Finite Group Databasebeta

← #29779427

Subgroups — #29779427

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450394C211yesmaximal, characteristic, complemented
2#21449669C211yesmaximal, characteristic
3#21447317C211yesmaximal, characteristic, complemented
4#21447224C211yesmaximal, characteristic
5#21447137C211yesmaximal, characteristic
6#21446961C211yesmaximal, characteristic, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446673C211yesmaximal, characteristic
9#21446516C211yesmaximal, characteristic
10#21446509C211yesmaximal, characteristic, complemented
11#21446509C211yesmaximal, characteristic, complemented
12#21446405C211yesmaximal, characteristic, complemented
13#21446152C211yesmaximal, characteristic
14#21445645C211yesmaximal, characteristic
15C61×#1771C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519755—21no—
19C2×C4×D244C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27#16839704—21no—
28#16839701—21no—
29#16839684—21no—
30#16839675—21no—
31#16839668C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839648—21no—
40#16839645—21no—
41#16839636—21no—
42#16839632C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C2×#97×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758190—22no—
64C2×#11758190—22no—
65C2×#11758190—41no—
66#14519777—22no—
67#14519777—41no—
68#14519755—44no—
69C4×D244—22no—
70C4×D244—41no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89#14519709—22no—
90#14519702—22no—
91#14519702—22no—
92#14519702—41no—
93#14519699—22no—
94#14519699—22no—
95#14519699—22no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115C4⋊C8×C61—22no—
116#97×C61—22no—
117#97×C61—41no—
118C4×C8×C61—41no—
119#95×C61D411yescharacteristic
120#95×C61D411yescharacteristic
121C23×D61—21no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
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—21no—
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×Dic122—81no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61#4112yes—
165C2×C8×C61#4112yes—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174#11758198—42no—
175#11758198—42no—
176#11758198—42no—
177#11758198—42no—
178#11758198—81no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—42no—
191C4⋊Dic61—42no—
192C4⋊Dic61—81no—
193C4⋊Dic61#4112yes—
194#11758196—42no—
195#11758196—42no—
196#11758196—81no—
197C4×Dic61—81no—
198#11758190—44no—
199C23×C61C22×D411yescharacteristic
200C22×D61—22no—
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—21no—
214C2×C4×C61—21no—
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×C61AGL(1,ℤ/8)12yes—
224C2×C4×C61#13712yes—
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—22no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252D244—22no—
253D244—22no—
254D244—41no—
255D244—41no—
256C4×D61—84no—
257Dic122—41no—
258Dic122—41no—
259Dic122—41no—
260Dic122—41no—
261Dic122—84no—
262C8×C61—22no—
263C8×C61—22no—
264C8×C61—41no—
265C8×C61—41no—
266#29099—611nomaximal, Sylow
267C22×C61#53211yescharacteristic
268C22×C61#54611yescharacteristic
269C22×C61#54811yescharacteristic
270C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C22×C61#57711yescharacteristic
273C22×C61#57711yescharacteristic
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61#52911yescharacteristic
288C4×C61#53911yescharacteristic
289C4×C61#55111yescharacteristic
290C4×C61#55611yescharacteristic
291Dic61—42no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295Dic61—81no—
296Dic61—81no—
297Dic61—81no—
298#2511—611no—
299#2452—611no—
300#2088—611no—
301#2071—611no—
302#2055—611no—
303#2024—611no—
304#1990—611no—
305#1968—611no—
306#1936—611no—
307#1935—611no—
308#1935—611no—
309#1920—611no—
310#1877—611no—
311#1771D6111yescharacteristic, complemented
312#1762—611no—
313C2×C61#256011yescharacteristic
314C2×C61#317311yescharacteristic
315C2×C61#318211yescharacteristic
316C2×C61#331011yescharacteristic
317C2×C61#331611yescharacteristic
318C2×C61#337211yescharacteristic
319C2×C61#337311yescharacteristic
320D61—82no—
321C22×SD16—611no—
322C2×C4⋊Q8—611no—
323C2×C22⋊Q8—1221no—
324C2×C4×D4—611no—
325C2×#106—611no—
326C2×C4⋊C8C2×D6111yescharacteristic
327C2×#103—611no—
328C2×#103—611no—
329C2×#103—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#102—611no—
333C2×#97C2×D6111yescharacteristic
334C2×#97C2×D6111yescharacteristic
335C2×C4×C8C2×D6111yescharacteristic
336#400—1221no—
337#400—1221no—
338#396—1221no—
339#393—1221no—
340#391—611no—
341#391—611no—
342#391—611no—
343#390—611no—
344#390—611no—
345#389—611no—
346#389—611no—
347#389—611no—
348#387—1221no—
349#387—1221no—
350#385—1221no—
351#384—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4C2×D6111yescharacteristic
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#336C2×D6111yescharacteristic
363#336C2×D6111yescharacteristic
364C61#2909911yescharacteristic, complemented, Sylow
365C22×Q8—611no—
366C22×Q8—1221no—
367C22×D4—611no—
368C23×C4—1221no—
369C2×SD16—1222no—
370C2×SD16—1222no—
371C2×SD16—2441no—
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C22×C8C22×D6111yescharacteristic
376C4⋊Q8—1222no—
377C4⋊Q8—2441no—
378C22⋊Q8—2444no—
379C4×D4—1222no—
380C4×D4—2441no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4D24411yescharacteristic
394C2×C4⋊C4D24411yescharacteristic
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—1221no—
401C2×C42C22×D6111yescharacteristic
402C2×C42C22×D6111yescharacteristic
403C2×C42C22×D6111yescharacteristic
404#106—1222no—
405C4⋊C8—22no—
406#103—1222no—
407#103—1222no—
408#103—2441no—
409#102—1222no—
410#102—1222no—
411#102—1222no—
412#97—22no—
413#97—41no—
414C4×C8—41no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95D24411yescharacteristic
426#95D24411yescharacteristic
427C24—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—4881no—
435C2×Q8—2442no—
436C2×Q8—2442no—
437C2×D4—612no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—4881no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4C2×D24411yescharacteristic
454C22×C4C2×D24411yescharacteristic
455C22×C4C2×D24411yescharacteristic
456C22×C4C2×D24411yescharacteristic
457C22×C4C2×D24411yescharacteristic
458C22×C4C2×D24411yescharacteristic
459C22×C4C23×D6111yescharacteristic
460SD16—2444no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8#1175822212yes—
472C2×C8#1175822212yes—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—612no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—2442no—
490C4⋊C4—2442no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C22⋊C4—4881no—
494C22⋊C4—2442no—
495C22⋊C4—2442no—
496C22⋊C4—2442no—
497C22⋊C4—2442no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—4881no—
505C23—1222no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23—2441no—
512C23C22×D24411yescharacteristic
513Q8—4884no—
514Q8—2441no—
515Q8—2441no—
516Q8—2441no—
517Q8—2441no—
518D4—1222no—
519D4—1222no—
520D4—2441no—
521D4—2441no—
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—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—1222no—
539C2×C4—4884no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
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—2441no—
559C2×C4—2441no—
560C2×C4#1451980512yes—
561C2×C4#1451981012yes—
562C2×C4#41×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4C2×#1175822211yescharacteristic
565C2×C4C2×#1175822211yescharacteristic
566C2×C4#1451982211yescharacteristic
567C2×C4#1451982511yescharacteristic
568C8—22no—
569C8—22no—
570C8—41no—
571C8—41no—
572C22—4881no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—2442no—
577C22—2442no—
578C22—2442no—
579C22#1684031211yescharacteristic
580C22#1684040711yescharacteristic
581C22#1684042411yescharacteristic
582C22C2×#1451980511yescharacteristic
583C22C2×#1451981011yescharacteristic
584C22#1684057411yescharacteristic
585C22#1684057411yescharacteristic
586C4—22no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—4881no—
593C4—4881no—
594C4—4881no—
595C4—4881no—
596C4—4881no—
597C4—4881no—
598C4—2442no—
599C4#1684029011yescharacteristic
600C4#1684035911yescharacteristic
601C4#1684044111yescharacteristic
602C4#1684048411yescharacteristic
603C2—4882no—
604C2#2145095711yescharacteristic
605C2#2145643611yescharacteristic
606C2#2145653711yescharacteristic
607C2#2145784211yescharacteristic
608C2#2145793211yescharacteristic
609C2#2145860211yescharacteristic
610C2#2145861311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1771C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1771D61
hypocenterC61#29099
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
1. #21450394C2semidirect
3. #21447317C2semidirect
6. #21446961C2semidirect
7. #21446780C2semidirect
10. #21446509C2semidirect
11. #21446509C2semidirect
12. #21446405C2semidirect
15. C61×#1771C2semidirect
311. #1771D61semidirect
364. C61#29099semidirect

Maximal subgroups

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

subgroupindex
#214503942
#214496692
#214473172
#214472242
#214471372
#214469612
#214467802
#214466732
#214465162
#214465092
#214464052
#214461522
#214456452
C61×#17712
#2909961