Finite Group Databasebeta

← #29688148

Subgroups — #29688148

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840125C211yesmaximal, characteristic
2C4×#14519773C211yesmaximal, characteristic
3C4×#14519768C211yesmaximal, characteristic, complemented
4#21446786C211yesmaximal, characteristic, complemented
5#21446641C211yesmaximal, characteristic
6#21446504C211yesmaximal, characteristic, complemented
7#21446503C211yesmaximal, characteristic, complemented
8#21446500C211yesmaximal, characteristic, complemented
9#21446499C211yesmaximal, characteristic, complemented
10#21445951C211yesmaximal, characteristic, complemented
11#21445647C211yesmaximal, characteristic, complemented
12#21445645C211yesmaximal, characteristic, complemented
13#21445644C211yesmaximal, characteristic, complemented
14#97×Dic61C211yesmaximal, characteristic, complemented
15C61×#1762C211yesmaximal, characteristic, complemented
16C2×#14519773C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×C4×C22⋊D61—21no—
20C2×C4×C22⋊D61—21no—
21#16840125C418yescomplemented
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839663C2211yescharacteristic
35#16839662—21no—
36#16839662—21no—
37#16839661C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839649—21no—
40#16839645—21no—
41#16839631C2211yescharacteristic
42#16839627C2211yescharacteristic
43C4×#11758196—21no—
44C4×C4⋊Dic61—21no—
45C4×C4⋊Dic61C2211yescharacteristic
46C42×Dic61C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C22⋊D61—21no—
63C22×C4×D61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519773—22no—
67#14519773C2×C414yes—
68#14519768—22no—
69#14519768C2×C414yescomplemented
70C4×D244—22no—
71C4×D244—22no—
72C4×C22⋊D61—44no—
73C4×C22⋊D61—44no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519699—22no—
96#14519699C2×C414yescomplemented
97#14519698—22no—
98#14519698C2×C414yescomplemented
99#14519697—22no—
100#14519697C2×C414yescomplemented
101#14519688—22no—
102#14519687—22no—
103#14519687C2×C414yescomplemented
104C8×Dic61—22no—
105C8×Dic61—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61—21no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C4×D4×C61—22no—
116C4×D4×C61—22no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123#102×C61—22no—
124#102×C61C2×C414yescomplemented
125#97×C61—22no—
126#95×C61—21no—
127C24×C61—21no—
128C23×D61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C22×C412yes—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—24no—
145C2×C22⋊D61—24no—
146C2×C22⋊D61—42no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C412yes—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C22×C412yes—
171C2×C8×C61#4112yes—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C22×C412yes—
178C22⋊C4×C61—42no—
179C42×C61—42no—
180C42×C61#4112yes—
181C42×C61#4112yes—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184#11758198—42no—
185#11758198—42no—
186#11758198—42no—
187C4⋊Dic61—21no—
188C4⋊Dic61—24no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61C22×C412yes—
195#11758196—24no—
196#11758196—42no—
197#11758196—42no—
198#11758196—42no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—41no—
204C4×Dic61—41no—
205C4×Dic61—42no—
206C4×Dic61C22×C412yes—
207C4×Dic61#4112yes—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C22×C412yes—
213C2×#92418#4112yes—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—41no—
232D4×C61—41no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—84no—
247C2×C4×C61#12412yes—
248C2×C4×C61#12412yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C22⋊D61—48no—
256C22⋊D61—48no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61C4×D412yes—
275C2×Dic61C4×D412yes—
276C2×Dic61AGL(1,ℤ/8)12yes—
277C2×Dic61AGL(1,ℤ/8)12yes—
278D244—22no—
279D244—22no—
280D244—41no—
281D244—41no—
282C4×D61—84no—
283C8×C61—22no—
284C8×C61—22no—
285#92418—22no—
286#92418—22no—
287#21656—611nomaximal, Sylow
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61C2×C4×D411yescharacteristic
296C22×C61C2×#12411yescharacteristic
297C22×C61#53211yescharacteristic
298C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61#57511yescharacteristic
301C22×C61#57511yescharacteristic
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61C4×#4111yescharacteristic
315C4×C61C4×#4111yescharacteristic
316C4×C61#52911yescharacteristic
317C4×C61#52911yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—22no—
321Dic61—22no—
322Dic61—42no—
323Dic61—81no—
324Dic61—81no—
325C2×#490—611no—
326C4×#127—611no—
327C4×#125—611no—
328#1992—611no—
329#1961—611no—
330#1934—611no—
331#1934—611no—
332#1934—611no—
333#1934—611no—
334#1836—611no—
335#1762—611no—
336#1762—611no—
337#1762—611no—
338#1762D6111yescharacteristic, complemented
339C4×#97—611no—
340C2×C61—82no—
341C2×C61C4×#12411yescharacteristic
342C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
343C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
344C2×C61#313811yescharacteristic
345C2×C61#314211yescharacteristic
346C2×C61#318011yescharacteristic
347C2×C61#318011yescharacteristic
348D61—82no—
349C2×#127—611no—
350C2×#125—611no—
351C2×C4×D4—611no—
352C2×C4×D4—1221no—
353C2×C4×D4—1221no—
354C2×C4×D4C2×D6111yescharacteristic
355#490—618no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×#102C2×D6111yescharacteristic
363C2×#102C2×D6111yescharacteristic
364C2×#97—611no—
365C2×#97—611no—
366C2×#97—611no—
367C2×#97C2×D6111yescharacteristic
368C2×C4×C8—611no—
369C2×C4×C8—611no—
370#390—611no—
371#390—611no—
372#390—1221no—
373#390—1221no—
374#389—611no—
375#389—611no—
376#387—1221no—
377#387—1221no—
378#384—611no—
379#382—611no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—1221no—
382C4×C4⋊C4—1221no—
383C4×C4⋊C4C2×D6111yescharacteristic
384C43—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340C2×D6111yescharacteristic
392#340C2×D6111yescharacteristic
393C61#2165611yescharacteristic, complemented, Sylow
394C22×D4—611no—
395C22×D4—1221no—
396C22×D4—1221no—
397C22×D4C22×D6111yescharacteristic
398C23×C4—21no—
399C23×C4—1221no—
400C22×C8—611no—
401C22×C8—611no—
402C22×C8C22×D6111yescharacteristic
403C22×C8C22×D6111yescharacteristic
404#127—614no—
405#127—1222no—
406#125—614no—
407#125—1222no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—2444no—
413C4×D4—2444no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C42—21no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—1221no—
440C2×C42C22×D6111yescharacteristic
441#102—22no—
442#102—614no—
443#102—614no—
444#102—614no—
445#102—1222no—
446#102—1222no—
447#102—1222no—
448#102C4×D6114yescomplemented
449#97—22no—
450#97—614no—
451#97—1222no—
452#97—1222no—
453C4×C8—1222no—
454C4×C8—1222no—
455#95—21no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461C24—21no—
462C24—1221no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—612no—
469C2×D4—1224no—
470C2×D4—1224no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4—2442no—
479C2×D4—2442no—
480C2×D4C2×C4×D6112yes—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—612no—
505C2×C8—612no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8C2×C4×D6112yes—
511C2×C8#1175822212yes—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—612no—
518C4⋊C4—1224no—
519C4⋊C4—1224no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C4⋊C4C2×C4×D6112yes—
530C22⋊C4—42no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C22⋊C4—2442no—
536C42—42no—
537C42—612no—
538C42—612no—
539C42—1222no—
540C42—1222no—
541C42—1221no—
542C42—1221no—
543C42—2441no—
544C42—2441no—
545C42—2442no—
546C42#1175822212yes—
547C42#1175822412yes—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—1222no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23C2×D4×D6111yescharacteristic
563D4—22no—
564D4—22no—
565D4—41no—
566D4—41no—
567D4—1222no—
568D4—1222no—
569D4—2448no—
570D4—2448no—
571D4—2441no—
572D4—2441no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1222no—
600C2×C4—4884no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4#1451976312yes—
610C2×C4#1451976312yes—
611C2×C4C22×C4×D6111yescharacteristic
612C2×C4#41×D6111yescharacteristic
613C2×C4#41×D6111yescharacteristic
614C2×C4C2×#1175822411yescharacteristic
615C2×C4C2×#1175822211yescharacteristic
616C2×C4C2×#1175822211yescharacteristic
617C8—22no—
618C8—22no—
619C8—1222no—
620C8—1222no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—4881no—
629C22—4881no—
630C22—4881no—
631C22—4881no—
632C22—2442no—
633C22—2442no—
634C22—2442no—
635C22C4×D4×D6111yescharacteristic
636C22C2×#1451976311yescharacteristic
637C22#1684030911yescharacteristic
638C22AGL(1,ℤ/8)×D6111yescharacteristic
639C22AGL(1,ℤ/8)×D6111yescharacteristic
640C22#1684055811yescharacteristic
641C22#1684056111yescharacteristic
642C4—22no—
643C4—22no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—1222no—
648C4—1222no—
649C4—1222no—
650C4—1222no—
651C4—4881no—
652C4—4881no—
653C4—2442no—
654C4C4×#1175822411yescharacteristic
655C4C4×#1175822211yescharacteristic
656C4#1684029011yescharacteristic
657C4#1684029211yescharacteristic
658C2—82no—
659C2—4882no—
660C2C4×#1451976311yescharacteristic
661C2#2145462411yescharacteristic
662C2#2145463011yescharacteristic
663C2#2145619111yescharacteristic
664C2#2145621711yescharacteristic
665C2#2145651311yescharacteristic
666C2#2145651511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#21656
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
3. C4×#14519768C2semidirect
4. #21446786C2semidirect
6. #21446504C2semidirect
7. #21446503C2semidirect
8. #21446500C2semidirect
9. #21446499C2semidirect
10. #21445951C2semidirect
11. #21445647C2semidirect
12. #21445645C2semidirect
13. #21445644C2semidirect
14. #97×Dic61C2semidirect
15. C61×#1762C2semidirect
21. #16840125C4semidirect
69. #14519768C2×C4semidirect
96. #14519699C2×C4semidirect
98. #14519698C2×C4semidirect
100. #14519697C2×C4semidirect
103. #14519687C2×C4semidirect
124. #102×C61C2×C4semidirect
338. #1762D61semidirect
393. C61#21656semidirect
448. #102C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401252
C4×#145197732
C4×#145197682
#214467862
#214466412
#214465042
#214465032
#214465002
#214464992
#214459512
#214456472
#214456452
#214456442
#97×Dic612
C61×#17622
#2165661