Finite Group Databasebeta

← C41×#18243 #23794444

Subgroups — C41×#18243 #23794444

Order 10496.

928 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3561C211yesmaximal, characteristic
2C41×#3117C211yesmaximal, characteristic, complemented
3C41×#3104C211yesmaximal, characteristic, complemented
4C41×#2632C211yesmaximal, characteristic, complemented
5C41×#1618C212yesmaximal, complemented
6C41×#1614C212yesmaximal, complemented
7C41×#1606C212yesmaximal, complemented
8C41×#1602C212yesmaximal, complemented
9C41×#1521C211yesmaximal, characteristic, complemented
10C41×#1515C211yesmaximal, characteristic, complemented
11C41×#1469C211yesmaximal, characteristic, complemented
12C41×#563—21no—
13C41×#562—21no—
14C41×#556—21no—
15C41×#551—22no—
16D4×Q8×C41—21no—
17C41×#548—21no—
18C41×#546—21no—
19C41×#543C2212yes—
20C41×#542—21no—
21C41×#542—22no—
22C41×#538C2212yes—
23C41×#536—21no—
24C41×#533C2211yescharacteristic
25C41×#529C2211yescharacteristic
26C2×#124×C41C2211yescharacteristic
27C2×C4×Q8×C41C2211yescharacteristic
28C41×#484C2212yes—
29C41×#484C2211yescharacteristic
30C41×#481C2211yescharacteristic
31C41×#479—21no—
32C41×#477—21no—
33C41×#464—21no—
34C41×#462—21no—
35C41×#451C2212yes—
36C41×#451C2211yescharacteristic
37C41×#448C2211yescharacteristic
38C41×#441—21no—
39C41×#440—21no—
40C4×Q16×C41—22no—
41C41×#432C2211yescharacteristic
42C41×#431C2211yescharacteristic
43C41×#399C2212yes—
44C41×#395—21no—
45C41×#394—21no—
46C41×#393C2212yes—
47C41×#388—22no—
48C41×#383C2211yescharacteristic
49C41×#328C2212yes—
50C41×#328C2212yes—
51C41×#328C2212yes—
52C41×#327C2212yes—
53C41×#323C2212yes—
54C41×#323C2212yes—
55C2×#41×C41—21no—
56C22×Q8×C41—21no—
57C22×Q8×C41D411yescharacteristic
58C22×Q8×C41C2311yescharacteristic
59C22×D4×C41D411yescharacteristic
60C2×Q16×C41—22no—
61C2×Q16×C41—21no—
62C2×SD16×C41—21no—
63C4⋊Q8×C41—21no—
64C4⋊Q8×C41—22no—
65C4⋊Q8×C41D411yescharacteristic
66C4⋊Q8×C41D411yescharacteristic
67#126×C41—22no—
68#126×C41—22no—
69#126×C41—22no—
70#126×C41—21no—
71#125×C41—21no—
72#125×C41D411yescharacteristic
73#124×C41—21no—
74#124×C41—22no—
75#124×C41—22no—
76#124×C41—21no—
77#124×C41—21no—
78#124×C41—21no—
79#124×C41—21no—
80#124×C41D411yescharacteristic
81#123×C41—22no—
82#123×C41—22no—
83#123×C41—22no—
84#123×C41—21no—
85#123×C41—22no—
86C22⋊Q8×C41—22no—
87C22⋊Q8×C41—21no—
88C22⋊Q8×C41—22no—
89C22⋊Q8×C41—22no—
90C22⋊Q8×C41—22no—
91C22⋊Q8×C41—21no—
92C22⋊Q8×C41—22no—
93C22⋊Q8×C41—22no—
94C22⋊Q8×C41—22no—
95C22⋊Q8×C41C2311yescharacteristic
96C22⋊Q8×C41C2312yes—
97#121×C41—21no—
98#121×C41—22no—
99#121×C41—22no—
100#121×C41C2311yescharacteristic
101#120×C41—22no—
102C4×Q8×C41—21no—
103C4×Q8×C41—21no—
104C4×Q8×C41—21no—
105C4×Q8×C41—21no—
106C4×Q8×C41—21no—
107C4×Q8×C41—21no—
108C4×Q8×C41D411yescharacteristic
109C4×Q8×C41D412yes—
110C4×D4×C41—21no—
111C4×D4×C41—22no—
112C4×D4×C41—21no—
113C4×D4×C41—22no—
114C4×D4×C41D411yescharacteristic
115#117×C41—22no—
116#117×C41—21no—
117#117×C41—21no—
118C2×C4⋊C4×C41—21no—
119C2×C4⋊C4×C41—22no—
120C2×C4⋊C4×C41C2311yescharacteristic
121C2×C22⋊C4×C41—22no—
122C2×C22⋊C4×C41—22no—
123C2×C22⋊C4×C41—22no—
124C2×C22⋊C4×C41D412yes—
125C2×C22⋊C4×C41C2312yes—
126C2×C42×C41—21no—
127C2×C42×C41C2311yescharacteristic
128#107×C41—22no—
129#107×C41—21no—
130#106×C41—21no—
131C4⋊C8×C41—21no—
132C4⋊C8×C41—21no—
133#103×C41—22no—
134#103×C41—21no—
135#103×C41—22no—
136#103×C41—21no—
137#103×C41—21no—
138#102×C41—21no—
139C22⋊C8×C41C2312yes—
140C22⋊C8×C41C2311yescharacteristic
141C22⋊C8×C41C2311yescharacteristic
142#97×C41—21no—
143C4×C8×C41—21no—
144#95×C41—21no—
145#95×C41—21no—
146#95×C41—22no—
147#95×C41C2312yes—
148C24×C41C2×D412yes—
149#41×C41—82no—
150C2×Q8×C41—22no—
151C2×Q8×C41—21no—
152C2×Q8×C41—21no—
153C2×Q8×C41—21no—
154C2×Q8×C41—21no—
155C2×Q8×C41—21no—
156C2×Q8×C41—42no—
157C2×Q8×C41—81no—
158C2×Q8×C41—82no—
159C2×Q8×C41C2×D411yescharacteristic
160C2×Q8×C41C2×D412yes—
161C2×D4×C41—21no—
162C2×D4×C41—22no—
163C2×D4×C41—22no—
164C2×D4×C41—21no—
165C2×D4×C41—21no—
166C2×D4×C41—81no—
167C2×D4×C41C2×D411yescharacteristic
168C22×C4×C41—21no—
169C22×C4×C41—22no—
170C22×C4×C41—22no—
171C22×C4×C41—22no—
172C22×C4×C41—21no—
173C22×C4×C41—22no—
174C22×C4×C41—21no—
175C22×C4×C41C2×D411yescharacteristic
176C22×C4×C41C2×D412yes—
177C22×C4×C41C2×D411yescharacteristic
178C22×C4×C41C2411yescharacteristic
179Q16×C41—42no—
180Q16×C41—44no—
181SD16×C41—42no—
182C2×C8×C41—22no—
183C2×C8×C41—21no—
184C2×C8×C41—21no—
185C4⋊C4×C41—22no—
186C4⋊C4×C41—22no—
187C4⋊C4×C41—22no—
188C4⋊C4×C41—22no—
189C4⋊C4×C41—21no—
190C4⋊C4×C41—21no—
191C4⋊C4×C41—21no—
192C4⋊C4×C41—21no—
193C4⋊C4×C41—21no—
194C4⋊C4×C41—21no—
195C4⋊C4×C41—22no—
196C4⋊C4×C41—21no—
197C4⋊C4×C41—22no—
198C4⋊C4×C41—21no—
199C4⋊C4×C41—21no—
200C4⋊C4×C41—41no—
201C4⋊C4×C41—41no—
202C4⋊C4×C41—82no—
203C4⋊C4×C41C2×D411yescharacteristic
204C4⋊C4×C41C2×D411yescharacteristic
205C4⋊C4×C41C2×D412yes—
206C22⋊C4×C41—22no—
207C22⋊C4×C41—22no—
208C22⋊C4×C41—22no—
209C22⋊C4×C41—22no—
210C22⋊C4×C41—22no—
211C22⋊C4×C41—21no—
212C22⋊C4×C41—22no—
213C22⋊C4×C41—22no—
214C22⋊C4×C41—22no—
215C22⋊C4×C41—21no—
216C22⋊C4×C41—21no—
217C22⋊C4×C41—21no—
218C22⋊C4×C41—22no—
219C22⋊C4×C41—22no—
220C22⋊C4×C41—21no—
221C22⋊C4×C41—44no—
222C22⋊C4×C41—82no—
223C42×C41—22no—
224C42×C41—22no—
225C42×C41—21no—
226C42×C41—41no—
227C42×C41—41no—
228C42×C41C2×D411yescharacteristic
229C42×C41C2×D411yescharacteristic
230C42×C41C2×D411yescharacteristic
231C42×C41C2×D411yescharacteristic
232C23×C41—22no—
233C23×C41—21no—
234C23×C41—24no—
235C23×C41—42no—
236C23×C41—42no—
237C23×C41C22×D411yescharacteristic
238Q8×C41—22no—
239Q8×C41—22no—
240Q8×C41—41no—
241Q8×C41—42no—
242Q8×C41—41no—
243Q8×C41—41no—
244Q8×C41—42no—
245Q8×C41—42no—
246Q8×C41—81no—
247Q8×C41—82no—
248D4×C41—41no—
249D4×C41—42no—
250D4×C41—42no—
251D4×C41—81no—
252D4×C41—82no—
253D4×C41—82no—
254C2×C4×C41—22no—
255C2×C4×C41—22no—
256C2×C4×C41—21no—
257C2×C4×C41—22no—
258C2×C4×C41—21no—
259C2×C4×C41—22no—
260C2×C4×C41—22no—
261C2×C4×C41—21no—
262C2×C4×C41—41no—
263C2×C4×C41—41no—
264C2×C4×C41—42no—
265C2×C4×C41—41no—
266C2×C4×C41—41no—
267C2×C4×C41—42no—
268C2×C4×C41—42no—
269C2×C4×C41—42no—
270C2×C4×C41—42no—
271C2×C4×C41—41no—
272C2×C4×C41—42no—
273C2×C4×C41—82no—
274C2×C4×C41—82no—
275C2×C4×C41—82no—
276C2×C4×C41#12011yescharacteristic
277C2×C4×C41#12011yescharacteristic
278C2×C4×C41#12011yescharacteristic
279C2×C4×C41#12011yescharacteristic
280C2×C4×C41C22×D411yescharacteristic
281C2×C4×C41C22×D411yescharacteristic
282C8×C41—41no—
283C8×C41—41no—
284C8×C41—42no—
285#18243C4111yesmaximal, characteristic, direct product factor, Sylow
286C22×C41—21no—
287C22×C41—21no—
288C22×C41—41no—
289C22×C41—42no—
290C22×C41—41no—
291C22×C41—42no—
292C22×C41—42no—
293C22×C41—41no—
294C22×C41—84no—
295C22×C41C2×#12011yescharacteristic
296C22×C41#57812yes—
297C4×C41—21no—
298C4×C41—21no—
299C4×C41—21no—
300C4×C41—21no—
301C4×C41—21no—
302C4×C41—21no—
303C4×C41—41no—
304C4×C41—41no—
305C4×C41—42no—
306C4×C41—42no—
307C4×C41—42no—
308C4×C41—41no—
309C4×C41—42no—
310C4×C41—82no—
311C4×C41—81no—
312#3561C2×C4111yescharacteristic
313#3117C2×C4111yescharacteristic, complemented
314#3104C2×C4111yescharacteristic, complemented
315#2632C2×C4111yescharacteristic, complemented
316#1618C2×C4112yescomplemented
317#1614C2×C4112yescomplemented
318#1606C2×C4112yescomplemented
319#1602C2×C4112yescomplemented
320#1521C2×C4111yescharacteristic, complemented
321#1515C2×C4111yescharacteristic, complemented
322#1469C2×C4111yescharacteristic, complemented
323C2×C41—22no—
324C2×C41—81no—
325C2×C41—82no—
326C2×C41#301111yescharacteristic
327C2×C41#301911yescharacteristic
328C2×C41#302511yescharacteristic
329#563—21no—
330#562—21no—
331#556—21no—
332#551—22no—
333D4×Q8—21no—
334#548—21no—
335#546—21no—
336#543C22×C4112yes—
337#542—21no—
338#542—22no—
339#538C22×C4112yes—
340#536—21no—
341#533C22×C4111yescharacteristic
342#529C22×C4111yescharacteristic
343C2×#124C22×C4111yescharacteristic
344C2×C4×Q8C22×C4111yescharacteristic
345#484C22×C4112yes—
346#484C22×C4111yescharacteristic
347#481C22×C4111yescharacteristic
348#479—21no—
349#477—21no—
350#464—21no—
351#462—21no—
352#451C22×C4112yes—
353#451C22×C4111yescharacteristic
354#448C22×C4111yescharacteristic
355#441—21no—
356#440—21no—
357C4×Q16—22no—
358#432C22×C4111yescharacteristic
359#431C22×C4111yescharacteristic
360#399C22×C4112yes—
361#395—21no—
362#394—21no—
363#393C22×C4112yes—
364#388—22no—
365#383C22×C4111yescharacteristic
366#328C22×C4112yes—
367#328C22×C4112yes—
368#328C22×C4112yes—
369#327C22×C4112yes—
370#323C22×C4112yes—
371#323C22×C4112yes—
372C41#1824311yescharacteristic, direct product factor, Sylow
373C2×#41—21no—
374C22×Q8—21no—
375C22×Q8D4×C4111yescharacteristic
376C22×Q8C23×C4111yescharacteristic
377C22×D4D4×C4111yescharacteristic
378C2×Q16—22no—
379C2×Q16—21no—
380C2×SD16—21no—
381C4⋊Q8—21no—
382C4⋊Q8—22no—
383C4⋊Q8D4×C4111yescharacteristic
384C4⋊Q8D4×C4111yescharacteristic
385#126—22no—
386#126—22no—
387#126—22no—
388#126—21no—
389#125—21no—
390#125D4×C4111yescharacteristic
391#124—21no—
392#124—22no—
393#124—22no—
394#124—21no—
395#124—21no—
396#124—21no—
397#124—21no—
398#124D4×C4111yescharacteristic
399#123—22no—
400#123—22no—
401#123—22no—
402#123—21no—
403#123—22no—
404C22⋊Q8—22no—
405C22⋊Q8—21no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—21no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8—22no—
413C22⋊Q8C23×C4111yescharacteristic
414C22⋊Q8C23×C4112yes—
415#121—21no—
416#121—22no—
417#121—22no—
418#121C23×C4111yescharacteristic
419#120—22no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8D4×C4111yescharacteristic
427C4×Q8D4×C4112yes—
428C4×D4—21no—
429C4×D4—22no—
430C4×D4—21no—
431C4×D4—22no—
432C4×D4D4×C4111yescharacteristic
433#117—22no—
434#117—21no—
435#117—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—22no—
438C2×C4⋊C4C23×C4111yescharacteristic
439C2×C22⋊C4—22no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4—22no—
442C2×C22⋊C4D4×C4112yes—
443C2×C22⋊C4C23×C4112yes—
444C2×C42—21no—
445C2×C42C23×C4111yescharacteristic
446#107—22no—
447#107—21no—
448#106—21no—
449C4⋊C8—21no—
450C4⋊C8—21no—
451#103—22no—
452#103—21no—
453#103—22no—
454#103—21no—
455#103—21no—
456#102—21no—
457C22⋊C8C23×C4112yes—
458C22⋊C8C23×C4111yescharacteristic
459C22⋊C8C23×C4111yescharacteristic
460#97—21no—
461C4×C8—21no—
462#95—21no—
463#95—21no—
464#95—22no—
465#95C23×C4112yes—
466C24C2×D4×C4112yes—
467#41—82no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—42no—
475C2×Q8—81no—
476C2×Q8—82no—
477C2×Q8C2×D4×C4111yescharacteristic
478C2×Q8C2×D4×C4112yes—
479C2×D4—21no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—81no—
485C2×D4C2×D4×C4111yescharacteristic
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4C2×D4×C4111yescharacteristic
494C22×C4C2×D4×C4112yes—
495C22×C4C2×D4×C4111yescharacteristic
496C22×C4C24×C4111yescharacteristic
497Q16—42no—
498Q16—44no—
499SD16—42no—
500C2×C8—22no—
501C2×C8—21no—
502C2×C8—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—82no—
521C4⋊C4C2×D4×C4111yescharacteristic
522C4⋊C4C2×D4×C4111yescharacteristic
523C4⋊C4C2×D4×C4112yes—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C42—22no—
542C42—22no—
543C42—21no—
544C42—41no—
545C42—41no—
546C42C2×D4×C4111yescharacteristic
547C42C2×D4×C4111yescharacteristic
548C42C2×D4×C4111yescharacteristic
549C42C2×D4×C4111yescharacteristic
550C23—22no—
551C23—21no—
552C23—24no—
553C23—42no—
554C23—42no—
555C23C22×D4×C4111yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—42no—
560Q8—41no—
561Q8—41no—
562Q8—42no—
563Q8—42no—
564Q8—81no—
565Q8—82no—
566D4—41no—
567D4—42no—
568D4—42no—
569D4—81no—
570D4—82no—
571D4—82no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4#120×C4111yescharacteristic
595C2×C4#120×C4111yescharacteristic
596C2×C4#120×C4111yescharacteristic
597C2×C4#120×C4111yescharacteristic
598C2×C4C22×D4×C4111yescharacteristic
599C2×C4C22×D4×C4111yescharacteristic
600C8—41no—
601C8—41no—
602C8—42no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—84no—
612C22C2×#120×C4111yescharacteristic
613C22C41×#57812yes—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—82no—
628C4—81no—
629C2—22no—
630C2—82no—
631C2—81no—
632C2C41×#301111yescharacteristic
633C2C41×#301911yescharacteristic
634C2C41×#302511yescharacteristic
subgroupG/H
center, socleC22×C41C2×#120
commutator, Frattini, cosocleC22×C4C24×C41
prime radicalC41×#3561C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C41 ◁ C2×C42×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#3117C2semidirect
3. C41×#3104C2semidirect
4. C41×#2632C2semidirect
5. C41×#1618C2semidirect
6. C41×#1614C2semidirect
7. C41×#1606C2semidirect
8. C41×#1602C2semidirect
9. C41×#1521C2semidirect
10. C41×#1515C2semidirect
11. C41×#1469C2semidirect
285. #18243C41direct (N × H)
313. #3117C2×C41semidirect
314. #3104C2×C41semidirect
315. #2632C2×C41semidirect
316. #1618C2×C41semidirect
317. #1614C2×C41semidirect
318. #1606C2×C41semidirect
319. #1602C2×C41semidirect
320. #1521C2×C41semidirect
321. #1515C2×C41semidirect
322. #1469C2×C41semidirect
372. C41#18243direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#35612
C41×#31172
C41×#31042
C41×#26322
C41×#16182
C41×#16142
C41×#16062
C41×#16022
C41×#15212
C41×#15152
C41×#14692
#1824341