Finite Group Databasebeta

← C41×#24985 #23801186

Subgroups — C41×#24985 #23801186

Order 10496.

646 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3067C211yesmaximal, characteristic
2C4×#135×C41C211yesmaximal, characteristic, complemented
3C41×#2080C212yesmaximal, complemented
4C41×#2053C212yesmaximal, complemented
5C41×#1976C211yesmaximal, characteristic, complemented
6C41×#1945C211yesmaximal, characteristic, complemented
7C41×#1906C212yesmaximal, complemented
8C41×#1833C211yesmaximal, characteristic, complemented
9C41×#1827C212yesmaximal, complemented
10C41×#1812C211yesmaximal, characteristic, complemented
11C41×#1764C211yesmaximal, characteristic, complemented
12C41×#575C2212yes—
13C2×#135×C41C2211yescharacteristic
14C2×#131×C41C2211yescharacteristic
15C4×#41×C41C2212yes—
16C41×#473—21no—
17C41×#472—21no—
18C41×#471—22no—
19C4×Q16×C41—21no—
20C4×SD16×C41—22no—
21C4×D8×C41—21no—
22C2×#108×C41C2211yescharacteristic
23C41×#427C2211yescharacteristic
24C41×#427C2211yescharacteristic
25C41×#424C2212yes—
26C41×#421C2212yes—
27C2×#104×C41C2211yescharacteristic
28C41×#416C2211yescharacteristic
29C41×#416C2212yes—
30C41×#413C2212yes—
31C41×#410C2212yes—
32C41×#410C2212yes—
33C41×#405C2211yescharacteristic
34C2×C4×C8×C41C2211yescharacteristic
35C41×#344C2212yes—
36C41×#341C2211yescharacteristic
37C41×#339C2211yescharacteristic
38C41×#339C2212yes—
39C41×H(3,ℤ/4)C2212yes—
40C41×H(3,ℤ/4)C2212yes—
41C2×#41×C41C2312yes—
42C2×#41×C41C2311yescharacteristic
43#137×C41—22no—
44#137×C41—22no—
45AGL(1,ℤ/8)×C41—22no—
46AGL(1,ℤ/8)×C41—22no—
47#135×C41—21no—
48#135×C41—21no—
49#135×C41—21no—
50#135×C41—21no—
51#135×C41—42no—
52C2×Q16×C41D411yescharacteristic
53C2×SD16×C41D412yes—
54C2×D8×C41D411yescharacteristic
55#131×C41—21no—
56#131×C41—21no—
57#131×C41—21no—
58#131×C41—21no—
59C2×M16×C41—21no—
60C2×M16×C41C2311yescharacteristic
61C2×M16×C41C2312yes—
62C2×M16×C41C2311yescharacteristic
63C22×C8×C41—21no—
64C22×C8×C41C2311yescharacteristic
65C22×C8×C41C2311yescharacteristic
66C4×Q8×C41—22no—
67C4×D4×C41—22no—
68C4×D4×C41—42no—
69#117×C41—22no—
70#117×C41C2312yes—
71#117×C41C2311yescharacteristic
72#117×C41C2312yes—
73C2×C42×C41—22no—
74C2×C42×C41C2311yescharacteristic
75#108×C41—21no—
76#108×C41—21no—
77#107×C41—21no—
78#107×C41—21no—
79#106×C41—21no—
80#106×C41—21no—
81C4⋊C8×C41—42no—
82#104×C41—21no—
83#104×C41—21no—
84#104×C41—42no—
85#103×C41—22no—
86#103×C41—21no—
87#102×C41—22no—
88#102×C41—21no—
89#101×C41—22no—
90#100×C41—22no—
91#99×C41—42no—
92#99×C41—42no—
93#97×C41—21no—
94C4×C8×C41—21no—
95C4×C8×C41—21no—
96C4×C8×C41—21no—
97#41×C41—21no—
98#41×C41—21no—
99#41×C41—21no—
100#41×C41—22no—
101#41×C41—22no—
102#41×C41—22no—
103#41×C41—22no—
104#41×C41—21no—
105C2×Q8×C41C2×D412yes—
106C2×Q8×C41#4111yescharacteristic
107C2×D4×C41—21no—
108C2×D4×C41—22no—
109C2×D4×C41C2×D412yes—
110C2×D4×C41#4111yescharacteristic
111C22×C4×C41—21no—
112C22×C4×C41—22no—
113C22×C4×C41—21no—
114C22×C4×C41—42no—
115C22×C4×C41C2411yescharacteristic
116Q16×C41—21no—
117Q16×C41—21no—
118Q16×C41—42no—
119SD16×C41—22no—
120SD16×C41—22no—
121SD16×C41—42no—
122SD16×C41—42no—
123D8×C41—21no—
124D8×C41—21no—
125D8×C41—42no—
126M16×C41—21no—
127M16×C41—22no—
128M16×C41—21no—
129M16×C41—21no—
130M16×C41—22no—
131M16×C41—21no—
132M16×C41—41no—
133M16×C41—41no—
134C2×C8×C41—21no—
135C2×C8×C41—21no—
136C2×C8×C41—21no—
137C2×C8×C41—21no—
138C2×C8×C41—21no—
139C2×C8×C41—22no—
140C2×C8×C41—41no—
141C2×C8×C41—41no—
142C2×C8×C41C2×D411yescharacteristic
143C2×C8×C41C2×D411yescharacteristic
144C2×C8×C41#4111yescharacteristic
145C2×C8×C41#4111yescharacteristic
146C2×C8×C41#4111yescharacteristic
147C2×C8×C41#4111yescharacteristic
148C4⋊C4×C41—22no—
149C4⋊C4×C41—21no—
150C4⋊C4×C41—22no—
151C4⋊C4×C41—42no—
152C22⋊C4×C41—22no—
153C22⋊C4×C41—21no—
154C22⋊C4×C41—22no—
155C22⋊C4×C41—42no—
156C42×C41—21no—
157C42×C41—21no—
158C42×C41—22no—
159C42×C41—22no—
160C42×C41—42no—
161C42×C41#4111yescharacteristic
162C42×C41#4111yescharacteristic
163C23×C41—22no—
164C23×C41—21no—
165C23×C41H(5,𝔽2)11yescharacteristic
166Q8×C41—22no—
167Q8×C41—21no—
168Q8×C41—21no—
169Q8×C41—22no—
170D4×C41—22no—
171D4×C41—21no—
172D4×C41—21no—
173D4×C41—22no—
174D4×C41—41no—
175D4×C41—42no—
176D4×C41—41no—
177D4×C41—42no—
178C2×C4×C41—21no—
179C2×C4×C41—21no—
180C2×C4×C41—22no—
181C2×C4×C41—21no—
182C2×C4×C41—42no—
183C2×C4×C41—41no—
184C2×C4×C41—42no—
185C2×C4×C41—42no—
186C2×C4×C41—41no—
187C2×C4×C41—42no—
188C2×C4×C41—42no—
189C2×C4×C41—41no—
190C2×C4×C41—84no—
191C2×C4×C41C22×D411yescharacteristic
192C2×C4×C41C2×#4111yescharacteristic
193C2×C4×C41C2×#4111yescharacteristic
194C2×C4×C41C2×#4111yescharacteristic
195C2×C4×C41C2×#4111yescharacteristic
196C2×C4×C41H(5,𝔽2)11yescharacteristic
197C8×C41—41no—
198C8×C41—41no—
199C8×C41—42no—
200C8×C41—41no—
201C8×C41#12311yescharacteristic
202C8×C41#12311yescharacteristic
203C8×C41#12311yescharacteristic
204C8×C41#12311yescharacteristic
205#24985C4111yesmaximal, characteristic, direct product factor, Sylow
206C22×C41—41no—
207C22×C41—41no—
208C22×C41—42no—
209C22×C41—82no—
210C22×C41—81no—
211C22×C41#54011yescharacteristic
212C22×C41#54611yescharacteristic
213C22×C41#55311yescharacteristic
214C4×C41—22no—
215C4×C41—41no—
216C4×C41—42no—
217C4×C41—41no—
218C4×C41—82no—
219C4×C41—81no—
220C4×C41—82no—
221C4×C41C2×#12311yescharacteristic
222C4×C41#52911yescharacteristic
223C4×C41#54611yescharacteristic
224C4×C41#55511yescharacteristic
225#3067C2×C4111yescharacteristic
226C4×#135C2×C4111yescharacteristic, complemented
227#2080C2×C4112yescomplemented
228#2053C2×C4112yescomplemented
229#1976C2×C4111yescharacteristic, complemented
230#1945C2×C4111yescharacteristic, complemented
231#1906C2×C4112yescomplemented
232#1833C2×C4111yescharacteristic, complemented
233#1827C2×C4112yescomplemented
234#1812C2×C4111yescharacteristic, complemented
235#1764C2×C4111yescharacteristic, complemented
236C2×C41—21no—
237C2×C41—21no—
238C2×C41—21no—
239C2×C41—82no—
240C2×C41—81no—
241C2×C41#242011yescharacteristic
242#575C22×C4112yes—
243C2×#135C22×C4111yescharacteristic
244C2×#131C22×C4111yescharacteristic
245C4×#41C22×C4112yes—
246#473—21no—
247#472—21no—
248#471—22no—
249C4×Q16—21no—
250C4×SD16—22no—
251C4×D8—21no—
252C2×#108C22×C4111yescharacteristic
253#427C22×C4111yescharacteristic
254#427C22×C4111yescharacteristic
255#424C22×C4112yes—
256#421C22×C4112yes—
257C2×#104C22×C4111yescharacteristic
258#416C22×C4111yescharacteristic
259#416C22×C4112yes—
260#413C22×C4112yes—
261#410C22×C4112yes—
262#410C22×C4112yes—
263#405C22×C4111yescharacteristic
264C2×C4×C8C22×C4111yescharacteristic
265#344C22×C4112yes—
266#341C22×C4111yescharacteristic
267#339C22×C4111yescharacteristic
268#339C22×C4112yes—
269H(3,ℤ/4)C22×C4112yes—
270H(3,ℤ/4)C22×C4112yes—
271C41#2498511yescharacteristic, direct product factor, Sylow
272C2×#41C23×C4112yes—
273C2×#41C23×C4111yescharacteristic
274#137—22no—
275#137—22no—
276AGL(1,ℤ/8)—22no—
277AGL(1,ℤ/8)—22no—
278#135—21no—
279#135—21no—
280#135—21no—
281#135—21no—
282#135—42no—
283C2×Q16D4×C4111yescharacteristic
284C2×SD16D4×C4112yes—
285C2×D8D4×C4111yescharacteristic
286#131—21no—
287#131—21no—
288#131—21no—
289#131—21no—
290C2×M16—21no—
291C2×M16C23×C4111yescharacteristic
292C2×M16C23×C4112yes—
293C2×M16C23×C4111yescharacteristic
294C22×C8—21no—
295C22×C8C23×C4111yescharacteristic
296C22×C8C23×C4111yescharacteristic
297C4×Q8—22no—
298C4×D4—22no—
299C4×D4—42no—
300#117—22no—
301#117C23×C4112yes—
302#117C23×C4111yescharacteristic
303#117C23×C4112yes—
304C2×C42—22no—
305C2×C42C23×C4111yescharacteristic
306#108—21no—
307#108—21no—
308#107—21no—
309#107—21no—
310#106—21no—
311#106—21no—
312C4⋊C8—42no—
313#104—21no—
314#104—21no—
315#104—42no—
316#103—22no—
317#103—21no—
318#102—22no—
319#102—21no—
320#101—22no—
321#100—22no—
322#99—42no—
323#99—42no—
324#97—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—21no—
336C2×Q8C2×D4×C4112yes—
337C2×Q8#41×C4111yescharacteristic
338C2×D4—21no—
339C2×D4—22no—
340C2×D4C2×D4×C4112yes—
341C2×D4#41×C4111yescharacteristic
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—42no—
346C22×C4C24×C4111yescharacteristic
347Q16—21no—
348Q16—21no—
349Q16—42no—
350SD16—22no—
351SD16—22no—
352SD16—42no—
353SD16—42no—
354D8—21no—
355D8—21no—
356D8—42no—
357M16—21no—
358M16—22no—
359M16—21no—
360M16—21no—
361M16—22no—
362M16—21no—
363M16—41no—
364M16—41no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8C2×D4×C4111yescharacteristic
374C2×C8C2×D4×C4111yescharacteristic
375C2×C8#41×C4111yescharacteristic
376C2×C8#41×C4111yescharacteristic
377C2×C8#41×C4111yescharacteristic
378C2×C8#41×C4111yescharacteristic
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C22⋊C4—22no—
384C22⋊C4—21no—
385C22⋊C4—22no—
386C22⋊C4—42no—
387C42—21no—
388C42—21no—
389C42—22no—
390C42—22no—
391C42—42no—
392C42#41×C4111yescharacteristic
393C42#41×C4111yescharacteristic
394C23—22no—
395C23—21no—
396C23H(5,𝔽2)×C4111yescharacteristic
397Q8—22no—
398Q8—21no—
399Q8—21no—
400Q8—22no—
401D4—22no—
402D4—21no—
403D4—21no—
404D4—22no—
405D4—41no—
406D4—42no—
407D4—41no—
408D4—42no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—84no—
422C2×C4C22×D4×C4111yescharacteristic
423C2×C4C2×#41×C4111yescharacteristic
424C2×C4C2×#41×C4111yescharacteristic
425C2×C4C2×#41×C4111yescharacteristic
426C2×C4C2×#41×C4111yescharacteristic
427C2×C4H(5,𝔽2)×C4111yescharacteristic
428C8—41no—
429C8—41no—
430C8—42no—
431C8—41no—
432C8#123×C4111yescharacteristic
433C8#123×C4111yescharacteristic
434C8#123×C4111yescharacteristic
435C8#123×C4111yescharacteristic
436C22—41no—
437C22—41no—
438C22—42no—
439C22—82no—
440C22—81no—
441C22C41×#54011yescharacteristic
442C22C41×#54611yescharacteristic
443C22C41×#55311yescharacteristic
444C4—22no—
445C4—41no—
446C4—42no—
447C4—41no—
448C4—82no—
449C4—82no—
450C4—81no—
451C4C2×#123×C4111yescharacteristic
452C4C41×#52911yescharacteristic
453C4C41×#54611yescharacteristic
454C4C41×#55511yescharacteristic
455C2—21no—
456C2—21no—
457C2—21no—
458C2—82no—
459C2—81no—
460C2C41×#242011yescharacteristic
subgroupG/H
centerC4×C41#555
commutator, Frattini, cosocleC22×C4C24×C41
socleC2×C41#2420
prime radicalC41×#3067C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C41 ◁ C22×C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#135×C41C2semidirect
3. C41×#2080C2semidirect
4. C41×#2053C2semidirect
5. C41×#1976C2semidirect
6. C41×#1945C2semidirect
7. C41×#1906C2semidirect
8. C41×#1833C2semidirect
9. C41×#1827C2semidirect
10. C41×#1812C2semidirect
11. C41×#1764C2semidirect
205. #24985C41direct (N × H)
226. C4×#135C2×C41semidirect
227. #2080C2×C41semidirect
228. #2053C2×C41semidirect
229. #1976C2×C41semidirect
230. #1945C2×C41semidirect
231. #1906C2×C41semidirect
232. #1833C2×C41semidirect
233. #1827C2×C41semidirect
234. #1812C2×C41semidirect
235. #1764C2×C41semidirect
271. C41#24985direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#30672
C4×#135×C412
C41×#20802
C41×#20532
C41×#19762
C41×#19452
C41×#19062
C41×#18332
C41×#18272
C41×#18122
C41×#17642
#2498541