Finite Group Databasebeta

← #29853067

Subgroups — #29853067

Order 15616.

1978 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454127C212yesmaximal, complemented
2#21454103C212yesmaximal
3#21451933C211yesmaximal, characteristic, complemented
4#21451627C212yesmaximal
5#21451096C214yesmaximal, complemented
6#21450906C214yesmaximal, complemented
7#21450540C214yesmaximal, complemented
8#21449843C214yesmaximal, complemented
9#21448554C212yesmaximal, complemented
10#21448416C212yesmaximal
11#21448347C212yesmaximal, complemented
12C2×#16839609C211yesmaximal, characteristic
13C61×#2659C211yesmaximal, characteristic, complemented
14C23×C4×D61C2211yescharacteristic
15C2×#14519768C2212yescomplemented
16C2×#14519756C2212yes—
17C2×#14519753C2212yes—
18C2×#14519753C2212yes—
19C2×#14519742C2212yescomplemented
20C22×#11758198C2211yescharacteristic
21C22×#11758198C2211yescharacteristic
22C22×C4×Dic61C2211yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24#16839699C2214yes—
25#16839694C2214yescomplemented
26#16839684C2214yes—
27#16839684C2214yes—
28#16839683C2214yescomplemented
29#16839675C2214yes—
30#16839669C2214yescomplemented
31#16839667C2212yescomplemented
32#16839661C2212yescomplemented
33#16839661C2214yescomplemented
34#16839658C2212yescomplemented
35#16839658C2212yescomplemented
36#16839654C2214yes—
37#16839654C2214yes—
38#16839652C2214yes—
39#16839652C2214yes—
40#16839649C2214yescomplemented
41#16839648C2214yescomplemented
42#16839648C2214yescomplemented
43#16839645C2214yescomplemented
44#16839644C2214yescomplemented
45#16839637C2214yes—
46#16839634C2214yes—
47#16839633C2214yes—
48#16839633C2214yes—
49#16839627C2212yescomplemented
50#16839622C2212yes—
51#16839622C2212yes—
52#16839619C2212yes—
53#16839619C2212yes—
54#16839617C2214yes—
55C4⋊C4×Dic61C2212yescomplemented
56#16839613C2214yescomplemented
57C22⋊C4×Dic61C2214yes—
58#16839609C2214yes—
59#16839609C2214yes—
60C2×#14519684C2211yescharacteristic
61C2×C22⋊Q8×C61C2212yes—
62C61×#398C2214yescomplemented
63C61×#395C2214yescomplemented
64C61×#380C2212yes—
65C61×#380C2212yes—
66C2×#95×C61C2211yescharacteristic
67C24×D61C2311yescharacteristic
68C22×Dic122C2312yes—
69C23×Dic61C2311yescharacteristic
70C23×Dic61C2311yescharacteristic
71C23×Dic61C2311yescharacteristic
72C22×C4×D61—24no—
73C22×C4×D61—42no—
74C22×C4×D61—42no—
75C22×C4×D61C2312yes—
76C22×C4×D61C2312yes—
77#14519768—44no—
78#14519756—28no—
79#14519753—44no—
80#14519753—44no—
81#14519742—44no—
82C2×C4⋊Dic61C2312yes—
83C2×C4⋊Dic61C2312yes—
84C2×C4⋊Dic61C2312yes—
85C2×C4⋊Dic61C2314yes—
86C2×C4⋊Dic61C2314yes—
87C2×C4⋊Dic61C2312yes—
88C2×C4⋊Dic61C2312yes—
89C2×#11758196C2314yes—
90C2×#11758196C2314yes—
91C2×#11758196C2314yes—
92C2×C22⋊Dic61C2314yes—
93C2×#11758198—24no—
94C2×#11758198Q818yescomplemented
95C2×#11758198C2314yes—
96C2×#11758198C2314yes—
97C2×#11758198C2314yes—
98C2×#11758198C2314yes—
99C2×#11758198C2314yes—
100C2×#11758198C2314yes—
101C2×C4×Dic61—42no—
102C2×C4×Dic61—42no—
103C2×C4×Dic61C2312yes—
104C2×C4×Dic61C2312yes—
105C2×C4×Dic61C2312yes—
106C2×C4×Dic61C2312yes—
107C2×C4×Dic61C2314yes—
108C2×C4×Dic61C2314yes—
109#14519684C2314yes—
110#14519684C2314yes—
111#14519684C2314yes—
112#14519684C2314yes—
113#14519684C2314yes—
114#14519684C2314yes—
115#14519684C2314yes—
116#14519684C2314yes—
117#14519684C2314yes—
118C22×Q8×C61C2312yes—
119C23×C4×C61C2311yescharacteristic
120C23×C4×C61C2311yescharacteristic
121C23×C4×C61C2311yescharacteristic
122C22⋊Q8×C61—28no—
123C2×C4⋊C4×C61C2314yes—
124C2×C4⋊C4×C61C2312yes—
125C2×C4⋊C4×C61C2312yes—
126C2×C4⋊C4×C61C2312yes—
127C2×C4⋊C4×C61C2312yes—
128C2×C4⋊C4×C61C2312yes—
129C2×C22⋊C4×C61C2314yes—
130#95×C61C2314yes—
131#95×C61C2314yes—
132#95×C61C2314yes—
133C24×C61C2411yescharacteristic
134C23×D61—22no—
135C23×D61—22no—
136C23×D61—41no—
137C23×D61—41no—
138C23×D61—41no—
139C23×D61—41no—
140C23×D61C2×Q814yes—
141C23×D61C2412yes—
142C2×Q8×C61—24no—
143C2×Q8×C61—42no—
144C2×Q8×C61—42no—
145C22×C4×C61—41no—
146C22×C4×C61—41no—
147C22×C4×C61—41no—
148C22×C4×C61—41no—
149C22×C4×C61—41no—
150C22×C4×C61—41no—
151C22×C4×C61C2×Q814yes—
152C22×C4×C61#4114yes—
153C22×C4×C61#4114yes—
154C22×C4×C61C2414yes—
155C22×C4×C61C2414yes—
156C22×C4×C61C2412yes—
157C22×C4×C61C2412yes—
158C22×C4×C61C2412yes—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—41no—
164C22×Dic61—41no—
165C22×Dic61—41no—
166C22×Dic61—41no—
167C22×Dic61C2×Q814yes—
168C22×Dic61C2412yes—
169C22×Dic61C2414yes—
170C22×Dic61C2414yes—
171C22×Dic61C2412yes—
172C22×Dic61C2411yescharacteristic
173C22×Dic61C2411yescharacteristic
174C2×C4×D61—24no—
175C2×C4×D61—24no—
176C2×C4×D61—24no—
177C2×C4×D61—24no—
178C2×C4×D61—42no—
179C2×C4×D61—42no—
180C2×C4×D61—416no—
181C2×Dic122—24no—
182C2×Dic122—42no—
183C2×Dic122—42no—
184C4⋊C4×C61—24no—
185C4⋊C4×C61—28no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61—42no—
188C4⋊C4×C61—42no—
189C4⋊C4×C61—42no—
190C22⋊C4×C61—28no—
191C22⋊Dic61—28no—
192#11758198—28no—
193#11758198—44no—
194#11758198—44no—
195#11758198—44no—
196#11758198—44no—
197#11758198C2×Q8116yes—
198C4⋊Dic61—24no—
199C4⋊Dic61—44no—
200C4⋊Dic61—42no—
201C4⋊Dic61—42no—
202C4⋊Dic61—42no—
203C4⋊Dic61—44no—
204C4⋊Dic61—42no—
205#11758196—28no—
206#11758196—28no—
207#11758196—28no—
208C4×Dic61—24no—
209C4×Dic61—24no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C4×Dic61—44no—
213C4×Dic61—44no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61C22×Q812yes—
219C23×C61C2×#4112yes—
220C23×C61C2×#4112yes—
221C23×C61C2511yescharacteristic
222C22×D61—28no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—44no—
230C22×D61—44no—
231C22×D61—44no—
232C22×D61—44no—
233C22×D61—44no—
234C22×D61—44no—
235C22×D61C22×Q814yes—
236Q8×C61—84no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—44no—
250C2×C4×C61—44no—
251C2×C4×C61—44no—
252C2×C4×C61C22×Q814yes—
253C2×C4×C61C2×#4114yes—
254C2×C4×C61C2×#4114yes—
255C2×Dic61—28no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—24no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—24no—
267C2×Dic61—24no—
268C2×Dic61—24no—
269C2×Dic61—22no—
270C2×Dic61—24no—
271C2×Dic61—44no—
272C2×Dic61—44no—
273C2×Dic61C22×Q814yes—
274C2×Dic61H(5,𝔽2)12yes—
275C2×Dic61H(5,𝔽2)12yes—
276C4×D61—48no—
277C4×D61—48no—
278Dic122—48no—
279#35012—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61#53314yes—
287C22×C61C23×Q811yescharacteristic
288C22×C61C22×#4111yescharacteristic
289C22×C61C22×#4111yescharacteristic
290C22×C61C2×H(5,𝔽2)11yescharacteristic
291C22×C61C2×H(5,𝔽2)11yescharacteristic
292C22×C61#58511yescharacteristic
293C22×C61#58511yescharacteristic
294C2×D61—24no—
295C2×D61—24no—
296C2×D61—24no—
297C2×D61—416no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C4×C61—24no—
303C4×C61—24no—
304C4×C61—24no—
305C4×C61—44no—
306C4×C61—44no—
307Dic61—22no—
308Dic61—22no—
309Dic61—24no—
310Dic61—44no—
311Dic61—44no—
312Dic61—42no—
313#2851—612no—
314#2848—612no—
315#2659—611no—
316#2659D6111yescharacteristic, complemented
317#2626—612no—
318#2574—614no—
319#2556—614no—
320#2522—614no—
321#2465—614no—
322#2329—612no—
323#2312—612no—
324C4×C22⋊Q8—612no—
325C2×C4×C22⋊C4—611no—
326C2×C61—24no—
327C2×C61C2×#53311yescharacteristic
328C2×C61#345911yescharacteristic
329C2×C61#346011yescharacteristic
330C2×C61#349011yescharacteristic
331C2×C61#349311yescharacteristic
332C2×C61#350111yescharacteristic
333C2×C61#350111yescharacteristic
334D61—44no—
335C24×C4—611no—
336C2×#125—612no—
337C2×C22⋊Q8—612no—
338C2×C22⋊Q8—612no—
339C2×C22⋊Q8—612no—
340C2×C22⋊Q8C2×D6112yes—
341C2×C4×Q8—612no—
342C22×C22⋊C4—611no—
343C22×C22⋊C4—611no—
344C22×C42—611no—
345C22×C42—611no—
346#399—614no—
347#398—614no—
348#398C2×D6114yescomplemented
349#396—614no—
350#396—614no—
351#395—614no—
352#395C2×D6114yescomplemented
353#393—614no—
354#391—614no—
355#391—612no—
356#389—614no—
357#389—612no—
358#389—612no—
359#389—612no—
360#388—614no—
361#388—614no—
362#388—614no—
363#388—614no—
364#387—614no—
365#387—614no—
366#387—614no—
367#387—614no—
368#387—614no—
369#385—614no—
370#385—614no—
371#385—614no—
372#385—614no—
373#382—612no—
374#380—612no—
375#380—612no—
376#380—612no—
377#380—612no—
378#380C2×D6112yes—
379#380C2×D6112yes—
380#379—614no—
381C4×C4⋊C4—612no—
382C4×C4⋊C4—614no—
383C4×C22⋊C4—614no—
384C4×C22⋊C4—614no—
385C4×C22⋊C4—614no—
386C2×#95—611no—
387C2×#95C2×D6111yescharacteristic
388C61#3501211yescharacteristic, complemented, Sylow
389C25—611no—
390C22×Q8—612no—
391C22×Q8C22×D6112yes—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—612no—
396C23×C4—612no—
397C23×C4—1224no—
398C23×C4—2442no—
399C23×C4—2442no—
400C23×C4C22×D6111yescharacteristic
401C23×C4C22×D6111yescharacteristic
402C23×C4C22×D6111yescharacteristic
403#125—2444no—
404C22⋊Q8—28no—
405C22⋊Q8—1228no—
406C22⋊Q8—2444no—
407C22⋊Q8—2444no—
408C4×Q8—2444no—
409C2×C4⋊C4—614no—
410C2×C4⋊C4—614no—
411C2×C4⋊C4—614no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—612no—
414C2×C4⋊C4—612no—
415C2×C4⋊C4—612no—
416C2×C4⋊C4—614no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—614no—
419C2×C4⋊C4C22×D6112yes—
420C2×C4⋊C4C22×D6114yes—
421C2×C4⋊C4C22×D6112yes—
422C2×C4⋊C4C22×D6112yes—
423C2×C4⋊C4C22×D6112yes—
424C2×C4⋊C4C22×D6112yes—
425C2×C22⋊C4—618no—
426C2×C22⋊C4—614no—
427C2×C22⋊C4—614no—
428C2×C22⋊C4—614no—
429C2×C22⋊C4—614no—
430C2×C22⋊C4—614no—
431C2×C22⋊C4—614no—
432C2×C22⋊C4—614no—
433C2×C22⋊C4—1224no—
434C2×C22⋊C4C22×D6114yes—
435C2×C42—612no—
436C2×C42—612no—
437C2×C42—614no—
438C2×C42—612no—
439C2×C42—612no—
440C2×C42—614no—
441C2×C42—2442no—
442C2×C42—2442no—
443#95—614no—
444#95—614no—
445#95—614no—
446#95—614no—
447#95—614no—
448#95—614no—
449#95—614no—
450#95—614no—
451#95—614no—
452#95C22×D6114yes—
453#95C22×D6114yes—
454#95C22×D6114yes—
455C24—612no—
456C24—614no—
457C24—1222no—
458C24—1222no—
459C24—2441no—
460C24—2441no—
461C24—2441no—
462C24—2441no—
463C24C23×D6111yescharacteristic
464C2×Q8—24no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—1224no—
468C2×Q8—2442no—
469C2×Q8—2442no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—612no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—614no—
480C22×C4—612no—
481C22×C4—614no—
482C22×C4—614no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1224no—
490C22×C4—1224no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—24416no—
494C22×C4—2441no—
495C22×C4—2441no—
496C22×C4—2441no—
497C22×C4—2441no—
498C22×C4#1175822414yes—
499C22×C4Q8×D6114yes—
500C22×C4#1175822614yes—
501C22×C4C23×D6114yes—
502C22×C4C23×D6114yes—
503C22×C4C23×D6112yes—
504C22×C4C23×D6112yes—
505C22×C4C23×D6112yes—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—1228no—
513C4⋊C4—1228no—
514C4⋊C4—1228no—
515C4⋊C4—1224no—
516C4⋊C4—2442no—
517C4⋊C4—2444no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2444no—
522C22⋊C4—28no—
523C22⋊C4—6116no—
524C22⋊C4—1228no—
525C22⋊C4—1228no—
526C22⋊C4—2444no—
527C22⋊C4—2444no—
528C22⋊C4—2444no—
529C22⋊C4—2444no—
530C42—1224no—
531C42—1224no—
532C42—2442no—
533C42—2442no—
534C42—2444no—
535C42—2444no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—614no—
541C23—1228no—
542C23—1222no—
543C23—1222no—
544C23—1222no—
545C23—1222no—
546C23—1222no—
547C23—1222no—
548C23—2444no—
549C23—2444no—
550C23—2444no—
551C23—2444no—
552C23—2444no—
553C23—2444no—
554C23C2×Q8×D6112yes—
555C23C2×#1175822412yes—
556C23C2×#1175822612yes—
557C23C24×D6111yescharacteristic
558Q8—84no—
559Q8—2448no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—614no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1228no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1222no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1222no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1222no—
594C2×C4—2448no—
595C2×C4—2448no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4C2×Q8×D6114yes—
599C2×C4C2×#1175822414yes—
600C2×C4C2×#1175822614yes—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—1224no—
608C22—1224no—
609C22—1224no—
610C22—24416no—
611C22—2442no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22#1684031714yes—
616C22C22×Q8×D6111yescharacteristic
617C22C22×#1175822411yescharacteristic
618C22C22×#1175822611yescharacteristic
619C22H(5,𝔽2)×D6111yescharacteristic
620C22H(5,𝔽2)×D6111yescharacteristic
621C22#1684059711yescharacteristic
622C22#1684059911yescharacteristic
623C4—24no—
624C4—24no—
625C4—24no—
626C4—44no—
627C4—44no—
628C4—1222no—
629C4—1222no—
630C4—1224no—
631C4—2444no—
632C4—2444no—
633C4—2442no—
634C2—24no—
635C2—2444no—
636C2C2×#1684031711yescharacteristic
637C2#2145923111yescharacteristic
638C2#2145924211yescharacteristic
639C2#2145950711yescharacteristic
640C2#2145953811yescharacteristic
641C2#2145961911yescharacteristic
642C2#2145962111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2659C2
prime radicalC24×D61C23
hypercenter#2659D61
hypocenterC61#35012
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2659

Complements

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

NH ≅ G/Ntype
1. #21454127C2semidirect
3. #21451933C2semidirect
5. #21451096C2semidirect
6. #21450906C2semidirect
7. #21450540C2semidirect
8. #21449843C2semidirect
9. #21448554C2semidirect
11. #21448347C2semidirect
13. C61×#2659C2semidirect
15. C2×#14519768C22semidirect
19. C2×#14519742C22semidirect
25. #16839694C22semidirect
28. #16839683C22semidirect
30. #16839669C22semidirect
31. #16839667C22semidirect
32. #16839661C22semidirect
33. #16839661C22semidirect
34. #16839658C22semidirect
35. #16839658C22semidirect
40. #16839649C22semidirect
41. #16839648C22semidirect
42. #16839648C22semidirect
43. #16839645C22semidirect
44. #16839644C22semidirect
49. #16839627C22semidirect
55. C4⋊C4×Dic61C22semidirect
56. #16839613C22semidirect
62. C61×#398C22semidirect
63. C61×#395C22semidirect
94. C2×#11758198Q8semidirect
316. #2659D61semidirect
348. #398C2×D61semidirect
352. #395C2×D61semidirect
388. C61#35012semidirect

Maximal subgroups

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

subgroupindex
#214541272
#214541032
#214519332
#214516272
#214510962
#214509062
#214505402
#214498432
#214485542
#214484162
#214483472
C2×#168396092
C61×#26592
#3501261