Finite Group Databasebeta

← #29736180

Subgroups — #29736180

Order 15616.

2026 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#127×D61C211yesmaximal, characteristic
2C2×#16840110C211yesmaximal, characteristic, complemented
3#21454166C211yesmaximal, characteristic, complemented
4#21447051C214yesmaximal, complemented
5#21446626C211yesmaximal, characteristic, complemented
6#21446625C211yesmaximal, characteristic, complemented
7#21445748C214yesmaximal, complemented
8C4×#14519689C211yesmaximal, characteristic, complemented
9C2×C61×#486C211yesmaximal, characteristic, complemented
10C22×D4×D61—21no—
11C22×D4×D61—21no—
12C22×D4×D61C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519773—21no—
15C2×#14519773—21no—
16C2×#14519773C2212yes—
17#127×D61—24no—
18#127×D61—24no—
19C2×#14519753—21no—
20C2×C42×D61C2211yescharacteristic
21#16840110—24no—
22C2×#14519699C2214yescomplemented
23C2×#14519698C2214yescomplemented
24C2×#14519697C2214yescomplemented
25C2×#14519689C2212yes—
26C2×C4×#92418C2211yescharacteristic, complemented
27#16839669—21no—
28#16839658—21no—
29#16839658C2212yescomplemented
30#16839632C2211yescharacteristic, complemented
31#16839619—21no—
32#16839619C2212yes—
33#16839516C2212yescomplemented
34C2×C4⋊Q8×C61C2211yescharacteristic, complemented
35C2×#127×C61C2211yescharacteristic
36C61×#486—24no—
37C2×#102×C61C2214yescomplemented
38C2×C4×C8×C61C2211yescharacteristic, complemented
39C24×D61—22no—
40C22×D244—21no—
41C22×D244—21no—
42C22×D244C2312yes—
43C22×C22⋊D61—22no—
44C22×C22⋊D61—22no—
45C22×C22⋊D61—22no—
46C2×D4×D61—28no—
47C2×D4×D61—28no—
48C2×D4×D61—24no—
49C2×D4×D61—24no—
50C2×D4×D61—28no—
51C2×D4×D61—44no—
52C2×D4×D61—44no—
53C2×D4×D61—44no—
54C2×D4×D61—44no—
55C2×D4×D61—44no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C22×#92418C2312yes—
60#14519774—22no—
61#14519774D414yes—
62#14519773—28no—
63#14519773—28no—
64#14519773—24no—
65#14519773—24no—
66#14519753—44no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—22no—
71C2×#11758198—22no—
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61C2311yescharacteristic
75C42×D61—22no—
76C42×D61—22no—
77#14519699—44no—
78#14519698—44no—
79#14519697—28no—
80#14519689—24no—
81C4×#92418—22no—
82#14519684—22no—
83#14519684—22no—
84C22×Q8×C61—21no—
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61C2312yes—
88C22×C8×C61C2312yes—
89C4⋊Q8×C61—22no—
90C4⋊Q8×C61—22no—
91#127×C61—22no—
92#127×C61D414yes—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2312yes—
95C2×C42×C61C2311yescharacteristic
96#102×C61—28no—
97C4×C8×C61—22no—
98C24×C61—22no—
99C23×D61—28no—
100C23×D61—22no—
101C23×D61—44no—
102C23×D61—44no—
103C23×D61—44no—
104C23×D61—44no—
105C23×D61—44no—
106C23×D61—44no—
107C23×D61C2×D411yescharacteristic
108C2×Q8×C61—42no—
109C2×Q8×C61—42no—
110C2×Q8×C61—42no—
111C2×D4×C61—24no—
112C2×D4×C61—24no—
113C2×D4×C61—22no—
114C2×D4×C61—24no—
115C2×D4×C61—24no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61C2×D414yes—
121C22×C4×C61—22no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—28no—
126C2×C22⋊D61—28no—
127C2×C22⋊D61—28no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C2×C22⋊D61—44no—
133C2×C22⋊D61—44no—
134C22×Dic61—22no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138D4×D61—416no—
139D4×D61—416no—
140D4×D61—432no—
141D4×D61—432no—
142C2×D244—24no—
143C2×D244—24no—
144C2×D244—24no—
145C2×D244—24no—
146C2×D244—22no—
147C2×D244—42no—
148C2×D244—42no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244C2×D414yes—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—22no—
159C2×C4×D61D814yes—
160C2×C4×D61SD1614yes—
161C2×C4×D61#4114yes—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—42no—
168C42×C61C2×D412yes—
169C42×C61C2×D412yes—
170#11758198—44no—
171C4⋊Dic61—24no—
172C4⋊Dic61—42no—
173C4⋊Dic61—42no—
174#11758196—44no—
175C4×Dic61—22no—
176C4×Dic61—22no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C23×C61—24no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61C22×D411yescharacteristic
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—24no—
195C22×D61—416no—
196C22×D61—416no—
197C22×D61—416no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—88no—
205C22×D61—88no—
206C22×D61—88no—
207C22×D61—88no—
208C22×D61#12412yes—
209C22×D61C2×D812yes—
210C22×D61C2×SD1612yes—
211Q8×C61—84no—
212D4×C61—28no—
213D4×C61—44no—
214D4×C61—48no—
215D4×C61—48no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61#12012yes—
220C2×C4×C61#12012yes—
221C2×C4×C61C2×D812yes—
222C2×C4×C61C2×SD1612yes—
223C2×C4×C61AGL(1,ℤ/8)12yes—
224C2×C4×C61AGL(1,ℤ/8)12yes—
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4112yes—
228C2×C4×C61H(5,𝔽2)12yes—
229C22⋊D61—416no—
230C22⋊D61—416no—
231C22⋊D61—416no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61#12412yes—
242C2×Dic61C2×D812yes—
243C2×Dic61C2×SD1612yes—
244D244—28no—
245D244—44no—
246D244—48no—
247D244—48no—
248C4×D61—24no—
249C4×D61—24no—
250C4×D61—28no—
251C4×D61—28no—
252C8×C61—42no—
253#92418—42no—
254#25855—611nomaximal, Sylow
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—82no—
259C22×C61—82no—
260C22×C61—82no—
261C22×C61—82no—
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×#12411yescharacteristic
264C22×C61#53511yescharacteristic
265C22×C61C22×D811yescharacteristic
266C22×C61C22×SD1611yescharacteristic
267C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—44no—
276C2×D61—44no—
277C2×D61—44no—
278C2×D61—832no—
279C2×D61—82no—
280C2×D61—82no—
281C2×D61—82no—
282C2×D61—82no—
283C2×D61#48614yescomplemented
284C4×C61—82no—
285C4×C61#44714yes—
286C4×C61#45014yes—
287C4×C61#54612yes—
288C4×C61#54612yes—
289Dic61—22no—
290Dic61—22no—
291Dic61—22no—
292Dic61—82no—
293C22×#127—611no—
294C2×#486—611no—
295C2×#486D6111yescharacteristic, complemented
296#2857—611no—
297#2041—614no—
298#1957—611no—
299#1957—611no—
300#1786—614no—
301C4×C22⋊C8—611no—
302C2×C61—84no—
303C2×C61#243511yescharacteristic
304C2×C61C2×#44711yescharacteristic
305C2×C61C2×#45011yescharacteristic
306C2×C61C2×#48611yescharacteristic
307C2×C61#313811yescharacteristic
308C2×C61#320511yescharacteristic
309C2×C61#320811yescharacteristic
310D61—24no—
311D61—84no—
312C23×D4—612no—
313C23×D4—1221no—
314C23×D4—1221no—
315C2×C4⋊Q8C2×D6111yescharacteristic, complemented
316C2×#127—612no—
317C2×#127—611no—
318C2×#127—1221no—
319C2×#127—1221no—
320C2×#127—1224no—
321C2×#127—1224no—
322C2×#127C2×D6111yescharacteristic
323C2×C22⋊Q8—1221no—
324C22×C42—611no—
325#486—24no—
326#486—1224no—
327C2×#102—614no—
328C2×#102—614no—
329C2×#102—614no—
330C2×#102C2×D6114yescomplemented
331C2×C22⋊C8—612no—
332C2×C4×C8—611no—
333C2×C4×C8C2×D6111yescharacteristic, complemented
334#391—1221no—
335#389—612no—
336#389—1221no—
337#384—611no—
338#380—612no—
339#380—1221no—
340#336—612no—
341C61#2585511yescharacteristic, complemented, Sylow
342C25—1222no—
343C22×Q8—21no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—612no—
347C22×D4—1228no—
348C22×D4—1222no—
349C22×D4—1222no—
350C22×D4—1222no—
351C22×D4—1224no—
352C22×D4—1228no—
353C22×D4—1228no—
354C22×D4—1224no—
355C22×D4—1221no—
356C22×D4—1221no—
357C22×D4—2444no—
358C22×D4—2444no—
359C22×D4—2444no—
360C22×D4—2444no—
361C22×D4—2444no—
362C22×D4C22×D6112yes—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C22×C8—612no—
367C22×C8C22×D6112yes—
368C4⋊Q8—22no—
369C4⋊Q8—22no—
370#127—22no—
371#127—614no—
372#127—1224no—
373#127—1228no—
374#127—1228no—
375#127—1224no—
376#127—1222no—
377#127C22⋊D6114yes—
378C22⋊Q8—2444no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4C22×D6112yes—
385C2×C22⋊C4—1222no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—1222no—
390C2×C42—1222no—
391C2×C42C22×D6111yescharacteristic
392#102—28no—
393#102—1228no—
394#102—2444no—
395#102—2444no—
396C22⋊C8—1224no—
397C4×C8—22no—
398C4×C8—1222no—
399#95—1222no—
400#95—1222no—
401C24—22no—
402C24—611no—
403C24—1228no—
404C24—1222no—
405C24—2444no—
406C24—2444no—
407C24—2444no—
408C24—2444no—
409C24—2444no—
410C24—2444no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—614no—
424C2×D4—1228no—
425C2×D4—1228no—
426C2×D4—1228no—
427C2×D4—1222no—
428C2×D4—1224no—
429C2×D4—1224no—
430C2×D4—1224no—
431C2×D4—1224no—
432C2×D4—2444no—
433C2×D4—2444no—
434C2×D4—2444no—
435C2×D4—24416no—
436C2×D4—24416no—
437C2×D4—24432no—
438C2×D4—24432no—
439C2×D4—2444no—
440C2×D4—2444no—
441C2×D4—2444no—
442C2×D4—2442no—
443C2×D4—2442no—
444C2×D4—2442no—
445C2×D4—2442no—
446C2×D4C2×C22⋊D6114yes—
447C22×C4—22no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—614no—
452C22×C4—614no—
453C22×C4—614no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4D4×D6111yescharacteristic
463C22×C4D4×D6111yescharacteristic
464C22×C4C23×D6111yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—1224no—
475C4⋊C4—2444no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C22⋊C4—2444no—
479C42—1221no—
480C42—1221no—
481C42—1222no—
482C42—1222no—
483C42D4×D6112yes—
484C42C2×C22⋊D6112yes—
485C23—24no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—612no—
493C23—612no—
494C23—612no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1224no—
500C23—4888no—
501C23—4888no—
502C23—4888no—
503C23—4888no—
504C23—24416no—
505C23—24416no—
506C23—24416no—
507C23—2442no—
508C23—2442no—
509C23—2442no—
510C23—2442no—
511C23—2442no—
512C23—2442no—
513C23C2×D4×D6111yescharacteristic
514Q8—84no—
515D4—28no—
516D4—44no—
517D4—48no—
518D4—48no—
519D4—1228no—
520D4—24416no—
521D4—24416no—
522D4—24416no—
523D4—2444no—
524D4—2448no—
525D4—2448no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1228no—
541C2×C4—1228no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4#1451974512yes—
546C2×C4#1451974512yes—
547C2×C4D8×D6112yes—
548C2×C4SD16×D6112yes—
549C2×C4#1451980112yes—
550C2×C4#1451980412yes—
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C22×C22⋊D6111yescharacteristic
553C2×C4#41×D6112yes—
554C2×C4#1451982312yes—
555C8—42no—
556C8—2442no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—614no—
565C22—1222no—
566C22—1222no—
567C22—1222no—
568C22—1222no—
569C22—1222no—
570C22—1222no—
571C22—48832no—
572C22—4882no—
573C22—4882no—
574C22—4882no—
575C22—4882no—
576C22—2444no—
577C22—2444no—
578C22—2444no—
579C22C2×#1451974511yescharacteristic
580C22#124×D6111yescharacteristic
581C22#1684033011yescharacteristic
582C22C2×D8×D6111yescharacteristic
583C22C2×SD16×D6111yescharacteristic
584C22C2×#1451980111yescharacteristic
585C22C2×#1451980411yescharacteristic
586C4—82no—
587C4—1222no—
588C4—1222no—
589C4—1222no—
590C4—4882no—
591C4#1683989414yes—
592C4#1683991114yes—
593C4#1684040012yes—
594C4#1684040512yes—
595C2—84no—
596C2—1224no—
597C2—4884no—
598C2#2144946711yescharacteristic
599C2C2×#1683989411yescharacteristic
600C2C2×#1683991111yescharacteristic
601C2#486×D6111yescharacteristic
602C2#2145619011yescharacteristic
603C2#2145673811yescharacteristic
604C2#2145676111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#486C2
socleC23×C61C22×D4
prime radicalC2×#127×D61C2
hypercenterC2×#486D61
hypocenterC61#25855
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
2. C2×#16840110C2semidirect
3. #21454166C2semidirect
4. #21447051C2semidirect
5. #21446626C2semidirect
6. #21446625C2semidirect
7. #21445748C2semidirect
8. C4×#14519689C2semidirect
9. C2×C61×#486C2semidirect
22. C2×#14519699C22semidirect
23. C2×#14519698C22semidirect
24. C2×#14519697C22semidirect
26. C2×C4×#92418C22semidirect
29. #16839658C22semidirect
30. #16839632C22semidirect
33. #16839516C22semidirect
34. C2×C4⋊Q8×C61C22semidirect
37. C2×#102×C61C22semidirect
38. C2×C4×C8×C61C22semidirect
283. C2×D61#486semidirect
295. C2×#486D61semidirect
315. C2×C4⋊Q8C2×D61semidirect
330. C2×#102C2×D61semidirect
333. C2×C4×C8C2×D61semidirect
341. C61#25855semidirect

Maximal subgroups

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

subgroupindex
C2×#127×D612
C2×#168401102
#214541662
#214470512
#214466262
#214466252
#214457482
C4×#145196892
C2×C61×#4862
#2585561