Finite Group Databasebeta

← #29748655

Subgroups — #29748655

Order 15616.

732 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448888C211yesmaximal, characteristic
2#21447223C211yesmaximal, characteristic
3#21447190C211yesmaximal, characteristic
4#21447060C211yesmaximal, characteristic
5#21447060C211yesmaximal, characteristic
6#21446494C211yesmaximal, characteristic, complemented
7#21446453C211yesmaximal, characteristic, complemented
8#21446408C211yesmaximal, characteristic, complemented
9#21446401C211yesmaximal, characteristic, complemented
10#21446378C211yesmaximal, characteristic, complemented
11#21446072C211yesmaximal, characteristic, complemented
12#21446065C211yesmaximal, characteristic
13#21445821C211yesmaximal, characteristic, complemented
14C4×#14519697C211yesmaximal, characteristic
15C61×#2506C211yesmaximal, characteristic, complemented
16C2×#14519765C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839652—21no—
34#16839651—21no—
35#16839633—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839609—21no—
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839517C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×#125×C61—21no—
49C61×#400—21no—
50C61×#398—21no—
51C61×#389C2211yescharacteristic
52C61×#389C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#387—21no—
56C61×#384C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519765—22no—
67#14519765—22no—
68C4×Dic122—22no—
69C4×Dic122—41no—
70C4×D244—22no—
71C4×D244—41no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82#14519710—22no—
83#14519708—22no—
84C4⋊#92418—41no—
85#14519700—22no—
86#14519700—22no—
87#14519700—22no—
88#14519700—41no—
89#14519697—22no—
90#14519697—22no—
91#14519697—22no—
92#14519697—41no—
93C4×#92418—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#125×C61—44no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C42×C61—21no—
111C2×C42×C61C2311yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114#95×C61—21no—
115#95×C61—21no—
116#95×C61—21no—
117#95×C61—21no—
118#95×C61—21no—
119#95×C61D411yescharacteristic
120#95×C61D411yescharacteristic
121C23×D61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244#4112yes—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—81no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122#4112yes—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61#4112yes—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—42no—
174#11758198—24no—
175#11758198—24no—
176#11758198—42no—
177#11758198—42no—
178#11758198—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61#4112yes—
183#11758196—42no—
184C4×Dic61—42no—
185C4×Dic61—81no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C23×C61C22×D411yescharacteristic
199C22×D61—22no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61#13512yes—
235C2×C4×C61AGL(1,ℤ/8)12yes—
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C2×C4×C612−1+411yescharacteristic
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251D244—22no—
252D244—22no—
253D244—41no—
254D244—41no—
255C4×D61—84no—
256Dic122—22no—
257Dic122—22no—
258Dic122—41no—
259Dic122—41no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#26761—611nomaximal, Sylow
265C22×C61#52511yescharacteristic
266C22×C61#54611yescharacteristic
267C22×C61#54811yescharacteristic
268C22×C61C2×#13511yescharacteristic
269C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C61#57711yescharacteristic
271C22×C61#57811yescharacteristic
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61#52911yescharacteristic
290C4×C61#53911yescharacteristic
291C4×C61#55211yescharacteristic
292C4×C61#55311yescharacteristic
293Dic61—42no—
294Dic61—42no—
295Dic61—81no—
296#2506D6111yescharacteristic, complemented
297#2367—611no—
298#2071—611no—
299#2065—611no—
300#2042—611no—
301#2042—611no—
302#1933—611no—
303#1927—611no—
304#1921—611no—
305#1920—611no—
306#1916—611no—
307#1863—611no—
308#1862—611no—
309#1803—611no—
310C4×#102—611no—
311C2×C61#246011yescharacteristic
312C2×C61#310611yescharacteristic
313C2×C61#312011yescharacteristic
314C2×C61#330011yescharacteristic
315C2×C61#330611yescharacteristic
316C2×C61#336511yescharacteristic
317C2×C61#336711yescharacteristic
318D61—82no—
319C2×#125—21no—
320C2×#124—611no—
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C2×#107—611no—
324C2×#106—611no—
325C2×C4⋊C8—611no—
326C2×#103—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#102—611no—
333C2×#102—611no—
334C2×C4×C8—611no—
335#400—21no—
336#398—21no—
337#391—611no—
338#390—611no—
339#389C2×D6111yescharacteristic
340#389C2×D6111yescharacteristic
341#389C2×D6111yescharacteristic
342#389C2×D6111yescharacteristic
343#388—1221no—
344#388—1221no—
345#387—21no—
346#385—1221no—
347#384C2×D6111yescharacteristic
348C4×C4⋊C4—611no—
349C4×C4⋊C4C2×D6111yescharacteristic
350C4×C4⋊C4C2×D6111yescharacteristic
351C4×C22⋊C4—1221no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#336—611no—
359#336—611no—
360#336—611no—
361#336—611no—
362C61#2676111yescharacteristic, complemented, Sylow
363C22×Q8—611no—
364C22×D4—611no—
365C23×C4—1221no—
366C22×C8—611no—
367C22×C8—611no—
368C22×C8—611no—
369C22×C8—611no—
370#125—44no—
371#124—1222no—
372#124—1222no—
373C4×Q8—1222no—
374C4×Q8—2441no—
375C4×D4—1222no—
376C4×D4—2441no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4C22⋊D6111yescharacteristic
387C2×C4⋊C4C22⋊D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C42—21no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401C2×C42C22×D6111yescharacteristic
402C2×C42C22×D6111yescharacteristic
403#107—1222no—
404#106—1222no—
405C4⋊C8—2441no—
406#103—1222no—
407#103—1222no—
408#103—1222no—
409#103—2441no—
410#102—1222no—
411#102—1222no—
412#102—1222no—
413#102—2441no—
414C4×C8—2441no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95C22⋊D6111yescharacteristic
424#95C22⋊D6111yescharacteristic
425C24—1221no—
426C2×Q8—612no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×D4—612no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4—1221no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—4881no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C23×D6111yescharacteristic
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—612no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2442no—
487C4⋊C4#1175822612yes—
488C22⋊C4—1224no—
489C22⋊C4—1224no—
490C22⋊C4—4881no—
491C22⋊C4—2442no—
492C22⋊C4—2442no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—4881no—
501C42—2442no—
502C23—1222no—
503C23—2441no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23C22×C22⋊D6111yescharacteristic
510Q8—1222no—
511Q8—1222no—
512Q8—2441no—
513Q8—2441no—
514D4—1222no—
515D4—1222no—
516D4—2441no—
517D4—2441no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—4884no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4#1451979712yes—
557C2×C4#1451980312yes—
558C2×C4#41×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C2×C4#41×D6111yescharacteristic
561C2×C4C2×#1175822611yescharacteristic
562C2×C4#1451982311yescharacteristic
563C2×C4#1451982511yescharacteristic
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C22—4881no—
569C22—4881no—
570C22—4881no—
571C22—4881no—
572C22—2442no—
573C22—2442no—
574C22—2442no—
575C22#1684026911yescharacteristic
576C22#1684040011yescharacteristic
577C22#1684042311yescharacteristic
578C22C2×#1451979711yescharacteristic
579C22C2×#1451980311yescharacteristic
580C22#1684057011yescharacteristic
581C22#1684057711yescharacteristic
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—4881no—
593C4—2442no—
594C4—2442no—
595C4#1684029711yescharacteristic
596C4#1684036011yescharacteristic
597C4#1684044411yescharacteristic
598C4#1684046511yescharacteristic
599C2—4882no—
600C2#2144976711yescharacteristic
601C2#2145587311yescharacteristic
602C2#2145602711yescharacteristic
603C2#2145770011yescharacteristic
604C2#2145777411yescharacteristic
605C2#2145851511yescharacteristic
606C2#2145853711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2506C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2506D61
hypocenterC61#26761
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2506

Complements

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

NH ≅ G/Ntype
6. #21446494C2semidirect
7. #21446453C2semidirect
8. #21446408C2semidirect
9. #21446401C2semidirect
10. #21446378C2semidirect
11. #21446072C2semidirect
13. #21445821C2semidirect
15. C61×#2506C2semidirect
296. #2506D61semidirect
362. C61#26761semidirect

Maximal subgroups

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

subgroupindex
#214488882
#214472232
#214471902
#214470602
#214464942
#214464532
#214464082
#214464012
#214463782
#214460722
#214460652
#214458212
C4×#145196972
C61×#25062
#2676161