Finite Group Databasebeta

← C3×#25522 #10644129

Subgroups — C3×#25522 #10644129

Order 768.

1084 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3493C211yesmaximal, characteristic, complemented
2C3×#3055C212yesmaximal, complemented
3C3×#2500C211yesmaximal, characteristic, complemented
4C3×#2060C211yesmaximal, characteristic, complemented
5C3×#2039C211yesmaximal, characteristic, complemented
6C3×#2027C211yesmaximal, characteristic, complemented
7C3×#2006C211yesmaximal, characteristic, complemented
8C3×#1883C212yesmaximal, complemented
9C3×#1871C212yesmaximal, complemented
10C3×#1860C212yesmaximal, complemented
11C3×#1742C211yesmaximal, characteristic, complemented
12#25522C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C22×C3×M16C2212yes—
15C3×#556—22no—
16C3×#555—22no—
17C3×#553—22no—
18C3×#552—22no—
19C3×#544C2211yescharacteristic
20C3×#538—22no—
21C3×#537C2211yescharacteristic
22C3×#529—22no—
23C3×#525C2212yes—
24C2×C3×#121C2211yescharacteristic
25C3×#520C2212yes—
26C2×C3×#117C2211yescharacteristic
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×#394—22no—
37C3×#390C2211yescharacteristic
38C3×#390C2211yescharacteristic
39C3×#388—22no—
40C3×#384C2211yescharacteristic
41C3×#382C2211yescharacteristic
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#343C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2212yes—
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3493C2×C311yescharacteristic, complemented
49#3055C2×C312yescomplemented
50#2500C2×C311yescharacteristic, complemented
51#2060C2×C311yescharacteristic, complemented
52#2039C2×C311yescharacteristic, complemented
53#2027C2×C311yescharacteristic, complemented
54#2006C2×C311yescharacteristic, complemented
55#1883C2×C312yescomplemented
56#1871C2×C312yescomplemented
57#1860C2×C312yescomplemented
58#1742C2×C311yescharacteristic, complemented
59C2×C3×#41C2312yes—
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C3×AGL(1,ℤ/8)—24no—
64C3×AGL(1,ℤ/8)—42no—
65C2×C3×SD16—22no—
66C2×C3×D8—22no—
67C2×C3×M16—22no—
68C2×C3×M16—24no—
69C2×C3×M16—42no—
70C2×C3×M16C2312yes—
71C2×C3×M16C2312yes—
72C22×C3×C8—22no—
73C3×C4⋊Q8—22no—
74C3×#126—24no—
75C3×#126—24no—
76C3×#125—22no—
77C3×#125—22no—
78C3×#124—22no—
79C3×#124—24no—
80C3×#123—24no—
81C3×#123—22no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8D412yes—
85C3×C22⋊Q8D412yes—
86C3×#121—24no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121D412yes—
92C3×#121D412yes—
93C3×#120—22no—
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—24no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×#117—24no—
102C3×#117—22no—
103C3×#117—22no—
104C3×#117—22no—
105C3×#117C2312yes—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4—22no—
115C2×C3×C22⋊C4D412yes—
116C2×C3×C42D412yes—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C3×#107—22no—
120C3×#106—22no—
121C3×#104—24no—
122C3×#104—42no—
123C3×#103—22no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#102—22no—
127C3×#95—21no—
128C3×#95—22no—
129C2×AGL(1,ℤ/8)C22×C312yes—
130C22×M16C22×C312yes—
131#556—22no—
132#555—22no—
133#553—22no—
134#552—22no—
135#544C22×C311yescharacteristic
136#538—22no—
137#537C22×C311yescharacteristic
138#529—22no—
139#525C22×C312yes—
140C2×#121C22×C311yescharacteristic
141#520C22×C312yes—
142C2×#117C22×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#394—22no—
153#390C22×C311yescharacteristic
154#390C22×C311yescharacteristic
155#388—22no—
156#384C22×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—
170C2×C3×Q8—22no—
171C2×C3×Q8C2×D412yes—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—42no—
181C2×C3×D4—82no—
182C2×C3×D4C2×D412yes—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2411yescharacteristic
197C3×SD16—44no—
198C3×D8—44no—
199C3×M16—22no—
200C3×M16—22no—
201C3×M16—22no—
202C3×M16—44no—
203C3×M16#4114yes—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—44no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—82no—
217C3×C4⋊C4C2×D412yes—
218C3×C4⋊C4C2×D412yes—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—44no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—41no—
230C3×C22⋊C4—41no—
231C3×C22⋊C4—82no—
232C3×C42—24no—
233C3×C42—22no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—21no—
238C3×C42—21no—
239C2×#41C23×C312yes—
240C22×D4—21no—
241C22×D4C23×C311yescharacteristic
242C23×C4C23×C311yescharacteristic
243AGL(1,ℤ/8)—24no—
244AGL(1,ℤ/8)—42no—
245C2×SD16—22no—
246C2×D8—22no—
247C2×M16—22no—
248C2×M16—24no—
249C2×M16—42no—
250C2×M16C23×C312yes—
251C2×M16C23×C312yes—
252C22×C8—22no—
253C4⋊Q8—22no—
254#126—24no—
255#126—24no—
256#125—22no—
257#125—22no—
258#124—22no—
259#124—24no—
260#123—24no—
261#123—22no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8C3×D412yes—
265C22⋊Q8C3×D412yes—
266#121—24no—
267#121—22no—
268#121—22no—
269#121—22no—
270#121—22no—
271#121C3×D412yes—
272#121C3×D412yes—
273#120—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—22no—
281#117—24no—
282#117—22no—
283#117—22no—
284#117—22no—
285#117C23×C312yes—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4C3×D412yes—
296C2×C42C3×D412yes—
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—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—22no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3#12012yes—
320C23×C3C22×D411yescharacteristic
321C3×Q8—22no—
322C3×Q8—22no—
323C3×Q8—82no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—42no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—82no—
331C3×D4—82no—
332C3×D4—82no—
333C3×D4—84no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
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—22no—
345C2×C3×C4—22no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—84no—
358C2×C3×C4#12011yescharacteristic
359C2×C3×C4#12011yescharacteristic
360C2×C3×C4C22×D412yes—
361C2×C3×C4C2×#4112yes—
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C42−1+411yescharacteristic
364C3×C8—24no—
365C3×C8—42no—
366C24—21no—
367C24C2×C3×D411yescharacteristic
368#41—22no—
369#41—22no—
370#41—22no—
371#41—22no—
372C2×Q8—22no—
373C2×Q8C2×C3×D412yes—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—82no—
384C2×D4C2×C3×D412yes—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D412yes—
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D412yes—
397C22×C4C2×C3×D412yes—
398C22×C4C24×C311yescharacteristic
399SD16—44no—
400D8—44no—
401M16—22no—
402M16—22no—
403M16—22no—
404M16—44no—
405M16C3×#4114yes—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—44no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—82no—
419C4⋊C4C2×C3×D412yes—
420C4⋊C4C2×C3×D412yes—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—44no—
430C22⋊C4—42no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—82no—
434C42—24no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—21no—
440C42—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#53211yescharacteristic
458C22×C3#53711yescharacteristic
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4#54612yes—
470C3×C4#54712yes—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23C3×#12012yes—
482C23C22×C3×D411yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—82no—
486D4—22no—
487D4—22no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—82no—
493D4—82no—
494D4—82no—
495D4—84no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—84no—
520C2×C4C3×#12011yescharacteristic
521C2×C4C3×#12011yescharacteristic
522C2×C4C22×C3×D412yes—
523C2×C4C2×C3×#4112yes—
524C2×C4C3×H(5,𝔽2)11yescharacteristic
525C2×C4C3×2−1+411yescharacteristic
526C8—24no—
527C8—42no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—42no—
531C2×C3—82no—
532C2×C3—161no—
533C2×C3#243011yescharacteristic
534C2×C3#301811yescharacteristic
535C2×C3#302111yescharacteristic
536C22—21no—
537C22—21no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22C2×C3×#12011yescharacteristic
552C22C3×#53211yescharacteristic
553C22C3×#53711yescharacteristic
554C4—21no—
555C4—21no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—81no—
563C4—81no—
564C4C3×#54612yes—
565C4C3×#54712yes—
566C3#2552211yescharacteristic, direct product factor, Sylow
567C2—21no—
568C2—21no—
569C2—42no—
570C2—82no—
571C2—161no—
572C2C3×#243011yescharacteristic
573C2C3×#301811yescharacteristic
574C2C3×#302111yescharacteristic
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×#3493C2semidirect
2. C3×#3055C2semidirect
3. C3×#2500C2semidirect
4. C3×#2060C2semidirect
5. C3×#2039C2semidirect
6. C3×#2027C2semidirect
7. C3×#2006C2semidirect
8. C3×#1883C2semidirect
9. C3×#1871C2semidirect
10. C3×#1860C2semidirect
11. C3×#1742C2semidirect
12. #25522C3direct (N × H)
48. #3493C2×C3semidirect
49. #3055C2×C3semidirect
50. #2500C2×C3semidirect
51. #2060C2×C3semidirect
52. #2039C2×C3semidirect
53. #2027C2×C3semidirect
54. #2006C2×C3semidirect
55. #1883C2×C3semidirect
56. #1871C2×C3semidirect
57. #1860C2×C3semidirect
58. #1742C2×C3semidirect
566. C3#25522direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34932
C3×#30552
C3×#25002
C3×#20602
C3×#20392
C3×#20272
C3×#20062
C3×#18832
C3×#18712
C3×#18602
C3×#17422
#255223