Finite Group Databasebeta

← C3×#33568 #10652175

Subgroups — C3×#33568 #10652175

Order 768.

1110 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3545C211yesmaximal, characteristic, complemented
2C3×#3020C212yesmaximal, complemented
3C2×C3×#454C211yesmaximal, characteristic, complemented
4C3×#2204C214yesmaximal, complemented
5C3×#2203C212yesmaximal, complemented
6C3×#2202C212yesmaximal, complemented
7C3×#2133C212yesmaximal, complemented
8C2×C3×#355C211yesmaximal, characteristic, complemented
9#33568C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×2−1+4C2211yescharacteristic, complemented
11C3×#575—22no—
12C2×C3×AGL(1,ℤ/8)—21no—
13C3×#563C2211yescharacteristic, complemented
14C3×#562—21no—
15C3×H(3,𝔽4)C2211yescharacteristic, complemented
16C3×#555C2212yescomplemented
17C3×#551—21no—
18C3×#551—22no—
19C3×#546—22no—
20C3×#542—22no—
21C3×#538—22no—
22C3×#536C2212yes—
23C3×#534—21no—
24C3×#529—21no—
25C2×C3×#124C2211yescharacteristic
26C3×#456—22no—
27C3×#456C2214yescomplemented
28C3×#454—21no—
29C3×#454—21no—
30C3×#454—22no—
31C3×#454C2212yes—
32C3×#454C2212yescomplemented
33C3×#454C2214yescomplemented
34C3×#421—22no—
35C2×C3×#104—21no—
36C2×C3×#101C2212yescomplemented
37C2×C3×#101C2211yescharacteristic, complemented
38C3×#356C2214yescomplemented
39C3×#355C2212yescomplemented
40C3×#355C2212yescomplemented
41C3×#355C2214yescomplemented
42#3545C2×C311yescharacteristic, complemented
43#3020C2×C312yescomplemented
44C2×#454C2×C311yescharacteristic, complemented
45#2204C2×C314yescomplemented
46#2203C2×C312yescomplemented
47#2202C2×C312yescomplemented
48#2133C2×C312yescomplemented
49C2×#355C2×C311yescharacteristic, complemented
50C3×2−1+4—21no—
51C3×2−1+4—22no—
52C3×2−1+4—42no—
53C3×2−1+4C2312yes—
54C2×C3×#41—21no—
55C2×C3×#41—22no—
56C2×C3×#41—41no—
57C2×C3×#41D412yes—
58C22×C3×Q8—41no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4D411yescharacteristic
62C3×#137—24no—
63C3×#137—42no—
64C3×AGL(1,ℤ/8)—22no—
65C3×AGL(1,ℤ/8)—22no—
66C3×AGL(1,ℤ/8)—24no—
67C3×AGL(1,ℤ/8)—42no—
68C3×AGL(1,ℤ/8)—42no—
69C3×#135—44no—
70C2×C3×SD16—41no—
71C2×C3×D8—41no—
72C2×C3×M16—21no—
73C2×C3×M16—22no—
74C3×C4⋊Q8D412yes—
75C3×C4⋊Q8C2312yes—
76C3×#126—22no—
77C3×#126—22no—
78C3×#126—22no—
79C3×#125—21no—
80C3×#124—22no—
81C3×#124—22no—
82C3×#124—21no—
83C3×#124—22no—
84C3×#124D411yescharacteristic
85C3×#124D411yescharacteristic
86C3×#124D412yes—
87C3×#124C2311yescharacteristic
88C3×#124C2312yes—
89C3×#124C2311yescharacteristic
90C3×#123—22no—
91C3×#123—22no—
92C3×#123—21no—
93C3×#123—22no—
94C3×#123—21no—
95C3×#123—22no—
96C3×#123—22no—
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—21no—
99C3×C22⋊Q8—22no—
100C3×C22⋊Q8—22no—
101C3×C22⋊Q8—22no—
102C3×#121—22no—
103C3×#121—21no—
104C3×#121—22no—
105C3×#121—22no—
106C3×#121—22no—
107C3×#120—21no—
108C3×#120—22no—
109C3×#120—21no—
110C3×#120—22no—
111C3×C4×Q8—21no—
112C3×C4×D4—21no—
113C3×C4×D4—22no—
114C3×C4×D4—22no—
115C3×C4×D4—22no—
116C3×#117—22no—
117C3×#117—22no—
118C3×#117D412yes—
119C2×C3×C22⋊C4—21no—
120C2×C3×C22⋊C4—22no—
121C2×C3×C22⋊C4—22no—
122C2×C3×C42D411yescharacteristic
123C3×#104—22no—
124C3×#104—22no—
125C3×#104—24no—
126C3×#104—24no—
127C3×#101—21no—
128C3×#101—22no—
129C3×#101C2312yes—
130C3×#101C2314yes—
131C2×2−1+4C22×C311yescharacteristic, complemented
132#575—22no—
133C2×AGL(1,ℤ/8)—21no—
134#563C22×C311yescharacteristic, complemented
135#562—21no—
136H(3,𝔽4)C22×C311yescharacteristic, complemented
137#555C22×C312yescomplemented
138#551—21no—
139#551—22no—
140#546—22no—
141#542—22no—
142#538—22no—
143#536C22×C312yes—
144#534—21no—
145#529—21no—
146C2×#124C22×C311yescharacteristic
147#456—22no—
148#456C22×C314yescomplemented
149#454—21no—
150#454—21no—
151#454—22no—
152#454C22×C312yes—
153#454C22×C314yescomplemented
154#454C22×C312yescomplemented
155#421—22no—
156C2×#104—21no—
157C2×#101C22×C312yescomplemented
158C2×#101C22×C311yescharacteristic, complemented
159#356C22×C314yescomplemented
160#355C22×C312yescomplemented
161#355C22×C312yescomplemented
162#355C22×C314yescomplemented
163C24×C3—21no—
164C24×C3—21no—
165C3×#41—24no—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—24no—
169C3×#41—42no—
170C3×#41—41no—
171C3×#41—42no—
172C3×#41—42no—
173C3×#41—42no—
174C3×#41—42no—
175C3×#41—44no—
176C3×#41—41no—
177C3×#41—42no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8—22no—
181C2×C3×Q8—41no—
182C2×C3×Q8—41no—
183C2×C3×Q8—42no—
184C2×C3×Q8—42no—
185C2×C3×Q8—44no—
186C2×C3×Q8C2×D412yes—
187C2×C3×Q8C2×D411yescharacteristic
188C2×C3×Q8C2×D412yes—
189C2×C3×Q8C2411yescharacteristic
190C2×C3×D4—21no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—21no—
196C2×C3×D4—22no—
197C2×C3×D4—21no—
198C2×C3×D4—41no—
199C2×C3×D4—41no—
200C2×C3×D4—41no—
201C2×C3×D4—81no—
202C2×C3×D4—81no—
203C2×C3×D4C2×D412yes—
204C2×C3×D4C2×D411yescharacteristic
205C2×C3×D4C2×D411yescharacteristic
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—22no—
211C22×C3×C4—41no—
212C22×C3×C4—42no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D412yes—
215C3×Q16—44no—
216C3×SD16—44no—
217C3×SD16—42no—
218C3×SD16—42no—
219C3×SD16—44no—
220C3×D8—42no—
221C3×D8—42no—
222C3×D8—44no—
223C3×M16—22no—
224C3×M16—24no—
225C3×M16—41no—
226C3×M16—42no—
227C2×C3×C8—41no—
228C2×C3×C8—42no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C22⋊C4—22no—
246C3×C22⋊C4—22no—
247C3×C22⋊C4—22no—
248C3×C22⋊C4—22no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C42—22no—
254C3×C42—21no—
255C3×C42C2×D411yescharacteristic
256C3×C42C2×D411yescharacteristic
257C3×C42C2×D412yes—
258C3×C42C2×D412yes—
2592−1+4—21no—
2602−1+4—22no—
2612−1+4—42no—
2622−1+4C23×C312yes—
263C2×#41—21no—
264C2×#41—22no—
265C2×#41—41no—
266C2×#41C3×D412yes—
267C22×Q8—41no—
268C22×Q8C23×C311yescharacteristic
269C22×D4—21no—
270C22×D4C3×D411yescharacteristic
271#137—24no—
272#137—42no—
273AGL(1,ℤ/8)—22no—
274AGL(1,ℤ/8)—22no—
275AGL(1,ℤ/8)—24no—
276AGL(1,ℤ/8)—42no—
277AGL(1,ℤ/8)—42no—
278#135—44no—
279C2×SD16—41no—
280C2×D8—41no—
281C2×M16—21no—
282C2×M16—22no—
283C4⋊Q8C3×D412yes—
284C4⋊Q8C23×C312yes—
285#126—22no—
286#126—22no—
287#126—22no—
288#125—21no—
289#124—22no—
290#124—22no—
291#124—21no—
292#124—22no—
293#124C3×D411yescharacteristic
294#124C3×D411yescharacteristic
295#124C3×D412yes—
296#124C23×C311yescharacteristic
297#124C23×C311yescharacteristic
298#124C23×C312yes—
299#123—21no—
300#123—22no—
301#123—22no—
302#123—22no—
303#123—21no—
304#123—22no—
305#123—22no—
306C22⋊Q8—21no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311#121—21no—
312#121—22no—
313#121—22no—
314#121—22no—
315#121—22no—
316#120—21no—
317#120—22no—
318#120—21no—
319#120—22no—
320C4×Q8—21no—
321C4×D4—21no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325#117—22no—
326#117—22no—
327#117C3×D412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C42C3×D411yescharacteristic
332#104—22no—
333#104—22no—
334#104—24no—
335#104—24no—
336#101—21no—
337#101—22no—
338#101C23×C314yes—
339#101C23×C312yes—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—22no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—81no—
347C23×C3—82no—
348C23×C3—82no—
349C23×C3#12011yescharacteristic
350C3×Q8—22no—
351C3×Q8—24no—
352C3×Q8—41no—
353C3×Q8—42no—
354C3×Q8—41no—
355C3×Q8—42no—
356C3×Q8—42no—
357C3×Q8—41no—
358C3×Q8—81no—
359C3×D4—22no—
360C3×D4—24no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—42no—
364C3×D4—42no—
365C3×D4—41no—
366C3×D4—42no—
367C3×D4—42no—
368C3×D4—42no—
369C3×D4—42no—
370C3×D4—81no—
371C3×D4—82no—
372C3×D4—82no—
373C3×D4—81no—
374C3×D4—82no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—42no—
385C2×C3×C4—44no—
386C2×C3×C4—44no—
387C2×C3×C4—42no—
388C2×C3×C4—41no—
389C2×C3×C4—82no—
390C2×C3×C4—82no—
391C2×C3×C4—81no—
392C2×C3×C4—82no—
393C2×C3×C4—81no—
394C2×C3×C4#12011yescharacteristic
395C2×C3×C4#12012yes—
396C2×C3×C4C22×D412yes—
397C2×C3×C4C22×D411yescharacteristic
398C3×C8—42no—
399C3×C8—44no—
400C24—21no—
401C24—21no—
402#41—24no—
403#41—22no—
404#41—22no—
405#41—24no—
406#41—42no—
407#41—41no—
408#41—42no—
409#41—42no—
410#41—42no—
411#41—42no—
412#41—44no—
413#41—41no—
414#41—42no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—41no—
419C2×Q8—41no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8C2×C3×D412yes—
424C2×Q8C2×C3×D411yescharacteristic
425C2×Q8C2×C3×D412yes—
426C2×Q8C24×C311yescharacteristic
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4C2×C3×D412yes—
441C2×D4C2×C3×D411yescharacteristic
442C2×D4C2×C3×D411yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4C2×C3×D411yescharacteristic
451C22×C4C2×C3×D412yes—
452Q16—44no—
453SD16—44no—
454SD16—42no—
455SD16—42no—
456SD16—44no—
457D8—42no—
458D8—42no—
459D8—44no—
460M16—22no—
461M16—24no—
462M16—41no—
463M16—42no—
464C2×C8—41no—
465C2×C8—42no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C42—22no—
491C42—21no—
492C42C2×C3×D411yescharacteristic
493C42C2×C3×D411yescharacteristic
494C42C2×C3×D412yes—
495C42C2×C3×D412yes—
496C22×C3—41no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—82no—
506C22×C3—161no—
507C22×C3—161no—
508C22×C3—162no—
509C22×C3B(4,𝔽2)11yescharacteristic
510C22×C3B(4,𝔽2)11yescharacteristic
511C22×C3C2×#12011yescharacteristic
512C3×C4—21no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—42no—
521C3×C4—42no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—81no—
529C23—82no—
530C23—82no—
531C23C3×#12011yescharacteristic
532Q8—22no—
533Q8—24no—
534Q8—41no—
535Q8—42no—
536Q8—41no—
537Q8—42no—
538Q8—42no—
539Q8—41no—
540Q8—81no—
541D4—22no—
542D4—24no—
543D4—41no—
544D4—41no—
545D4—42no—
546D4—42no—
547D4—41no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—81no—
553D4—82no—
554D4—82no—
555D4—82no—
556D4—81no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4C3×#12011yescharacteristic
577C2×C4C3×#12012yes—
578C2×C4C22×C3×D412yes—
579C2×C4C22×C3×D411yescharacteristic
580C8—42no—
581C8—44no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—81no—
586C2×C3—82no—
587C2×C3—81no—
588C2×C3—82no—
589C2×C3C2×B(4,𝔽2)11yescharacteristic
590C22—41no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—82no—
600C22—161no—
601C22—161no—
602C22—162no—
603C22C3×B(4,𝔽2)11yescharacteristic
604C22C3×B(4,𝔽2)11yescharacteristic
605C22C2×C3×#12011yescharacteristic
606C4—21no—
607C4—22no—
608C4—22no—
609C4—21no—
610C4—41no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—42no—
615C4—42no—
616C3#3356811yescharacteristic, direct product factor, Sylow
617C2—21no—
618C2—21no—
619C2—21no—
620C2—81no—
621C2—81no—
622C2—82no—
623C2—82no—
624C2C2×C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C3
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3545C2semidirect
2. C3×#3020C2semidirect
3. C2×C3×#454C2semidirect
4. C3×#2204C2semidirect
5. C3×#2203C2semidirect
6. C3×#2202C2semidirect
7. C3×#2133C2semidirect
8. C2×C3×#355C2semidirect
9. #33568C3direct (N × H)
10. C2×C3×2−1+4C22semidirect
13. C3×#563C22semidirect
15. C3×H(3,𝔽4)C22semidirect
16. C3×#555C22semidirect
27. C3×#456C22semidirect
32. C3×#454C22semidirect
33. C3×#454C22semidirect
36. C2×C3×#101C22semidirect
37. C2×C3×#101C22semidirect
38. C3×#356C22semidirect
39. C3×#355C22semidirect
40. C3×#355C22semidirect
41. C3×#355C22semidirect
42. #3545C2×C3semidirect
43. #3020C2×C3semidirect
44. C2×#454C2×C3semidirect
45. #2204C2×C3semidirect
46. #2203C2×C3semidirect
47. #2202C2×C3semidirect
48. #2133C2×C3semidirect
49. C2×#355C2×C3semidirect
131. C2×2−1+4C22×C3semidirect
134. #563C22×C3semidirect
136. H(3,𝔽4)C22×C3semidirect
137. #555C22×C3semidirect
148. #456C22×C3semidirect
153. #454C22×C3semidirect
154. #454C22×C3semidirect
157. C2×#101C22×C3semidirect
158. C2×#101C22×C3semidirect
159. #356C22×C3semidirect
160. #355C22×C3semidirect
161. #355C22×C3semidirect
162. #355C22×C3semidirect
616. C3#33568direct (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×#30202
C2×C3×#4542
C3×#22042
C3×#22032
C3×#22022
C3×#21332
C2×C3×#3552
#335683