Finite Group Databasebeta

← C3×#25521 #10644128

Subgroups — C3×#25521 #10644128

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3492C211yesmaximal, characteristic, complemented
2C3×#3055C212yesmaximal, complemented
3C3×#2504C211yesmaximal, characteristic, complemented
4C3×#2061C211yesmaximal, characteristic, complemented
5C3×#2040C211yesmaximal, characteristic, complemented
6C3×#2027C211yesmaximal, characteristic, complemented
7C3×#2006C211yesmaximal, characteristic, complemented
8C3×#1883C212yesmaximal, complemented
9C3×#1871C212yesmaximal, complemented
10C3×#1860C212yesmaximal, complemented
11C2×C3×#343C211yesmaximal, characteristic, complemented
12#25521C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C22×C3×M16C2212yes—
15C3×#554—22no—
16C3×#553—22no—
17C3×#552—22no—
18C3×#547—22no—
19C3×#544C2211yescharacteristic
20C3×#539—22no—
21C3×#537C2211yescharacteristic
22C3×#525—22no—
23C3×#525C2212yes—
24C2×C3×#121C2211yescharacteristic
25C3×#520C2212yes—
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#469—22no—
28C3×#468—22no—
29C3×#466—22no—
30C3×#465—22no—
31C3×#428C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#418C2212yes—
35C3×#417C2212yes—
36C3×#397—22no—
37C3×#390C2211yescharacteristic
38C3×#390C2211yescharacteristic
39C3×#386—22no—
40C3×#382C2211yescharacteristic
41C3×#382C2211yescharacteristic
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#343C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2212yes—
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3492C2×C311yescharacteristic, complemented
49#3055C2×C312yescomplemented
50#2504C2×C311yescharacteristic, complemented
51#2061C2×C311yescharacteristic, complemented
52#2040C2×C311yescharacteristic, complemented
53#2027C2×C311yescharacteristic, complemented
54#2006C2×C311yescharacteristic, complemented
55#1883C2×C312yescomplemented
56#1871C2×C312yescomplemented
57#1860C2×C312yescomplemented
58C2×#343C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×D4—21no—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4D412yes—
64C23×C3×C4C2311yescharacteristic
65C3×AGL(1,ℤ/8)—24no—
66C3×AGL(1,ℤ/8)—42no—
67C2×C3×SD16—22no—
68C2×C3×D8—22no—
69C2×C3×M16—22no—
70C2×C3×M16—24no—
71C2×C3×M16—42no—
72C2×C3×M16C2312yes—
73C2×C3×M16C2312yes—
74C22×C3×C8—22no—
75C3×C4⋊Q8—22no—
76C3×#126—24no—
77C3×#125—24no—
78C3×#125—22no—
79C3×#123—24no—
80C3×#123—24no—
81C3×#123—22no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—24no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8D412yes—
87C3×C22⋊Q8D412yes—
88C3×#121—22no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—22no—
93C3×#121D412yes—
94C3×#121D412yes—
95C3×#120—22no—
96C3×C4×Q8—22no—
97C3×C4×D4—24no—
98C3×C4×D4—22no—
99C3×C4×D4—24no—
100C3×C4×D4—22no—
101C3×C4×D4—22no—
102C3×#117—22no—
103C3×#117—22no—
104C3×#117C2312yes—
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4D412yes—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4—22no—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×#107—22no—
119C3×#106—22no—
120C3×#104—24no—
121C3×#104—42no—
122C3×#103—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#95—21no—
127C3×#95—22no—
128C3×#95—22no—
129C2×AGL(1,ℤ/8)C22×C312yes—
130C22×M16C22×C312yes—
131#554—22no—
132#553—22no—
133#552—22no—
134#547—22no—
135#544C22×C311yescharacteristic
136#539—22no—
137#537C22×C311yescharacteristic
138#525—22no—
139#525C22×C312yes—
140C2×#121C22×C311yescharacteristic
141#520C22×C312yes—
142C22×C4⋊C4C22×C311yescharacteristic
143#469—22no—
144#468—22no—
145#466—22no—
146#465—22no—
147#428C22×C312yes—
148C2×#104C22×C312yes—
149C2×#104C22×C312yes—
150#418C22×C312yes—
151#417C22×C312yes—
152#397—22no—
153#390C22×C311yescharacteristic
154#390C22×C311yescharacteristic
155#386—22no—
156#382C22×C311yescharacteristic
157#382C22×C311yescharacteristic
158C4×C22⋊C4C22×C311yescharacteristic
159#343C22×C311yescharacteristic
160#343C22×C312yes—
161#343C22×C311yescharacteristic
162#339C22×C312yes—
163#339C22×C312yes—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—84no—
171C2×C3×Q8—22no—
172C2×C3×Q8C2×D412yes—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—44no—
182C2×C3×D4—41no—
183C2×C3×D4—41no—
184C2×C3×D4C2×D412yes—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—82no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2411yescharacteristic
202C3×SD16—44no—
203C3×D8—44no—
204C3×M16—22no—
205C3×M16—22no—
206C3×M16—22no—
207C3×M16—44no—
208C3×M16#4114yescomplemented
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—44no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4C2×D412yes—
223C3×C4⋊C4C2×D412yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—42no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—82no—
234C3×C22⋊C4—82no—
235C3×C42—24no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—21no—
239C3×C42—21no—
240C2×#41—22no—
241C2×#41C23×C312yes—
242C22×D4—21no—
243C22×D4C23×C311yescharacteristic
244C23×C4C3×D412yes—
245C23×C4C23×C311yescharacteristic
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—42no—
248C2×SD16—22no—
249C2×D8—22no—
250C2×M16—22no—
251C2×M16—24no—
252C2×M16—42no—
253C2×M16C23×C312yes—
254C2×M16C23×C312yes—
255C22×C8—22no—
256C4⋊Q8—22no—
257#126—24no—
258#125—24no—
259#125—22no—
260#123—24no—
261#123—24no—
262#123—22no—
263C22⋊Q8—22no—
264C22⋊Q8—24no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8C3×D412yes—
268C22⋊Q8C3×D412yes—
269#121—22no—
270#121—22no—
271#121—22no—
272#121—22no—
273#121—22no—
274#121C3×D412yes—
275#121C3×D412yes—
276#120—22no—
277C4×Q8—22no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—22no—
283#117—22no—
284#117—22no—
285#117C23×C312yes—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C3×D412yes—
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C42C23×C311yescharacteristic
298C2×C42C23×C311yescharacteristic
299#107—22no—
300#106—22no—
301#104—24no—
302#104—42no—
303#103—22no—
304#102—22no—
305#102—22no—
306#102—22no—
307#95—21no—
308#95—22no—
309#95—22no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—41no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3#12011yescharacteristic
321C23×C3#12011yescharacteristic
322C23×C3C22×D411yescharacteristic
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—82no—
326C3×D4—22no—
327C3×D4—22no—
328C3×D4—44no—
329C3×D4—42no—
330C3×D4—42no—
331C3×D4—44no—
332C3×D4—42no—
333C3×D4—42no—
334C3×D4—82no—
335C3×D4—82no—
336C3×D4—82no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—44no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—42no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—82no—
362C2×C3×C4—84no—
363C2×C3×C4#12012yes—
364C2×C3×C4C22×D412yes—
365C2×C3×C4C2×#4112yes—
366C2×C3×C4H(5,𝔽2)11yescharacteristic
367C2×C3×C42−1+411yescharacteristic
368C3×C8—24no—
369C3×C8—42no—
370C24—21no—
371C24C2×C3×D411yescharacteristic
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—84no—
377C2×Q8—22no—
378C2×Q8C2×C3×D412yes—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—44no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4C2×C3×D412yes—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—82no—
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D412yes—
404C22×C4C2×C3×D412yes—
405C22×C4C2×C3×D412yes—
406C22×C4C2×C3×D412yes—
407C22×C4C24×C311yescharacteristic
408SD16—44no—
409D8—44no—
410M16—22no—
411M16—22no—
412M16—22no—
413M16—44no—
414M16C3×#4114yescomplemented
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—44no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4C2×C3×D412yes—
429C4⋊C4C2×C3×D412yes—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C42—24no—
442C42—22no—
443C42—22no—
444C42—21no—
445C42—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3#45412yes—
464C22×C3C2×#12011yescharacteristic
465C22×C3#53211yescharacteristic
466C22×C3#53711yescharacteristic
467C3×C4—22no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—42no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4—81no—
476C3×C4#54612yes—
477C3×C4#54712yes—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23C3×#12011yescharacteristic
489C23C3×#12011yescharacteristic
490C23C22×C3×D411yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—82no—
494D4—22no—
495D4—22no—
496D4—44no—
497D4—44no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—82no—
503D4—82no—
504D4—82no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—82no—
530C2×C4—84no—
531C2×C4C3×#12012yes—
532C2×C4C22×C3×D412yes—
533C2×C4C2×C3×#4112yes—
534C2×C4C3×H(5,𝔽2)11yescharacteristic
535C2×C4C3×2−1+411yescharacteristic
536C8—24no—
537C8—42no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—41no—
542C2×C3—82no—
543C2×C3—161no—
544C2×C3#243011yescharacteristic
545C2×C3C2×#45411yescharacteristic
546C2×C3#301911yescharacteristic
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—82no—
560C22—82no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22C3×#45412yes—
565C22C2×C3×#12011yescharacteristic
566C22C3×#53211yescharacteristic
567C22C3×#53711yescharacteristic
568C4—22no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4C3×#54612yes—
578C4C3×#54712yes—
579C3#2552111yescharacteristic, direct product factor, Sylow
580C2—21no—
581C2—21no—
582C2—22no—
583C2—41no—
584C2—82no—
585C2—161no—
586C2C3×#243011yescharacteristic
587C2C2×C3×#45411yescharacteristic
588C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3492C2semidirect
2. C3×#3055C2semidirect
3. C3×#2504C2semidirect
4. C3×#2061C2semidirect
5. C3×#2040C2semidirect
6. C3×#2027C2semidirect
7. C3×#2006C2semidirect
8. C3×#1883C2semidirect
9. C3×#1871C2semidirect
10. C3×#1860C2semidirect
11. C2×C3×#343C2semidirect
12. #25521C3direct (N × H)
48. #3492C2×C3semidirect
49. #3055C2×C3semidirect
50. #2504C2×C3semidirect
51. #2061C2×C3semidirect
52. #2040C2×C3semidirect
53. #2027C2×C3semidirect
54. #2006C2×C3semidirect
55. #1883C2×C3semidirect
56. #1871C2×C3semidirect
57. #1860C2×C3semidirect
58. C2×#343C2×C3semidirect
208. C3×M16#41semidirect
414. M16C3×#41semidirect
579. C3#25521direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34922
C3×#30552
C3×#25042
C3×#20612
C3×#20402
C3×#20272
C3×#20062
C3×#18832
C3×#18712
C3×#18602
C2×C3×#3432
#255213