Finite Group Databasebeta

← #29695627

Subgroups — #29695627

Order 15616.

1910 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840397C211yesmaximal, characteristic
2C2×#16839841C211yesmaximal, characteristic, complemented
3C2×#16839765C212yesmaximal, complemented
4#21448770C211yesmaximal, characteristic, complemented
5#21446747C211yesmaximal, characteristic, complemented
6#21446254C212yesmaximal, complemented
7#21446138C212yesmaximal, complemented
8#21445816C211yesmaximal, characteristic, complemented
9#21445748C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C61×#1957C211yesmaximal, characteristic, complemented
12C22×D4×D61—21no—
13C22×D4×D61C2212yes—
14C22×#11758189C2212yescomplemented
15#16840397—24no—
16#16840397C418yescomplemented
17C2×#14519748—21no—
18C2×#14519748C2212yes—
19C2×#14519745—22no—
20C2×C4×D244C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519735C2212yescomplemented
23#16839841—24no—
24C2×C4⋊#92418C2211yescharacteristic, complemented
25#16839765—28no—
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519698C2212yescomplemented
29C2×#14519698C2212yescomplemented
30C2×#14519697C2212yes—
31C2×#14519689C2212yescomplemented
32#16839663—21no—
33#16839652—22no—
34#16839619C2212yescomplemented
35C4⋊C4×Dic61—21no—
36C4×C4⋊Dic61C2211yescharacteristic, complemented
37#16839525C2211yescharacteristic, complemented
38#16839525C2211yescharacteristic, complemented
39#16839524C2212yescomplemented
40C2×#127×C61C2211yescharacteristic
41C2×#102×C61C2212yescomplemented
42C2×#102×C61C2212yescomplemented
43C2×C4×C8×C61C2211yescharacteristic, complemented
44C61×#384C2211yescharacteristic, complemented
45C24×D61—22no—
46C22×D244D411yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—22no—
49C22×C22⋊D61—22no—
50C2×D4×D61—28no—
51C2×D4×D61—24no—
52C2×D4×D61—24no—
53C2×D4×D61—24no—
54C2×D4×D61—42no—
55C2×D4×D61—42no—
56C2×D4×D61—44no—
57C2×D4×D61—44no—
58C2×D4×D61C2×C418yes—
59C22×C4×D61—21no—
60C22×C4×D61C2312yes—
61C2×#11758189—24no—
62C2×#11758189—24no—
63C2×#11758189—42no—
64C22×#92418C2312yes—
65#14519748—24no—
66#14519748—28no—
67#14519748C2×C418yes—
68#14519745—216no—
69C4×D244—22no—
70C4×D244C2×C414yes—
71C4×D244C2×C414yes—
72C4×D244D414yes—
73#14519735—24no—
74#14519735—42no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C22⋊Dic61—22no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×#11758198D412yes—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—22no—
86C4⋊#92418—22no—
87#14519699—22no—
88#14519699D414yescomplemented
89#14519698—24no—
90#14519698—42no—
91#14519697—24no—
92#14519689—24no—
93#14519684—21no—
94#14519684—22no—
95C22×D4×C61—21no—
96C22×D4×C61—21no—
97C22×D4×C61C2312yes—
98C22×C8×C61C2311yescharacteristic
99C22×C8×C61C2311yescharacteristic
100#127×C61—22no—
101#127×C61C2×C414yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C42×C61C2311yescharacteristic
106#102×C61—24no—
107#102×C61—42no—
108C4×C8×C61—22no—
109C24×C61—22no—
110C23×D61—21no—
111C23×D61—28no—
112C23×D61—44no—
113C23×D61—44no—
114C23×D61—44no—
115C23×D61—44no—
116C23×D61—82no—
117C23×D61—82no—
118C23×D61C2×D412yes—
119C2×D4×C61—24no—
120C2×D4×C61—24no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61—81no—
129C2×D4×C61—81no—
130C2×D4×C61C22×C414yes—
131C22×C4×C61—22no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—28no—
136C2×C22⋊D61—28no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61—44no—
139C2×C22⋊D61—44no—
140C2×C22⋊D61—44no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D412yes—
145D4×D61—216no—
146D4×D61—48no—
147D4×D61—48no—
148D4×D61—48no—
149D4×D61—432no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244C22⋊C414yes—
156C2×D244C22×C412yes—
157C2×D244C2×D412yes—
158C2×D244#4112yes—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—24no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61C22×C414yes—
167C2×C4×D61C2×D414yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D412yescomplemented
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61—81no—
178C42×C61C22×C412yes—
179C42×C61C2×D412yes—
180C22⋊Dic61—28no—
181#11758198—24no—
182#11758198—44no—
183#11758198—82no—
184#11758198C22⋊C418yes—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C22⋊C414yes—
190C4⋊Dic61C22×C412yes—
191C4⋊Dic61C2×D412yescomplemented
192C4×Dic61—44no—
193C4×Dic61—42no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197#11758189—48no—
198C23×C61—24no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61C22×D411yescharacteristic
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—416no—
216C22×D61—416no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—88no—
222C22×D61—88no—
223C22×D61—88no—
224C22×D61—88no—
225C22×D61—88no—
226C22×D61C2×C22⋊C414yes—
227D4×C61—24no—
228D4×C61—24no—
229D4×C61—42no—
230D4×C61—42no—
231D4×C61—48no—
232D4×C61—84no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61AGL(1,ℤ/8)12yes—
241C2×C4×C61AGL(1,ℤ/8)12yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C22⋊D61—416no—
249C22⋊D61—416no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61C2×C22⋊C414yes—
262D244—22no—
263D244—22no—
264D244—22no—
265D244—28no—
266D244C4×D414yes—
267C4×D61—24no—
268C4×D61—24no—
269C4×D61—24no—
270C4×D61—24no—
271C4×D61—48no—
272C8×C61—22no—
273C8×C61—41no—
274#92418—42no—
275#22428—611nomaximal, Sylow
276C22×C61—44no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—82no—
280C22×C61—82no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61D4211yescharacteristic
285C22×C61#54611yescharacteristic
286C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C61#57611yescharacteristic
289C22×C61#57711yescharacteristic
290C2×D61—24no—
291C2×D61—24no—
292C2×D61—24no—
293C2×D61—24no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—832no—
305C4×C61—22no—
306C4×C61—82no—
307C4×C61#41814yescomplemented
308C4×C61C2×C4×D411yescharacteristic
309C4×C61#51811yescharacteristic
310C4×C61D4211yescharacteristic
311C4×C61#54611yescharacteristic
312Dic61—24no—
313Dic61—42no—
314Dic61—42no—
315Dic61—81no—
316C2×D42—611no—
317C2×#436—611no—
318C2×#418—612no—
319#2354—611no—
320#1982—611no—
321#1957D6111yescharacteristic, complemented
322#1895—612no—
323#1875—612no—
324#1802—611no—
325#1786—612no—
326C4×#102—611no—
327C2×C61—84no—
328C2×C61D4×C22⋊C411yescharacteristic
329C2×C61C2×#41811yescharacteristic
330C2×C61#289711yescharacteristic
331C2×C61#327211yescharacteristic
332C2×C61#327411yescharacteristic
333C2×C61#332411yescharacteristic
334C2×C61#332511yescharacteristic
335D61—44no—
336D61—44no—
337C23×D4—612no—
338C23×D4—1221no—
339C22×M16—612no—
340D42—618no—
341D42—1224no—
342C2×#127C2×D6111yescharacteristic
343C2×#121—612no—
344C2×#121—1221no—
345C2×#120—1222no—
346C2×C4×D4—611no—
347C2×C4×D4—611no—
348C2×#117—612no—
349#436—1224no—
350C2×C4⋊C8—611no—
351#418—1228no—
352C2×#102—611no—
353C2×#102—612no—
354C2×#102—611no—
355C2×#102—612no—
356C2×#102—612no—
357C2×#102C2×D6112yescomplemented
358C2×#102C2×D6112yescomplemented
359C2×C22⋊C8—612no—
360C2×C4×C8C2×D6111yescharacteristic, complemented
361#390—1221no—
362#388—1222no—
363#384C2×D6111yescharacteristic, complemented
364#380—612no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4—1221no—
367#340—611no—
368#340—611no—
369#340—612no—
370C61#2242811yescharacteristic, complemented, Sylow
371C25—1222no—
372C22×D4—21no—
373C22×D4—21no—
374C22×D4—618no—
375C22×D4—611no—
376C22×D4—611no—
377C22×D4—1224no—
378C22×D4—1222no—
379C22×D4—1224no—
380C22×D4—1224no—
381C22×D4—1228no—
382C22×D4—1222no—
383C22×D4—2442no—
384C22×D4—2442no—
385C22×D4—2444no—
386C22×D4—2444no—
387C22×D4C22×D6112yes—
388C23×C4—612no—
389C23×C4—1221no—
390C2×M16—1224no—
391C2×M16—1224no—
392C2×M16—2442no—
393C22×C8—612no—
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396#127—22no—
397#127C4×D6114yes—
398#121—618no—
399#121—1228no—
400#121—1224no—
401#120—12216no—
402C4×D4—614no—
403C4×D4—614no—
404C4×D4—614no—
405C4×D4—1222no—
406#117—1224no—
407#117—2442no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4C22×D6112yes—
415C2×C22⋊C4—612no—
416C2×C22⋊C4—1222no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1222no—
420C2×C42—1222no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423C4⋊C8—1222no—
424#102—24no—
425#102—42no—
426#102—614no—
427#102—1224no—
428#102—1224no—
429#102—1222no—
430#102—2442no—
431C22⋊C8—1224no—
432C4×C8—22no—
433#95—1222no—
434#95—1221no—
435C24—22no—
436C24—612no—
437C24—1228no—
438C24—1221no—
439C24—4882no—
440C24—4882no—
441C24—2444no—
442C24—2444no—
443C24—2444no—
444C24—2444no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—81no—
455C2×D4—81no—
456C2×D4—612no—
457C2×D4—612no—
458C2×D4—612no—
459C2×D4—614no—
460C2×D4—1228no—
461C2×D4—12216no—
462C2×D4—1228no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1222no—
467C2×D4—1222no—
468C2×D4—2448no—
469C2×D4—2448no—
470C2×D4—2448no—
471C2×D4—2444no—
472C2×D4—2444no—
473C2×D4—24432no—
474C2×D4—2444no—
475C2×D4—2444no—
476C2×D4C2×C4×D6114yes—
477C22×C4—22no—
478C22×C4—612no—
479C22×C4—614no—
480C22×C4—614no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1224no—
488C22×C4—1221no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494M16—2448no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8D4×D6112yescomplemented
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—614no—
509C4⋊C4—612no—
510C4⋊C4—612no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C22⋊C4—618no—
516C22⋊C4—1228no—
517C22⋊C4—1224no—
518C22⋊C4—4882no—
519C22⋊C4—2444no—
520C42—2444no—
521C42—2442no—
522C42C2×C4×D6112yes—
523C42D4×D6112yes—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—614no—
532C23—1222no—
533C23—1222no—
534C23—1222no—
535C23—1222no—
536C23—1222no—
537C23—1222no—
538C23—1222no—
539C23—1222no—
540C23—1222no—
541C23—4888no—
542C23—4888no—
543C23—4888no—
544C23—4888no—
545C23—4888no—
546C23—24416no—
547C23—24416no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23C2×D4×D6111yescharacteristic
553D4—24no—
554D4—24no—
555D4—48no—
556D4—42no—
557D4—42no—
558D4—84no—
559D4—614no—
560D4—1222no—
561D4—1228no—
562D4—1222no—
563D4—1222no—
564D4—24416no—
565D4—24416no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—614no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2448no—
588C2×C4C22⋊C4×D6112yes—
589C2×C4C22⋊C4×D6112yes—
590C2×C4#1451980112yes—
591C2×C4#1451980412yes—
592C2×C4C22×C4×D6111yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4#1451982211yescharacteristic
597C2×C4#1451982311yescharacteristic
598C8—22no—
599C8—41no—
600C8—2442no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—1224no—
609C22—1224no—
610C22—1224no—
611C22—1224no—
612C22—48832no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22—2442no—
622C22—2442no—
623C22C2×C22⋊C4×D6111yescharacteristic
624C22#1684039711yescharacteristic
625C22#1684040511yescharacteristic
626C22C2×#1451980111yescharacteristic
627C22C2×#1451980411yescharacteristic
628C22#1684056311yescharacteristic
629C22#1684056811yescharacteristic
630C4—22no—
631C4—82no—
632C4—1224no—
633C4—4881no—
634C4—2442no—
635C4—2442no—
636C4#1683976514yescomplemented
637C4C4×D4×D6111yescharacteristic
638C4#1684023411yescharacteristic
639C4#1684039611yescharacteristic
640C4#1684040911yescharacteristic
641C2—84no—
642C2—2444no—
643C2—2444no—
644C2#2144864011yescharacteristic
645C2C2×#1683976511yescharacteristic
646C2#2145434011yescharacteristic
647C2#2145741811yescharacteristic
648C2#2145743811yescharacteristic
649C2#2145804911yescharacteristic
650C2#2145806011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1957C2
socleC23×C61C22×D4
prime radicalC2×#16840397C2
cosocleC22×C4×C61C24
hypercenter#1957D61
hypocenterC61#22428
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. C2×#16839841C2semidirect
3. C2×#16839765C2semidirect
4. #21448770C2semidirect
5. #21446747C2semidirect
6. #21446254C2semidirect
7. #21446138C2semidirect
8. #21445816C2semidirect
9. #21445748C2semidirect
10. C4×#14519699C2semidirect
11. C61×#1957C2semidirect
14. C22×#11758189C22semidirect
16. #16840397C4semidirect
22. C2×#14519735C22semidirect
24. C2×C4⋊#92418C22semidirect
28. C2×#14519698C22semidirect
29. C2×#14519698C22semidirect
31. C2×#14519689C22semidirect
34. #16839619C22semidirect
36. C4×C4⋊Dic61C22semidirect
37. #16839525C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
41. C2×#102×C61C22semidirect
42. C2×#102×C61C22semidirect
43. C2×C4×C8×C61C22semidirect
44. C61×#384C22semidirect
88. #14519699D4semidirect
173. C2×C8×C61C2×D4semidirect
191. C4⋊Dic61C2×D4semidirect
307. C4×C61#418semidirect
321. #1957D61semidirect
357. C2×#102C2×D61semidirect
358. C2×#102C2×D61semidirect
360. C2×C4×C8C2×D61semidirect
363. #384C2×D61semidirect
370. C61#22428semidirect
503. C2×C8D4×D61semidirect
636. C4#16839765semidirect

Maximal subgroups

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

subgroupindex
C2×#168403972
C2×#168398412
C2×#168397652
#214487702
#214467472
#214462542
#214461382
#214458162
#214457482
C4×#145196992
C61×#19572
#2242861