Finite Group Databasebeta

← C41×#12766 #23788967

Subgroups — C41×#12766 #23788967

Order 10496.

760 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C41×#3529C211yesmaximal, characteristic
2C41×#2132C211yesmaximal, characteristic, complemented
3C41×#2120C212yesmaximal, complemented
4C41×#1891C212yesmaximal, complemented
5C41×#1884C211yesmaximal, characteristic, complemented
6C41×#568C2211yescharacteristic
7C41×#563—21no—
8C41×#563C411yescharacteristic, complemented
9C41×#562—21no—
10C41×#562—22no—
11C41×H(3,𝔽4)C411yescharacteristic, complemented
12C41×#560—22no—
13C41×#560C412yescomplemented
14C41×#532—21no—
15C41×#532C2211yescharacteristic
16C41×#529—22no—
17C41×#529—21no—
18C41×#525—22no—
19C41×#525C2212yes—
20C41×#421—21no—
21C41×#413C2212yes—
22C41×#410C2211yescharacteristic
23C41×#407—22no—
24C41×#356—21no—
25C41×#355—21no—
26C41×#352—22no—
27C41×AΣL(1,𝔽16)—22no—
28C41×#344—22no—
29C41×#339—21no—
30C2×#41×C41—21no—
31C2×#41×C41D411yescharacteristic
32C2×#41×C41C2311yescharacteristic
33C23×C4×C41D412yes—
34#131×C41—42no—
35C2×M16×C41—22no—
36C2×M16×C41—21no—
37C2×M16×C41—22no—
38C4⋊Q8×C41—21no—
39C4⋊Q8×C41D411yescharacteristic
40#127×C41—21no—
41#127×C41C2×C411yescharacteristic
42#126×C41—22no—
43#126×C41—22no—
44#126×C41—22no—
45#125×C41—22no—
46#125×C41—21no—
47#124×C41—21no—
48#124×C41—21no—
49#124×C41—22no—
50#124×C41—21no—
51#124×C41C2×C411yescharacteristic
52#124×C41D411yescharacteristic
53#123×C41—22no—
54#123×C41—22no—
55#123×C41—22no—
56#123×C41—22no—
57#123×C41—22no—
58#123×C41C2×C412yes—
59#123×C41D412yes—
60C22⋊Q8×C41—22no—
61C22⋊Q8×C41—22no—
62C22⋊Q8×C41—21no—
63C22⋊Q8×C41—22no—
64C22⋊Q8×C41—22no—
65#121×C41—22no—
66#121×C41—21no—
67#121×C41—22no—
68#121×C41—22no—
69#121×C41—22no—
70#121×C41—22no—
71#121×C41—22no—
72#121×C41—22no—
73#120×C41—22no—
74#120×C41—22no—
75#120×C41C2×C412yes—
76#120×C41D412yes—
77C4×Q8×C41—21no—
78C4×D4×C41—21no—
79C4×D4×C41—22no—
80C4×D4×C41—22no—
81C4×D4×C41—22no—
82C4×D4×C41—22no—
83#117×C41—22no—
84#117×C41—21no—
85#117×C41D412yes—
86C2×C42×C41—21no—
87C2×C42×C41D411yescharacteristic
88#104×C41—42no—
89#101×C41—22no—
90#100×C41—22no—
91#99×C41—22no—
92C22⋊C8×C41—44no—
93C24×C41C22⋊C412yes—
94#41×C41—41no—
95#41×C41—41no—
96#41×C41—41no—
97#41×C41—41no—
98#41×C41—81no—
99#41×C41—81no—
100C2×Q8×C41—21no—
101C2×Q8×C41—21no—
102C2×Q8×C41C22⋊C411yescharacteristic
103C2×Q8×C41C2×D411yescharacteristic
104C2×D4×C41—21no—
105C2×D4×C41—22no—
106C2×D4×C41—21no—
107C2×D4×C41—22no—
108C2×D4×C41—22no—
109C2×D4×C41—22no—
110C2×D4×C41—22no—
111C2×D4×C41—21no—
112C2×D4×C41C22⋊C411yescharacteristic
113C2×D4×C41C22×C411yescharacteristic
114C2×D4×C41C2×D412yes—
115C22×C4×C41—22no—
116C22×C4×C41—21no—
117C22×C4×C41—21no—
118C22×C4×C41—22no—
119C22×C4×C41—22no—
120C22×C4×C41—42no—
121C22×C4×C41—42no—
122C22×C4×C41C22⋊C412yes—
123C22×C4×C41C2×D411yescharacteristic
124C22×C4×C41C2×D412yes—
125M16×C41—22no—
126M16×C41—42no—
127M16×C41—41no—
128M16×C41—42no—
129M16×C41—42no—
130C2×C8×C41—24no—
131C2×C8×C41—41no—
132C2×C8×C41—42no—
133C4⋊C4×C41—22no—
134C4⋊C4×C41—22no—
135C4⋊C4×C41—22no—
136C4⋊C4×C41—21no—
137C4⋊C4×C41—22no—
138C4⋊C4×C41—21no—
139C4⋊C4×C41—22no—
140C4⋊C4×C41—22no—
141C4⋊C4×C41—22no—
142C4⋊C4×C41—22no—
143C4⋊C4×C41—41no—
144C22⋊C4×C41—22no—
145C22⋊C4×C41—22no—
146C22⋊C4×C41—22no—
147C22⋊C4×C41—22no—
148C22⋊C4×C41—22no—
149C22⋊C4×C41—22no—
150C22⋊C4×C41—22no—
151C22⋊C4×C41—22no—
152C22⋊C4×C41—22no—
153C22⋊C4×C41—22no—
154C22⋊C4×C41—22no—
155C22⋊C4×C41—41no—
156C22⋊C4×C41C22⋊C412yes—
157C22⋊C4×C41C22⋊C412yes—
158C42×C41—21no—
159C42×C41—21no—
160C42×C41—22no—
161C42×C41—21no—
162C42×C41—41no—
163C42×C41C22⋊C411yescharacteristic
164C42×C41C22⋊C411yescharacteristic
165C23×C41—22no—
166C23×C41—21no—
167C23×C41—42no—
168C23×C41—82no—
169C23×C41C2×C22⋊C412yes—
170C23×C41#12011yescharacteristic
171Q8×C41—41no—
172Q8×C41—41no—
173Q8×C41—81no—
174Q8×C41—81no—
175D4×C41—41no—
176D4×C41—42no—
177D4×C41—41no—
178D4×C41—81no—
179D4×C41—82no—
180D4×C41—82no—
181D4×C41—82no—
182D4×C41—82no—
183D4×C41—81no—
184D4×C41—82no—
185D4×C41—81no—
186C2×C4×C41—21no—
187C2×C4×C41—22no—
188C2×C4×C41—21no—
189C2×C4×C41—22no—
190C2×C4×C41—22no—
191C2×C4×C41—21no—
192C2×C4×C41—22no—
193C2×C4×C41—21no—
194C2×C4×C41—21no—
195C2×C4×C41—22no—
196C2×C4×C41—42no—
197C2×C4×C41—41no—
198C2×C4×C41—42no—
199C2×C4×C41—41no—
200C2×C4×C41—42no—
201C2×C4×C41—82no—
202C2×C4×C41—81no—
203C2×C4×C41—82no—
204C2×C4×C41—81no—
205C2×C4×C41C2×C22⋊C411yescharacteristic
206C2×C4×C41#12012yes—
207C2×C4×C41#12011yescharacteristic
208C8×C41—42no—
209C8×C41—82no—
210#12766C4111yesmaximal, characteristic, direct product factor, Sylow
211C22×C41—22no—
212C22×C41—21no—
213C22×C41—41no—
214C22×C41—42no—
215C22×C41—42no—
216C22×C41—42no—
217C22×C41—42no—
218C22×C41—44no—
219C22×C41—42no—
220C22×C41—41no—
221C22×C41—82no—
222C22×C41#37911yescharacteristic
223C4×C41—22no—
224C4×C41—21no—
225C4×C41—42no—
226C4×C41—41no—
227C4×C41—42no—
228C4×C41—41no—
229C4×C41—42no—
230C4×C41—41no—
231C4×C41—161no—
232C4×C41B(4,𝔽2)11yescharacteristic
233C4×C41#45811yescharacteristic
234#3529C2×C4111yescharacteristic
235#2132C2×C4111yescharacteristic, complemented
236#2120C2×C4112yescomplemented
237#1891C2×C4112yescomplemented
238#1884C2×C4111yescharacteristic, complemented
239C2×C41—21no—
240C2×C41—41no—
241C2×C41—42no—
242C2×C41—81no—
243C2×C41—82no—
244C2×C41#179011yescharacteristic
245#568C22×C4111yescharacteristic
246#563—21no—
247#563C4×C4111yescharacteristic, complemented
248#562—21no—
249#562—22no—
250H(3,𝔽4)C4×C4111yescharacteristic, complemented
251#560—22no—
252#560C4×C4112yescomplemented
253#532—21no—
254#532C22×C4111yescharacteristic
255#529—22no—
256#529—21no—
257#525—22no—
258#525C22×C4112yes—
259#421—21no—
260#413C22×C4112yes—
261#410C22×C4111yescharacteristic
262#407—22no—
263#356—21no—
264#355—21no—
265#352—22no—
266AΣL(1,𝔽16)—22no—
267#344—22no—
268#339—21no—
269C41#1276611yescharacteristic, direct product factor, Sylow
270C2×#41—21no—
271C2×#41D4×C4111yescharacteristic
272C2×#41C23×C4111yescharacteristic
273C23×C4D4×C4112yes—
274#131—42no—
275C2×M16—22no—
276C2×M16—21no—
277C2×M16—22no—
278C4⋊Q8—21no—
279C4⋊Q8D4×C4111yescharacteristic
280#127—21no—
281#127C2×C4×C4111yescharacteristic
282#126—22no—
283#126—22no—
284#126—22no—
285#125—22no—
286#125—21no—
287#124—21no—
288#124—21no—
289#124—22no—
290#124—21no—
291#124C2×C4×C4111yescharacteristic
292#124D4×C4111yescharacteristic
293#123—22no—
294#123—22no—
295#123—22no—
296#123—22no—
297#123—22no—
298#123C2×C4×C4112yes—
299#123D4×C4112yes—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—21no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305#121—22no—
306#121—21no—
307#121—22no—
308#121—22no—
309#121—22no—
310#121—22no—
311#121—22no—
312#121—22no—
313#120—22no—
314#120—22no—
315#120C2×C4×C4112yes—
316#120D4×C4112yes—
317C4×Q8—21no—
318C4×D4—21no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323#117—22no—
324#117—21no—
325#117D4×C4112yes—
326C2×C42—21no—
327C2×C42D4×C4111yescharacteristic
328#104—42no—
329#101—22no—
330#100—22no—
331#99—22no—
332C22⋊C8—44no—
333C24C22⋊C4×C4112yes—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—81no—
339#41—81no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8C22⋊C4×C4111yescharacteristic
343C2×Q8C2×D4×C4111yescharacteristic
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4C22⋊C4×C4111yescharacteristic
353C2×D4C22×C4×C4111yescharacteristic
354C2×D4C2×D4×C4112yes—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4C22⋊C4×C4112yes—
363C22×C4C2×D4×C4111yescharacteristic
364C22×C4C2×D4×C4112yes—
365M16—22no—
366M16—42no—
367M16—41no—
368M16—42no—
369M16—42no—
370C2×C8—24no—
371C2×C8—41no—
372C2×C8—42no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—41no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—41no—
396C22⋊C4C22⋊C4×C4112yes—
397C22⋊C4C22⋊C4×C4112yes—
398C42—21no—
399C42—21no—
400C42—22no—
401C42—21no—
402C42—41no—
403C42C22⋊C4×C4111yescharacteristic
404C42C22⋊C4×C4111yescharacteristic
405C23—22no—
406C23—21no—
407C23—42no—
408C23—82no—
409C23C2×C22⋊C4×C4112yes—
410C23#120×C4111yescharacteristic
411Q8—41no—
412Q8—41no—
413Q8—81no—
414Q8—81no—
415D4—41no—
416D4—42no—
417D4—41no—
418D4—81no—
419D4—82no—
420D4—81no—
421D4—82no—
422D4—82no—
423D4—82no—
424D4—82no—
425D4—81no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4C2×C22⋊C4×C4111yescharacteristic
446C2×C4#120×C4112yes—
447C2×C4#120×C4111yescharacteristic
448C8—42no—
449C8—82no—
450C22—22no—
451C22—21no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22—42no—
459C22—41no—
460C22—82no—
461C22C41×#37911yescharacteristic
462C4—22no—
463C4—21no—
464C4—42no—
465C4—41no—
466C4—42no—
467C4—41no—
468C4—42no—
469C4—41no—
470C4—161no—
471C4C41×B(4,𝔽2)11yescharacteristic
472C4C41×#45811yescharacteristic
473C2—21no—
474C2—41no—
475C2—42no—
476C2—82no—
477C2—81no—
478C2C41×#179011yescharacteristic
subgroupG/H
centerC4×C41B(4,𝔽2)
commutatorC2×D4C22×C4×C41
Frattini, cosocleC2×#41C23×C41
socleC2×C41#1790
prime radicalC41×#3529C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C41 ◁ C2×C4×C41 ◁ C2×#41×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#2132C2semidirect
3. C41×#2120C2semidirect
4. C41×#1891C2semidirect
5. C41×#1884C2semidirect
8. C41×#563C4semidirect
11. C41×H(3,𝔽4)C4semidirect
13. C41×#560C4semidirect
210. #12766C41direct (N × H)
235. #2132C2×C41semidirect
236. #2120C2×C41semidirect
237. #1891C2×C41semidirect
238. #1884C2×C41semidirect
247. #563C4×C41semidirect
250. H(3,𝔽4)C4×C41semidirect
252. #560C4×C41semidirect
269. C41#12766direct (N × H)

Maximal subgroups

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

subgroupindex
C41×#35292
C41×#21322
C41×#21202
C41×#18912
C41×#18842
#1276641