Finite Group Databasebeta

← C41×#18177 #23794378

Subgroups — C41×#18177 #23794378

Order 10496.

950 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3491C211yesmaximal, characteristic
2C41×#3117C212yesmaximal, complemented
3C41×#2396C211yesmaximal, characteristic, complemented
4C41×#1615C212yesmaximal, complemented
5C41×#1614C211yesmaximal, characteristic, complemented
6C41×#1613C211yesmaximal, characteristic
7C41×#1603C212yesmaximal, complemented
8C41×#1602C211yesmaximal, characteristic, complemented
9C41×#1601C211yesmaximal, characteristic
10C41×#1509C212yesmaximal, complemented
11C41×#1464C211yesmaximal, characteristic, complemented
12C41×#552—22no—
13C41×#552—22no—
14C41×#551—22no—
15C41×#551—22no—
16C41×#543C2211yescharacteristic
17C41×#539—21no—
18C41×#539C2212yes—
19C41×#534C2211yescharacteristic
20C41×#529—21no—
21C41×#529C2212yes—
22C2×#123×C41—21no—
23C41×#518—21no—
24C2×C4×Q8×C41C2211yescharacteristic, complemented
25C2×#117×C41C2211yescharacteristic
26C41×#484C2212yescomplemented
27C41×#481C2212yes—
28C41×#479—22no—
29C41×#464—22no—
30C41×#451C2212yescomplemented
31C41×#448C2212yes—
32C41×#441—22no—
33C41×#440—22no—
34C41×#432C2212yescomplemented
35C41×#400C2211yescharacteristic, complemented
36C41×#399C2211yescharacteristic
37C41×#398—21no—
38C41×#396—21no—
39C41×#395C2211yescharacteristic, complemented
40C41×#393C2211yescharacteristic
41C4×C4⋊C4×C41—21no—
42C4×C4⋊C4×C41C2211yescharacteristic, complemented
43C4×C22⋊C4×C41—21no—
44C41×#328C2212yescomplemented
45C41×#328C2212yescomplemented
46C41×#327C2212yes—
47C41×#327C2212yes—
48C41×#324C2212yescomplemented
49C41×#323C2211yescharacteristic
50C41×#323C2211yescharacteristic
51C22×Q8×C41C2311yescharacteristic
52C22×D4×C41—21no—
53C23×C4×C41D411yescharacteristic
54C2×Q16×C41—22no—
55C2×SD16×C41—22no—
56#126×C41—22no—
57#126×C41—22no—
58#126×C41—22no—
59#126×C41—22no—
60#125×C41—21no—
61#125×C41—22no—
62#125×C41—22no—
63#125×C41D412yes—
64#124×C41—21no—
65#124×C41D412yes—
66#123×C41—22no—
67#123×C41—22no—
68#123×C41—22no—
69#123×C41—22no—
70#123×C41—22no—
71#123×C41—22no—
72#123×C41—22no—
73C22⋊Q8×C41—22no—
74C22⋊Q8×C41—22no—
75C22⋊Q8×C41—22no—
76C22⋊Q8×C41—21no—
77C22⋊Q8×C41—22no—
78C22⋊Q8×C41C2312yes—
79#121×C41—21no—
80#121×C41C2312yes—
81#120×C41—22no—
82C4×Q8×C41—21no—
83C4×Q8×C41—22no—
84C4×Q8×C41—22no—
85C4×Q8×C41D412yes—
86C4×D4×C41—22no—
87C4×D4×C41—22no—
88C4×D4×C41—21no—
89C4×D4×C41—22no—
90C4×D4×C41—22no—
91C4×D4×C41D412yes—
92#117×C41—22no—
93#117×C41—22no—
94#117×C41—22no—
95#117×C41—22no—
96#117×C41—22no—
97C2×C4⋊C4×C41—21no—
98C2×C4⋊C4×C41—21no—
99C2×C4⋊C4×C41—21no—
100C2×C4⋊C4×C41—21no—
101C2×C4⋊C4×C41—21no—
102C2×C4⋊C4×C41—21no—
103C2×C4⋊C4×C41D411yescharacteristic
104C2×C4⋊C4×C41C2311yescharacteristic
105C2×C4⋊C4×C41C2311yescharacteristic
106C2×C22⋊C4×C41—22no—
107C2×C22⋊C4×C41—21no—
108C2×C22⋊C4×C41—21no—
109C2×C22⋊C4×C41—21no—
110C2×C22⋊C4×C41—21no—
111C2×C22⋊C4×C41—21no—
112C2×C22⋊C4×C41D411yescharacteristic
113C2×C22⋊C4×C41C2311yescharacteristic
114C2×C42×C41—21no—
115C2×C42×C41—21no—
116C2×C42×C41D411yescharacteristic
117C2×C42×C41C2311yescharacteristic
118#107×C41—22no—
119#106×C41—22no—
120C4⋊C8×C41—22no—
121#103×C41—22no—
122#103×C41—22no—
123#103×C41—22no—
124#102×C41—22no—
125C22⋊C8×C41C2312yes—
126C22⋊C8×C41C2312yes—
127#97×C41—22no—
128#95×C41—21no—
129#95×C41—21no—
130#95×C41—21no—
131#95×C41—21no—
132#95×C41C2311yescharacteristic
133#95×C41C2311yescharacteristic
134C24×C41—21no—
135C24×C41C2×D411yescharacteristic
136C2×Q8×C41—21no—
137C2×Q8×C41—22no—
138C2×Q8×C41—42no—
139C2×Q8×C41C2×D412yes—
140C2×D4×C41—22no—
141C2×D4×C41—22no—
142C2×D4×C41—21no—
143C2×D4×C41—42no—
144C2×D4×C41—42no—
145C2×D4×C41C2×D412yes—
146C22×C4×C41—21no—
147C22×C4×C41—22no—
148C22×C4×C41—21no—
149C22×C4×C41—21no—
150C22×C4×C41—21no—
151C22×C4×C41—21no—
152C22×C4×C41—22no—
153C22×C4×C41—21no—
154C22×C4×C41—21no—
155C22×C4×C41—22no—
156C22×C4×C41—21no—
157C22×C4×C41—21no—
158C22×C4×C41—41no—
159C22×C4×C41C2×D411yescharacteristic
160C22×C4×C41C2×D411yescharacteristic
161C22×C4×C41C2×D411yescharacteristic
162C22×C4×C41C2×D411yescharacteristic
163C22×C4×C41C2×D411yescharacteristic
164C22×C4×C41C2411yescharacteristic
165Q16×C41—44no—
166SD16×C41—44no—
167C2×C8×C41—22no—
168C2×C8×C41—22no—
169C4⋊C4×C41—22no—
170C4⋊C4×C41—22no—
171C4⋊C4×C41—22no—
172C4⋊C4×C41—22no—
173C4⋊C4×C41—21no—
174C4⋊C4×C41—21no—
175C4⋊C4×C41—22no—
176C4⋊C4×C41—22no—
177C4⋊C4×C41—22no—
178C4⋊C4×C41—22no—
179C4⋊C4×C41—22no—
180C4⋊C4×C41—22no—
181C4⋊C4×C41—22no—
182C4⋊C4×C41—42no—
183C4⋊C4×C41—42no—
184C4⋊C4×C41—42no—
185C4⋊C4×C41—42no—
186C4⋊C4×C41C2×D412yes—
187C4⋊C4×C41C2×D412yes—
188C22⋊C4×C41—22no—
189C22⋊C4×C41—22no—
190C22⋊C4×C41—22no—
191C22⋊C4×C41—22no—
192C22⋊C4×C41—22no—
193C22⋊C4×C41—22no—
194C22⋊C4×C41—22no—
195C22⋊C4×C41—22no—
196C22⋊C4×C41—22no—
197C22⋊C4×C41—22no—
198C22⋊C4×C41—22no—
199C22⋊C4×C41—22no—
200C22⋊C4×C41—42no—
201C22⋊C4×C41—42no—
202C42×C41—22no—
203C42×C41—21no—
204C42×C41—21no—
205C42×C41—22no—
206C42×C41—42no—
207C42×C41—41no—
208C42×C41—41no—
209C42×C41C2×D412yes—
210C42×C41C2×D412yes—
211C23×C41—21no—
212C23×C41—21no—
213C23×C41—22no—
214C23×C41—41no—
215C23×C41—41no—
216C23×C41—42no—
217C23×C41—81no—
218C23×C41—81no—
219C23×C41#13712yes—
220C23×C41C22×D411yescharacteristic
221Q8×C41—24no—
222Q8×C41—42no—
223Q8×C41—42no—
224Q8×C41—81no—
225D4×C41—42no—
226D4×C41—44no—
227D4×C41—42no—
228D4×C41—42no—
229D4×C41—81no—
230C2×C4×C41—21no—
231C2×C4×C41—21no—
232C2×C4×C41—22no—
233C2×C4×C41—22no—
234C2×C4×C41—21no—
235C2×C4×C41—21no—
236C2×C4×C41—22no—
237C2×C4×C41—22no—
238C2×C4×C41—22no—
239C2×C4×C41—21no—
240C2×C4×C41—21no—
241C2×C4×C41—21no—
242C2×C4×C41—41no—
243C2×C4×C41—41no—
244C2×C4×C41—42no—
245C2×C4×C41—42no—
246C2×C4×C41—41no—
247C2×C4×C41—42no—
248C2×C4×C41—41no—
249C2×C4×C41—41no—
250C2×C4×C41—41no—
251C2×C4×C41—41no—
252C2×C4×C41—41no—
253C2×C4×C41—41no—
254C2×C4×C41—41no—
255C2×C4×C41—41no—
256C2×C4×C41—41no—
257C2×C4×C41—41no—
258C2×C4×C41—41no—
259C2×C4×C41—82no—
260C2×C4×C41—81no—
261C2×C4×C41—82no—
262C2×C4×C41—81no—
263C2×C4×C41#12011yescharacteristic
264C2×C4×C41#12011yescharacteristic
265C2×C4×C41#12012yes—
266C2×C4×C41C22×D412yes—
267C8×C41—42no—
268C8×C41—42no—
269#18177C4111yesmaximal, characteristic, direct product factor, Sylow
270C22×C41—21no—
271C22×C41—21no—
272C22×C41—22no—
273C22×C41—42no—
274C22×C41—41no—
275C22×C41—41no—
276C22×C41—42no—
277C22×C41—42no—
278C22×C41—42no—
279C22×C41—81no—
280C22×C41—162no—
281C22×C41C2×#12011yescharacteristic
282C22×C41C2×#13711yescharacteristic
283C22×C41#57511yescharacteristic
284C4×C41—21no—
285C4×C41—21no—
286C4×C41—22no—
287C4×C41—22no—
288C4×C41—42no—
289C4×C41—41no—
290C4×C41—42no—
291C4×C41—41no—
292C4×C41—42no—
293C4×C41—42no—
294C4×C41—41no—
295C4×C41—42no—
296C4×C41—81no—
297C4×C41—81no—
298#3491C2×C4111yescharacteristic
299#3117C2×C4112yescomplemented
300#2396C2×C4111yescharacteristic, complemented
301#1615C2×C4112yescomplemented
302#1614C2×C4111yescharacteristic, complemented
303#1613C2×C4111yescharacteristic
304#1603C2×C4112yescomplemented
305#1602C2×C4111yescharacteristic, complemented
306#1601C2×C4111yescharacteristic
307#1509C2×C4112yescomplemented
308#1464C2×C4111yescharacteristic, complemented
309C2×C41—21no—
310C2×C41—21no—
311C2×C41—42no—
312C2×C41—82no—
313C2×C41#300711yescharacteristic
314C2×C41#301811yescharacteristic
315C2×C41#302511yescharacteristic
316#552—22no—
317#552—22no—
318#551—22no—
319#551—22no—
320#543C22×C4111yescharacteristic
321#539—21no—
322#539C22×C4112yes—
323#534C22×C4111yescharacteristic
324#529—21no—
325#529C22×C4112yes—
326C2×#123—21no—
327#518—21no—
328C2×C4×Q8C22×C4111yescharacteristic, complemented
329C2×#117C22×C4111yescharacteristic
330#484C22×C4112yescomplemented
331#481C22×C4112yes—
332#479—22no—
333#464—22no—
334#451C22×C4112yescomplemented
335#448C22×C4112yes—
336#441—22no—
337#440—22no—
338#432C22×C4112yescomplemented
339#400C22×C4111yescharacteristic, complemented
340#399C22×C4111yescharacteristic
341#398—21no—
342#396—21no—
343#395C22×C4111yescharacteristic, complemented
344#393C22×C4111yescharacteristic
345C4×C4⋊C4—21no—
346C4×C4⋊C4C22×C4111yescharacteristic, complemented
347C4×C22⋊C4—21no—
348#328C22×C4112yescomplemented
349#328C22×C4112yescomplemented
350#327C22×C4112yes—
351#327C22×C4112yes—
352#324C22×C4112yescomplemented
353#323C22×C4111yescharacteristic
354#323C22×C4111yescharacteristic
355C41#1817711yescharacteristic, direct product factor, Sylow
356C22×Q8C23×C4111yescharacteristic
357C22×D4—21no—
358C23×C4D4×C4111yescharacteristic
359C2×Q16—22no—
360C2×SD16—22no—
361#126—22no—
362#126—22no—
363#126—22no—
364#126—22no—
365#125—21no—
366#125—22no—
367#125—22no—
368#125D4×C4112yes—
369#124—21no—
370#124D4×C4112yes—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375#123—22no—
376#123—22no—
377#123—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—21no—
382C22⋊Q8—22no—
383C22⋊Q8C23×C4112yes—
384#121—21no—
385#121C23×C4112yes—
386#120—22no—
387C4×Q8—21no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×Q8D4×C4112yes—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4D4×C4112yes—
397#117—22no—
398#117—22no—
399#117—22no—
400#117—22no—
401#117—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4D4×C4111yescharacteristic
409C2×C4⋊C4C23×C4111yescharacteristic
410C2×C4⋊C4C23×C4111yescharacteristic
411C2×C22⋊C4—22no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4D4×C4111yescharacteristic
418C2×C22⋊C4C23×C4111yescharacteristic
419C2×C42—21no—
420C2×C42—21no—
421C2×C42D4×C4111yescharacteristic
422C2×C42C23×C4111yescharacteristic
423#107—22no—
424#106—22no—
425C4⋊C8—22no—
426#103—22no—
427#103—22no—
428#103—22no—
429#102—22no—
430C22⋊C8C23×C4112yes—
431C22⋊C8C23×C4112yes—
432#97—22no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95C23×C4111yescharacteristic
438#95C23×C4111yescharacteristic
439C24—21no—
440C24C2×D4×C4111yescharacteristic
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—42no—
444C2×Q8C2×D4×C4112yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4C2×D4×C4112yes—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—41no—
464C22×C4C2×D4×C4111yescharacteristic
465C22×C4C2×D4×C4111yescharacteristic
466C22×C4C2×D4×C4111yescharacteristic
467C22×C4C2×D4×C4111yescharacteristic
468C22×C4C2×D4×C4111yescharacteristic
469C22×C4C24×C4111yescharacteristic
470Q16—44no—
471SD16—44no—
472C2×C8—22no—
473C2×C8—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4C2×D4×C4112yes—
492C4⋊C4C2×D4×C4112yes—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—22no—
511C42—42no—
512C42—41no—
513C42—41no—
514C42C2×D4×C4112yes—
515C42C2×D4×C4112yes—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—81no—
523C23—81no—
524C23#137×C4112yes—
525C23C22×D4×C4111yescharacteristic
526Q8—24no—
527Q8—42no—
528Q8—42no—
529Q8—81no—
530D4—42no—
531D4—44no—
532D4—42no—
533D4—42no—
534D4—81no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4—81no—
568C2×C4#120×C4111yescharacteristic
569C2×C4#120×C4111yescharacteristic
570C2×C4#120×C4112yes—
571C2×C4C22×D4×C4112yes—
572C8—42no—
573C8—42no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—162no—
585C22C2×#120×C4111yescharacteristic
586C22C2×#137×C4111yescharacteristic
587C22C41×#57511yescharacteristic
588C4—21no—
589C4—21no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—81no—
601C4—81no—
602C2—21no—
603C2—21no—
604C2—42no—
605C2—82no—
606C2C41×#300711yescharacteristic
607C2C41×#301811yescharacteristic
608C2C41×#302511yescharacteristic
subgroupG/H
center, socleC22×C41C2×#120
commutator, Frattini, cosocleC22×C4C24×C41
prime radicalC41×#534C22
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×#2396C2semidirect
4. C41×#1615C2semidirect
5. C41×#1614C2semidirect
7. C41×#1603C2semidirect
8. C41×#1602C2semidirect
10. C41×#1509C2semidirect
11. C41×#1464C2semidirect
24. C2×C4×Q8×C41C22semidirect
26. C41×#484C22semidirect
30. C41×#451C22semidirect
34. C41×#432C22semidirect
35. C41×#400C22semidirect
39. C41×#395C22semidirect
42. C4×C4⋊C4×C41C22semidirect
44. C41×#328C22semidirect
45. C41×#328C22semidirect
48. C41×#324C22semidirect
269. #18177C41direct (N × H)
299. #3117C2×C41semidirect
300. #2396C2×C41semidirect
301. #1615C2×C41semidirect
302. #1614C2×C41semidirect
304. #1603C2×C41semidirect
305. #1602C2×C41semidirect
307. #1509C2×C41semidirect
308. #1464C2×C41semidirect
328. C2×C4×Q8C22×C41semidirect
330. #484C22×C41semidirect
334. #451C22×C41semidirect
338. #432C22×C41semidirect
339. #400C22×C41semidirect
343. #395C22×C41semidirect
346. C4×C4⋊C4C22×C41semidirect
348. #328C22×C41semidirect
349. #328C22×C41semidirect
352. #324C22×C41semidirect
355. C41#18177direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#34912
C41×#31172
C41×#23962
C41×#16152
C41×#16142
C41×#16132
C41×#16032
C41×#16022
C41×#16012
C41×#15092
C41×#14642
#1817741