Finite Group Databasebeta

← #29853066

Subgroups — #29853066

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#21454129C212yesmaximal, complemented
2#21454106C212yesmaximal
3#21451933C211yesmaximal, characteristic, complemented
4#21451933C211yesmaximal, characteristic, complemented
5#21451624C212yesmaximal
6#21451093C214yesmaximal, complemented
7#21450903C214yesmaximal, complemented
8#21450537C214yesmaximal, complemented
9#21449840C214yesmaximal, complemented
10#21448552C212yesmaximal, complemented
11#21448417C212yesmaximal
12C4×#14519756C212yesmaximal, complemented
13C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
14C2×#14519769C2212yescomplemented
15C2×#14519756C2212yes—
16C2×#14519756C2212yes—
17C2×#14519756C2212yes—
18C2×#14519756C2212yes—
19C2×C4×Dic122C2212yescomplemented
20#16839698C2214yes—
21#16839694C2214yescomplemented
22#16839694C2214yescomplemented
23#16839685C2214yes—
24#16839685C2214yes—
25#16839683C2214yescomplemented
26#16839683C2214yescomplemented
27#16839674C2214yes—
28#16839668C2212yescomplemented
29#16839666C2214yescomplemented
30#16839660C2212yescomplemented
31#16839660C2212yescomplemented
32#16839660C2212yescomplemented
33#16839659C2214yescomplemented
34#16839655C2214yes—
35#16839655C2214yes—
36#16839655C2214yes—
37#16839655C2214yes—
38#16839649C2214yescomplemented
39#16839646C2214yescomplemented
40#16839646C2214yescomplemented
41#16839646C2214yescomplemented
42#16839645C2214yescomplemented
43#16839635C2214yes—
44#16839635C2214yes—
45#16839635C2214yes—
46#16839635C2214yes—
47#16839628C2212yescomplemented
48#16839622C2212yes—
49#16839622C2212yes—
50#16839622C2212yes—
51#16839622C2212yes—
52#16839622C2212yes—
53#16839622C2212yes—
54#16839618C2214yes—
55C4×#11758196C2214yescomplemented
56C4×C4⋊Dic61C2212yescomplemented
57C4×C22⋊Dic61C2214yes—
58C24×C4×C61C2211yescharacteristic
59C22×C22⋊C4×C61C2211yescharacteristic
60C22×C22⋊C4×C61C2211yescharacteristic
61C22×C42×C61C2211yescharacteristic
62C22×C42×C61C2211yescharacteristic
63C4×C22⋊C4×C61C2214yes—
64C4×C22⋊C4×C61C2214yes—
65C2×#95×C61C2211yescharacteristic
66C2×#95×C61C2211yescharacteristic
67C22×Dic122C2312yes—
68C22×Dic122C2312yes—
69#14519769—44no—
70#14519756—28no—
71#14519756—28no—
72#14519756—44no—
73#14519756—44no—
74C4×Dic122—44no—
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2312yes—
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2312yes—
80C2×C4⋊Dic61C2312yes—
81C2×C4⋊Dic61C2312yes—
82C2×C4⋊Dic61C2312yes—
83C2×C4⋊Dic61C2312yes—
84C2×C4⋊Dic61C2312yes—
85C2×#11758196C2314yes—
86C2×#11758196C2314yes—
87C2×#11758196C2314yes—
88C2×#11758196C2314yes—
89C2×#11758196C2314yes—
90C2×#11758196C2314yes—
91C2×C22⋊Dic61C2314yes—
92C2×C22⋊Dic61C2314yes—
93C2×C22⋊Dic61C2314yes—
94C2×C22⋊Dic61C2314yes—
95C2×C22⋊Dic61C2314yes—
96C2×C22⋊Dic61C2314yes—
97C2×C4×Dic61C2314yes—
98C2×C4×Dic61C2314yes—
99#14519684C2314yes—
100#14519684C2314yes—
101#14519684C2314yes—
102#14519684C2314yes—
103#14519684C2314yes—
104#14519684C2314yes—
105#14519684C2314yes—
106#14519684C2314yes—
107#14519684C2314yes—
108#14519684C2314yes—
109C25×C61C2311yescharacteristic
110C23×C4×C61—24no—
111C23×C4×C61—42no—
112C23×C4×C61—42no—
113C23×C4×C61C2311yescharacteristic
114C23×C4×C61C2312yes—
115C23×C4×C61C2312yes—
116C23×C4×C61C2311yescharacteristic
117C23×C4×C61C2311yescharacteristic
118C23×C4×C61C2311yescharacteristic
119C23×C4×C61C2311yescharacteristic
120C23×C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—24no—
122C2×C22⋊C4×C61Q818yescomplemented
123C2×C22⋊C4×C61C2314yes—
124C2×C22⋊C4×C61C2314yes—
125C2×C42×C61—42no—
126C2×C42×C61—42no—
127C2×C42×C61C2312yes—
128C2×C42×C61C2312yes—
129C2×C42×C61C2312yes—
130C2×C42×C61C2312yes—
131#95×C61C2314yes—
132#95×C61C2314yes—
133C24×C61—22no—
134C24×C61—22no—
135C24×C61—41no—
136C24×C61—41no—
137C24×C61—41no—
138C24×C61—41no—
139C24×C61C2×Q814yes—
140C24×C61C2412yes—
141C24×C61C2411yescharacteristic
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—24no—
147C22×C4×C61—24no—
148C22×C4×C61—24no—
149C22×C4×C61—24no—
150C22×C4×C61—41no—
151C22×C4×C61—41no—
152C22×C4×C61—41no—
153C22×C4×C61—41no—
154C22×C4×C61—41no—
155C22×C4×C61—41no—
156C22×C4×C61—42no—
157C22×C4×C61—42no—
158C22×C4×C61—41no—
159C22×C4×C61—41no—
160C22×C4×C61—41no—
161C22×C4×C61—41no—
162C22×C4×C61—416no—
163C22×C4×C61C2×Q814yes—
164C22×C4×C61C2×Q814yes—
165C22×C4×C61#4114yes—
166C22×C4×C61#4114yes—
167C22×C4×C61C2412yes—
168C22×C4×C61C2412yes—
169C22×C4×C61C2412yes—
170C22×C4×C61C2412yes—
171C22×C4×C61C2412yes—
172C22×C4×C61C2411yescharacteristic
173C22×C4×C61C2411yescharacteristic
174C22×Dic61C2414yes—
175C22×Dic61C2414yes—
176C22×Dic61C2414yes—
177C22×Dic61C2414yes—
178C2×Dic122—24no—
179C2×Dic122—24no—
180C2×Dic122—42no—
181C2×Dic122—42no—
182C2×Dic122—42no—
183C2×Dic122—42no—
184C22⋊C4×C61—28no—
185C22⋊C4×C61C2×Q8116yes—
186C42×C61—24no—
187C42×C61—24no—
188C42×C61—42no—
189C42×C61—42no—
190C22⋊Dic61—28no—
191C22⋊Dic61—28no—
192C22⋊Dic61—44no—
193C22⋊Dic61—44no—
194C22⋊Dic61—44no—
195C22⋊Dic61—44no—
196C4⋊Dic61—24no—
197C4⋊Dic61—24no—
198C4⋊Dic61—42no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61—42no—
202C4⋊Dic61—42no—
203C4⋊Dic61—42no—
204C4⋊Dic61—42no—
205C4⋊Dic61—42no—
206#11758196—28no—
207#11758196—28no—
208#11758196—28no—
209#11758196—28no—
210#11758196—44no—
211#11758196—44no—
212C4×Dic61—44no—
213C4×Dic61—44no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—28no—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—44no—
226C23×C61—44no—
227C23×C61—44no—
228C23×C61—44no—
229C23×C61—44no—
230C23×C61—44no—
231C23×C61C22×Q814yes—
232C23×C61C22×Q812yes—
233C23×C61C2×#4112yes—
234C23×C61C2×#4112yes—
235C23×C61C2511yescharacteristic
236C2×C4×C61—28no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—44no—
253C2×C4×C61—44no—
254C2×C4×C61—44no—
255C2×C4×C61—44no—
256C2×C4×C61—48no—
257C2×C4×C61—48no—
258C2×C4×C61—44no—
259C2×C4×C61C22×Q814yes—
260C2×C4×C61C22×Q814yes—
261C2×C4×C61C2×#4114yes—
262C2×C4×C61C2×#4114yes—
263C2×C4×C61H(5,𝔽2)12yes—
264C2×C4×C61H(5,𝔽2)12yes—
265C2×Dic61—24no—
266C2×Dic61—24no—
267C2×Dic61—24no—
268C2×Dic61—24no—
269C2×Dic61—24no—
270C2×Dic61—24no—
271C2×Dic61—24no—
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—24no—
275C2×Dic61—24no—
276C2×Dic61—24no—
277Dic122—48no—
278Dic122—84no—
279#35012—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—24no—
287C22×C61—24no—
288C22×C61—24no—
289C22×C61—416no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61#53314yes—
295C22×C61C23×Q811yescharacteristic
296C22×C61C22×#4111yescharacteristic
297C22×C61C22×#4111yescharacteristic
298C22×C61C2×H(5,𝔽2)11yescharacteristic
299C22×C61C2×H(5,𝔽2)11yescharacteristic
300C22×C61#58511yescharacteristic
301C22×C61#58511yescharacteristic
302C4×C61—24no—
303C4×C61—24no—
304C4×C61—22no—
305C4×C61—22no—
306C4×C61—24no—
307C4×C61—24no—
308C4×C61—42no—
309Dic61—44no—
310Dic61—44no—
311Dic61—44no—
312Dic61—44no—
313#2851—612no—
314#2848—612no—
315#2659—611no—
316#2659—611no—
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⋊C4D6111yescharacteristic
326C2×C61—24no—
327C2×C61—44no—
328C2×C61C2×#53311yescharacteristic
329C2×C61#345911yescharacteristic
330C2×C61#346011yescharacteristic
331C2×C61#349011yescharacteristic
332C2×C61#349311yescharacteristic
333C2×C61#350111yescharacteristic
334C2×C61#350111yescharacteristic
335C24×C4C2×D6111yescharacteristic
336C2×#125—612no—
337C2×C22⋊Q8—612no—
338C2×C22⋊Q8—612no—
339C2×C22⋊Q8—612no—
340C2×C22⋊Q8—612no—
341C2×C4×Q8—612no—
342C22×C22⋊C4C2×D6111yescharacteristic
343C22×C22⋊C4C2×D6111yescharacteristic
344C22×C42C2×D6111yescharacteristic
345C22×C42C2×D6111yescharacteristic
346#399—614no—
347#398—614no—
348#398—614no—
349#396—614no—
350#396—614no—
351#395—614no—
352#395—614no—
353#393—614no—
354#391—612no—
355#391—614no—
356#389—612no—
357#389—612no—
358#389—612no—
359#389—614no—
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#380—612no—
379#380—612no—
380#379—614no—
381C4×C4⋊C4—614no—
382C4×C4⋊C4—612no—
383C4×C22⋊C4—614no—
384C4×C22⋊C4C2×D6114yes—
385C4×C22⋊C4C2×D6114yes—
386C2×#95C2×D6111yescharacteristic
387C2×#95C2×D6111yescharacteristic
388C61#3501211yescharacteristic, complemented, Sylow
389C25C22×D6111yescharacteristic
390C22×Q8—612no—
391C22×Q8—612no—
392C23×C4—24no—
393C23×C4—42no—
394C23×C4—42no—
395C23×C4C22×D6111yescharacteristic
396C23×C4C22×D6112yes—
397C23×C4C22×D6112yes—
398C23×C4C22×D6111yescharacteristic
399C23×C4C22×D6111yescharacteristic
400C23×C4C22×D6111yescharacteristic
401C23×C4C22×D6111yescharacteristic
402C23×C4C22×D6111yescharacteristic
403#125—2444no—
404C22⋊Q8—1228no—
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—614no—
415C2×C4⋊C4—612no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—612no—
419C2×C4⋊C4—612no—
420C2×C4⋊C4—614no—
421C2×C4⋊C4—612no—
422C2×C4⋊C4—612no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4—614no—
425C2×C22⋊C4—24no—
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⋊C4Dic12218yescomplemented
433C2×C22⋊C4C22×D6114yes—
434C2×C22⋊C4C22×D6114yes—
435C2×C42—42no—
436C2×C42—42no—
437C2×C42—614no—
438C2×C42—614no—
439C2×C42C22×D6112yes—
440C2×C42C22×D6112yes—
441C2×C42C22×D6112yes—
442C2×C42C22×D6112yes—
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#95—614no—
453#95C22×D6114yes—
454#95C22×D6114yes—
455C24—22no—
456C24—22no—
457C24—41no—
458C24—41no—
459C24—41no—
460C24—41no—
461C24C2×Dic12214yes—
462C24C23×D6112yes—
463C24C23×D6111yescharacteristic
464C2×Q8—1224no—
465C2×Q8—1224no—
466C2×Q8—2442no—
467C2×Q8—2442no—
468C2×Q8—2442no—
469C2×Q8—2442no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—416no—
491C22×C4—614no—
492C22×C4—614no—
493C22×C4—614no—
494C22×C4—614no—
495C22×C4C2×Dic12214yes—
496C22×C4C2×Dic12214yes—
497C22×C4#1175822214yes—
498C22×C4#1175822214yes—
499C22×C4C23×D6112yes—
500C22×C4C23×D6112yes—
501C22×C4C23×D6112yes—
502C22×C4C23×D6112yes—
503C22×C4C23×D6112yes—
504C22×C4C23×D6111yescharacteristic
505C22×C4C23×D6111yescharacteristic
506C4⋊C4—1228no—
507C4⋊C4—1228no—
508C4⋊C4—1224no—
509C4⋊C4—1224no—
510C4⋊C4—1228no—
511C4⋊C4—1228no—
512C4⋊C4—2444no—
513C4⋊C4—2444no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C22⋊C4—28no—
523C22⋊C4—1228no—
524C22⋊C4—1228no—
525C22⋊C4—2444no—
526C22⋊C4—2444no—
527C22⋊C4—2444no—
528C22⋊C4—2444no—
529C22⋊C4C2×Dic122116yes—
530C42—24no—
531C42—24no—
532C42—42no—
533C42—42no—
534C42—2444no—
535C42—2444no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C22×Dic12214yes—
554C23C22×Dic12212yes—
555C23C2×#1175822212yes—
556C23C2×#1175822212yes—
557C23C24×D6111yescharacteristic
558Q8—4884no—
559Q8—2448no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—1224no—
591C2×C4—1224no—
592C2×C4—1224no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4C22×Dic12214yes—
596C2×C4C22×Dic12214yes—
597C2×C4C2×#1175822214yes—
598C2×C4C2×#1175822214yes—
599C2×C4#1451982212yes—
600C2×C4#1451982212yes—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—416no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1684031914yes—
616C22C23×Dic12211yescharacteristic
617C22C22×#1175822211yescharacteristic
618C22C22×#1175822211yescharacteristic
619C22C2×#1451982211yescharacteristic
620C22C2×#1451982211yescharacteristic
621C22#1684059811yescharacteristic
622C22#1684059811yescharacteristic
623C4—24no—
624C4—22no—
625C4—22no—
626C4—24no—
627C4—24no—
628C4—24no—
629C4—42no—
630C4—2444no—
631C4—2444no—
632C4—2444no—
633C4—2444no—
634C2—24no—
635C2—44no—
636C2C2×#1684031911yescharacteristic
637C2#2145922811yescharacteristic
638C2#2145924111yescharacteristic
639C2#2145950811yescharacteristic
640C2#2145953911yescharacteristic
641C2#2145962211yescharacteristic
642C2#2145962211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC2×C4×C22⋊C4×C61C2
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#35012
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #21454129C2semidirect
3. #21451933C2semidirect
4. #21451933C2semidirect
6. #21451093C2semidirect
7. #21450903C2semidirect
8. #21450537C2semidirect
9. #21449840C2semidirect
10. #21448552C2semidirect
12. C4×#14519756C2semidirect
14. C2×#14519769C22semidirect
19. C2×C4×Dic122C22semidirect
21. #16839694C22semidirect
22. #16839694C22semidirect
25. #16839683C22semidirect
26. #16839683C22semidirect
28. #16839668C22semidirect
29. #16839666C22semidirect
30. #16839660C22semidirect
31. #16839660C22semidirect
32. #16839660C22semidirect
33. #16839659C22semidirect
38. #16839649C22semidirect
39. #16839646C22semidirect
40. #16839646C22semidirect
41. #16839646C22semidirect
42. #16839645C22semidirect
47. #16839628C22semidirect
55. C4×#11758196C22semidirect
56. C4×C4⋊Dic61C22semidirect
122. C2×C22⋊C4×C61Q8semidirect
388. C61#35012semidirect
432. C2×C22⋊C4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214541292
#214541062
#214519332
#214516242
#214510932
#214509032
#214505372
#214498402
#214485522
#214484172
C4×#145197562
C2×C4×C22⋊C4×C612
#3501261