Finite Group Databasebeta

← #29852764

Subgroups — #29852764

Order 15616.

2214 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454132C214yesmaximal, complemented
2#21451774C214yesmaximal, complemented
3#21450383C214yesmaximal, complemented
4#21450208C214yesmaximal, complemented
5#21450085C214yesmaximal, complemented
6#21450062C214yesmaximal, complemented
7#21448433C211yesmaximal, characteristic, complemented
8#21448431C211yesmaximal, characteristic, complemented
9#21448397C211yesmaximal, characteristic
10C2×#16839642C211yesmaximal, characteristic
11C2×#16839634C211yesmaximal, characteristic, complemented
12C2×C22⋊C4×Dic61C211yesmaximal, characteristic, complemented
13C61×#2308C211yesmaximal, characteristic
14C24×Dic61C2211yescharacteristic
15C2×#14519770C2214yescomplemented
16C22×#11758196C2212yescomplemented
17C22×C22⋊Dic61C2211yescharacteristic
18C22×C22⋊Dic61C2211yescharacteristic
19C22×C22⋊Dic61C2211yescharacteristic
20C22×C22⋊Dic61C2212yes—
21C22×C4×Dic61C2212yescomplemented
22C61⋊#401C2214yescomplemented
23#16839704C2214yescomplemented
24#16839703C2214yescomplemented
25#16839702C2214yescomplemented
26#16839690C2214yescomplemented
27#16839690C2214yescomplemented
28#16839687C2214yescomplemented
29#16839685C2214yescomplemented
30#16839656C2214yescomplemented
31#16839655C2214yescomplemented
32#16839650C2214yescomplemented
33#16839649C2214yescomplemented
34#16839647C2214yescomplemented
35#16839644C2214yescomplemented
36#16839642C2212yes—
37#16839642C2214yes—
38#16839642C2212yes—
39#16839642C2214yes—
40#16839642C2214yes—
41#16839637C2214yes—
42#16839637C2214yescomplemented
43#16839635C2214yescomplemented
44#16839634C2214yes—
45#16839634C2212yescomplemented
46#16839634C2212yescomplemented
47#16839634C2214yescomplemented
48#16839629C2214yescomplemented
49#16839625C2212yes—
50#16839622C2212yescomplemented
51#16839622C2212yescomplemented
52#16839620C2214yes—
53#16839618C2212yes—
54C22⋊C4×Dic61C2214yes—
55C22⋊C4×Dic61C2214yescomplemented
56C2×#14519684C2212yes—
57C2×#14519684C2212yescomplemented
58C2×#14519684C2212yescomplemented
59C23×D4×C61C2211yescharacteristic
60C22×C22⋊C4×C61C2211yescharacteristic
61C22×C22⋊C4×C61C2211yescharacteristic
62C61×#386C2214yes—
63C61×#386C2212yes—
64C61×#386C2212yes—
65C61×#381C2212yes—
66C61×#379C2212yes—
67C23×Dic61—24no—
68C23×Dic61—44no—
69C23×Dic61C2311yescharacteristic
70C23×Dic61C2312yes—
71C23×Dic61C2312yes—
72C23×Dic61C2312yes—
73C23×Dic61C2312yes—
74C23×Dic61C2312yes—
75C23×Dic61C2311yescharacteristic
76#14519770—48no—
77C2×C4⋊Dic61C2314yes—
78C2×C4⋊Dic61C2314yes—
79C2×#11758196—44no—
80C2×#11758196C2314yes—
81C2×#11758196C2314yes—
82C2×#11758196C2314yes—
83C2×#11758196C2314yes—
84C2×C22⋊Dic61—24no—
85C2×C22⋊Dic61—24no—
86C2×C22⋊Dic61—24no—
87C2×C22⋊Dic61—44no—
88C2×C22⋊Dic61C2314yes—
89C2×C22⋊Dic61C2312yes—
90C2×C22⋊Dic61C2312yes—
91C2×C22⋊Dic61C2312yes—
92C2×C22⋊Dic61C2312yes—
93C2×C22⋊Dic61C2312yes—
94C2×C22⋊Dic61C2312yes—
95C2×C22⋊Dic61C2314yes—
96C2×C22⋊Dic61C2314yes—
97C2×C22⋊Dic61C2314yes—
98C2×C4×Dic61—44no—
99C2×C4×Dic61C2314yes—
100C2×C4×Dic61C2314yes—
101#14519684C2314yes—
102#14519684C2314yes—
103#14519684C2314yes—
104#14519684C2314yes—
105#14519684C2314yes—
106#14519684C2314yes—
107#14519684C2314yes—
108#14519684C2314yes—
109#14519684C2314yes—
110#14519684C2314yes—
111#14519684C2314yes—
112#14519684C2314yes—
113C25×C61C2311yescharacteristic
114C25×C61C2311yescharacteristic
115C22×D4×C61—24no—
116C22×D4×C61—24no—
117C22×D4×C61—24no—
118C22×D4×C61C2312yes—
119C22×D4×C61C2312yes—
120C23×C4×C61C2311yescharacteristic
121C23×C4×C61C2312yes—
122C23×C4×C61C2311yescharacteristic
123C23×C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—24no—
125C2×C22⋊C4×C61—24no—
126C2×C22⋊C4×C61C2312yes—
127C2×C22⋊C4×C61C2312yes—
128C2×C22⋊C4×C61C2312yes—
129C2×C22⋊C4×C61C2312yes—
130C2×C22⋊C4×C61C2314yes—
131C2×C22⋊C4×C61C2314yes—
132#95×C61C2314yes—
133#95×C61C2314yes—
134C24×C61—22no—
135C24×C61—22no—
136C24×C61—22no—
137C24×C61—42no—
138C24×C61—42no—
139C24×C61—42no—
140C24×C61—42no—
141C24×C61#4114yes—
142C24×C61#4114yes—
143C24×C61#4114yes—
144C24×C61C2412yes—
145C24×C61C2411yescharacteristic
146C24×C61C2411yescharacteristic
147C24×C61C2411yescharacteristic
148C2×D4×C61—24no—
149C2×D4×C61—24no—
150C2×D4×C61—24no—
151C2×D4×C61—24no—
152C2×D4×C61—24no—
153C2×D4×C61—24no—
154C2×D4×C61—416no—
155C22×C4×C61—24no—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—22no—
161C22×C4×C61—42no—
162C22×C4×C61—42no—
163C22×C4×C61—42no—
164C22×C4×C61—42no—
165C22×C4×C61C2412yes—
166C22×C4×C61C2412yes—
167C22×C4×C61C2414yes—
168C22×C4×C61C2412yes—
169C22×Dic61—22no—
170C22×Dic61—24no—
171C22×Dic61—24no—
172C22×Dic61—24no—
173C22×Dic61—22no—
174C22×Dic61—24no—
175C22×Dic61—24no—
176C22×Dic61—24no—
177C22×Dic61—24no—
178C22×Dic61—24no—
179C22×Dic61—42no—
180C22×Dic61—42no—
181C22×Dic61—42no—
182C22×Dic61—42no—
183C22×Dic61—42no—
184C22×Dic61—42no—
185C22×Dic61—42no—
186C22×Dic61—42no—
187C22×Dic61—42no—
188C22×Dic61—416no—
189C22×Dic61C2412yes—
190C22×Dic61C2414yes—
191C22×Dic61C2414yes—
192C22×Dic61C2414yes—
193C22×Dic61C2412yes—
194C22⋊C4×C61—24no—
195C22⋊C4×C61—24no—
196C22⋊C4×C61—24no—
197C22⋊C4×C61—24no—
198C22⋊C4×C61—44no—
199C22⋊C4×C61—44no—
200C22⋊Dic61—24no—
201C22⋊Dic61—24no—
202C22⋊Dic61—24no—
203C22⋊Dic61—24no—
204C22⋊Dic61—24no—
205C22⋊Dic61—24no—
206C22⋊Dic61—28no—
207C22⋊Dic61—44no—
208C22⋊Dic61—44no—
209C22⋊Dic61—44no—
210C4⋊Dic61—44no—
211C4⋊Dic61—44no—
212#11758196—28no—
213#11758196—44no—
214#11758196—44no—
215#11758196—44no—
216C4×Dic61—28no—
217C4×Dic61—44no—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—24no—
228C23×C61—24no—
229C23×C61—44no—
230C23×C61—48no—
231C23×C61—44no—
232C23×C61—48no—
233C23×C61—48no—
234C23×C61—44no—
235C23×C61—82no—
236C23×C61—82no—
237C23×C61—84no—
238C23×C61#126116yescomplemented
239C23×C61C2×#4112yes—
240C23×C61C2×#4112yes—
241C23×C61C2×#4112yes—
242C23×C61C2×#4112yes—
243C23×C61C2×#4112yes—
244C23×C61C2×#4112yes—
245C23×C61C2×#4112yes—
246C23×C61C2×#4112yes—
247C23×C61C2×#4112yes—
248C23×C61C2511yescharacteristic
249D4×C61—48no—
250D4×C61—48no—
251C2×C4×C61—24no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—24no—
258C2×C4×C61—24no—
259C2×C4×C61—24no—
260C2×C4×C61—24no—
261C2×C4×C61—48no—
262C2×C4×C61—44no—
263C2×C4×C61—82no—
264C2×C4×C61—82no—
265C2×Dic61—28no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—24no—
269C2×Dic61—24no—
270C2×Dic61—24no—
271C2×Dic61—22no—
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—22no—
275C2×Dic61—24no—
276C2×Dic61—22no—
277C2×Dic61—24no—
278C2×Dic61—24no—
279C2×Dic61—24no—
280C2×Dic61—48no—
281C2×Dic61—48no—
282C2×Dic61—48no—
283C2×Dic61—48no—
284C2×Dic61—44no—
285C2×Dic61—84no—
286C2×Dic61H(5,𝔽2)14yes—
287#34995—611nomaximal, Sylow
288C22×C61—24no—
289C22×C61—24no—
290C22×C61—24no—
291C22×C61—22no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—24no—
298C22×C61—24no—
299C22×C61—24no—
300C22×C61—416no—
301C22×C61—42no—
302C22×C61—44no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—44no—
306C22×C61—88no—
307C22×C61C2×#12614yes—
308C22×C61C2×#12614yes—
309C22×C61C2×#12614yes—
310C22×C61C22×#4111yescharacteristic
311C22×C61C22×#4111yescharacteristic
312C22×C61C22×#4111yescharacteristic
313C22×C61C2×H(5,𝔽2)12yes—
314C22×C61#58512yes—
315C4×C61—42no—
316C4×C61—44no—
317C4×C61—42no—
318C4×C61—42no—
319Dic61—24no—
320Dic61—44no—
321Dic61—42no—
322Dic61—44no—
323Dic61—44no—
324#2852—614no—
325#2643—614no—
326#2510—614no—
327#2493—614no—
328#2481—614no—
329#2480—614no—
330#2315—611no—
331#2315—611no—
332#2308—611no—
333#2308D6111yescharacteristic
334C2×#386—611no—
335C2×#385—611no—
336C2×C4×C22⋊C4—611no—
337C2×C61—24no—
338C2×C61—24no—
339C2×C61—24no—
340C2×C61—82no—
341C2×C61C22×#12611yescharacteristic
342C2×C61#345912yes—
343C2×C61#349011yescharacteristic
344C2×C61#349111yescharacteristic
345C2×C61#350512yes—
346C23×D4C2×D6111yescharacteristic
347C24×C4—611no—
348C2×#126—614no—
349C22×C4⋊C4—612no—
350C22×C22⋊C4—611no—
351C22×C22⋊C4—611no—
352C22×C22⋊C4—611no—
353C22×C22⋊C4—612no—
354C22×C22⋊C4C2×D6111yescharacteristic
355C22×C22⋊C4C2×D6111yescharacteristic
356C22×C42—612no—
357#401—614no—
358#400—614no—
359#400—614no—
360#400—614no—
361#397—614no—
362#397—614no—
363#397—614no—
364#396—614no—
365#388—614no—
366#388—614no—
367#388—614no—
368#387—614no—
369#387—614no—
370#387—614no—
371#386—614no—
372#386—614no—
373#386—612no—
374#386—612no—
375#386—614no—
376#386C2×D6114yes—
377#386C2×D6112yes—
378#386C2×D6112yes—
379#385—614no—
380#385—614no—
381#385—614no—
382#385—614no—
383#385—612no—
384#385—612no—
385#385—614no—
386#383—614no—
387#381—612no—
388#381C2×D6112yes—
389#380—612no—
390#380—612no—
391#380—614no—
392#379—612no—
393#379C2×D6112yes—
394C4×C22⋊C4—614no—
395C4×C22⋊C4—614no—
396C2×#95—612no—
397C2×#95—612no—
398C2×#95—612no—
399C61#3499511yescharacteristic, complemented, Sylow
400C25C22×D6111yescharacteristic
401C25C22×D6111yescharacteristic
402C22×D4—24no—
403C22×D4—24no—
404C22×D4—24no—
405C22×D4C22×D6112yes—
406C22×D4C22×D6112yes—
407C23×C4—611no—
408C23×C4—612no—
409C23×C4—612no—
410C23×C4—612no—
411C23×C4—612no—
412C23×C4—612no—
413C23×C4—611no—
414C23×C4—1224no—
415C23×C4—2444no—
416C23×C4C22×D6111yescharacteristic
417C23×C4C22×D6111yescharacteristic
418C23×C4C22×D6111yescharacteristic
419C23×C4C22×D6112yes—
420#126—2448no—
421C2×C4⋊C4—614no—
422C2×C4⋊C4—614no—
423C2×C4⋊C4—614no—
424C2×C4⋊C4—614no—
425C2×C4⋊C4—614no—
426C2×C4⋊C4—614no—
427C2×C4⋊C4—2444no—
428C2×C22⋊C4—24no—
429C2×C22⋊C4—24no—
430C2×C22⋊C4—614no—
431C2×C22⋊C4—612no—
432C2×C22⋊C4—612no—
433C2×C22⋊C4—612no—
434C2×C22⋊C4—612no—
435C2×C22⋊C4—612no—
436C2×C22⋊C4—612no—
437C2×C22⋊C4—614no—
438C2×C22⋊C4—614no—
439C2×C22⋊C4—614no—
440C2×C22⋊C4—1224no—
441C2×C22⋊C4—1224no—
442C2×C22⋊C4—1224no—
443C2×C22⋊C4—2444no—
444C2×C22⋊C4C22×D6112yes—
445C2×C22⋊C4C22×D6112yes—
446C2×C22⋊C4C22×D6112yes—
447C2×C22⋊C4C22×D6112yes—
448C2×C22⋊C4C22×D6114yes—
449C2×C22⋊C4C22×D6114yes—
450C2×C42—614no—
451C2×C42—614no—
452C2×C42—2444no—
453#95—614no—
454#95—614no—
455#95—614no—
456#95—614no—
457#95—614no—
458#95—614no—
459#95—614no—
460#95—614no—
461#95—614no—
462#95—614no—
463#95—614no—
464#95—614no—
465#95C22×D6114yes—
466#95C22×D6114yes—
467C24—22no—
468C24—22no—
469C24—22no—
470C24—42no—
471C24—42no—
472C24—42no—
473C24—42no—
474C24#1175822214yes—
475C24#1175822414yes—
476C24#1175822414yes—
477C24C23×D6112yes—
478C24C23×D6111yescharacteristic
479C24C23×D6111yescharacteristic
480C24C23×D6111yescharacteristic
481C2×D4—24no—
482C2×D4—24no—
483C2×D4—24no—
484C2×D4—24no—
485C2×D4—24no—
486C2×D4—24no—
487C2×D4—416no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—612no—
499C22×C4—612no—
500C22×C4—614no—
501C22×C4—614no—
502C22×C4—614no—
503C22×C4—1224no—
504C22×C4—1224no—
505C22×C4—1224no—
506C22×C4—1222no—
507C22×C4—1224no—
508C22×C4—1224no—
509C22×C4—1224no—
510C22×C4—1224no—
511C22×C4—1222no—
512C22×C4—1224no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4—24416no—
523C22×C4C23×D6112yes—
524C22×C4C23×D6112yes—
525C22×C4C23×D6114yes—
526C22×C4C23×D6112yes—
527C4⋊C4—1228no—
528C4⋊C4—2444no—
529C4⋊C4—2444no—
530C4⋊C4—2444no—
531C4⋊C4—2444no—
532C4⋊C4—2444no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—1228no—
540C22⋊C4—1224no—
541C22⋊C4—1224no—
542C22⋊C4—1224no—
543C22⋊C4—1224no—
544C22⋊C4—1224no—
545C22⋊C4—1224no—
546C22⋊C4—2444no—
547C22⋊C4—2444no—
548C22⋊C4—2444no—
549C42—1228no—
550C42—2444no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—44no—
563C23—48no—
564C23—48no—
565C23—44no—
566C23—48no—
567C23—44no—
568C23—82no—
569C23—82no—
570C23—84no—
571C23#14519770116yescomplemented
572C23C2×#1175822412yes—
573C23C2×#1175822412yes—
574C23C2×#1175822412yes—
575C23C2×#1175822412yes—
576C23C2×#1175822412yes—
577C23C2×#1175822412yes—
578C23C2×#1175822212yes—
579C23C2×#1175822212yes—
580C23C2×#1175822212yes—
581C23C24×D6111yescharacteristic
582D4—48no—
583D4—48no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—614no—
599C2×C4—1228no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1224no—
603C2×C4—1222no—
604C2×C4—1224no—
605C2×C4—1224no—
606C2×C4—1224no—
607C2×C4—1224no—
608C2×C4—1222no—
609C2×C4—1224no—
610C2×C4—1222no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1224no—
614C2×C4—4884no—
615C2×C4—2448no—
616C2×C4—2448no—
617C2×C4—2448no—
618C2×C4—2448no—
619C2×C4—2444no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—416no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—88no—
639C22C2×#1451977014yes—
640C22C2×#1451977014yes—
641C22C2×#1451977014yes—
642C22C22×#1175822411yescharacteristic
643C22C22×#1175822411yescharacteristic
644C22C22×#1175822211yescharacteristic
645C22H(5,𝔽2)×D6112yes—
646C22#1684059712yes—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—1224no—
652C4—2444no—
653C4—2442no—
654C4—2444no—
655C4—2444no—
656C2—24no—
657C2—24no—
658C2—24no—
659C2—82no—
660C2C22×#1451977011yescharacteristic
661C2#2145922512yes—
662C2#2145950711yescharacteristic
663C2#2145951611yescharacteristic
664C2#2145964912yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2308C2
prime radicalC23×D4×C61C22
hypercenter#2308D61
hypocenterC61#34995
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2308

Complements

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

NH ≅ G/Ntype
1. #21454132C2semidirect
2. #21451774C2semidirect
3. #21450383C2semidirect
4. #21450208C2semidirect
5. #21450085C2semidirect
6. #21450062C2semidirect
7. #21448433C2semidirect
8. #21448431C2semidirect
11. C2×#16839634C2semidirect
12. C2×C22⋊C4×Dic61C2semidirect
15. C2×#14519770C22semidirect
16. C22×#11758196C22semidirect
21. C22×C4×Dic61C22semidirect
22. C61⋊#401C22semidirect
23. #16839704C22semidirect
24. #16839703C22semidirect
25. #16839702C22semidirect
26. #16839690C22semidirect
27. #16839690C22semidirect
28. #16839687C22semidirect
29. #16839685C22semidirect
30. #16839656C22semidirect
31. #16839655C22semidirect
32. #16839650C22semidirect
33. #16839649C22semidirect
34. #16839647C22semidirect
35. #16839644C22semidirect
42. #16839637C22semidirect
43. #16839635C22semidirect
45. #16839634C22semidirect
46. #16839634C22semidirect
47. #16839634C22semidirect
48. #16839629C22semidirect
50. #16839622C22semidirect
51. #16839622C22semidirect
55. C22⋊C4×Dic61C22semidirect
57. C2×#14519684C22semidirect
58. C2×#14519684C22semidirect
238. C23×C61#126semidirect
399. C61#34995semidirect
571. C23#14519770semidirect

Maximal subgroups

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

subgroupindex
#214541322
#214517742
#214503832
#214502082
#214500852
#214500622
#214484332
#214484312
#214483972
C2×#168396422
C2×#168396342
C2×C22⋊C4×Dic612
C61×#23082
#3499561