Finite Group Databasebeta

← #29651068

Subgroups — #29651068

Order 15616.

662 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#21455880C211yesmaximal, characteristic
2#21454834C211yesmaximal, characteristic, complemented
3#21454565C211yesmaximal, characteristic, complemented
4#21444723C211yesmaximal, characteristic, complemented
5#21444711C211yesmaximal, characteristic, complemented
6#21444685C211yesmaximal, characteristic, complemented
7#21444678C211yesmaximal, characteristic, complemented
8#21444657C211yesmaximal, characteristic, complemented
9#21444650C211yesmaximal, characteristic, complemented
10#21444644C211yesmaximal, characteristic, complemented
11C8×D488C211yesmaximal, characteristic, complemented
12#21444604C211yesmaximal, characteristic, complemented
13#21444289C211yesmaximal, characteristic, complemented
14#21444207C211yesmaximal, characteristic, complemented
15C61×#1453C211yesmaximal, characteristic, complemented
16C2×#14519785—21no—
17#16840312C2211yescharacteristic
18C4×#11758222C2211yescharacteristic
19#16840087C411yescharacteristic
20#16840087C411yescharacteristic
21#16840058C411yescharacteristic
22#16840053C411yescharacteristic
23#16839963C411yescharacteristic
24#16839958C411yescharacteristic
25#16839909C411yescharacteristic
26#16839909C411yescharacteristic
27#16839892C2211yescharacteristic
28#16839892C2211yescharacteristic
29C4×#11758190C2211yescharacteristic
30C4×#11758190C2211yescharacteristic
31C4×D488C2211yescharacteristic
32C4×D488C2211yescharacteristic
33#16839841C2211yescharacteristic
34#16839841C2211yescharacteristic
35#16839835—21no—
36#16839835—21no—
37#16839831—21no—
38#16839831—21no—
39C8×D244C2211yescharacteristic
40C8×D244C2211yescharacteristic
41C8×D244C2211yescharacteristic
42C8×D244C2211yescharacteristic
43#16839819C2211yescharacteristic
44#16839817—21no—
45C2×C4⋊#92418C2211yescharacteristic
46#16839730—21no—
47#97×D61—21no—
48#16839515C2211yescharacteristic
49#16839515C2211yescharacteristic
50#16839513C2211yescharacteristic
51#16839513C2211yescharacteristic
52#16839494C2211yescharacteristic
53#16839494C2211yescharacteristic
54#16839491C2211yescharacteristic
55#16839491C2211yescharacteristic
56#16839491C2211yescharacteristic
57#16839491C2211yescharacteristic
58#16839491C2211yescharacteristic
59#16839491C2211yescharacteristic
60C61×#431C2211yescharacteristic
61C61×#431C2211yescharacteristic
62C4×M16×C61C2211yescharacteristic
63C61×#322C2211yescharacteristic
64C61×#322C2211yescharacteristic
65C61×#322C2211yescharacteristic
66C82×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758222C2×C411yescharacteristic
69C2×#11758190C2×C411yescharacteristic
70C2×#11758190C2×C411yescharacteristic
71C2×D488C2×C411yescharacteristic
72C2×D488C2×C411yescharacteristic
73#14519785—44no—
74C2×#11758189—21no—
75C2×#11758189—21no—
76C22×#92418—21no—
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81#14519777C2×C411yescharacteristic
82#14519774C2×C411yescharacteristic
83#14519765—21no—
84#14519751—21no—
85#14519751C2×C411yescharacteristic
86#14519751C2×C411yescharacteristic
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122C2311yescharacteristic
89C4×D244—21no—
90C4×D244C2×C411yescharacteristic
91C4×D244C2311yescharacteristic
92C4×D244C2311yescharacteristic
93C4×D244C2311yescharacteristic
94C4×C22⋊D61—21no—
95C4×C22⋊D61—21no—
96#14519736C2×C411yescharacteristic
97#14519734—21no—
98C42×D61—21no—
99#14519711C2×C411yescharacteristic
100#14519711C2×C411yescharacteristic
101#14519709C2×C411yescharacteristic
102#14519709C2×C411yescharacteristic
103C4⋊#92418—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108#14519706—21no—
109#14519706—21no—
110#14519702C2×C411yescharacteristic
111#14519702C2×C411yescharacteristic
112#14519699C2×C411yescharacteristic
113#14519699C2×C411yescharacteristic
114#14519699C2×C411yescharacteristic
115#14519699C2×C411yescharacteristic
116#14519699C2×C411yescharacteristic
117#14519699C2×C411yescharacteristic
118C22⋊#92418—21no—
119#14519689—21no—
120#14519689—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125#14519688—21no—
126#14519687—21no—
127#14519687—21no—
128C2×M16×C61D411yescharacteristic
129C2×M16×C61D411yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—21no—
132C4⋊C8×C61C2×C411yescharacteristic
133C4⋊C8×C61C2×C411yescharacteristic
134C22⋊C8×C61—21no—
135C22⋊C8×C61C2×C411yescharacteristic
136C22⋊C8×C61C2×C411yescharacteristic
137#97×C61D411yescharacteristic
138#97×C61D411yescharacteristic
139C4×C8×C61C2311yescharacteristic
140C4×C8×C61C2311yescharacteristic
141C4×C8×C61C2311yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61#4111yescharacteristic
148C2×C22⋊D61—21no—
149C2×C22⋊D61—21no—
150C2×C22⋊D61—21no—
151#11758222—22no—
152#11758222—41no—
153#11758222—42no—
154#11758222—42no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C22×C411yescharacteristic
158C2×D244C22×C411yescharacteristic
159C2×D244C22×C411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×Dic122—21no—
166C2×Dic122C22×C411yescharacteristic
167M16×C61—22no—
168M16×C61—41no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C22×C411yescharacteristic
172C2×C8×C61C22×C411yescharacteristic
173C2×C8×C61C22×C411yescharacteristic
174C2×C8×C61C22×C411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C42×C61C22×C411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61#4111yescharacteristic
182C42×C61C2411yescharacteristic
183C22⋊Dic61—21no—
184#11758198—21no—
185#11758198—21no—
186#11758198—21no—
187#11758198—21no—
188#11758198—21no—
189#11758198—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C22×C411yescharacteristic
192C4⋊Dic61C22×C411yescharacteristic
193C4⋊Dic61C22×C411yescharacteristic
194C4⋊Dic61C22×C411yescharacteristic
195#11758196—21no—
196#11758196—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—42no—
205D488—22no—
206D488—22no—
207#11758190—41no—
208#11758190—41no—
209#11758189—42no—
210#11758189—42no—
211C8×D61—42no—
212C8×D61—42no—
213C8×D61—42no—
214C8×D61—42no—
215C23×C61C4×D411yescharacteristic
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C4×D411yescharacteristic
226C2×C4×C61C4×D411yescharacteristic
227C2×C4×C61C23×C411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C22⋊D61—42no—
231C22⋊D61—42no—
232C22⋊D61—42no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238D244—21no—
239D244—21no—
240D244—21no—
241D244—21no—
242D244—21no—
243D244—42no—
244D244—81no—
245D244#13112yescomplemented
246C4×D61—42no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—81no—
254Dic122—21no—
255Dic122—21no—
256Dic122—42no—
257C8×C61—21no—
258C8×C61—21no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61AGL(1,ℤ/8)12yes—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#18007—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61C2×C4×D411yescharacteristic
276C2×D61—22no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61C2×#13111yescharacteristic
290C4×C61#56811yescharacteristic
291C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C61#57511yescharacteristic
293Dic61—41no—
294Dic61—41no—
295Dic61—41no—
296Dic61—41no—
297Dic61—41no—
298#3106—611no—
299#2970—611no—
300#2935—611no—
301#1592—611no—
302#1590—611no—
303#1586—611no—
304#1585—611no—
305#1582—611no—
306#1581—611no—
307#1580—611no—
308C8×D8—611no—
309#1573—611no—
310#1496—611no—
311#1477—611no—
312#1453D6111yescharacteristic, complemented
313C2×C61—41no—
314C2×C61#293311yescharacteristic
315C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C61#295711yescharacteristic
317D61—42no—
318D61—81no—
319D61—81no—
320C2×#131—1221no—
321#532—611no—
322C4×#41—611no—
323#482—611no—
324#482—611no—
325#477—611no—
326#476—611no—
327#460—611no—
328#459—611no—
329#449—611no—
330#449—611no—
331#446—611no—
332#446—611no—
333C4×SD16—611no—
334C4×SD16—611no—
335C4×D8—611no—
336C4×D8—611no—
337#436—611no—
338#436—611no—
339#435—1221no—
340#435—1221no—
341#435—1221no—
342#435—1221no—
343C8×D4—611no—
344C8×D4—611no—
345C8×D4—611no—
346C8×D4—611no—
347#432—611no—
348#432—1221no—
349#431C2×D6111yescharacteristic
350#431C2×D6111yescharacteristic
351C2×C4⋊C8—611no—
352#408—1221no—
353C4×M16C2×D6111yescharacteristic
354C2×#97—1221no—
355#335—611no—
356#335—611no—
357#334—611no—
358#334—611no—
359Q8⋊C8—611no—
360Q8⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—611no—
367#322C2×D6111yescharacteristic
368#322C2×D6111yescharacteristic
369#322C2×D6111yescharacteristic
370C82C2×D6111yescharacteristic
371C61#1800711yescharacteristic, complemented, Sylow
372C2×#41—611no—
373C2×#41—1221no—
374C2×SD16—611no—
375C2×SD16—611no—
376C2×D8—611no—
377C2×D8—611no—
378#131—2444no—
379C2×M16—1221no—
380C2×M16—1221no—
381C2×M16D24411yescharacteristic
382C2×M16D24411yescharacteristic
383C22×C8—1221no—
384C22×C8—1221no—
385C22×C8—1221no—
386C22×C8—1221no—
387C22×C8—1221no—
388C4⋊Q8—611no—
389#127—611no—
390#124—1221no—
391#121—611no—
392#121—611no—
393#121—1221no—
394C4×Q8—611no—
395C4×Q8—611no—
396C4×D4—611no—
397C4×D4—611no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4—1221no—
403#117—611no—
404#117—1221no—
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407#107—611no—
408#107—611no—
409#106—611no—
410#106—611no—
411C4⋊C8—21no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8C4×D6111yescharacteristic
420C4⋊C8C4×D6111yescharacteristic
421#103—611no—
422#103—611no—
423#102—611no—
424#102—611no—
425#102—611no—
426#102—611no—
427#102—611no—
428#102—611no—
429C22⋊C8—21no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8C4×D6111yescharacteristic
438C22⋊C8C4×D6111yescharacteristic
439#97—1221no—
440#97—1221no—
441#97—1221no—
442#97D24411yescharacteristic
443#97D24411yescharacteristic
444C4×C8C22×D6111yescharacteristic
445C4×C8C22×D6111yescharacteristic
446C4×C8C22×D6111yescharacteristic
447C4×C8C22×D6111yescharacteristic
448C4×C8C22×D6111yescharacteristic
449C4×C8C22×D6111yescharacteristic
450#41—1222no—
451#41—2441no—
452#41—2442no—
453#41—2442no—
454C2×Q8—611no—
455C2×Q8—1221no—
456C2×D4—611no—
457C2×D4—611no—
458C2×D4—611no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4C2×C4×D6111yescharacteristic
470C22×C4C2×D24411yescharacteristic
471C22×C4#1175822211yescharacteristic
472SD16—2441no—
473SD16—2441no—
474D8—1222no—
475D8—1222no—
476M16—22no—
477M16—41no—
478M16—2442no—
479M16—2442no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—2442no—
487C2×C8—2442no—
488C2×C8—2442no—
489C2×C8—2442no—
490C2×C8—2442no—
491C2×C8C2×C4×D6111yescharacteristic
492C2×C8C2×C4×D6111yescharacteristic
493C2×C8C2×C4×D6111yescharacteristic
494C2×C8C2×C4×D6111yescharacteristic
495C2×C8C2×D24411yescharacteristic
496C2×C8C2×D24411yescharacteristic
497C2×C8C2×D24411yescharacteristic
498C2×C8C2×D24411yescharacteristic
499C4⋊C4—611no—
500C4⋊C4—611no—
501C4⋊C4—611no—
502C4⋊C4—611no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—1221no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C42—1221no—
514C42—1221no—
515C42—1221no—
516C42C2×C4×D6111yescharacteristic
517C42C2×D24411yescharacteristic
518C42#1175822211yescharacteristic
519C42C23×D6111yescharacteristic
520C23—1221no—
521C23—1221no—
522C23—1221no—
523C23C4×D24411yescharacteristic
524Q8—1221no—
525Q8—1221no—
526Q8—2442no—
527D4—612no—
528D4—1221no—
529D4—1221no—
530D4—1221no—
531D4—1221no—
532D4—1221no—
533D4—4881no—
534D4—2442no—
535D4—2442no—
536D4—2442no—
537D4—2442no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—4881no—
549C2×C4—2442no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4C4×D24411yescharacteristic
557C2×C4C4×D24411yescharacteristic
558C2×C4C4×D24411yescharacteristic
559C2×C4C22×C4×D6111yescharacteristic
560C2×C4C22×D24411yescharacteristic
561C2×C4C2×#1175822211yescharacteristic
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—41no—
570C8—41no—
571C8—2441no—
572C8—2441no—
573C8—2441no—
574C8—2441no—
575C8#1451980512yes—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—1222no—
580C22—2441no—
581C22—2441no—
582C22—2441no—
583C22—2441no—
584C22—2441no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22C2×C4×D24411yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—2441no—
594C4—2441no—
595C4—2441no—
596C4—2441no—
597C4—2441no—
598C4#131×D6111yescharacteristic
599C4#1684052711yescharacteristic
600C4C2×#1451980511yescharacteristic
601C4#1684056211yescharacteristic
602C2—41no—
603C2—4881no—
604C2—4881no—
605C2—2442no—
606C2#2145454711yescharacteristic
607C2C4×#1451980511yescharacteristic
608C2#2145473711yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1453C2
socleC22×C61C2×C4×D4
prime radical#21455880C2
cosocleC42×C61C24
hypercenter#1453D61
hypocenterC61#18007
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1453

Complements

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

NH ≅ G/Ntype
2. #21454834C2semidirect
3. #21454565C2semidirect
4. #21444723C2semidirect
5. #21444711C2semidirect
6. #21444685C2semidirect
7. #21444678C2semidirect
8. #21444657C2semidirect
9. #21444650C2semidirect
10. #21444644C2semidirect
11. C8×D488C2semidirect
12. #21444604C2semidirect
13. #21444289C2semidirect
14. #21444207C2semidirect
15. C61×#1453C2semidirect
245. D244#131semidirect
312. #1453D61semidirect
371. C61#18007semidirect

Maximal subgroups

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

subgroupindex
#214558802
#214548342
#214545652
#214447232
#214447112
#214446852
#214446782
#214446572
#214446502
#214446442
C8×D4882
#214446042
#214442892
#214442072
C61×#14532
#1800761