Finite Group Databasebeta

← #29779469

Subgroups — #29779469

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#21450399C211yesmaximal, characteristic
2#21449659C211yesmaximal, characteristic, complemented
3#21447318C211yesmaximal, characteristic, complemented
4#21447231C211yesmaximal, characteristic, complemented
5#21447138C211yesmaximal, characteristic
6#21446954C211yesmaximal, characteristic
7#21446778C211yesmaximal, characteristic, complemented
8#21446671C211yesmaximal, characteristic
9#21446515C211yesmaximal, characteristic
10#21446514C211yesmaximal, characteristic
11#21446507C211yesmaximal, characteristic, complemented
12#21446173C211yesmaximal, characteristic
13#21445692C211yesmaximal, characteristic, complemented
14#21445652C211yesmaximal, characteristic, complemented
15C61×#1919C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519771—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839693—21no—
31#16839691—21no—
32#16839686—21no—
33#16839682—21no—
34#16839667C2211yescharacteristic
35#16839667C2211yescharacteristic
36#16839666—21no—
37#16839664C2211yescharacteristic
38#16839663C2211yescharacteristic
39#16839663C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839653—21no—
43#16839644—21no—
44#16839631C2211yescharacteristic
45C4×#11758196C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#106×C61C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C2×#11758190—22no—
63C2×#11758190—22no—
64C2×#11758190—41no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519776—22no—
68#14519776—41no—
69#14519771—44no—
70C4×Dic122—22no—
71C4×Dic122—41no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519706—22no—
92#14519702—22no—
93#14519700—22no—
94#14519700—22no—
95#14519699—22no—
96#14519697—22no—
97#14519697—41no—
98#14519688—41no—
99#14519687—22no—
100C8×Dic61—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118#106×C61—22no—
119#95×C61—21no—
120#95×C61—21no—
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×C61C2411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122—81no—
150C2×Dic122#4112yes—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61#4112yes—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—81no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61—42no—
169#11758198—42no—
170#11758198—42no—
171#11758198—42no—
172#11758198—42no—
173#11758198—81no—
174#11758198—81no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61#4112yes—
179#11758196—22no—
180#11758196—22no—
181#11758196—42no—
182#11758196—42no—
183#11758196—42no—
184#11758196—42no—
185#11758196—81no—
186#11758196—81no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—42no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418#4112yes—
198#11758190—44no—
199C23×C61C22×D411yescharacteristic
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
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—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61AGL(1,ℤ/8)12yes—
251C2×Dic61#13712yes—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256Dic122—22no—
257Dic122—22no—
258Dic122—41no—
259Dic122—41no—
260Dic122—84no—
261C8×C61—22no—
262C8×C61—41no—
263#92418—22no—
264#92418—41no—
265#29101—611nomaximal, Sylow
266C22×C61#53211yescharacteristic
267C22×C61#54611yescharacteristic
268C22×C61#54811yescharacteristic
269C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C61C2×#13711yescharacteristic
271C22×C61#57711yescharacteristic
272C22×C61#57711yescharacteristic
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61#52911yescharacteristic
287C4×C61#53911yescharacteristic
288C4×C61#55111yescharacteristic
289C4×C61#55611yescharacteristic
290Dic61—22no—
291Dic61—22no—
292Dic61—42no—
293Dic61—41no—
294Dic61—41no—
295Dic61—42no—
296Dic61—81no—
297#2512—611no—
298#2451—611no—
299#2088—611no—
300#2072—611no—
301#2055—611no—
302#2023—611no—
303#1990—611no—
304#1968—611no—
305#1936—611no—
306#1936—611no—
307#1935—611no—
308#1919D6111yescharacteristic, complemented
309#1880—611no—
310#1771—611no—
311#1763—611no—
312C2×C61#256011yescharacteristic
313C2×C61#317311yescharacteristic
314C2×C61#318211yescharacteristic
315C2×C61#330611yescharacteristic
316C2×C61#332011yescharacteristic
317C2×C61#336811yescharacteristic
318C2×C61#336911yescharacteristic
319D61—161no—
320C22×SD16—611no—
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×#103—611no—
329C2×#102—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#97—611no—
333C2×#97—611no—
334C2×C4×C8—611no—
335#398—1221no—
336#397—1221no—
337#396—1221no—
338#395—1221no—
339#391—611no—
340#391—611no—
341#391—1221no—
342#390—611no—
343#390—611no—
344#390—611no—
345#389—611no—
346#389—611no—
347#389C2×D6111yescharacteristic
348#388—1221no—
349#387—1221no—
350#384—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4C2×D6111yescharacteristic
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340C2×D6111yescharacteristic
359#340C2×D6111yescharacteristic
360#340C2×D6111yescharacteristic
361#336—611no—
362#336C2×D6111yescharacteristic
363C61#2910111yescharacteristic, complemented, Sylow
364C22×Q8—611no—
365C22×Q8—1221no—
366C22×D4—611no—
367C2×SD16—1222no—
368C2×SD16—1222no—
369C2×SD16—2441no—
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—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—21no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—1221no—
404C2×C42C22×D6111yescharacteristic
405#106—22no—
406C4⋊C8—1222no—
407#103—1222no—
408#103—1222no—
409#103—1222no—
410#102—1222no—
411#102—1222no—
412#102—2441no—
413#97—1222no—
414#97—2441no—
415C4×C8—2441no—
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—1221no—
427C2×Q8—612no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—4881no—
434C2×Q8—2442no—
435C2×Q8—2442no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—1221no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456SD16—2444no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—612no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8#1175822212yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—81no—
478C4⋊C4—612no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—4881no—
485C4⋊C4—4881no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C4⋊C4—2442no—
489C4⋊C4—2442no—
490C22⋊C4—4881no—
491C22⋊C4—4881no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C22⋊C4—2442no—
495C22⋊C4—2442no—
496C42—21no—
497C42—21no—
498C42—42no—
499C42—1221no—
500C42—1221no—
501C42—1221no—
502C42—1221no—
503C42—2442no—
504C23—2441no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23C2×D4×D6111yescharacteristic
512Q8—1222no—
513Q8—1222no—
514Q8—4884no—
515Q8—2441no—
516Q8—2441no—
517D4—2441no—
518D4—2441no—
519D4—2441no—
520D4—2441no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4#41×D6111yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4#41×D6111yescharacteristic
563C2×C4C2×#1175822211yescharacteristic
564C2×C4#1451982311yescharacteristic
565C2×C4#1451982511yescharacteristic
566C8—22no—
567C8—41no—
568C8—1222no—
569C8—2441no—
570C22—4881no—
571C22—4881no—
572C22—4881no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22#1684030911yescharacteristic
578C22#1684040411yescharacteristic
579C22#1684042211yescharacteristic
580C22AGL(1,ℤ/8)×D6111yescharacteristic
581C22#137×D6111yescharacteristic
582C22#1684056811yescharacteristic
583C22#1684056911yescharacteristic
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—1222no—
591C4—1222no—
592C4—4881no—
593C4—2441no—
594C4—2442no—
595C4—2441no—
596C4—2442no—
597C4#1684029511yescharacteristic
598C4#1684036011yescharacteristic
599C4#1684043611yescharacteristic
600C4#1684048411yescharacteristic
601C2—9761no—
602C2#2145096911yescharacteristic
603C2#2145643211yescharacteristic
604C2#2145653411yescharacteristic
605C2#2145778311yescharacteristic
606C2#2145799311yescharacteristic
607C2#2145855111yescharacteristic
608C2#2145856211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1919C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1919D61
hypocenterC61#29101
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
2. #21449659C2semidirect
3. #21447318C2semidirect
4. #21447231C2semidirect
7. #21446778C2semidirect
11. #21446507C2semidirect
13. #21445692C2semidirect
14. #21445652C2semidirect
15. C61×#1919C2semidirect
308. #1919D61semidirect
363. C61#29101semidirect

Maximal subgroups

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

subgroupindex
#214503992
#214496592
#214473182
#214472312
#214471382
#214469542
#214467782
#214466712
#214465152
#214465142
#214465072
#214461732
#214456922
#214456522
C61×#19192
#2910161