Finite Group Databasebeta

← C3×#33569 #10652176

Subgroups — C3×#33569 #10652176

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3544C211yesmaximal, characteristic, complemented
2C3×#3020C212yesmaximal, complemented
3C2×C3×#456C211yesmaximal, characteristic, complemented
4C3×#2205C212yesmaximal, complemented
5C3×#2204C212yesmaximal, complemented
6C3×#2203C214yesmaximal, complemented
7C3×#2133C212yesmaximal, complemented
8C2×C3×#356C211yesmaximal, characteristic, complemented
9#33569C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C3×#575—22no—
12C2×C3×#137—21no—
13C3×#563—21no—
14C3×#563C2212yescomplemented
15C3×Q82C2211yescharacteristic, complemented
16C3×#555C2211yescharacteristic, complemented
17C3×#554—21no—
18C3×#552—22no—
19C3×#547—22no—
20C3×#544—22no—
21C3×#542—22no—
22C3×#536—21no—
23C3×#536C2212yes—
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#529—21no—
26C3×#456—21no—
27C3×#456—21no—
28C3×#456—22no—
29C3×#456C2212yes—
30C3×#456C2212yes—
31C3×#456C2214yes—
32C3×#454—22no—
33C3×#454C2214yes—
34C3×#421—22no—
35C2×C3×#104—21no—
36C2×C3×#101C2212yes—
37C2×C3×#101C2211yescharacteristic
38C3×#356C2212yescomplemented
39C3×#356C2212yescomplemented
40C3×#356C2214yescomplemented
41C3×#355C2214yescomplemented
42#3544C2×C311yescharacteristic, complemented
43#3020C2×C312yescomplemented
44C2×#456C2×C311yescharacteristic, complemented
45#2205C2×C312yescomplemented
46#2204C2×C312yescomplemented
47#2203C2×C314yescomplemented
48#2133C2×C312yescomplemented
49C2×#356C2×C311yescharacteristic, complemented
50C3×2−1+4—22no—
51C3×2−1+4—21no—
52C3×2−1+4—42no—
53C3×2−1+4C2312yes—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41—22no—
57C2×C3×#41—41no—
58C2×C3×#41D412yes—
59C22×C3×Q8—41no—
60C22×C3×Q8D411yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C3×#137—22no—
63C3×#137—22no—
64C3×#137—24no—
65C3×#137—42no—
66C3×#137—42no—
67C3×AGL(1,ℤ/8)—24no—
68C3×AGL(1,ℤ/8)—42no—
69C3×#135—44no—
70C2×C3×Q16—41no—
71C2×C3×SD16—41no—
72C2×C3×M16—21no—
73C2×C3×M16—22no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8D411yescharacteristic
79C3×C4⋊Q8D411yescharacteristic
80C3×C4⋊Q8D412yes—
81C3×C4⋊Q8C2312yes—
82C3×C4⋊Q8C2311yescharacteristic
83C3×C4⋊Q8C2311yescharacteristic
84C3×#126—22no—
85C3×#126—22no—
86C3×#126—22no—
87C3×#125—22no—
88C3×#125—21no—
89C3×#125—22no—
90C3×#125—22no—
91C3×#124—21no—
92C3×#124—21no—
93C3×#124—21no—
94C3×#124D412yes—
95C3×#124C2312yes—
96C3×#123—22no—
97C3×#123—22no—
98C3×#123—22no—
99C3×#123—22no—
100C3×#123—22no—
101C3×C22⋊Q8—22no—
102C3×C22⋊Q8—21no—
103C3×C22⋊Q8—22no—
104C3×C22⋊Q8—22no—
105C3×C22⋊Q8—22no—
106C3×C22⋊Q8—22no—
107C3×C22⋊Q8—22no—
108C3×C22⋊Q8—22no—
109C3×#121—21no—
110C3×#121—22no—
111C3×C4×Q8—21no—
112C3×C4×Q8—22no—
113C3×C4×D4—22no—
114C3×C4×D4—22no—
115C3×C4×D4—21no—
116C3×#117—22no—
117C3×#117—21no—
118C3×#117—22no—
119C3×#117D412yes—
120C2×C3×C4⋊C4—22no—
121C2×C3×C4⋊C4—22no—
122C2×C3×C42D411yescharacteristic
123C3×#104—22no—
124C3×#104—22no—
125C3×#104—24no—
126C3×#104—24no—
127C3×#101—22no—
128C3×#101—21no—
129C3×#101C2314yes—
130C3×#101C2312yes—
131C2×2−1+4C22×C311yescharacteristic
132#575—22no—
133C2×#137—21no—
134#563—21no—
135#563C22×C312yescomplemented
136Q82C22×C311yescharacteristic, complemented
137#555C22×C311yescharacteristic, complemented
138#554—21no—
139#552—22no—
140#547—22no—
141#544—22no—
142#542—22no—
143#536—21no—
144#536C22×C312yes—
145C2×C4⋊Q8C22×C311yescharacteristic
146#529—21no—
147#456—21no—
148#456—21no—
149#456—22no—
150#456C22×C312yes—
151#456C22×C314yes—
152#456C22×C312yes—
153#454—22no—
154#454C22×C314yes—
155#421—22no—
156C2×#104—21no—
157C2×#101C22×C312yes—
158C2×#101C22×C311yescharacteristic
159#356C22×C312yescomplemented
160#356C22×C312yescomplemented
161#356C22×C314yescomplemented
162#355C22×C314yescomplemented
163C3×#41—24no—
164C3×#41—22no—
165C3×#41—24no—
166C3×#41—22no—
167C3×#41—42no—
168C3×#41—41no—
169C3×#41—42no—
170C3×#41—42no—
171C3×#41—42no—
172C3×#41—44no—
173C3×#41—42no—
174C3×#41—41no—
175C3×#41—42no—
176C3×#41—81no—
177C3×#41—81no—
178C2×C3×Q8—21no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—22no—
183C2×C3×Q8—41no—
184C2×C3×Q8—41no—
185C2×C3×Q8—41no—
186C2×C3×Q8—41no—
187C2×C3×Q8—42no—
188C2×C3×Q8—44no—
189C2×C3×Q8—42no—
190C2×C3×Q8C2×D411yescharacteristic
191C2×C3×Q8C2×D412yes—
192C2×C3×Q8C2×D411yescharacteristic
193C2×C3×Q8C2×D411yescharacteristic
194C2×C3×Q8C2×D412yes—
195C2×C3×Q8C2411yescharacteristic
196C2×C3×D4—21no—
197C2×C3×D4—22no—
198C2×C3×D4—21no—
199C2×C3×D4—22no—
200C2×C3×D4—21no—
201C2×C3×D4—41no—
202C2×C3×D4C2×D412yes—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—41no—
209C22×C3×C4—42no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D412yes—
212C3×Q16—42no—
213C3×Q16—42no—
214C3×Q16—44no—
215C3×SD16—42no—
216C3×SD16—42no—
217C3×SD16—44no—
218C3×SD16—44no—
219C3×D8—44no—
220C3×M16—24no—
221C3×M16—22no—
222C3×M16—42no—
223C3×M16—41no—
224C2×C3×C8—41no—
225C2×C3×C8—42no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—22no—
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×C42—21no—
252C3×C42—22no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42C2×D411yescharacteristic
256C3×C42C2×D411yescharacteristic
257C3×C42C2×D412yes—
258C3×C42C2×D412yes—
2592−1+4—22no—
2602−1+4—21no—
2612−1+4—42no—
2622−1+4C23×C312yes—
263C2×#41—21no—
264C2×#41—21no—
265C2×#41—22no—
266C2×#41—41no—
267C2×#41C3×D412yes—
268C22×Q8—41no—
269C22×Q8C3×D411yescharacteristic
270C22×Q8C23×C311yescharacteristic
271#137—22no—
272#137—22no—
273#137—24no—
274#137—42no—
275#137—42no—
276AGL(1,ℤ/8)—24no—
277AGL(1,ℤ/8)—42no—
278#135—44no—
279C2×Q16—41no—
280C2×SD16—41no—
281C2×M16—21no—
282C2×M16—22no—
283C4⋊Q8—21no—
284C4⋊Q8—22no—
285C4⋊Q8—21no—
286C4⋊Q8—22no—
287C4⋊Q8C3×D411yescharacteristic
288C4⋊Q8C3×D411yescharacteristic
289C4⋊Q8C3×D412yes—
290C4⋊Q8C23×C311yescharacteristic
291C4⋊Q8C23×C311yescharacteristic
292C4⋊Q8C23×C312yes—
293#126—22no—
294#126—22no—
295#126—22no—
296#125—21no—
297#125—22no—
298#125—22no—
299#125—22no—
300#124—21no—
301#124—21no—
302#124—21no—
303#124C3×D412yes—
304#124C23×C312yes—
305#123—22no—
306#123—22no—
307#123—22no—
308#123—22no—
309#123—22no—
310C22⋊Q8—21no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318#121—21no—
319#121—22no—
320C4×Q8—21no—
321C4×Q8—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—21no—
325#117—22no—
326#117—21no—
327#117—22no—
328#117C3×D412yes—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C42C3×D411yescharacteristic
332#104—22no—
333#104—22no—
334#104—24no—
335#104—24no—
336#101—22no—
337#101—21no—
338#101C23×C312yes—
339#101C23×C314yes—
340C23×C3—22no—
341C23×C3—41no—
342C23×C3#12011yescharacteristic
343C3×Q8—24no—
344C3×Q8—22no—
345C3×Q8—42no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×Q8—41no—
349C3×Q8—41no—
350C3×Q8—42no—
351C3×Q8—42no—
352C3×Q8—42no—
353C3×Q8—41no—
354C3×Q8—81no—
355C3×Q8—81no—
356C3×D4—24no—
357C3×D4—22no—
358C3×D4—42no—
359C3×D4—42no—
360C3×D4—42no—
361C3×D4—41no—
362C3×D4—42no—
363C3×D4—42no—
364C3×D4—42no—
365C3×D4—42no—
366C3×D4—81no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—42no—
379C2×C3×C4—44no—
380C2×C3×C4—44no—
381C2×C3×C4—42no—
382C2×C3×C4—44no—
383C2×C3×C4—42no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—82no—
388C2×C3×C4—82no—
389C2×C3×C4#12011yescharacteristic
390C2×C3×C4#12012yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D412yes—
393C3×C8—44no—
394C3×C8—42no—
395#41—24no—
396#41—22no—
397#41—24no—
398#41—22no—
399#41—42no—
400#41—41no—
401#41—42no—
402#41—42no—
403#41—42no—
404#41—44no—
405#41—42no—
406#41—41no—
407#41—42no—
408#41—81no—
409#41—81no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8C2×C3×D411yescharacteristic
423C2×Q8C2×C3×D412yes—
424C2×Q8C2×C3×D411yescharacteristic
425C2×Q8C2×C3×D411yescharacteristic
426C2×Q8C2×C3×D412yes—
427C2×Q8C24×C311yescharacteristic
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—41no—
434C2×D4C2×C3×D412yes—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4C2×C3×D411yescharacteristic
443C22×C4C2×C3×D412yes—
444Q16—42no—
445Q16—42no—
446Q16—44no—
447SD16—42no—
448SD16—42no—
449SD16—44no—
450SD16—44no—
451D8—44no—
452M16—24no—
453M16—22no—
454M16—42no—
455M16—41no—
456C2×C8—41no—
457C2×C8—42no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊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—22no—
482C22⋊C4—22no—
483C42—21no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42C2×C3×D411yescharacteristic
488C42C2×C3×D411yescharacteristic
489C42C2×C3×D412yes—
490C42C2×C3×D412yes—
491C22×C3—41no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—82no—
497C22×C3B(4,𝔽2)11yescharacteristic
498C22×C3B(4,𝔽2)11yescharacteristic
499C22×C3C2×#12011yescharacteristic
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—21no—
503C3×C4—22no—
504C3×C4—21no—
505C3×C4—42no—
506C3×C4—41no—
507C3×C4—42no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4—42no—
512C23—22no—
513C23—41no—
514C23C3×#12011yescharacteristic
515Q8—24no—
516Q8—22no—
517Q8—42no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—42no—
523Q8—42no—
524Q8—41no—
525Q8—42no—
526Q8—81no—
527Q8—81no—
528D4—24no—
529D4—22no—
530D4—42no—
531D4—42no—
532D4—41no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—81no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4C3×#12011yescharacteristic
562C2×C4C3×#12012yes—
563C2×C4C22×C3×D411yescharacteristic
564C2×C4C22×C3×D412yes—
565C8—44no—
566C8—42no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—82no—
571C2×C3—82no—
572C2×C3C2×B(4,𝔽2)11yescharacteristic
573C22—41no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—82no—
579C22C3×B(4,𝔽2)11yescharacteristic
580C22C3×B(4,𝔽2)11yescharacteristic
581C22C2×C3×#12011yescharacteristic
582C4—22no—
583C4—22no—
584C4—21no—
585C4—22no—
586C4—21no—
587C4—42no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—42no—
594C3#3356911yescharacteristic, direct product factor, Sylow
595C2—21no—
596C2—21no—
597C2—21no—
598C2—82no—
599C2—82no—
600C2C2×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×#3544C2semidirect
2. C3×#3020C2semidirect
3. C2×C3×#456C2semidirect
4. C3×#2205C2semidirect
5. C3×#2204C2semidirect
6. C3×#2203C2semidirect
7. C3×#2133C2semidirect
8. C2×C3×#356C2semidirect
9. #33569C3direct (N × H)
14. C3×#563C22semidirect
15. C3×Q82C22semidirect
16. C3×#555C22semidirect
38. C3×#356C22semidirect
39. C3×#356C22semidirect
40. C3×#356C22semidirect
41. C3×#355C22semidirect
42. #3544C2×C3semidirect
43. #3020C2×C3semidirect
44. C2×#456C2×C3semidirect
45. #2205C2×C3semidirect
46. #2204C2×C3semidirect
47. #2203C2×C3semidirect
48. #2133C2×C3semidirect
49. C2×#356C2×C3semidirect
135. #563C22×C3semidirect
136. Q82C22×C3semidirect
137. #555C22×C3semidirect
159. #356C22×C3semidirect
160. #356C22×C3semidirect
161. #356C22×C3semidirect
162. #355C22×C3semidirect
594. C3#33569direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35442
C3×#30202
C2×C3×#4562
C3×#22052
C3×#22042
C3×#22032
C3×#21332
C2×C3×#3562
#335693