Finite Group Databasebeta

← #29714316

Subgroups — #29714316

Order 15616.

1478 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×#16840310C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3C4×#14519748C211yesmaximal, characteristic, complemented
4#21446999C212yesmaximal, complemented
5#21446814C212yesmaximal, complemented
6#21446770C211yesmaximal, characteristic, complemented
7#21446747C211yesmaximal, characteristic, complemented
8#21446169C212yesmaximal, complemented
9#21446064C212yesmaximal, complemented
10#21445796C211yesmaximal, characteristic, complemented
11C2×C61×#459C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13#16840310—24no—
14#16840310—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519764—21no—
17C2×#14519748—21no—
18C2×#14519748C2212yes—
19C2×#14519748C2211yescharacteristic
20C2×D4×Dic61—21no—
21C2×D4×Dic61C2212yes—
22C2×C4×C22⋊D61—21no—
23C2×C42×D61C2211yescharacteristic
24#16840042—24no—
25C2×#14519710C2212yes—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519698C2212yes—
28C2×#14519698C2212yes—
29C2×#14519698C2212yes—
30C2×#14519689C2212yes—
31#16839665C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839650—21no—
34#16839638—21no—
35#16839619C2211yescharacteristic
36C4×C4⋊Dic61C2211yescharacteristic
37C4×C22⋊Dic61—21no—
38#16839525C2212yes—
39#16839525C2212yes—
40#16839516C2212yes—
41C22×D8×C61C2212yes—
42C2×#127×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C61×#459—24no—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#102×C61C2212yes—
47C2×#11758224—28no—
48C2×#11758224—44no—
49C2×#11758224—44no—
50C22×Dic122—21no—
51C22×C22⋊D61—21no—
52C22×C22⋊D61—22no—
53C23×Dic61—22no—
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58#14519777—22no—
59#14519777D414yescomplemented
60#14519764—28no—
61#14519748—28no—
62#14519748—24no—
63#14519748—22no—
64#14519748—24no—
65#14519748—22no—
66D4×Dic61—28no—
67D4×Dic61—24no—
68D4×Dic61—24no—
69C4×C22⋊D61—44no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×C22⋊Dic61—22no—
76C2×C22⋊Dic61—22no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61—22no—
86C42×D61—22no—
87#14519710—24no—
88C4⋊#92418—22no—
89#14519698—24no—
90#14519698—24no—
91#14519698—42no—
92#14519689—42no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96C22×D4×C61—21no—
97C22×D4×C61—21no—
98C22×D4×C61C2312yes—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C2×D8×C61—24no—
102C2×D8×C61—24no—
103C2×D8×C61—42no—
104C22×C8×C61C2312yes—
105#127×C61—22no—
106#127×C61D414yescomplemented
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C42×C61C2311yescharacteristic
112C4⋊C8×C61—22no—
113#102×C61—24no—
114C24×C61—22no—
115C24×C61—21no—
116C23×D61C2×D411yescharacteristic
117C2×D4×C61—24no—
118C2×D4×C61—24no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—21no—
129C2×D4×C61—42no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C2×D4×C61—42no—
133C2×D4×C61C2×D414yes—
134C2×D4×C61#4112yescomplemented
135C22×C4×C61—21no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—28no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61—42no—
145C2×C22⋊D61—44no—
146C2×C22⋊D61—44no—
147C2×C22⋊D61—42no—
148C22×Dic61—28no—
149C22×Dic61—22no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—44no—
153C22×Dic61—44no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157#11758224—432no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—41no—
168C2×C4×D61—41no—
169C2×C4×D61D814yescomplemented
170C2×Dic122—24no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173D8×C61—48no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C22⋊C4×C61—42no—
180C42×C61C2×D412yes—
181C42×C61C2×D412yes—
182C22⋊Dic61—28no—
183C22⋊Dic61—28no—
184C22⋊Dic61—24no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187#11758198—42no—
188C4⋊Dic61—24no—
189C4⋊Dic61—22no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61C2×D414yes—
194C4⋊Dic61#4112yescomplemented
195#11758196—42no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—42no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C23×C61—24no—
205C23×C61—22no—
206C23×C61—41no—
207C23×C61—42no—
208C23×C61—41no—
209C23×C61—42no—
210C23×C61—41no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61C22×D411yescharacteristic
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61C2×D812yes—
226D4×C61—22no—
227D4×C61—24no—
228D4×C61—24no—
229D4×C61—22no—
230D4×C61—48no—
231D4×C61—41no—
232D4×C61—42no—
233D4×C61—41no—
234D4×C61—42no—
235D4×C61—48no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—84no—
242C2×C4×C61#12012yes—
243C2×C4×C61#12012yes—
244C2×C4×C61C2×D812yes—
245C2×C4×C61AGL(1,ℤ/8)12yes—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C22⋊D61—416no—
253C22⋊D61—48no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—24no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—416no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61C2×D812yes—
273C4×D61—28no—
274C4×D61—44no—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—41no—
279Dic122—48no—
280C8×C61—42no—
281#92418—42no—
282#24150—611nomaximal, Sylow
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—44no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61—82no—
290C22×C61—82no—
291C22×C61—81no—
292C22×C61—82no—
293C22×C61—82no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61C2×#12011yescharacteristic
298C22×C61#52511yescharacteristic
299C22×C61#53711yescharacteristic
300C22×C61C22×D811yescharacteristic
301C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C61#57511yescharacteristic
303C22×C61#57511yescharacteristic
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C4×C61—22no—
312C4×C61—81no—
313C4×C61#44714yes—
314C4×C61#54611yescharacteristic
315C4×C61#54711yescharacteristic
316C4×C61#54811yescharacteristic
317C4×C61#55011yescharacteristic
318Dic61—22no—
319Dic61—42no—
320Dic61—41no—
321Dic61—44no—
322Dic61—41no—
323Dic61—81no—
324C2×#532—611no—
325C2×#474—611no—
326C2×#459D6111yescharacteristic, complemented
327C4×#121—611no—
328#2033—612no—
329#1999—612no—
330#1988—611no—
331#1982—611no—
332#1879—612no—
333#1862—612no—
334#1797—611no—
335C2×C61—82no—
336C2×C61—84no—
337C2×C61#240711yescharacteristic
338C2×C61C2×#44711yescharacteristic
339C2×C61#300511yescharacteristic
340C2×C61#310011yescharacteristic
341C2×C61#311511yescharacteristic
342C2×C61#324511yescharacteristic
343C2×C61#325111yescharacteristic
344D61—42no—
345C22×#41—1221no—
346C22×D8C2×D6112yes—
347#532—1224no—
348#532—1224no—
349C2×C4⋊Q8—611no—
350C2×#127C2×D6111yescharacteristic
351C2×#124—1221no—
352C2×#121—612no—
353C2×#121—611no—
354C2×#121—1221no—
355C2×C4×D4—612no—
356C2×C4×D4—1221no—
357C2×C4×D4—1221no—
358C2×C4×D4C2×D6111yescharacteristic
359C22×C42—611no—
360#474—1224no—
361#459—24no—
362C2×#107—612no—
363C2×C4⋊C8—611no—
364C2×C4⋊C8C2×D6111yescharacteristic
365C2×#102—612no—
366C2×#102—612no—
367C2×#102—612no—
368C2×#102C2×D6112yes—
369C2×C22⋊C8—612no—
370#390—611no—
371#389—611no—
372#388—1221no—
373#386—1221no—
374#380—611no—
375C4×C4⋊C4—611no—
376C4×C22⋊C4—1221no—
377#340—612no—
378#340—612no—
379#336—612no—
380C61#2415011yescharacteristic, complemented, Sylow
381C2×#41—1228no—
382C2×#41—2444no—
383C2×#41—2444no—
384C22×Q8—1221no—
385C22×D4—21no—
386C22×D4—21no—
387C22×D4—1221no—
388C22×D4—1222no—
389C22×D4C22×D6111yescharacteristic
390C22×D4C22×D6112yes—
391C23×C4—21no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—1222no—
396C2×D8—24no—
397C2×D8—24no—
398C2×D8—42no—
399C22×C8—612no—
400C22×C8C22×D6112yes—
401C4⋊Q8—614no—
402C4⋊Q8—1222no—
403#127—22no—
404#127C22⋊D6114yescomplemented
405#124—1228no—
406#121—1224no—
407#121—1222no—
408#121—1228no—
409#121—1222no—
410#121—1224no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—1224no—
414C4×D4—1228no—
415C4×D4—1224no—
416C4×D4—2444no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1222no—
425C2×C22⋊C4—1222no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—1222no—
434C2×C42—1222no—
435C2×C42—1221no—
436C2×C42C22×D6111yescharacteristic
437#107—1224no—
438C4⋊C8—22no—
439C4⋊C8—1222no—
440#102—24no—
441#102—1224no—
442#102—1224no—
443#102—2442no—
444C22⋊C8—2442no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448C24—21no—
449C24—22no—
450C24—611no—
451#41—24432no—
452C2×Q8—1224no—
453C2×Q8—2442no—
454C2×Q8—2442no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—1228no—
472C2×D4—1224no—
473C2×D4—2442no—
474C2×D4—2444no—
475C2×D4—2444no—
476C2×D4—2442no—
477C2×D4#1175822412yescomplemented
478C2×D4C2×C22⋊D6114yes—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—611no—
484C22×C4—614no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1228no—
490C22×C4—1221no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1221no—
500C22×C4—2441no—
501C22×C4—2441no—
502C22×C4—2444no—
503C22×C4—2444no—
504C22×C4D4×D6111yescharacteristic
505C22×C4D4×D6111yescharacteristic
506C22×C4C23×D6111yescharacteristic
507D8—48no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—1222no—
512C2×C8—1222no—
513C2×C8—1222no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—614no—
517C4⋊C4—612no—
518C4⋊C4—1222no—
519C4⋊C4—1221no—
520C4⋊C4—1224no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—2442no—
524C22⋊C4—42no—
525C22⋊C4—1228no—
526C22⋊C4—1228no—
527C22⋊C4—1224no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—1221no—
532C42—1221no—
533C42—1222no—
534C42—1222no—
535C42—2442no—
536C42D4×D6112yes—
537C42C2×C22⋊D6112yes—
538C23—24no—
539C23—22no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—612no—
553C23—1221no—
554C23—1221no—
555C23—1221no—
556C23—1221no—
557C23—1221no—
558C23—1221no—
559C23C2×D4×D6111yescharacteristic
560Q8—2448no—
561D4—22no—
562D4—24no—
563D4—24no—
564D4—22no—
565D4—48no—
566D4—48no—
567D4—41no—
568D4—42no—
569D4—41no—
570D4—42no—
571D4—2448no—
572D4—24416no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—612no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1228no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1224no—
586C2×C4—1221no—
587C2×C4—1222no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—2444no—
592C2×C4—24416no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4#1451974512yes—
605C2×C4#1451974512yes—
606C2×C4D8×D6112yes—
607C2×C4#1451980112yes—
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4C22×C22⋊D6111yescharacteristic
610C2×C4#41×D6111yescharacteristic
611C2×C4C2×#1175822411yescharacteristic
612C2×C4#1451982211yescharacteristic
613C2×C4#1451982411yescharacteristic
614C8—42no—
615C8—2442no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—44no—
622C22—81no—
623C22—82no—
624C22—82no—
625C22—81no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—82no—
630C22—1222no—
631C22—1222no—
632C22—1222no—
633C22—2441no—
634C22—2441no—
635C22—2441no—
636C22—2441no—
637C22C2×#1451974511yescharacteristic
638C22#1684027211yescharacteristic
639C22#1684034111yescharacteristic
640C22C2×D8×D6111yescharacteristic
641C22C2×#1451980111yescharacteristic
642C22#1684055811yescharacteristic
643C22#1684055811yescharacteristic
644C4—22no—
645C4—81no—
646C4—1222no—
647C4—4881no—
648C4—2444no—
649C4—2441no—
650C4—2442no—
651C4—2441no—
652C4#1683989414yes—
653C4#1684040611yescharacteristic
654C4#1684041111yescharacteristic
655C4#1684041811yescharacteristic
656C4#1684043011yescharacteristic
657C2—82no—
658C2—84no—
659C2—2442no—
660C2#2144916011yescharacteristic
661C2C2×#1683989411yescharacteristic
662C2#2145505811yescharacteristic
663C2#2145581811yescharacteristic
664C2#2145596711yescharacteristic
665C2#2145712711yescharacteristic
666C2#2145717711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#459C2
socleC23×C61C22×D4
hypercenterC2×#459D61
hypocenterC61#24150
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16840310C2semidirect
2. C2×#16840042C2semidirect
3. C4×#14519748C2semidirect
4. #21446999C2semidirect
5. #21446814C2semidirect
6. #21446770C2semidirect
7. #21446747C2semidirect
8. #21446169C2semidirect
9. #21446064C2semidirect
10. #21445796C2semidirect
11. C2×C61×#459C2semidirect
59. #14519777D4semidirect
106. #127×C61D4semidirect
134. C2×D4×C61#41semidirect
169. C2×C4×D61D8semidirect
194. C4⋊Dic61#41semidirect
326. C2×#459D61semidirect
380. C61#24150semidirect
404. #127C22⋊D61semidirect
477. C2×D4#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168403102
C2×#168400422
C4×#145197482
#214469992
#214468142
#214467702
#214467472
#214461692
#214460642
#214457962
C2×C61×#4592
#2415061