Finite Group Databasebeta

← C3×#25519 #10644126

Subgroups — C3×#25519 #10644126

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3492C211yesmaximal, characteristic
2C3×#3056C212yesmaximal, complemented
3C3×#2502C211yesmaximal, characteristic, complemented
4C3×#2060C211yesmaximal, characteristic, complemented
5C3×#2039C211yesmaximal, characteristic, complemented
6C3×#2028C211yesmaximal, characteristic, complemented
7C3×#2007C211yesmaximal, characteristic, complemented
8C3×#1882C212yesmaximal, complemented
9C3×#1871C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C2×C3×#343C211yesmaximal, characteristic, complemented
12#25519C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
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×C22⋊Q8C2211yescharacteristic
25C3×#520C2212yes—
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#470—22no—
28C3×#468—22no—
29C3×#467—22no—
30C3×#465—22no—
31C3×#428C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#419C2212yes—
35C3×#417C2212yes—
36C3×#398—22no—
37C3×#391C2211yescharacteristic
38C3×#391C2211yescharacteristic
39C3×#385—22no—
40C3×#384C2211yescharacteristic
41C3×#384C2211yescharacteristic
42C3×C4×C22⋊C4C2211yescharacteristic
43C3×#343C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2212yes—
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3492C2×C311yescharacteristic
49#3056C2×C312yescomplemented
50#2502C2×C311yescharacteristic, complemented
51#2060C2×C311yescharacteristic, complemented
52#2039C2×C311yescharacteristic, complemented
53#2028C2×C311yescharacteristic, complemented
54#2007C2×C311yescharacteristic, complemented
55#1882C2×C312yescomplemented
56#1871C2×C312yescomplemented
57#1859C2×C312yescomplemented
58C2×#343C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C23×C3×C4D412yes—
63C23×C3×C4C2311yescharacteristic
64C3×#137—24no—
65C3×#137—42no—
66C2×C3×Q16—22no—
67C2×C3×SD16—22no—
68C2×C3×M16—22no—
69C2×C3×M16—24no—
70C2×C3×M16—42no—
71C2×C3×M16C2312yes—
72C2×C3×M16C2312yes—
73C22×C3×C8—22no—
74C3×C4⋊Q8—22no—
75C3×#126—24no—
76C3×#125—24no—
77C3×#125—22no—
78C3×#123—24no—
79C3×#123—24no—
80C3×#123—22no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8D412yes—
88C3×C22⋊Q8D412yes—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121D412yes—
93C3×#121D412yes—
94C3×#120—22no—
95C3×C4×Q8—22no—
96C3×C4×D4—24no—
97C3×C4×D4—22no—
98C3×C4×D4—24no—
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×#117—22no—
102C3×#117—22no—
103C3×#117C2312yes—
104C2×C3×C4⋊C4—22no—
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4—22no—
110C2×C3×C4⋊C4D412yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×#107—22no—
119C3×#106—22no—
120C3×#104—24no—
121C3×#104—42no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#103—22no—
125C3×#102—22no—
126C3×#95—21no—
127C3×#95—22no—
128C3×#95—22no—
129C2×#137C22×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×C22⋊Q8C22×C311yescharacteristic
141#520C22×C312yes—
142C22×C4⋊C4C22×C311yescharacteristic
143#470—22no—
144#468—22no—
145#467—22no—
146#465—22no—
147#428C22×C312yes—
148C2×#104C22×C312yes—
149C2×#104C22×C312yes—
150#419C22×C312yes—
151#417C22×C312yes—
152#398—22no—
153#391C22×C311yescharacteristic
154#391C22×C311yescharacteristic
155#385—22no—
156#384C22×C311yescharacteristic
157#384C22×C311yescharacteristic
158C4×C22⋊C4C22×C311yescharacteristic
159#343C22×C311yescharacteristic
160#343C22×C312yes—
161#343C22×C311yescharacteristic
162#339C22×C312yes—
163#339C22×C312yes—
164C24×C3C2×D411yescharacteristic
165C3×#41—22no—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—84no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—21no—
175C2×C3×Q8C2×D412yes—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4C2×D412yes—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—82no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2411yescharacteristic
199C3×Q16—44no—
200C3×SD16—44no—
201C3×M16—22no—
202C3×M16—22no—
203C3×M16—22no—
204C3×M16—44no—
205C3×M16#4114yescomplemented
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—44no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4—82no—
221C3×C4⋊C4—82no—
222C3×C4⋊C4C2×D412yes—
223C3×C4⋊C4C2×D412yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—41no—
233C3×C22⋊C4—41no—
234C3×C42—24no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—21no—
238C3×C42—21no—
239C2×#41—22no—
240C2×#41C23×C312yes—
241C22×Q8C23×C311yescharacteristic
242C23×C4C3×D412yes—
243C23×C4C23×C311yescharacteristic
244#137—24no—
245#137—42no—
246C2×Q16—22no—
247C2×SD16—22no—
248C2×M16—22no—
249C2×M16—24no—
250C2×M16—42no—
251C2×M16C23×C312yes—
252C2×M16C23×C312yes—
253C22×C8—22no—
254C4⋊Q8—22no—
255#126—24no—
256#125—24no—
257#125—22no—
258#123—24no—
259#123—24no—
260#123—22no—
261C22⋊Q8—22no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8C3×D412yes—
268C22⋊Q8C3×D412yes—
269#121—22no—
270#121—22no—
271#121—22no—
272#121C3×D412yes—
273#121C3×D412yes—
274#120—22no—
275C4×Q8—22no—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—22no—
281#117—22no—
282#117—22no—
283#117C23×C312yes—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4C3×D412yes—
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C42C23×C311yescharacteristic
297C2×C42C23×C311yescharacteristic
298#107—22no—
299#106—22no—
300#104—24no—
301#104—42no—
302#103—22no—
303#103—22no—
304#103—22no—
305#102—22no—
306#95—21no—
307#95—22no—
308#95—22no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—41no—
314C23×C3#12011yescharacteristic
315C23×C3#12011yescharacteristic
316C23×C3C22×D411yescharacteristic
317C3×Q8—22no—
318C3×Q8—22no—
319C3×Q8—42no—
320C3×Q8—42no—
321C3×Q8—82no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—44no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—44no—
328C3×D4—82no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
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—41no—
343C2×C3×C4—41no—
344C2×C3×C4—44no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
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—41no—
355C2×C3×C4—41no—
356C2×C3×C4—82no—
357C2×C3×C4—84no—
358C2×C3×C4#12012yes—
359C2×C3×C4C22×D412yes—
360C2×C3×C4C2×#4112yes—
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×C3×C42−1+411yescharacteristic
363C3×C8—24no—
364C3×C8—42no—
365C24C2×C3×D411yescharacteristic
366#41—22no—
367#41—22no—
368#41—22no—
369#41—22no—
370#41—84no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8C2×C3×D412yes—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4C2×C3×D412yes—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—82no—
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D412yes—
397C22×C4C2×C3×D412yes—
398C22×C4C2×C3×D412yes—
399C22×C4C24×C311yescharacteristic
400Q16—44no—
401SD16—44no—
402M16—22no—
403M16—22no—
404M16—22no—
405M16—44no—
406M16C3×#4114yescomplemented
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—44no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4C2×C3×D412yes—
424C4⋊C4C2×C3×D412yes—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C42—24no—
436C42—22no—
437C42—22no—
438C42—21no—
439C42—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3#45612yes—
453C22×C3C2×#12011yescharacteristic
454C22×C3#53211yescharacteristic
455C22×C3#53711yescharacteristic
456C3×C4—22no—
457C3×C4—42no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—42no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4#54612yes—
467C3×C4#54712yes—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—41no—
473C23C3×#12011yescharacteristic
474C23C3×#12011yescharacteristic
475C23C22×C3×D411yescharacteristic
476Q8—22no—
477Q8—22no—
478Q8—42no—
479Q8—42no—
480Q8—82no—
481D4—22no—
482D4—22no—
483D4—44no—
484D4—44no—
485D4—42no—
486D4—42no—
487D4—82no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4C3×#12012yes—
518C2×C4C22×C3×D412yes—
519C2×C4C2×C3×#4112yes—
520C2×C4C3×H(5,𝔽2)11yescharacteristic
521C2×C4C3×2−1+411yescharacteristic
522C8—24no—
523C8—42no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—22no—
527C2×C3—41no—
528C2×C3—82no—
529C2×C3#243011yescharacteristic
530C2×C3C2×#45611yescharacteristic
531C2×C3#301911yescharacteristic
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22C3×#45612yes—
545C22C2×C3×#12011yescharacteristic
546C22C3×#53211yescharacteristic
547C22C3×#53711yescharacteristic
548C4—22no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—81no—
557C4—81no—
558C4C3×#54612yes—
559C4C3×#54712yes—
560C3#2551911yescharacteristic, direct product factor, Sylow
561C2—21no—
562C2—21no—
563C2—22no—
564C2—41no—
565C2—82no—
566C2C3×#243011yescharacteristic
567C2C2×C3×#45611yescharacteristic
568C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3492C2
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
2. C3×#3056C2semidirect
3. C3×#2502C2semidirect
4. C3×#2060C2semidirect
5. C3×#2039C2semidirect
6. C3×#2028C2semidirect
7. C3×#2007C2semidirect
8. C3×#1882C2semidirect
9. C3×#1871C2semidirect
10. C3×#1859C2semidirect
11. C2×C3×#343C2semidirect
12. #25519C3direct (N × H)
49. #3056C2×C3semidirect
50. #2502C2×C3semidirect
51. #2060C2×C3semidirect
52. #2039C2×C3semidirect
53. #2028C2×C3semidirect
54. #2007C2×C3semidirect
55. #1882C2×C3semidirect
56. #1871C2×C3semidirect
57. #1859C2×C3semidirect
58. C2×#343C2×C3semidirect
205. C3×M16#41semidirect
406. M16C3×#41semidirect
560. C3#25519direct (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×#30562
C3×#25022
C3×#20602
C3×#20392
C3×#20282
C3×#20072
C3×#18822
C3×#18712
C3×#18592
C2×C3×#3432
#255193