Finite Group Databasebeta

← C41×#18345 #23794546

Subgroups — C41×#18345 #23794546

Order 10496.

888 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3461C211yesmaximal, characteristic
2C41×#3239C212yesmaximal, complemented
3C41×#2568C211yesmaximal, characteristic, complemented
4C41×#1617C212yesmaximal, complemented
5C41×#1616C212yesmaximal
6C41×#1615C214yesmaximal, complemented
7C41×#1509C212yesmaximal, complemented
8C41×#1467C211yesmaximal, characteristic, complemented
9C41×#544C2211yescharacteristic
10C41×#542—22no—
11C41×#541—22no—
12C41×#539—21no—
13C41×#539C2212yes—
14C41×#537C2211yescharacteristic
15C41×#536—21no—
16C41×#533—22no—
17C2×C4⋊Q8×C41C2211yescharacteristic
18C41×#529—22no—
19C41×#529—21no—
20C41×#529C2212yes—
21C4×#41×C41—21no—
22C2×C4×Q8×C41C2211yescharacteristic
23C41×#501—22no—
24C41×#489—22no—
25C41×#484C2214yes—
26C41×#481C2214yes—
27C41×#475—22no—
28C41×#474—22no—
29C41×#432C2212yes—
30C41×#400C2212yes—
31C41×#400C2212yes—
32C41×#391—21no—
33C41×#391—21no—
34C41×#391—21no—
35C41×#382—21no—
36C4×C4⋊C4×C41C2211yescharacteristic
37C41×#328C2214yes—
38C41×#327C2214yes—
39C41×#324C2212yes—
40C41×#324C2212yes—
41C2×#41×C41—21no—
42C22×Q8×C41—21no—
43C22×Q8×C41D411yescharacteristic
44C4⋊Q8×C41—22no—
45C4⋊Q8×C41—22no—
46C4⋊Q8×C41—22no—
47C4⋊Q8×C41—22no—
48#126×C41—22no—
49#126×C41—22no—
50#126×C41—22no—
51#126×C41—22no—
52#125×C41—21no—
53#125×C41—22no—
54#125×C41—22no—
55#125×C41—22no—
56#125×C41D412yes—
57#124×C41—22no—
58#124×C41—21no—
59#124×C41D412yes—
60#123×C41—22no—
61#123×C41—22no—
62#123×C41—22no—
63#123×C41—22no—
64C22⋊Q8×C41—22no—
65C22⋊Q8×C41—22no—
66C22⋊Q8×C41—22no—
67C22⋊Q8×C41—22no—
68C22⋊Q8×C41—21no—
69C22⋊Q8×C41—22no—
70C22⋊Q8×C41C2312yes—
71#121×C41—22no—
72#121×C41—21no—
73#121×C41C2312yes—
74C4×Q8×C41—21no—
75C4×Q8×C41—21no—
76C4×Q8×C41—21no—
77C4×Q8×C41—22no—
78C4×Q8×C41—22no—
79C4×Q8×C41D412yes—
80C4×D4×C41—21no—
81C4×D4×C41—21no—
82C4×D4×C41—21no—
83C4×D4×C41—22no—
84C4×D4×C41—22no—
85C4×D4×C41D412yes—
86#117×C41—22no—
87#117×C41—22no—
88#117×C41—21no—
89#117×C41—22no—
90C2×C4⋊C4×C41—21no—
91C2×C4⋊C4×C41—21no—
92C2×C4⋊C4×C41D411yescharacteristic
93C2×C4⋊C4×C41C2312yes—
94C2×C4⋊C4×C41C2311yescharacteristic
95C2×C4⋊C4×C41C2311yescharacteristic
96C2×C42×C41—22no—
97C2×C42×C41—21no—
98C2×C42×C41D411yescharacteristic
99C2×C42×C41D411yescharacteristic
100C2×C42×C41C2311yescharacteristic
101#107×C41—24no—
102#106×C41—24no—
103C4⋊C8×C41—22no—
104#103×C41—24no—
105#102×C41—24no—
106C22⋊C8×C41C2314yes—
107#97×C41—22no—
108#95×C41—22no—
109#95×C41—22no—
110#95×C41—22no—
111#95×C41—22no—
112#95×C41C2312yes—
113#41×C41—42no—
114#41×C41—42no—
115C2×Q8×C41—21no—
116C2×Q8×C41—21no—
117C2×Q8×C41—22no—
118C2×Q8×C41—22no—
119C2×Q8×C41—41no—
120C2×Q8×C41—44no—
121C2×Q8×C41—41no—
122C2×Q8×C41—41no—
123C2×Q8×C41C2×D412yes—
124C2×D4×C41—21no—
125C2×D4×C41—22no—
126C2×D4×C41—21no—
127C2×D4×C41C2×D412yes—
128C22×C4×C41—22no—
129C22×C4×C41—22no—
130C22×C4×C41—22no—
131C22×C4×C41—22no—
132C22×C4×C41—22no—
133C22×C4×C41C2×D411yescharacteristic
134C22×C4×C41C2×D411yescharacteristic
135C22×C4×C41C2×D411yescharacteristic
136C22×C4×C41C2×D411yescharacteristic
137C22×C4×C41C2×D411yescharacteristic
138C22×C4×C41C2×D411yescharacteristic
139C22×C4×C41C2411yescharacteristic
140C2×C8×C41—24no—
141C4⋊C4×C41—21no—
142C4⋊C4×C41—22no—
143C4⋊C4×C41—21no—
144C4⋊C4×C41—22no—
145C4⋊C4×C41—22no—
146C4⋊C4×C41—21no—
147C4⋊C4×C41—21no—
148C4⋊C4×C41—22no—
149C4⋊C4×C41—22no—
150C4⋊C4×C41—22no—
151C4⋊C4×C41—22no—
152C4⋊C4×C41—22no—
153C4⋊C4×C41—22no—
154C4⋊C4×C41—22no—
155C4⋊C4×C41—22no—
156C4⋊C4×C41—24no—
157C4⋊C4×C41—42no—
158C4⋊C4×C41—42no—
159C4⋊C4×C41C2×D412yes—
160C4⋊C4×C41C2×D412yes—
161C22⋊C4×C41—22no—
162C22⋊C4×C41—22no—
163C22⋊C4×C41—22no—
164C22⋊C4×C41—22no—
165C22⋊C4×C41—22no—
166C22⋊C4×C41—22no—
167C22⋊C4×C41—22no—
168C22⋊C4×C41—22no—
169C22⋊C4×C41—22no—
170C42×C41—22no—
171C42×C41—22no—
172C42×C41—21no—
173C42×C41—22no—
174C42×C41—21no—
175C42×C41—21no—
176C42×C41—21no—
177C42×C41—42no—
178C42×C41C2×D412yes—
179C42×C41C2×D412yes—
180C23×C41—22no—
181C23×C41C22×D411yescharacteristic
182Q8×C41—44no—
183Q8×C41—42no—
184Q8×C41—42no—
185Q8×C41—82no—
186Q8×C41—81no—
187Q8×C41—82no—
188D4×C41—42no—
189D4×C41—42no—
190D4×C41—42no—
191D4×C41—42no—
192D4×C41—81no—
193C2×C4×C41—21no—
194C2×C4×C41—21no—
195C2×C4×C41—21no—
196C2×C4×C41—21no—
197C2×C4×C41—22no—
198C2×C4×C41—21no—
199C2×C4×C41—22no—
200C2×C4×C41—22no—
201C2×C4×C41—21no—
202C2×C4×C41—21no—
203C2×C4×C41—22no—
204C2×C4×C41—21no—
205C2×C4×C41—21no—
206C2×C4×C41—41no—
207C2×C4×C41—44no—
208C2×C4×C41—44no—
209C2×C4×C41—41no—
210C2×C4×C41—41no—
211C2×C4×C41—42no—
212C2×C4×C41—42no—
213C2×C4×C41—44no—
214C2×C4×C41—42no—
215C2×C4×C41—42no—
216C2×C4×C41—82no—
217C2×C4×C41#12011yescharacteristic
218C2×C4×C41#12012yes—
219C2×C4×C41#12011yescharacteristic
220C2×C4×C41#13712yes—
221C2×C4×C41C22×D412yes—
222C8×C41—44no—
223#18345C4111yesmaximal, characteristic, direct product factor, Sylow
224C22×C41—21no—
225C22×C41—21no—
226C22×C41—42no—
227C22×C41—42no—
228C22×C41—42no—
229C22×C41C2×#12011yescharacteristic
230C22×C41C2×#13711yescharacteristic
231C22×C41#57511yescharacteristic
232C4×C41—22no—
233C4×C41—22no—
234C4×C41—21no—
235C4×C41—22no—
236C4×C41—41no—
237C4×C41—42no—
238C4×C41—42no—
239C4×C41—41no—
240C4×C41—41no—
241C4×C41—42no—
242C4×C41—82no—
243C4×C41—82no—
244C4×C41#45412yes—
245#3461C2×C4111yescharacteristic
246#3239C2×C4112yescomplemented
247#2568C2×C4111yescharacteristic, complemented
248#1617C2×C4112yescomplemented
249#1616C2×C4112yes—
250#1615C2×C4114yescomplemented
251#1509C2×C4112yescomplemented
252#1467C2×C4111yescharacteristic, complemented
253C2×C41—21no—
254C2×C41—21no—
255C2×C41—82no—
256C2×C41#300711yescharacteristic
257C2×C41C2×#45411yescharacteristic
258C2×C41#302811yescharacteristic
259#544C22×C4111yescharacteristic
260#542—22no—
261#541—22no—
262#539—21no—
263#539C22×C4112yes—
264#537C22×C4111yescharacteristic
265#536—21no—
266#533—22no—
267C2×C4⋊Q8C22×C4111yescharacteristic
268#529—22no—
269#529—21no—
270#529C22×C4112yes—
271C4×#41—21no—
272C2×C4×Q8C22×C4111yescharacteristic
273#501—22no—
274#489—22no—
275#484C22×C4114yes—
276#481C22×C4114yes—
277#475—22no—
278#474—22no—
279#432C22×C4112yes—
280#400C22×C4112yes—
281#400C22×C4112yes—
282#391—21no—
283#391—21no—
284#391—21no—
285#382—21no—
286C4×C4⋊C4C22×C4111yescharacteristic
287#328C22×C4114yes—
288#327C22×C4114yes—
289#324C22×C4112yes—
290#324C22×C4112yes—
291C41#1834511yescharacteristic, direct product factor, Sylow
292C2×#41—21no—
293C22×Q8—21no—
294C22×Q8D4×C4111yescharacteristic
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299#126—22no—
300#126—22no—
301#126—22no—
302#126—22no—
303#125—21no—
304#125—22no—
305#125—22no—
306#125—22no—
307#125D4×C4112yes—
308#124—22no—
309#124—21no—
310#124D4×C4112yes—
311#123—22no—
312#123—22no—
313#123—22no—
314#123—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—21no—
320C22⋊Q8—22no—
321C22⋊Q8C23×C4112yes—
322#121—22no—
323#121—21no—
324#121C23×C4112yes—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8D4×C4112yes—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4D4×C4112yes—
337#117—22no—
338#117—22no—
339#117—21no—
340#117—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4D4×C4111yescharacteristic
344C2×C4⋊C4C23×C4112yes—
345C2×C4⋊C4C23×C4111yescharacteristic
346C2×C4⋊C4C23×C4111yescharacteristic
347C2×C42—22no—
348C2×C42—21no—
349C2×C42D4×C4111yescharacteristic
350C2×C42D4×C4111yescharacteristic
351C2×C42C23×C4111yescharacteristic
352#107—24no—
353#106—24no—
354C4⋊C8—22no—
355#103—24no—
356#102—24no—
357C22⋊C8C23×C4114yes—
358#97—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95C23×C4112yes—
364#41—42no—
365#41—42no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—41no—
371C2×Q8—44no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8C2×D4×C4112yes—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4C2×D4×C4112yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4C2×D4×C4111yescharacteristic
385C22×C4C2×D4×C4111yescharacteristic
386C22×C4C2×D4×C4111yescharacteristic
387C22×C4C2×D4×C4111yescharacteristic
388C22×C4C2×D4×C4111yescharacteristic
389C22×C4C2×D4×C4111yescharacteristic
390C22×C4C24×C4111yescharacteristic
391C2×C8—24no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4C2×D4×C4112yes—
411C4⋊C4C2×D4×C4112yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—22no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—42no—
429C42C2×D4×C4112yes—
430C42C2×D4×C4112yes—
431C23—22no—
432C23C22×D4×C4111yescharacteristic
433Q8—44no—
434Q8—42no—
435Q8—42no—
436Q8—82no—
437Q8—81no—
438Q8—82no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—81no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4#120×C4111yescharacteristic
469C2×C4#120×C4112yes—
470C2×C4#120×C4111yescharacteristic
471C2×C4#137×C4112yes—
472C2×C4C22×D4×C4112yes—
473C8—44no—
474C22—21no—
475C22—21no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22C2×#120×C4111yescharacteristic
480C22C2×#137×C4111yescharacteristic
481C22C41×#57511yescharacteristic
482C4—22no—
483C4—22no—
484C4—21no—
485C4—22no—
486C4—41no—
487C4—42no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—82no—
493C4—82no—
494C4C41×#45412yes—
495C2—21no—
496C2—21no—
497C2—82no—
498C2C41×#300711yescharacteristic
499C2C2×C41×#45411yescharacteristic
500C2C41×#302811yescharacteristic
subgroupG/H
center, socleC22×C41C2×#120
commutator, Frattini, cosocleC22×C4C24×C41
prime radicalC41×#537C22
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×#3239C2semidirect
3. C41×#2568C2semidirect
4. C41×#1617C2semidirect
6. C41×#1615C2semidirect
7. C41×#1509C2semidirect
8. C41×#1467C2semidirect
223. #18345C41direct (N × H)
246. #3239C2×C41semidirect
247. #2568C2×C41semidirect
248. #1617C2×C41semidirect
250. #1615C2×C41semidirect
251. #1509C2×C41semidirect
252. #1467C2×C41semidirect
291. C41#18345direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#34612
C41×#32392
C41×#25682
C41×#16172
C41×#16162
C41×#16152
C41×#15092
C41×#14672
#1834541