Finite Group Databasebeta

← C41×#24181 #23800382

Subgroups — C41×#24181 #23800382

Order 10496.

1050 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3507C211yesmaximal, characteristic, complemented
2C41×#3077C211yesmaximal, characteristic, complemented
3C41×#3036C211yesmaximal, characteristic
4C41×#2326C211yesmaximal, characteristic, complemented
5C41×#2038C212yesmaximal, complemented
6C41×#2003C212yesmaximal, complemented
7C41×#1995C211yesmaximal, characteristic, complemented
8C41×#1963C211yesmaximal, characteristic, complemented
9C41×#1883C212yesmaximal, complemented
10C41×#1859C212yesmaximal, complemented
11C41×#1801C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C41C2212yes—
13C41×#559—21no—
14C41×#557—21no—
15C41×#556—21no—
16C41×#556—22no—
17C41×#554—21no—
18C41×#552—22no—
19C41×#539—22no—
20C41×#536C2212yes—
21C41×#533C2211yescharacteristic
22C41×#532C2211yescharacteristic
23C41×#529—22no—
24C2×#124×C41C2211yescharacteristic
25C41×#520C2212yes—
26C2×C4×Q8×C41C2211yescharacteristic
27C2×C4×D4×C41C2211yescharacteristic
28C41×#478—22no—
29C41×#476—21no—
30C41×#474—21no—
31C41×#463—22no—
32C41×#461—21no—
33C41×#459—21no—
34C41×#428C2212yes—
35C41×#423C2211yescharacteristic
36C41×#423C2211yescharacteristic
37C2×#104×C41C2212yes—
38C2×#104×C41C2212yes—
39C41×#418C2212yes—
40C41×#417C2212yes—
41C2×#101×C41C2212yes—
42C41×#391C2211yescharacteristic
43C41×#390C2211yescharacteristic
44C41×#388—22no—
45C41×#385—22no—
46C41×#384C2211yescharacteristic
47C41×#376C2211yescharacteristic
48C41×#343C2212yes—
49C41×#339C2212yes—
50C41×#339C2212yes—
51C2×#41×C41C2312yes—
52C22×Q8×C41C2311yescharacteristic
53C22×D4×C41C2311yescharacteristic
54C23×C4×C41—21no—
55AGL(1,ℤ/8)×C41—24no—
56AGL(1,ℤ/8)×C41—42no—
57C2×SD16×C41—22no—
58C2×D8×C41—22no—
59C2×M16×C41C2312yes—
60C2×M16×C41C2312yes—
61C4⋊Q8×C41—22no—
62C4⋊Q8×C41D411yescharacteristic
63C4⋊Q8×C41D411yescharacteristic
64C4⋊Q8×C41D411yescharacteristic
65#127×C41D411yescharacteristic
66#126×C41—24no—
67#126×C41—24no—
68#125×C41—22no—
69#125×C41—24no—
70#125×C41D412yes—
71#124×C41—22no—
72#124×C41—22no—
73#124×C41—22no—
74#124×C41—22no—
75#124×C41D412yes—
76#123×C41—24no—
77#123×C41—24no—
78C22⋊Q8×C41—24no—
79C22⋊Q8×C41—21no—
80C22⋊Q8×C41—21no—
81C22⋊Q8×C41—22no—
82C22⋊Q8×C41—22no—
83#121×C41—21no—
84#121×C41—21no—
85C4×Q8×C41—22no—
86C4×Q8×C41—22no—
87C4×Q8×C41—21no—
88C4×Q8×C41—21no—
89C4×Q8×C41—21no—
90C4×Q8×C41—21no—
91C4×Q8×C41—21no—
92C4×Q8×C41—21no—
93C4×D4×C41—22no—
94C4×D4×C41—22no—
95C4×D4×C41—24no—
96C4×D4×C41—21no—
97C4×D4×C41—21no—
98C4×D4×C41—21no—
99C4×D4×C41—21no—
100C4×D4×C41—21no—
101C4×D4×C41—21no—
102#117×C41—24no—
103#117×C41—22no—
104#117×C41C2312yes—
105C2×C4⋊C4×C41—22no—
106C2×C4⋊C4×C41—22no—
107C2×C4⋊C4×C41D412yes—
108C2×C4⋊C4×C41C2311yescharacteristic
109C2×C4⋊C4×C41C2311yescharacteristic
110C2×C22⋊C4×C41—22no—
111C2×C22⋊C4×C41—21no—
112C2×C22⋊C4×C41—22no—
113C2×C42×C41D412yes—
114C2×C42×C41C2311yescharacteristic
115C2×C42×C41C2311yescharacteristic
116C2×C42×C41C2311yescharacteristic
117#107×C41—22no—
118#106×C41—22no—
119C4⋊C8×C41—21no—
120C4⋊C8×C41—21no—
121C4⋊C8×C41—21no—
122C4⋊C8×C41—21no—
123#104×C41—24no—
124#104×C41—42no—
125#103×C41—22no—
126#102×C41—22no—
127#102×C41—22no—
128#102×C41—22no—
129#101×C41—42no—
130#95×C41—21no—
131#95×C41—21no—
132#95×C41—22no—
133C24×C41—21no—
134#41×C41—22no—
135#41×C41—22no—
136#41×C41—22no—
137#41×C41—22no—
138C2×Q8×C41—21no—
139C2×Q8×C41—21no—
140C2×Q8×C41—21no—
141C2×Q8×C41—21no—
142C2×Q8×C41—21no—
143C2×Q8×C41—21no—
144C2×Q8×C41—21no—
145C2×Q8×C41—21no—
146C2×Q8×C41C2×D412yes—
147C2×D4×C41—22no—
148C2×D4×C41—21no—
149C2×D4×C41—21no—
150C2×D4×C41—21no—
151C2×D4×C41—21no—
152C2×D4×C41—21no—
153C2×D4×C41—21no—
154C2×D4×C41—21no—
155C2×D4×C41C2×D412yes—
156C2×D4×C41#4112yes—
157C22×C4×C41—21no—
158C22×C4×C41—24no—
159C22×C4×C41—22no—
160C22×C4×C41—21no—
161C22×C4×C41—21no—
162C22×C4×C41—41no—
163C22×C4×C41—41no—
164C22×C4×C41—82no—
165C22×C4×C41C2×D412yes—
166C22×C4×C41C2×D412yes—
167C22×C4×C41C2×D411yescharacteristic
168C22×C4×C41C2×D411yescharacteristic
169C22×C4×C41C2411yescharacteristic
170SD16×C41—44no—
171D8×C41—44no—
172M16×C41—22no—
173M16×C41—22no—
174M16×C41—22no—
175M16×C41—22no—
176C2×C8×C41—22no—
177C2×C8×C41—22no—
178C4⋊C4×C41—24no—
179C4⋊C4×C41—24no—
180C4⋊C4×C41—24no—
181C4⋊C4×C41—22no—
182C4⋊C4×C41—22no—
183C4⋊C4×C41—22no—
184C4⋊C4×C41—21no—
185C4⋊C4×C41—21no—
186C4⋊C4×C41—21no—
187C4⋊C4×C41—22no—
188C4⋊C4×C41—21no—
189C4⋊C4×C41—21no—
190C4⋊C4×C41—21no—
191C4⋊C4×C41—21no—
192C4⋊C4×C41—21no—
193C4⋊C4×C41—82no—
194C4⋊C4×C41C2×D412yes—
195C4⋊C4×C41C2×D412yes—
196C22⋊C4×C41—28no—
197C22⋊C4×C41—24no—
198C22⋊C4×C41—24no—
199C22⋊C4×C41—24no—
200C22⋊C4×C41—22no—
201C22⋊C4×C41—22no—
202C22⋊C4×C41—22no—
203C22⋊C4×C41—41no—
204C22⋊C4×C41—41no—
205C22⋊C4×C41—82no—
206C42×C41—24no—
207C42×C41—22no—
208C42×C41—22no—
209C42×C41—22no—
210C42×C41—21no—
211C42×C41—21no—
212C42×C41—21no—
213C42×C41C2×D411yescharacteristic
214C42×C41C2×D411yescharacteristic
215C42×C41C2×D411yescharacteristic
216C42×C41C2×D411yescharacteristic
217C42×C41#4112yes—
218C23×C41—22no—
219C23×C41—22no—
220C23×C41—42no—
221C23×C41—42no—
222C23×C41—41no—
223C23×C41—41no—
224C23×C41C22×D411yescharacteristic
225Q8×C41—22no—
226Q8×C41—22no—
227Q8×C41—42no—
228Q8×C41—41no—
229Q8×C41—41no—
230Q8×C41—41no—
231Q8×C41—41no—
232Q8×C41—81no—
233D4×C41—22no—
234D4×C41—22no—
235D4×C41—22no—
236D4×C41—22no—
237D4×C41—42no—
238D4×C41—42no—
239D4×C41—42no—
240D4×C41—41no—
241D4×C41—41no—
242D4×C41—81no—
243C2×C4×C41—22no—
244C2×C4×C41—22no—
245C2×C4×C41—22no—
246C2×C4×C41—22no—
247C2×C4×C41—22no—
248C2×C4×C41—22no—
249C2×C4×C41—22no—
250C2×C4×C41—22no—
251C2×C4×C41—41no—
252C2×C4×C41—41no—
253C2×C4×C41—41no—
254C2×C4×C41—41no—
255C2×C4×C41—42no—
256C2×C4×C41—42no—
257C2×C4×C41—42no—
258C2×C4×C41—41no—
259C2×C4×C41—42no—
260C2×C4×C41—41no—
261C2×C4×C41—41no—
262C2×C4×C41—41no—
263C2×C4×C41—82no—
264C2×C4×C41—82no—
265C2×C4×C41—84no—
266C2×C4×C41#12011yescharacteristic
267C2×C4×C41#12011yescharacteristic
268C2×C4×C41#12011yescharacteristic
269C2×C4×C41#12011yescharacteristic
270C2×C4×C41C22×D411yescharacteristic
271C2×C4×C41C22×D411yescharacteristic
272C2×C4×C41C2×#4111yescharacteristic
273C2×C4×C41C2×#4111yescharacteristic
274C2×C4×C41H(5,𝔽2)11yescharacteristic
275C2×C4×C412−1+411yescharacteristic
276C8×C41—42no—
277C8×C41—42no—
278#24181C4111yesmaximal, characteristic, direct product factor, Sylow
279C22×C41—21no—
280C22×C41—21no—
281C22×C41—44no—
282C22×C41—42no—
283C22×C41—42no—
284C22×C41—42no—
285C22×C41—42no—
286C22×C41—82no—
287C22×C41—81no—
288C22×C41—81no—
289C22×C41C2×#12011yescharacteristic
290C22×C41#52511yescharacteristic
291C22×C41#53711yescharacteristic
292C4×C41—21no—
293C4×C41—21no—
294C4×C41—21no—
295C4×C41—21no—
296C4×C41—42no—
297C4×C41—42no—
298C4×C41—42no—
299C4×C41—42no—
300C4×C41—42no—
301C4×C41—42no—
302C4×C41—81no—
303C4×C41—81no—
304C4×C41#54611yescharacteristic
305C4×C41#54711yescharacteristic
306C4×C41#54811yescharacteristic
307C4×C41#55011yescharacteristic
308#3507C2×C4111yescharacteristic, complemented
309#3077C2×C4111yescharacteristic, complemented
310#3036C2×C4111yescharacteristic
311#2326C2×C4111yescharacteristic, complemented
312#2038C2×C4112yescomplemented
313#2003C2×C4112yescomplemented
314#1995C2×C4111yescharacteristic, complemented
315#1963C2×C4111yescharacteristic, complemented
316#1883C2×C4112yescomplemented
317#1859C2×C4112yescomplemented
318#1801C2×C4111yescharacteristic, complemented
319C2×C41—21no—
320C2×C41—21no—
321C2×C41—82no—
322C2×C41—82no—
323C2×C41#240711yescharacteristic
324C2×C41#302011yescharacteristic
325C2×C41#302111yescharacteristic
326C2×AGL(1,ℤ/8)C22×C4112yes—
327#559—21no—
328#557—21no—
329#556—21no—
330#556—22no—
331#554—21no—
332#552—22no—
333#539—22no—
334#536C22×C4112yes—
335#533C22×C4111yescharacteristic
336#532C22×C4111yescharacteristic
337#529—22no—
338C2×#124C22×C4111yescharacteristic
339#520C22×C4112yes—
340C2×C4×Q8C22×C4111yescharacteristic
341C2×C4×D4C22×C4111yescharacteristic
342#478—22no—
343#476—21no—
344#474—21no—
345#463—22no—
346#461—21no—
347#459—21no—
348#428C22×C4112yes—
349#423C22×C4111yescharacteristic
350#423C22×C4111yescharacteristic
351C2×#104C22×C4112yes—
352C2×#104C22×C4112yes—
353#418C22×C4112yes—
354#417C22×C4112yes—
355C2×#101C22×C4112yes—
356#391C22×C4111yescharacteristic
357#390C22×C4111yescharacteristic
358#388—22no—
359#385—22no—
360#384C22×C4111yescharacteristic
361#376C22×C4111yescharacteristic
362#343C22×C4112yes—
363#339C22×C4112yes—
364#339C22×C4112yes—
365C41#2418111yescharacteristic, direct product factor, Sylow
366C2×#41C23×C4112yes—
367C22×Q8C23×C4111yescharacteristic
368C22×D4C23×C4111yescharacteristic
369C23×C4—21no—
370AGL(1,ℤ/8)—24no—
371AGL(1,ℤ/8)—42no—
372C2×SD16—22no—
373C2×D8—22no—
374C2×M16C23×C4112yes—
375C2×M16C23×C4112yes—
376C4⋊Q8—22no—
377C4⋊Q8D4×C4111yescharacteristic
378C4⋊Q8D4×C4111yescharacteristic
379C4⋊Q8D4×C4111yescharacteristic
380#127D4×C4111yescharacteristic
381#126—24no—
382#126—24no—
383#125—22no—
384#125—24no—
385#125D4×C4112yes—
386#124—22no—
387#124—22no—
388#124—22no—
389#124—22no—
390#124D4×C4112yes—
391#123—24no—
392#123—24no—
393C22⋊Q8—24no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398#121—21no—
399#121—21no—
400C4×Q8—22no—
401C4×Q8—22no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8—21no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—24no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417#117—24no—
418#117—22no—
419#117C23×C4112yes—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4D4×C4112yes—
423C2×C4⋊C4C23×C4111yescharacteristic
424C2×C4⋊C4C23×C4111yescharacteristic
425C2×C22⋊C4—22no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—22no—
428C2×C42D4×C4112yes—
429C2×C42C23×C4111yescharacteristic
430C2×C42C23×C4111yescharacteristic
431C2×C42C23×C4111yescharacteristic
432#107—22no—
433#106—22no—
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437C4⋊C8—21no—
438#104—24no—
439#104—42no—
440#103—22no—
441#102—22no—
442#102—22no—
443#102—22no—
444#101—42no—
445#95—21no—
446#95—21no—
447#95—22no—
448C24—21no—
449#41—22no—
450#41—22no—
451#41—22no—
452#41—22no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×D4×C4112yes—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×D4×C4112yes—
471C2×D4#41×C4112yes—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—82no—
480C22×C4C2×D4×C4112yes—
481C22×C4C2×D4×C4112yes—
482C22×C4C2×D4×C4111yescharacteristic
483C22×C4C2×D4×C4111yescharacteristic
484C22×C4C24×C4111yescharacteristic
485SD16—44no—
486D8—44no—
487M16—22no—
488M16—22no—
489M16—22no—
490M16—22no—
491C2×C8—22no—
492C2×C8—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—82no—
509C4⋊C4C2×D4×C4112yes—
510C4⋊C4C2×D4×C4112yes—
511C22⋊C4—28no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—82no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42C2×D4×C4111yescharacteristic
529C42C2×D4×C4111yescharacteristic
530C42C2×D4×C4111yescharacteristic
531C42C2×D4×C4111yescharacteristic
532C42#41×C4112yes—
533C23—22no—
534C23—22no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23C22×D4×C4111yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—42no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—81no—
548D4—22no—
549D4—22no—
550D4—22no—
551D4—22no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—41no—
556D4—41no—
557D4—81no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4#120×C4111yescharacteristic
582C2×C4#120×C4111yescharacteristic
583C2×C4#120×C4111yescharacteristic
584C2×C4#120×C4111yescharacteristic
585C2×C4C22×D4×C4111yescharacteristic
586C2×C4C22×D4×C4111yescharacteristic
587C2×C4C2×#41×C4111yescharacteristic
588C2×C4C2×#41×C4111yescharacteristic
589C2×C4H(5,𝔽2)×C4111yescharacteristic
590C2×C42−1+4×C4111yescharacteristic
591C8—42no—
592C8—42no—
593C22—21no—
594C22—21no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—82no—
601C22—81no—
602C22—81no—
603C22C2×#120×C4111yescharacteristic
604C22C41×#52511yescharacteristic
605C22C41×#53711yescharacteristic
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—81no—
617C4—81no—
618C4C41×#54611yescharacteristic
619C4C41×#54711yescharacteristic
620C4C41×#54811yescharacteristic
621C4C41×#55011yescharacteristic
622C2—21no—
623C2—21no—
624C2—82no—
625C2—82no—
626C2C41×#240711yescharacteristic
627C2C41×#302011yescharacteristic
628C2C41×#302111yescharacteristic
subgroupG/H
center, socleC22×C41C2×#120
commutator, Frattini, cosocleC22×C4C24×C41
prime radicalC41×#3036C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C2×C42×C41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C41×#3507C2semidirect
2. C41×#3077C2semidirect
4. C41×#2326C2semidirect
5. C41×#2038C2semidirect
6. C41×#2003C2semidirect
7. C41×#1995C2semidirect
8. C41×#1963C2semidirect
9. C41×#1883C2semidirect
10. C41×#1859C2semidirect
11. C41×#1801C2semidirect
278. #24181C41direct (N × H)
308. #3507C2×C41semidirect
309. #3077C2×C41semidirect
311. #2326C2×C41semidirect
312. #2038C2×C41semidirect
313. #2003C2×C41semidirect
314. #1995C2×C41semidirect
315. #1963C2×C41semidirect
316. #1883C2×C41semidirect
317. #1859C2×C41semidirect
318. #1801C2×C41semidirect
365. C41#24181direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#35072
C41×#30772
C41×#30362
C41×#23262
C41×#20382
C41×#20032
C41×#19952
C41×#19632
C41×#18832
C41×#18592
C41×#18012
#2418141