Finite Group Databasebeta

← #29695629

Subgroups — #29695629

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×D4×D244C211yesmaximal, characteristic
2C2×#16839768C212yesmaximal, complemented
3#21448768C211yesmaximal, characteristic, complemented
4#21446745C211yesmaximal, characteristic, complemented
5#21446625C211yesmaximal, characteristic, complemented
6#21446257C212yesmaximal, complemented
7#21446139C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9#21445751C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C2×C61×#436C211yesmaximal, characteristic, complemented
12C23×D244C2212yes—
13C22×D4×D61—21no—
14D4×D244—24no—
15D4×D244C418yescomplemented
16C2×#14519774C2211yescharacteristic
17C2×#14519752—21no—
18C2×#14519751C2212yes—
19C2×#14519747—22no—
20C2×C4×D244C2211yescharacteristic
21C2×#14519736C2212yescomplemented
22#16839768—28no—
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2212yes—
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2212yescomplemented
27C2×#14519699C2212yescomplemented
28C2×#14519699C2212yescomplemented
29C2×#14519699C2212yescomplemented
30#16839665—21no—
31#16839655—22no—
32#16839632C2211yescharacteristic, complemented
33#16839622C2212yescomplemented
34C4⋊C4×Dic61—21no—
35C4×C4⋊Dic61C2211yescharacteristic, complemented
36#16839525C2211yescharacteristic, complemented
37#16839525C2211yescharacteristic, complemented
38#16839525C2212yescomplemented
39C22×M16×C61C2212yescomplemented
40C2×C4×D4×C61C2211yescharacteristic
41C61×#436—24no—
42C2×C4⋊C8×C61C2211yescharacteristic, complemented
43C2×C22⋊C8×C61C2212yescomplemented
44C2×C4×C8×C61C2211yescharacteristic, complemented
45C24×D61—22no—
46C22×D244—24no—
47C22×D244—24no—
48C22×D244—22no—
49C22×D244—21no—
50C22×D244—24no—
51C22×D244—21no—
52C22×D244—42no—
53C22×D244—42no—
54C22×D244C2×C418yes—
55C22×D244C2312yes—
56C22×D244C2311yescharacteristic
57C22×C22⋊D61—22no—
58C2×D4×D61—28no—
59C2×D4×D61—44no—
60C2×D4×D61—44no—
61C22×C4×D61—21no—
62#14519774—22no—
63#14519774C2×C414yes—
64#14519752—28no—
65#14519751—24no—
66#14519751C2×C418yes—
67#14519747—216no—
68C4×D244—22no—
69C4×D244C2×C414yes—
70#14519736—24no—
71#14519736—42no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C22⋊Dic61—22no—
79C2×C22⋊Dic61—22no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—21no—
84#14519699—24no—
85#14519699—22no—
86#14519699—24no—
87#14519699—24no—
88#14519699—42no—
89#14519699—42no—
90#14519699D414yescomplemented
91#14519684—21no—
92#14519684—22no—
93C22×D4×C61D411yescharacteristic
94C23×C4×C61C2312yes—
95C2×M16×C61—24no—
96C2×M16×C61—24no—
97C2×M16×C61—42no—
98C22×C8×C61C2312yes—
99C22×C8×C61C2311yescharacteristic
100C22×C8×C61C2311yescharacteristic
101C4×D4×C61C2×C414yes—
102C4×D4×C61D414yes—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61D412yes—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—22no—
107C22⋊C8×C61—24no—
108C4×C8×C61—22no—
109C24×C61C2×D412yes—
110C23×D61—21no—
111C23×D61—22no—
112C23×D61—28no—
113C23×D61—44no—
114C23×D61—44no—
115C23×D61—44no—
116C23×D61—44no—
117C23×D61—82no—
118C23×D61—82no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61C22⋊C414yes—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61C22×C414yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D414yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C2×C22⋊D61—28no—
133C2×C22⋊D61—44no—
134C2×C22⋊D61—44no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139D4×D61—432no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—216no—
148C2×D244—24no—
149C2×D244—28no—
150C2×D244—22no—
151C2×D244—24no—
152C2×D244—48no—
153C2×D244—48no—
154C2×D244—48no—
155C2×D244—44no—
156C2×D244—44no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244—81no—
160C2×D244—81no—
161C2×D244C22×C414yes—
162C2×D244C22×C412yes—
163C2×D244C2×D412yes—
164C2×D244#4112yes—
165C2×C4×D61—24no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168M16×C61—48no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—21no—
175C2×C8×C61—22no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D412yescomplemented
178C4⋊C4×C61C22⋊C414yes—
179C22⋊C4×C61C22⋊C418yes—
180C42×C61C22×C412yes—
181C42×C61C2×D412yes—
182C22⋊Dic61—44no—
183C22⋊Dic61—82no—
184#11758198—24no—
185#11758198—28no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—22no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—42no—
193C4⋊Dic61—81no—
194C4⋊Dic61C22×C412yes—
195C4⋊Dic61C2×D412yescomplemented
196C4×Dic61—44no—
197C4×Dic61—42no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61C2×C22⋊C414yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—24no—
210C22×D61—416no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—416no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—88no—
223C22×D61—88no—
224C22×D61—88no—
225C22×D61—88no—
226C22×D61—88no—
227D4×C61—28no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C2×C22⋊C414yes—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61AGL(1,ℤ/8)12yes—
240C2×C4×C61AGL(1,ℤ/8)12yes—
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C22⋊D61—416no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—41no—
260D244—24no—
261D244—22no—
262D244—24no—
263D244—22no—
264D244—22no—
265D244—416no—
266D244—42no—
267D244—42no—
268D244—48no—
269D244—84no—
270D244C4×D414yes—
271C4×D61—48no—
272C8×C61—22no—
273C8×C61—42no—
274C8×C61—41no—
275#22428—611nomaximal, Sylow
276C22×C61—24no—
277C22×C61—24no—
278C22×C61—24no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
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—42no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—44no—
298C2×D61—44no—
299C2×D61—44no—
300C2×D61—832no—
301C2×D61—82no—
302C2×D61—82no—
303C2×D61—82no—
304C2×D61—82no—
305C4×C61—24no—
306C4×C61—22no—
307C4×C61#41814yescomplemented
308C4×C61C2×C4×D411yescharacteristic
309C4×C61#51811yescharacteristic
310C4×C61D4211yescharacteristic
311C4×C61#54611yescharacteristic
312Dic61—42no—
313Dic61—42no—
314Dic61—82no—
315Dic61—81no—
316C2×D42—611no—
317C2×#436D6111yescharacteristic, complemented
318C2×#418—612no—
319#2354—611no—
320#1982—611no—
321#1957—611no—
322#1895—612no—
323#1875—612no—
324#1802—611no—
325#1786—612no—
326C4×#102—611no—
327C2×C61—44no—
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—84no—
337C23×D4—612no—
338C23×D4—1221no—
339C22×M16C2×D6112yescomplemented
340D42—618no—
341D42—1224no—
342C2×#127—611no—
343C2×#121—612no—
344C2×#121—1221no—
345C2×#120—1222no—
346C2×C4×D4—611no—
347C2×C4×D4C2×D6111yescharacteristic
348C2×#117—612no—
349#436—24no—
350C2×C4⋊C8C2×D6111yescharacteristic, complemented
351#418—1228no—
352C2×#102—611no—
353C2×#102—612no—
354C2×#102—611no—
355C2×#102—612no—
356C2×#102—612no—
357C2×#102—612no—
358C2×#102—612no—
359C2×C22⋊C8C2×D6112yescomplemented
360C2×C4×C8C2×D6111yescharacteristic, complemented
361#390—1221no—
362#388—1222no—
363#384—611no—
364#380—612no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4—1221no—
367#340—612no—
368#340—611no—
369#340—611no—
370C61#2242811yescharacteristic, complemented, Sylow
371C25—1222no—
372C22×D4—618no—
373C22×D4—611no—
374C22×D4—612no—
375C22×D4—1224no—
376C22×D4—1224no—
377C22×D4—1222no—
378C22×D4—1228no—
379C22×D4—1221no—
380C22×D4—1222no—
381C22×D4—1221no—
382C22×D4—1224no—
383C22×D4—2442no—
384C22×D4—2442no—
385C22×D4—2444no—
386C22×D4—2444no—
387C22×D4D24411yescharacteristic
388C23×C4—1221no—
389C23×C4C22×D6112yes—
390C2×M16—24no—
391C2×M16—24no—
392C2×M16—42no—
393C22×C8C22×D6112yes—
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396#127—614no—
397#127—1222no—
398#121—618no—
399#121—1224no—
400#121—1228no—
401#120—12216no—
402C4×D4—614no—
403C4×D4—1222no—
404C4×D4C4×D6114yes—
405C4×D4C22⋊D6114yes—
406#117—1224no—
407#117—2442no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4D24411yescharacteristic
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1222no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4D24412yes—
420C2×C42—1222no—
421C2×C42—1221no—
422C2×C42C22×D6111yescharacteristic
423C4⋊C8—22no—
424#102—614no—
425#102—1224no—
426#102—1222no—
427#102—1224no—
428#102—1224no—
429#102—2442no—
430#102—2442no—
431C22⋊C8—24no—
432C4×C8—22no—
433#95—1221no—
434#95—1222no—
435C24—1221no—
436C24—1228no—
437C24—1222no—
438C24—4882no—
439C24—4882no—
440C24—2444no—
441C24—2444no—
442C24—2444no—
443C24—2444no—
444C24C2×D24412yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—612no—
448C2×D4—614no—
449C2×D4—612no—
450C2×D4—612no—
451C2×D4—12216no—
452C2×D4—1228no—
453C2×D4—1222no—
454C2×D4—1221no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1221no—
458C2×D4—1222no—
459C2×D4—1221no—
460C2×D4—1222no—
461C2×D4—1224no—
462C2×D4—1228no—
463C2×D4—1224no—
464C2×D4—4881no—
465C2×D4—4881no—
466C2×D4—2448no—
467C2×D4—2448no—
468C2×D4—2448no—
469C2×D4—2444no—
470C2×D4—2444no—
471C2×D4—24432no—
472C2×D4—2444no—
473C2×D4—2444no—
474C2×D4—2442no—
475C2×D4—2442no—
476C2×D4#1175819814yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1224no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4C2×C4×D6114yes—
489C22×C4C2×D24412yes—
490C22×C4C2×D24411yescharacteristic
491C22×C4C2×D24411yescharacteristic
492C22×C4C2×C22⋊D6114yes—
493C22×C4C23×D6111yescharacteristic
494M16—48no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—22no—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—21no—
501C2×C8—22no—
502C2×C8—21no—
503C2×C8D4×D6112yescomplemented
504C4⋊C4—612no—
505C4⋊C4—612no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1222no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1222no—
512C4⋊C4—4881no—
513C4⋊C4—2442no—
514C4⋊C4#1175819814yes—
515C22⋊C4—1224no—
516C22⋊C4—1228no—
517C22⋊C4—4882no—
518C22⋊C4—2444no—
519C22⋊C4#1175819818yes—
520C42—2444no—
521C42—2442no—
522C42C2×C4×D6112yes—
523C42C2×C22⋊D6112yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—1222no—
531C23—1222no—
532C23—1222no—
533C23—1224no—
534C23—4888no—
535C23—4888no—
536C23—4888no—
537C23—4888no—
538C23—4888no—
539C23—24416no—
540C23—24416no—
541C23—2442no—
542C23—2442no—
543C23—2442no—
544C23—2442no—
545C23—2442no—
546C23—2442no—
547C23—2441no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23C2×#1175819814yes—
552C23C22×D24411yescharacteristic
553D4—28no—
554D4—614no—
555D4—1224no—
556D4—1222no—
557D4—1224no—
558D4—1222no—
559D4—1222no—
560D4—4884no—
561D4—24416no—
562D4—24416no—
563D4—2448no—
564D4—2442no—
565D4—2442no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—2441no—
577C2×C4—2442no—
578C2×C4—2441no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2448no—
587C2×C4C2×#1175819812yes—
588C2×C4C2×#1175819814yes—
589C2×C4C2×#1175819812yes—
590C2×C4#1451980512yes—
591C2×C4#1451980512yes—
592C2×C4C22×C4×D6111yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4#1451982211yescharacteristic
597C2×C4#1451982311yescharacteristic
598C8—22no—
599C8—42no—
600C8—41no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—1224no—
609C22—48832no—
610C22—4882no—
611C22—4882no—
612C22—4882no—
613C22—4882no—
614C22—2444no—
615C22—2444no—
616C22—2444no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22—2442no—
622C22—2442no—
623C22C22×#1175819811yescharacteristic
624C22D4×D24411yescharacteristic
625C22#1684040711yescharacteristic
626C22C2×#1451980511yescharacteristic
627C22C2×#1451980511yescharacteristic
628C22#1684056711yescharacteristic
629C22#1684057411yescharacteristic
630C4—22no—
631C4—24no—
632C4—4882no—
633C4—4881no—
634C4—2442no—
635C4—2442no—
636C4#1683976814yescomplemented
637C4C4×D4×D6111yescharacteristic
638C4#1684023411yescharacteristic
639C4D4×C22⋊D6111yescharacteristic
640C4#1684040011yescharacteristic
641C2—44no—
642C2—4884no—
643C2—2444no—
644C2D4×#1175819811yescharacteristic
645C2C2×#1683976811yescharacteristic
646C2#2145434611yescharacteristic
647C2#2145742111yescharacteristic
648C2#2145744111yescharacteristic
649C2#2145805111yescharacteristic
650C2#2145806211yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#436C2
socleC23×C61C22×D4
prime radicalC2×D4×D244C2
cosocleC22×C4×C61C24
hypercenterC2×#436D61
hypocenterC61#22428
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
2. C2×#16839768C2semidirect
3. #21448768C2semidirect
4. #21446745C2semidirect
5. #21446625C2semidirect
6. #21446257C2semidirect
7. #21446139C2semidirect
8. #21445820C2semidirect
9. #21445751C2semidirect
10. C4×#14519699C2semidirect
11. C2×C61×#436C2semidirect
15. D4×D244C4semidirect
21. C2×#14519736C22semidirect
26. C2×#14519699C22semidirect
27. C2×#14519699C22semidirect
28. C2×#14519699C22semidirect
29. C2×#14519699C22semidirect
32. #16839632C22semidirect
33. #16839622C22semidirect
35. C4×C4⋊Dic61C22semidirect
36. #16839525C22semidirect
37. #16839525C22semidirect
38. #16839525C22semidirect
39. C22×M16×C61C22semidirect
42. C2×C4⋊C8×C61C22semidirect
43. C2×C22⋊C8×C61C22semidirect
44. C2×C4×C8×C61C22semidirect
90. #14519699D4semidirect
177. C2×C8×C61C2×D4semidirect
195. C4⋊Dic61C2×D4semidirect
307. C4×C61#418semidirect
317. C2×#436D61semidirect
339. C22×M16C2×D61semidirect
350. C2×C4⋊C8C2×D61semidirect
359. C2×C22⋊C8C2×D61semidirect
360. C2×C4×C8C2×D61semidirect
370. C61#22428semidirect
503. C2×C8D4×D61semidirect
636. C4#16839768semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D2442
C2×#168397682
#214487682
#214467452
#214466252
#214462572
#214461392
#214458202
#214457512
C4×#145196992
C2×C61×#4362
#2242861