C22⋊Q8C3×D412yes—
285#121—22no—
286#121—22no—
287#121—24no—
288#121—21no—
289#121—22no—
290#121—21no—
291#121—21no—
292#121C3×D412yes—
293#120—24no—
294C4×Q8—22no—
295C4×Q8—21no—
296C4×Q8C23×C312yes—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4C3×D412yes—
306C4×D4C3×D412yes—
307C4×D4C23×C312yes—
308#117—22no—
309#117—21no—
310#117—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C3×D411yescharacteristic
313C2×C22⋊C4—21no—
314C2×C42C23×C311yescharacteristic
315#107—22no—
316#106—22no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8C23×C312yes—
320C4⋊C8C23×C312yes—
321#103—22no—
322#103—22no—
323#102—22no—
324#102—22no—
325#102—22no—
326#102—22no—
327C4×C8—21no—
328C4×C8C23×C311yescharacteristic
329C4×C8C23×C311yescharacteristic
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—42no—
333C23×C3—41no—
334C23×C3—81no—
335C23×C3—81no—
336C23×C3#12011yescharacteristic
337C3×Q8—22no—
338C3×Q8—22no—
339C3×Q8—41no—
340C3×Q8—41no—
341C3×Q8—41no—
342C3×Q8—41no—
343C3×Q8—81no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—44no—
349C3×D4—41no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—41no—
356C3×D4—41no—
357C3×D4—42no—
358C3×D4—84no—
359C3×D4—84no—
360C3×D4—82no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—41no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—44no—
371C2×C3×C4—41no—
372C2×C3×C4—82no—
373C2×C3×C4—81no—
374C2×C3×C4—84no—
375C2×C3×C4—82no—
376C2×C3×C4#12011yescharacteristic
377C2×C3×C4#12011yescharacteristic
378C2×C3×C4#12011yescharacteristic
379C2×C3×C4#13512yes—
380C2×C3×C4AGL(1,ℤ/8)12yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C3×C8—22no—
385C3×C8—41no—
386C3×C8—42no—
387C3×C8—42no—
388C24—21no—
389#41—42no—
390#41—42no—
391#41—42no—
392#41—42no—
393#41—82no—
394#41—82no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8C2×C3×D412yes—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—41no—
411C2×D4C2×C3×D412yes—
412C2×D4C2×C3×D412yes—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424SD16—44no—
425SD16—82no—
426D8—44no—
427D8—44no—
428M16—42no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—42no—
434C2×C8—81no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4C2×C3×D412yes—
446C4⋊C4C2×C3×D412yes—
447C4⋊C4C2×C3×D412yes—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C42—24no—
459C42—22no—
460C42C2×C3×D411yescharacteristic
461C42C2×C3×D411yescharacteristic
462C42C2×C3×D411yescharacteristic
463C42C24×C311yescharacteristic
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—84no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3#44912yes—
477C22×C3C2×#12011yescharacteristic
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—81no—
486C3×C4#44912yes—
487C3×C4C2×#13511yescharacteristic
488C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
489C3×C4#57611yescharacteristic
490C3×C4#57711yescharacteristic
491C23—22no—
492C23—22no—
493C23—42no—
494C23—41no—
495C23—81no—
496C23—81no—
497C23C3×#12011yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—81no—
505D4—22no—
506D4—22no—
507D4—42no—
508D4—42no—
509D4—44no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—42no—
519D4—84no—
520D4—84no—
521D4—82no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4C3×#12011yescharacteristic
538C2×C4C3×#12011yescharacteristic
539C2×C4C3×#12011yescharacteristic
540C2×C4C3×#13512yes—
541C2×C4C3×AGL(1,ℤ/8)12yes—
542C2×C4C22×C3×D411yescharacteristic
543C2×C4C22×C3×D411yescharacteristic
544C2×C4C22×C3×D411yescharacteristic
545C8—22no—
546C8—41no—
547C8—42no—
548C8—42no—
549C2×C3—22no—
550C2×C3—82no—
551C2×C3—82no—
552C2×C3—161no—
553C2×C3C2×#44911yescharacteristic
554C2×C3#301211yescharacteristic
555C2×C3#301811yescharacteristic
556C22—21no—
557C22—21no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—84no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22C3×#44912yes—
569C22C2×C3×#12011yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—81no—
578C4C3×#44912yes—
579C4C2×C3×#13511yescharacteristic
580C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
581C4C3×#57611yescharacteristic
582C4C3×#57711yescharacteristic
583C3#1867811yescharacteristic, direct product factor, Sylow
584C2—22no—
585C2—82no—
586C2—82no—
587C2—161no—
588C2C2×C3×#44911yescharacteristic
589C2C3×#301211yescharacteristic
590C2C3×#301811yescharacteristic
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×#3506C2semidirect
2. C3×#3246C2semidirect
3. C3×#3169C2semidirect
4. C3×#3036C2semidirect
5. C3×#1650C2semidirect
6. C3×#1642C2semidirect
7. C3×#1628C2semidirect
8. C3×#1621C2semidirect
9. C3×#1547C2semidirect
10. C3×#1493C2semidirect
11. C2×C3×D4⋊C8C2semidirect
12. #18678C3direct (N × H)
28. C3×#500C22semidirect
29. C3×#493C22semidirect
30. C3×#486C22semidirect
31. C3×#486C22semidirect
33. C3×#475C22semidirect
39. C3×#460C22semidirect
46. C3×#332C22semidirect
47. C3×#331C22semidirect
48. C3×Q8⋊C8C22semidirect
49. C3×Q8⋊C8C22semidirect
52. #3506C2×C3semidirect
53. #3246C2×C3semidirect
54. #3169C2×C3semidirect
55. #3036C2×C3semidirect
56. #1650C2×C3semidirect
57. #1642C2×C3semidirect
58. #1628C2×C3semidirect
59. #1621C2×C3semidirect
60. #1547C2×C3semidirect
61. #1493C2×C3semidirect
62. C2×D4⋊C8C2×C3semidirect
154. #500C22×C3semidirect
155. #493C22×C3semidirect
156. #486C22×C3semidirect
157. #486C22×C3semidirect
159. #475C22×C3semidirect
165. #460C22×C3semidirect
172. #332C22×C3semidirect
173. #331C22×C3semidirect
174. Q8⋊C8C22×C3semidirect
175. Q8⋊C8C22×C3semidirect
583. C3#18678direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35062
C3×#32462
C3×#31692
C3×#30362
C3×#16502
C3×#16422
C3×#16282
C3×#16212
C3×#15472
C3×#14932
C2×C3×D4⋊C82
#186783