Finite Group Databasebeta

← C41×#25666 #23801867

Subgroups — C41×#25666 #23801867

Order 10496.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3511C211yesmaximal, characteristic
2C41×#3141C211yesmaximal, characteristic, complemented
3C41×#3079C211yesmaximal, characteristic, complemented
4C41×#2868C211yesmaximal, characteristic, complemented
5C41×#2039C214yesmaximal, complemented
6C41×#1997C211yesmaximal, characteristic, complemented
7C41×#1962C211yesmaximal, characteristic, complemented
8C41×#1859C214yesmaximal, complemented
9C4×#104×C41C211yesmaximal, characteristic, complemented
10C41×#556—21no—
11C41×#555—21no—
12C41×#554—21no—
13C41×#553—22no—
14C41×#548—22no—
15C41×#547—21no—
16C41×#542—22no—
17C41×#537C2212yes—
18C41×#533—21no—
19C41×#532—21no—
20C41×#532C2211yescharacteristic
21C2×C4⋊Q8×C41—22no—
22C2×C4⋊Q8×C41C2211yescharacteristic
23C2×C4⋊Q8×C41C2211yescharacteristic
24C41×#520C2212yes—
25C4×#41×C41C2211yescharacteristic
26C41×#489—22no—
27C41×#487—21no—
28C41×#486—21no—
29C41×#477—21no—
30C41×#476—21no—
31C41×#475—21no—
32C41×#474—21no—
33C41×#428C2212yes—
34C41×#423C2211yescharacteristic
35C2×#104×C41C2212yes—
36C2×#104×C41C2214yes—
37C41×#417C2212yes—
38C41×#417C2214yes—
39C4×M16×C41C2211yescharacteristic
40C41×#384—22no—
41C41×#384C2214yes—
42C43×C41C2211yescharacteristic
43C41×#343C2214yes—
44C41×#339C2212yes—
45C2×#41×C41—22no—
46C2×#41×C41C2311yescharacteristic
47C2×#41×C41C2312yes—
48C22×Q8×C41—21no—
49C22×Q8×C41D412yes—
50C2×M16×C41C2312yes—
51C2×M16×C41C2312yes—
52C4⋊Q8×C41—21no—
53C4⋊Q8×C41—21no—
54C4⋊Q8×C41—21no—
55C4⋊Q8×C41—21no—
56C4⋊Q8×C41—21no—
57C4⋊Q8×C41—21no—
58C4⋊Q8×C41—21no—
59C4⋊Q8×C41—21no—
60C4⋊Q8×C41—48no—
61C4⋊Q8×C41D411yescharacteristic
62#127×C41D411yescharacteristic
63#126×C41—24no—
64#125×C41—22no—
65#124×C41—22no—
66#124×C41—24no—
67#124×C41D412yes—
68#123×C41—24no—
69C22⋊Q8×C41—22no—
70C22⋊Q8×C41—24no—
71C22⋊Q8×C41—22no—
72C22⋊Q8×C41—22no—
73#121×C41—22no—
74#121×C41—22no—
75#121×C41—22no—
76#121×C41—21no—
77#121×C41—21no—
78#121×C41—24no—
79C4×Q8×C41—22no—
80C4×Q8×C41—22no—
81C4×Q8×C41D411yescharacteristic
82C4×Q8×C41D411yescharacteristic
83C4×D4×C41—22no—
84C4×D4×C41—22no—
85C4×D4×C41—21no—
86C4×D4×C41—21no—
87C4×D4×C41—24no—
88C4×D4×C41—21no—
89C4×D4×C41—21no—
90C4×D4×C41D411yescharacteristic
91C4×D4×C41D411yescharacteristic
92#117×C41—22no—
93#117×C41—22no—
94#117×C41—21no—
95#117×C41C2311yescharacteristic
96C2×C4⋊C4×C41—22no—
97C2×C4⋊C4×C41—28no—
98C2×C4⋊C4×C41—21no—
99C2×C4⋊C4×C41D412yes—
100C2×C4⋊C4×C41C2312yes—
101C2×C4⋊C4×C41C2312yes—
102C2×C42×C41—21no—
103C2×C42×C41—21no—
104C2×C42×C41—21no—
105C2×C42×C41C2312yes—
106C2×C42×C41C2311yescharacteristic
107#107×C41—22no—
108#106×C41—22no—
109C4⋊C8×C41—21no—
110C4⋊C8×C41—21no—
111#104×C41—24no—
112#104×C41—28no—
113#103×C41—22no—
114#103×C41—24no—
115#102×C41—22no—
116#102×C41—24no—
117#97×C41—21no—
118C4×C8×C41—21no—
119#41×C41—24no—
120#41×C41—22no—
121#41×C41—24no—
122#41×C41—22no—
123#41×C41—84no—
124C2×Q8×C41—22no—
125C2×Q8×C41—22no—
126C2×Q8×C41—44no—
127C2×Q8×C41—44no—
128C2×Q8×C41—41no—
129C2×Q8×C41—41no—
130C2×Q8×C41C2×D412yes—
131C2×Q8×C41C2×D411yescharacteristic
132C2×D4×C41—21no—
133C2×D4×C41—22no—
134C2×D4×C41—24no—
135C2×D4×C41—21no—
136C2×D4×C41—21no—
137C2×D4×C41C2×D412yes—
138C2×D4×C41C2×D411yescharacteristic
139C22×C4×C41—22no—
140C22×C4×C41—24no—
141C22×C4×C41—21no—
142C22×C4×C41—21no—
143C22×C4×C41—21no—
144C22×C4×C41—22no—
145C22×C4×C41—22no—
146C22×C4×C41C2×D411yescharacteristic
147C22×C4×C41C2×D411yescharacteristic
148C22×C4×C41C2×D414yes—
149C22×C4×C41C2411yescharacteristic
150M16×C41—24no—
151M16×C41—22no—
152M16×C41—22no—
153C2×C8×C41—22no—
154C2×C8×C41—22no—
155C4⋊C4×C41—24no—
156C4⋊C4×C41—24no—
157C4⋊C4×C41—22no—
158C4⋊C4×C41—22no—
159C4⋊C4×C41—21no—
160C4⋊C4×C41—21no—
161C4⋊C4×C41—21no—
162C4⋊C4×C41—21no—
163C4⋊C4×C41—22no—
164C4⋊C4×C41—22no—
165C4⋊C4×C41—22no—
166C4⋊C4×C41—22no—
167C4⋊C4×C41—416no—
168C4⋊C4×C41—41no—
169C4⋊C4×C41—41no—
170C4⋊C4×C41C2×D411yescharacteristic
171C4⋊C4×C41C2×D411yescharacteristic
172C22⋊C4×C41—24no—
173C22⋊C4×C41—24no—
174C22⋊C4×C41—24no—
175C22⋊C4×C41—21no—
176C22⋊C4×C41—21no—
177C22⋊C4×C41—21no—
178C22⋊C4×C41—22no—
179C22⋊C4×C41—22no—
180C42×C41—24no—
181C42×C41—24no—
182C42×C41—24no—
183C42×C41—21no—
184C42×C41—21no—
185C42×C41—21no—
186C42×C41—22no—
187C42×C41C2×D411yescharacteristic
188C42×C41C2×D411yescharacteristic
189C42×C41C2×D411yescharacteristic
190C42×C41C2×D411yescharacteristic
191C42×C41#4114yescomplemented
192C23×C41—21no—
193C23×C41—22no—
194C23×C41C22×D411yescharacteristic
195Q8×C41—21no—
196Q8×C41—21no—
197Q8×C41—24no—
198Q8×C41—44no—
199Q8×C41—82no—
200Q8×C41—82no—
201Q8×C41—82no—
202D4×C41—21no—
203D4×C41—21no—
204D4×C41—24no—
205D4×C41—44no—
206D4×C41—42no—
207D4×C41—42no—
208D4×C41—42no—
209D4×C41—84no—
210C2×C4×C41—24no—
211C2×C4×C41—24no—
212C2×C4×C41—24no—
213C2×C4×C41—22no—
214C2×C4×C41—22no—
215C2×C4×C41—21no—
216C2×C4×C41—22no—
217C2×C4×C41—21no—
218C2×C4×C41—21no—
219C2×C4×C41—24no—
220C2×C4×C41—42no—
221C2×C4×C41—42no—
222C2×C4×C41—42no—
223C2×C4×C41—42no—
224C2×C4×C41—42no—
225C2×C4×C41—44no—
226C2×C4×C41—44no—
227C2×C4×C41—42no—
228C2×C4×C41—84no—
229C2×C4×C41#12011yescharacteristic
230C2×C4×C41#12011yescharacteristic
231C2×C4×C41#12011yescharacteristic
232C2×C4×C41#12011yescharacteristic
233C2×C4×C41C22×D411yescharacteristic
234C2×C4×C41C22×D411yescharacteristic
235C2×C4×C41C2×#4112yes—
236C2×C4×C412−1+412yes—
237C8×C41—42no—
238C8×C41—42no—
239#25666C4111yesmaximal, characteristic, direct product factor, Sylow
240C22×C41—21no—
241C22×C41—21no—
242C22×C41—42no—
243C22×C41—41no—
244C22×C41—41no—
245C22×C41—42no—
246C22×C41—41no—
247C22×C41—42no—
248C22×C41C2×#12011yescharacteristic
249C22×C41#54812yes—
250C4×C41—28no—
251C4×C41—21no—
252C4×C41—21no—
253C4×C41—42no—
254C4×C41—41no—
255C4×C41—41no—
256C4×C41—44no—
257C4×C41—41no—
258C4×C41—82no—
259C4×C41#45312yes—
260C4×C41#45612yes—
261C4×C41#53211yescharacteristic
262C4×C41#53611yescharacteristic
263C4×C41#54712yes—
264#3511C2×C4111yescharacteristic
265#3141C2×C4111yescharacteristic, complemented
266#3079C2×C4111yescharacteristic, complemented
267#2868C2×C4111yescharacteristic, complemented
268#2039C2×C4114yescomplemented
269#1997C2×C4111yescharacteristic, complemented
270#1962C2×C4111yescharacteristic, complemented
271#1859C2×C4114yescomplemented
272C4×#104C2×C4111yescharacteristic, complemented
273C2×C41—22no—
274C2×C41—81no—
275C2×C41—82no—
276C2×C41#243211yescharacteristic
277C2×C41C2×#45311yescharacteristic
278C2×C41C2×#45611yescharacteristic
279#556—21no—
280#555—21no—
281#554—21no—
282#553—22no—
283#548—22no—
284#547—21no—
285#542—22no—
286#537C22×C4112yes—
287#533—21no—
288#532—21no—
289#532C22×C4111yescharacteristic
290C2×C4⋊Q8—22no—
291C2×C4⋊Q8C22×C4111yescharacteristic
292C2×C4⋊Q8C22×C4111yescharacteristic
293#520C22×C4112yes—
294C4×#41C22×C4111yescharacteristic
295#489—22no—
296#487—21no—
297#486—21no—
298#477—21no—
299#476—21no—
300#475—21no—
301#474—21no—
302#428C22×C4112yes—
303#423C22×C4111yescharacteristic
304C2×#104C22×C4114yes—
305C2×#104C22×C4112yes—
306#417C22×C4112yes—
307#417C22×C4114yes—
308C4×M16C22×C4111yescharacteristic
309#384—22no—
310#384C22×C4114yes—
311C43C22×C4111yescharacteristic
312#343C22×C4114yes—
313#339C22×C4112yes—
314C41#2566611yescharacteristic, direct product factor, Sylow
315C2×#41—22no—
316C2×#41C23×C4111yescharacteristic
317C2×#41C23×C4112yes—
318C22×Q8—21no—
319C22×Q8D4×C4112yes—
320C2×M16C23×C4112yes—
321C2×M16C23×C4112yes—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—48no—
331C4⋊Q8D4×C4111yescharacteristic
332#127D4×C4111yescharacteristic
333#126—24no—
334#125—22no—
335#124—22no—
336#124—24no—
337#124D4×C4112yes—
338#123—24no—
339C22⋊Q8—22no—
340C22⋊Q8—24no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343#121—22no—
344#121—22no—
345#121—22no—
346#121—21no—
347#121—21no—
348#121—24no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8D4×C4111yescharacteristic
352C4×Q8D4×C4111yescharacteristic
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—24no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4D4×C4111yescharacteristic
361C4×D4D4×C4111yescharacteristic
362#117—22no—
363#117—22no—
364#117—21no—
365#117C23×C4111yescharacteristic
366C2×C4⋊C4—22no—
367C2×C4⋊C4—28no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4D4×C4112yes—
370C2×C4⋊C4C23×C4112yes—
371C2×C4⋊C4C23×C4112yes—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42C23×C4112yes—
376C2×C42C23×C4111yescharacteristic
377#107—22no—
378#106—22no—
379C4⋊C8—21no—
380C4⋊C8—21no—
381#104—28no—
382#104—24no—
383#103—22no—
384#103—24no—
385#102—22no—
386#102—24no—
387#97—21no—
388C4×C8—21no—
389#41—24no—
390#41—22no—
391#41—24no—
392#41—22no—
393#41—84no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8C2×D4×C4112yes—
401C2×Q8C2×D4×C4111yescharacteristic
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C2×D4×C4112yes—
408C2×D4C2×D4×C4111yescharacteristic
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4C2×D4×C4111yescharacteristic
417C22×C4C2×D4×C4111yescharacteristic
418C22×C4C2×D4×C4114yes—
419C22×C4C24×C4111yescharacteristic
420M16—24no—
421M16—22no—
422M16—22no—
423C2×C8—22no—
424C2×C8—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—416no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4C2×D4×C4111yescharacteristic
441C4⋊C4C2×D4×C4111yescharacteristic
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C42—24no—
451C42—24no—
452C42—24no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—22no—
457C42C2×D4×C4111yescharacteristic
458C42C2×D4×C4111yescharacteristic
459C42C2×D4×C4111yescharacteristic
460C42C2×D4×C4111yescharacteristic
461C42#41×C4114yescomplemented
462C23—21no—
463C23—22no—
464C23C22×D4×C4111yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—24no—
468Q8—44no—
469Q8—82no—
470Q8—82no—
471Q8—82no—
472D4—21no—
473D4—21no—
474D4—24no—
475D4—44no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—84no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—84no—
499C2×C4#120×C4111yescharacteristic
500C2×C4#120×C4111yescharacteristic
501C2×C4#120×C4111yescharacteristic
502C2×C4#120×C4111yescharacteristic
503C2×C4C22×D4×C4111yescharacteristic
504C2×C4C22×D4×C4111yescharacteristic
505C2×C4C2×#41×C4112yes—
506C2×C42−1+4×C4112yes—
507C8—42no—
508C8—42no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—41no—
513C22—41no—
514C22—42no—
515C22—41no—
516C22—42no—
517C22C2×#120×C4111yescharacteristic
518C22C41×#54812yes—
519C4—28no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—41no—
524C4—41no—
525C4—44no—
526C4—41no—
527C4—82no—
528C4C41×#45312yes—
529C4C41×#45612yes—
530C4C41×#53211yescharacteristic
531C4C41×#53611yescharacteristic
532C4C41×#54712yes—
533C2—22no—
534C2—81no—
535C2—82no—
536C2C41×#243211yescharacteristic
537C2C2×C41×#45311yescharacteristic
538C2C2×C41×#45611yescharacteristic
subgroupG/H
center, socleC22×C41C2×#120
commutator, Frattini, cosocleC22×C4C24×C41
prime radicalC41×#3511C2
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
2. C41×#3141C2semidirect
3. C41×#3079C2semidirect
4. C41×#2868C2semidirect
5. C41×#2039C2semidirect
6. C41×#1997C2semidirect
7. C41×#1962C2semidirect
8. C41×#1859C2semidirect
9. C4×#104×C41C2semidirect
191. C42×C41#41semidirect
239. #25666C41direct (N × H)
265. #3141C2×C41semidirect
266. #3079C2×C41semidirect
267. #2868C2×C41semidirect
268. #2039C2×C41semidirect
269. #1997C2×C41semidirect
270. #1962C2×C41semidirect
271. #1859C2×C41semidirect
272. C4×#104C2×C41semidirect
314. C41#25666direct (N × H)
461. C42#41×C41semidirect

Maximal subgroups

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

subgroupindex
C41×#35112
C41×#31412
C41×#30792
C41×#28682
C41×#20392
C41×#19972
C41×#19622
C41×#18592
C4×#104×C412
#2566641