Finite Group Databasebeta

← #29779468

Subgroups — #29779468

Order 15616.

712 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450407C211yesmaximal, characteristic
2#21449660C211yesmaximal, characteristic, complemented
3#21447318C211yesmaximal, characteristic, complemented
4#21447227C211yesmaximal, characteristic, complemented
5#21447132C211yesmaximal, characteristic
6#21446955C211yesmaximal, characteristic
7#21446780C211yesmaximal, characteristic, complemented
8#21446518C211yesmaximal, characteristic
9#21446513C211yesmaximal, characteristic
10#21446506C211yesmaximal, characteristic, complemented
11#21446399C211yesmaximal, characteristic, complemented
12#21446176C211yesmaximal, characteristic
13#21445693C211yesmaximal, characteristic, complemented
14#21445651C211yesmaximal, characteristic, complemented
15C61×#1968C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519770—21no—
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519706C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519687C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26#16839694—21no—
27#16839687—21no—
28#16839685—21no—
29#16839683—21no—
30#16839669C2211yescharacteristic
31#16839666—21no—
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839655—21no—
38#16839649—21no—
39#16839632C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C22×Dic122—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519777—22no—
63#14519777—41no—
64#14519770—44no—
65Q8×Dic61—22no—
66Q8×Dic61—41no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196D411yescharacteristic
79C2×#11758196D411yescharacteristic
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519706—22no—
89#14519701—22no—
90#14519701—22no—
91#14519698—22no—
92#14519698—41no—
93#14519687—22no—
94#14519687—41no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C2×SD16×C61—22no—
106C2×SD16×C61—22no—
107C2×SD16×C61—41no—
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114C2×C42×C61C2311yescharacteristic
115#106×C61—22no—
116#103×C61—22no—
117#102×C61—22no—
118C4×C8×C61—41no—
119#95×C61—21no—
120#95×C61—21no—
121C24×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61#4112yes—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×Dic122—42no—
149C2×Dic122—42no—
150C2×Dic122—81no—
151SD16×C61—44no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61#4112yes—
157C2×C8×C61#4112yes—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C42×C61—21no—
165C42×C61—21no—
166C22⋊Dic61—42no—
167C22⋊Dic61—42no—
168C22⋊Dic61—81no—
169C22⋊Dic61—81no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—42no—
178C4⋊Dic61—42no—
179C4⋊Dic61—42no—
180C4⋊Dic61—81no—
181C4⋊Dic61—81no—
182C4⋊Dic61#4112yes—
183#11758196—22no—
184#11758196—22no—
185#11758196—42no—
186#11758196—81no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—42no—
192C4×Dic61—42no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61C22×D411yescharacteristic
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—41no—
210Q8×C61—41no—
211D4×C61—41no—
212D4×C61—41no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61AGL(1,ℤ/8)12yes—
229C2×C4×C61#13712yes—
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260Dic122—84no—
261C8×C61—22no—
262C8×C61—22no—
263#92418—41no—
264#92418—41no—
265#29101—611nomaximal, Sylow
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61#53211yescharacteristic
274C22×C61#54611yescharacteristic
275C22×C61#54811yescharacteristic
276C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57711yescharacteristic
279C22×C61#57711yescharacteristic
280C4×C61—22no—
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61#52911yescharacteristic
286C4×C61#53911yescharacteristic
287C4×C61#55111yescharacteristic
288C4×C61#55611yescharacteristic
289Dic61—41no—
290Dic61—41no—
291Dic61—42no—
292Dic61—41no—
293Dic61—41no—
294Dic61—81no—
295Dic61—81no—
296Dic61—81no—
297#2512—611no—
298#2451—611no—
299#2088—611no—
300#2072—611no—
301#2055—611no—
302#2023—611no—
303#1990—611no—
304#1968D6111yescharacteristic
305#1936—611no—
306#1936—611no—
307#1935—611no—
308#1919—611no—
309#1880—611no—
310#1771—611no—
311#1763—611no—
312C2×C61—161no—
313C2×C61#256011yescharacteristic
314C2×C61#317311yescharacteristic
315C2×C61#318211yescharacteristic
316C2×C61#330611yescharacteristic
317C2×C61#332011yescharacteristic
318C2×C61#336811yescharacteristic
319C2×C61#336911yescharacteristic
320C22×SD16C2×D6111yescharacteristic
321C2×C4⋊Q8—611no—
322C2×#126—1221no—
323C2×C4×Q8—611no—
324C2×#106C2×D6111yescharacteristic
325C2×C4⋊C8—611no—
326C2×#103—611no—
327C2×#103—611no—
328C2×#103C2×D6111yescharacteristic
329C2×#102—611no—
330C2×#102—611no—
331C2×#102C2×D6111yescharacteristic
332C2×#97—611no—
333C2×#97—611no—
334C2×C4×C8C2×D6111yescharacteristic
335#398—1221no—
336#397—1221no—
337#396—1221no—
338#395—1221no—
339#391—611no—
340#391—1221no—
341#391C2×D6111yescharacteristic
342#390—611no—
343#390—611no—
344#390C2×D6111yescharacteristic
345#389—611no—
346#389—611no—
347#389—611no—
348#388—1221no—
349#387—1221no—
350#384—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#336—611no—
362#336—611no—
363C61#2910111yescharacteristic, complemented, Sylow
364C22×Q8—1221no—
365C22×Q8C22×D6111yescharacteristic
366C22×D4C22×D6111yescharacteristic
367C2×SD16—22no—
368C2×SD16—22no—
369C2×SD16—41no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374C4⋊Q8—1222no—
375C4⋊Q8—2441no—
376#126—2444no—
377C4×Q8—1222no—
378C4×Q8—2441no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42C22×D6111yescharacteristic
405#106—22no—
406C4⋊C8—1222no—
407#103—22no—
408#103—1222no—
409#103—1222no—
410#102—22no—
411#102—1222no—
412#102—2441no—
413#97—1222no—
414#97—2441no—
415C4×C8—41no—
416#95—21no—
417#95—21no—
418#95—611no—
419#95—611no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426C24—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—4881no—
433C2×Q8—2442no—
434C2×Q8—2442no—
435C2×Q8#1175822212yes—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456SD16—44no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8#1175822412yes—
468C2×C8#1175822412yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—612no—
474C4⋊C4—1221no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—4881no—
484C4⋊C4—4881no—
485C4⋊C4—4881no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C4⋊C4—2442no—
489C4⋊C4—2442no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—4881no—
493C22⋊C4—4881no—
494C22⋊C4—2442no—
495C22⋊C4—2442no—
496C42—21no—
497C42—21no—
498C42—1221no—
499C42—1221no—
500C42—1221no—
501C42—1221no—
502C42—2442no—
503C42—2442no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D6111yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516Q8—4884no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4#1451980412yes—
559C2×C4#1451980612yes—
560C2×C4#41×D6111yescharacteristic
561C2×C4C2×#1175822411yescharacteristic
562C2×C4C2×#1175822411yescharacteristic
563C2×C4C2×#1175822211yescharacteristic
564C2×C4#1451982211yescharacteristic
565C2×C4#1451982411yescharacteristic
566C8—22no—
567C8—22no—
568C8—2441no—
569C8—2441no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22#1684031011yescharacteristic
578C22#1684040111yescharacteristic
579C22#1684042111yescharacteristic
580C22C2×#1451980411yescharacteristic
581C22C2×#1451980611yescharacteristic
582C22#1684056911yescharacteristic
583C22#1684056911yescharacteristic
584C4—22no—
585C4—22no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—4881no—
590C4—4881no—
591C4—4881no—
592C4—2441no—
593C4—2441no—
594C4—2441no—
595C4—2442no—
596C4—2441no—
597C4#1684029211yescharacteristic
598C4#1684035711yescharacteristic
599C4#1684043811yescharacteristic
600C4#1684048111yescharacteristic
601C2—161no—
602C2#2145096811yescharacteristic
603C2#2145643311yescharacteristic
604C2#2145653511yescharacteristic
605C2#2145778611yescharacteristic
606C2#2145799611yescharacteristic
607C2#2145854611yescharacteristic
608C2#2145855711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1968C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1968D61
hypocenterC61#29101
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #21449660C2semidirect
3. #21447318C2semidirect
4. #21447227C2semidirect
7. #21446780C2semidirect
10. #21446506C2semidirect
11. #21446399C2semidirect
13. #21445693C2semidirect
14. #21445651C2semidirect
363. C61#29101semidirect

Maximal subgroups

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

subgroupindex
#214504072
#214496602
#214473182
#214472272
#214471322
#214469552
#214467802
#214465182
#214465132
#214465062
#214463992
#214461762
#214456932
#214456512
C61×#19682
#2910161