Finite Group Databasebeta

← #29685660

Subgroups — #29685660

Order 15616.

1952 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D244C211yesmaximal, characteristic, complemented
2C2×#16839898C211yesmaximal, characteristic
3C2×#16839875C212yesmaximal, complemented
4C4×#14519751C211yesmaximal, characteristic, complemented
5#21446166C212yesmaximal, complemented
6#21446134C211yesmaximal, characteristic, complemented
7#21446083C212yesmaximal, complemented
8#21446014C211yesmaximal, characteristic, complemented
9#21445751C211yesmaximal, characteristic, complemented
10#21445645C212yesmaximal, complemented
11C61×#1749C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×C4×D61—21no—
14C22×D488C2212yes—
15C2×#14519751C2211yescharacteristic
16C2×C4×D244—28no—
17C2×C4×D244—21no—
18C2×C4×D244—21no—
19C2×C4×D244C2212yes—
20C2×C4×D244C2212yes—
21C2×C4×D244C2211yescharacteristic, complemented
22C2×C4×C22⋊D61—21no—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×#11758198—21no—
25#16839898C418yescomplemented
26#16839875—28no—
27C2×#14519709C2212yescomplemented
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30C2×#14519699C2212yes—
31C2×#14519699C2212yescomplemented
32#16839664C2211yescharacteristic, complemented
33#16839660C2211yescharacteristic, complemented
34#16839653—21no—
35#16839641—21no—
36#16839622C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic, complemented
38C4×#11758198—21no—
39#16839525C2212yescomplemented
40#16839525C2212yescomplemented
41C22×C42×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2211yescharacteristic
44C2×#97×C61C2212yescomplemented
45C61×#336C2212yescomplemented
46C24×D61—21no—
47C22×D244—21no—
48C22×D244—22no—
49C22×D244—24no—
50C22×D244—24no—
51C22×D244C2×C414yescomplemented
52C22×D244C2311yescharacteristic
53C22×D244C2312yes—
54C22×C22⋊D61—21no—
55C23×Dic61—21no—
56C22×C4×D61—28no—
57C22×C4×D61—22no—
58C22×C4×D61—22no—
59C22×C4×D61—21no—
60C22×C4×D61—44no—
61C22×C4×D61—44no—
62C2×D488—24no—
63C2×D488—24no—
64C2×D488C2×C418yescomplemented
65#14519751—22no—
66#14519751C2×C414yescomplemented
67C4×D244—28no—
68C4×D244—28no—
69C4×D244—24no—
70C4×D244—24no—
71C4×D244—24no—
72C4×D244—24no—
73C4×D244D418yescomplemented
74C4×C22⋊D61—44no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—24no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2312yes—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×#11758198—28no—
84C2×#11758198—22no—
85C2×#11758198—22no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91#14519709—24no—
92#14519699—24no—
93#14519699—24no—
94#14519699—24no—
95#14519699C2×C418yescomplemented
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99C23×C4×C61D411yescharacteristic
100C23×C4×C61D411yescharacteristic
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103C22×C8×C61C2312yes—
104C2×C42×C61—24no—
105C2×C42×C61D411yescharacteristic
106C2×C42×C61D411yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109C22⋊C8×C61—22no—
110C22⋊C8×C61C2×C414yescomplemented
111#97×C61—24no—
112C24×C61C2×D411yescharacteristic
113C23×D61—24no—
114C23×D61—24no—
115C23×D61—24no—
116C23×D61—22no—
117C23×D61—21no—
118C23×D61—42no—
119C23×D61—42no—
120C23×D61—42no—
121C23×D61—42no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—41no—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61#4112yes—
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—24no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C22×Dic61—24no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C2×D244—24no—
149C2×D244—216no—
150C2×D244—24no—
151C2×D244—24no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—216no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—21no—
159C2×D244C22×C414yes—
160C2×D244C22×C412yescomplemented
161C2×D244C2×D414yes—
162C2×D244#4114yes—
163C2×C4×D61—28no—
164C2×C4×D61—28no—
165C2×C4×D61—432no—
166C2×C4×D61—44no—
167C2×C4×D61—44no—
168C2×C4×D61—44no—
169C2×C4×D61—44no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C8×C61—22no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61C22×C414yescomplemented
179C42×C61—22no—
180C42×C61—22no—
181C42×C61—22no—
182C42×C61C2×D414yes—
183C22⋊Dic61—42no—
184#11758198—28no—
185#11758198—24no—
186#11758198—28no—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190C4⋊Dic61—24no—
191C4⋊Dic61—22no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22×C412yescomplemented
196C4⋊Dic61C2×D414yes—
197#11758196—42no—
198C4×Dic61—42no—
199D488—216no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61C4×D412yes—
205C23×C61AGL(1,ℤ/8)12yes—
206C23×C61AGL(1,ℤ/8)12yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—216no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61—24no—
212C22×D61—22no—
213C22×D61—48no—
214C22×D61—48no—
215C22×D61—48no—
216C22×D61—48no—
217C22×D61—48no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—48no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C2×C4×C61—28no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—44no—
244C2×C4×C61C4×D412yes—
245C2×C4×C61#12012yes—
246C2×C4×C61#12012yes—
247C2×C4×C61AGL(1,ℤ/8)12yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D412yes—
250C2×C4×C61C2×#4112yes—
251C2×C4×C61C2×#4111yescharacteristic
252C22⋊D61—48no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—48no—
256C2×Dic61—41no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266D244—216no—
267D244—24no—
268D244—24no—
269D244—24no—
270D244—24no—
271D244—42no—
272D244C4×D418yescomplemented
273C4×D61—416no—
274C4×D61—416no—
275C4×D61—84no—
276C8×C61—24no—
277C8×C61—42no—
278#21452—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61#44214yescomplemented
286C22×C61C2×C4×D411yescharacteristic
287C22×C61C2×#12011yescharacteristic
288C22×C61#52511yescharacteristic
289C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
291C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C61#57511yescharacteristic
293C2×D61—28no—
294C2×D61—432no—
295C2×D61—44no—
296C2×D61—44no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—44no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—41no—
313C4×C61C2×C4×D412yes—
314C4×C61#52512yes—
315Dic61—42no—
316Dic61—44no—
317Dic61—81no—
318Dic61—81no—
319C22×C4×D4—611no—
320C2×#447—611no—
321C2×#442—612no—
322C4×#121—611no—
323#1879—612no—
324#1874—611no—
325#1864—612no—
326#1853—611no—
327#1786—611no—
328#1762—612no—
329#1749D6111yescharacteristic, complemented
330C2×C61—24no—
331C2×C61C4×#12011yescharacteristic
332C2×C61C2×#44211yescharacteristic
333C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
334C2×C61#300311yescharacteristic
335C2×C61#300611yescharacteristic
336C2×C61#310911yescharacteristic
337C2×C61#311411yescharacteristic
338D61—48no—
339D61—82no—
340C23×D4—611no—
341C24×C4—1221no—
342C22×D8—612no—
343C2×#121—611no—
344C2×C4×D4—612no—
345C2×C4×D4—612no—
346C2×C4×D4—611no—
347C2×C4×D4—1228no—
348C2×C4×D4—1221no—
349C2×C4×D4—1221no—
350C2×C4×D4—1221no—
351C22×C4⋊C4—611no—
352C22×C22⋊C4—1221no—
353C22×C42C2×D6111yescharacteristic
354#447—618no—
355#442—1228no—
356C2×#106—612no—
357C2×#102—612no—
358C2×#102—612no—
359C2×#102—612no—
360C2×#102—612no—
361C2×C22⋊C8C2×D6111yescharacteristic
362C2×C22⋊C8C2×D6111yescharacteristic
363C2×#97C2×D6112yescomplemented
364#390—611no—
365#389—611no—
366#388—1221no—
367#386—1221no—
368#380—611no—
369C4×C4⋊C4—611no—
370C4×C22⋊C4—1221no—
371#340—612no—
372#340—612no—
373#336C2×D6112yescomplemented
374C61#2145211yescharacteristic, complemented, Sylow
375C25—1221no—
376C22×D4—614no—
377C22×D4—612no—
378C22×D4—611no—
379C22×D4—1224no—
380C22×D4—1224no—
381C22×D4—1221no—
382C22×D4—1221no—
383C22×D4—1222no—
384C23×C4—1228no—
385C23×C4—1221no—
386C23×C4—1222no—
387C23×C4—1222no—
388C23×C4—1221no—
389C23×C4—2444no—
390C23×C4—2444no—
391C23×C4D24411yescharacteristic
392C23×C4D24411yescharacteristic
393C23×C4C22×D6111yescharacteristic
394C2×D8—618no—
395C2×D8—1224no—
396C2×D8—1224no—
397C22×C8C22×D6112yes—
398C22×C8C22×D6112yes—
399#121—614no—
400#121—1222no—
401C4×D4—618no—
402C4×D4—1224no—
403C4×D4—1224no—
404C4×D4—1228no—
405C4×D4—1228no—
406C4×D4—1224no—
407C4×D4—1224no—
408C4×D4—2444no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—1224no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C22⋊C4—1228no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—24no—
425C2×C42—1221no—
426C2×C42D24411yescharacteristic
427C2×C42D24411yescharacteristic
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430#106—1224no—
431#102—618no—
432#102—1224no—
433#102—1224no—
434#102—1224no—
435C22⋊C8—22no—
436C22⋊C8C4×D6114yescomplemented
437#97—24no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441C24—1224no—
442C24—1224no—
443C24—1222no—
444C24—1224no—
445C24—1221no—
446C24—2442no—
447C24—2442no—
448C24—2442no—
449C24—2442no—
450C24C2×D24411yescharacteristic
451C2×D4—614no—
452C2×D4—612no—
453C2×D4—614no—
454C2×D4—614no—
455C2×D4—12216no—
456C2×D4—1224no—
457C2×D4—1224no—
458C2×D4—12216no—
459C2×D4—1224no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1221no—
466C2×D4—1224no—
467C2×D4—2442no—
468C2×D4—2442no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—41no—
477C22×C4—1228no—
478C22×C4—1224no—
479C22×C4—1221no—
480C22×C4—1222no—
481C22×C4—1228no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—2442no—
485C22×C4—2442no—
486C22×C4—24432no—
487C22×C4—2444no—
488C22×C4—2444no—
489C22×C4—2444no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2444no—
494C22×C4C2×C4×D6112yes—
495C22×C4C2×D24411yescharacteristic
496C22×C4C2×D24411yescharacteristic
497C22×C4C2×D24411yescharacteristic
498C22×C4C2×D24411yescharacteristic
499C22×C4C2×D24411yescharacteristic
500C22×C4#1175822212yes—
501C22×C4C23×D6111yescharacteristic
502D8—12216no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8C2×C4×D6114yescomplemented
509C4⋊C4—612no—
510C4⋊C4—614no—
511C4⋊C4—1224no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—2442no—
517C22⋊C4—1228no—
518C22⋊C4—1224no—
519C22⋊C4—1228no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—2442no—
528C42D4×D6114yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—12216no—
534C23—1224no—
535C23—1224no—
536C23—1224no—
537C23—1222no—
538C23—2448no—
539C23—2448no—
540C23—2448no—
541C23—2448no—
542C23—2448no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23—2441no—
549C23—2442no—
550C23—2442no—
551C23—2442no—
552C23—2442no—
553C23—2448no—
554C23C4×D24412yes—
555C23#1451980512yes—
556C23#1451980512yes—
557C23C22×D24411yescharacteristic
558D4—618no—
559D4—12216no—
560D4—1224no—
561D4—1224no—
562D4—1224no—
563D4—1224no—
564D4—2448no—
565D4—2442no—
566C2×C4—28no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—1222no—
582C2×C4—1224no—
583C2×C4—4884no—
584C2×C4—24416no—
585C2×C4—24416no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2448no—
593C2×C4—2441no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2441no—
597C2×C4C4×D24412yes—
598C2×C4#1451974712yes—
599C2×C4#1451974712yes—
600C2×C4#1451980512yes—
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C2×D4×D6112yes—
603C2×C4#41×D6112yes—
604C2×C4C2×#1175822211yescharacteristic
605C8—24no—
606C8—42no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—1228no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22—4881no—
618C22—24432no—
619C22—2442no—
620C22—2442no—
621C22—2442no—
622C22—2444no—
623C22—2444no—
624C22—2444no—
625C22—2444no—
626C22—2444no—
627C22—2444no—
628C22#1683987514yescomplemented
629C22C2×C4×D24411yescharacteristic
630C22C2×#1451974711yescharacteristic
631C22#1684027411yescharacteristic
632C22C2×#1451980511yescharacteristic
633C22C2×#1451980511yescharacteristic
634C22C2×#1451980511yescharacteristic
635C22#1684056211yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—4881no—
642C4—4881no—
643C4—2442no—
644C4—2444no—
645C4C4×D4×D6112yes—
646C4#1684027112yes—
647C2—24no—
648C2—4882no—
649C2—2448no—
650C2C4×#1451974711yescharacteristic
651C2C2×#1683987511yescharacteristic
652C2C4×#1451980511yescharacteristic
653C2#2145504911yescharacteristic
654C2#2145507311yescharacteristic
655C2#2145591111yescharacteristic
656C2#2145595911yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1749C2
socleC23×C61C22×D4
prime radicalC2×#16839898C2
cosocleC22×C4×C61C24
hypercenter#1749D61
hypocenterC61#21452
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
1. C22×C4×D244C2semidirect
3. C2×#16839875C2semidirect
4. C4×#14519751C2semidirect
5. #21446166C2semidirect
6. #21446134C2semidirect
7. #21446083C2semidirect
8. #21446014C2semidirect
9. #21445751C2semidirect
10. #21445645C2semidirect
11. C61×#1749C2semidirect
21. C2×C4×D244C22semidirect
25. #16839898C4semidirect
27. C2×#14519709C22semidirect
31. C2×#14519699C22semidirect
32. #16839664C22semidirect
33. #16839660C22semidirect
37. C4×C4⋊Dic61C22semidirect
39. #16839525C22semidirect
40. #16839525C22semidirect
44. C2×#97×C61C22semidirect
45. C61×#336C22semidirect
51. C22×D244C2×C4semidirect
64. C2×D488C2×C4semidirect
66. #14519751C2×C4semidirect
73. C4×D244D4semidirect
95. #14519699C2×C4semidirect
110. C22⋊C8×C61C2×C4semidirect
160. C2×D244C22×C4semidirect
178. C2×C8×C61C22×C4semidirect
195. C4⋊Dic61C22×C4semidirect
272. D244C4×D4semidirect
285. C22×C61#442semidirect
329. #1749D61semidirect
363. C2×#97C2×D61semidirect
373. #336C2×D61semidirect
374. C61#21452semidirect
436. C22⋊C8C4×D61semidirect
508. C2×C8C2×C4×D61semidirect
628. C22#16839875semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D2442
C2×#168398982
C2×#168398752
C4×#145197512
#214461662
#214461342
#214460832
#214460142
#214457512
#214456452
C61×#17492
#2145261