Finite Group Databasebeta

← #29729581

Subgroups — #29729581

Order 15616.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839975C211yesmaximal, characteristic
2C2×#16839891C211yesmaximal, characteristic, complemented
3#21451304C211yesmaximal, characteristic, complemented
4#21451175C211yesmaximal, characteristic, complemented
5#21447203C211yesmaximal, characteristic, complemented
6#21447199C211yesmaximal, characteristic, complemented
7#21447135C211yesmaximal, characteristic, complemented
8#21446699C211yesmaximal, characteristic, complemented
9#21446696C211yesmaximal, characteristic
10#21446629C211yesmaximal, characteristic, complemented
11#21446586C211yesmaximal, characteristic, complemented
12#21446519C211yesmaximal, characteristic, complemented
13#21446510C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2054C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16839975—24no—
21#16839891—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839698—21no—
36#16839690—21no—
37#16839674—21no—
38#16839674—21no—
39#16839668C2211yescharacteristic
40#16839667C2211yescharacteristic
41#16839665C2211yescharacteristic
42#16839665C2211yescharacteristic
43#16839661C2211yescharacteristic
44#16839660C2211yescharacteristic
45#16839655—21no—
46#16839650—21no—
47#16839637—21no—
48#16839634—21no—
49#16839627C2211yescharacteristic
50C4⋊C4×Dic61C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C22×D8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122D411yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—41no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519764—22no—
70#14519764—22no—
71C4×Dic122—22no—
72C4×Dic122—22no—
73D4×Dic61—22no—
74D4×Dic61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×#11758196D411yescharacteristic
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519710—41no—
94#14519709—41no—
95#14519706—22no—
96#14519706D414yes—
97#14519702—22no—
98#14519702—41no—
99#14519700—22no—
100#14519698—22no—
101#14519698—22no—
102#14519698—22no—
103#14519688—22no—
104#14519687—22no—
105C8×Dic61—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115C22×D4×C61C2311yescharacteristic
116C22×D4×C61C2311yescharacteristic
117C2×D8×C61—22no—
118C2×D8×C61—22no—
119C2×D8×C61—41no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—22no—
130#102×C61—22no—
131#102×C61—41no—
132C24×C61—21no—
133C24×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61#4112yes—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—42no—
156C22×Dic61—81no—
157C22×Dic61—81no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×Dic122—22no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—41no—
170C2×Dic122—41no—
171D8×C61—44no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D412yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C22⋊C4×C61—81no—
185C22⋊C4×C61—81no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—42no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—81no—
194C22⋊Dic61—81no—
195#11758200—44no—
196C4⋊Dic61—22no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—81no—
204#11758196—22no—
205#11758196—22no—
206#11758196—81no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—81no—
210C4×Dic61C2×D412yes—
211C4×Dic61#4112yes—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D412yes—
217C2×#92418#4112yes—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61C22×D411yescharacteristic
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—84no—
282C2×Dic61#12112yes—
283C2×Dic61#12112yes—
284Dic122—44no—
285Dic122—41no—
286Dic122—41no—
287Dic122—41no—
288Dic122—41no—
289C8×C61—22no—
290C8×C61—41no—
291#92418—22no—
292#92418—22no—
293#25298—611nomaximal, Sylow
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61C2×#12111yescharacteristic
309C22×C61#54811yescharacteristic
310C22×C61#55011yescharacteristic
311C22×C61#57511yescharacteristic
312C22×C61#57511yescharacteristic
313C22×C61#57611yescharacteristic
314C22×C61#57711yescharacteristic
315C4×C61—41no—
316C4×C61—41no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61#52911yescharacteristic
320C4×C61#54211yescharacteristic
321C4×C61#54611yescharacteristic
322C4×C61#54811yescharacteristic
323Dic61—22no—
324Dic61—22no—
325Dic61—41no—
326Dic61—41no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#463—611no—
332C2×#446—611no—
333#2592—611no—
334#2580—611no—
335#2067—611no—
336#2066—611no—
337#2055—611no—
338#2054D6111yescharacteristic
339#1973—611no—
340#1972—611no—
341#1958—611no—
342#1949—611no—
343#1936—611no—
344#1935—611no—
345C4×#102—611no—
346C2×C61—82no—
347C2×C61—161no—
348C2×C61#242611yescharacteristic
349C2×C61#305411yescharacteristic
350C2×C61#306211yescharacteristic
351C2×C61#337011yescharacteristic
352C2×C61#337611yescharacteristic
353C2×C61#340111yescharacteristic
354C2×C61#340311yescharacteristic
355C22×SD16—611no—
356C22×D8C2×D6111yescharacteristic
357C2×#124—611no—
358C2×C4×Q8—611no—
359C2×C4×D4—611no—
360#463—1224no—
361#446—1224no—
362C2×#107—611no—
363C2×#106—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C4⋊C8C2×D6111yescharacteristic
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102C2×D6111yescharacteristic
374C2×#102C2×D6111yescharacteristic
375C2×#97—611no—
376C2×#97—611no—
377C2×C4×C8—611no—
378#399—1221no—
379#397—1221no—
380#393—1221no—
381#393—1221no—
382#391—611no—
383#391—611no—
384#390—611no—
385#390—611no—
386#390C2×D6111yescharacteristic
387#390C2×D6111yescharacteristic
388#389—611no—
389#389—611no—
390#388—1221no—
391#388—1221no—
392#385—1221no—
393#385—1221no—
394#382—611no—
395C4×C4⋊C4—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340C2×D6111yescharacteristic
400C61#2529811yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×Q8—611no—
403C22×D4C22×D6111yescharacteristic
404C22×D4C22×D6111yescharacteristic
405C23×C4—1221no—
406C2×SD16—1222no—
407C2×SD16—1222no—
408C2×SD16—2441no—
409C2×D8—22no—
410C2×D8—22no—
411C2×D8—41no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416#124—1222no—
417#124—1222no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×D4—1222no—
421C4×D4—1222no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—1221no—
446C2×C42C22×D6111yescharacteristic
447#107—2441no—
448#106—2441no—
449C4⋊C8—22no—
450C4⋊C8—614no—
451C4⋊C8—1222no—
452#103—1222no—
453#103—1222no—
454#103—2441no—
455#102—22no—
456#102—41no—
457#102—1222no—
458#102—1222no—
459#102—1222no—
460#97—1222no—
461#97—1222no—
462C4×C8—1222no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—21no—
473C24—21no—
474C2×Q8—1222no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—2441no—
482C2×Q8—2441no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4#1175822412yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—4881no—
506C22×C4—4881no—
507C22×C4—2442no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—2444no—
512D8—44no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—612no—
519C2×C8—612no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8D4×D6112yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—4881no—
539C4⋊C4—4881no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—81no—
544C22⋊C4—4881no—
545C22⋊C4—4881no—
546C22⋊C4—2442no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C42—21no—
552C42—21no—
553C42—612no—
554C42—612no—
555C42—1221no—
556C42—1221no—
557C42—4881no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23C2×D4×D6111yescharacteristic
573Q8—2444no—
574Q8—2441no—
575Q8—2441no—
576Q8—2441no—
577Q8—2441no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4—41no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—612no—
599C2×C4—612no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—4884no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4#41×D6111yescharacteristic
626C2×C4C2×#1175822411yescharacteristic
627C2×C4#1451982211yescharacteristic
628C2×C4#1451982411yescharacteristic
629C8—22no—
630C8—41no—
631C8—1222no—
632C8—1222no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22#121×D6111yescharacteristic
648C22#1684041811yescharacteristic
649C22#1684043011yescharacteristic
650C22#1684055811yescharacteristic
651C22#1684055811yescharacteristic
652C22#1684056311yescharacteristic
653C22#1684056911yescharacteristic
654C4—41no—
655C4—41no—
656C4—81no—
657C4—81no—
658C4—1222no—
659C4—1222no—
660C4—4881no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—2441no—
665C4—2441no—
666C4#1684028911yescharacteristic
667C4#1684038011yescharacteristic
668C4#1684039911yescharacteristic
669C4#1684042011yescharacteristic
670C2—82no—
671C2—161no—
672C2#2144937811yescharacteristic
673C2#2145542611yescharacteristic
674C2#2145549811yescharacteristic
675C2#2145856511yescharacteristic
676C2#2145864311yescharacteristic
677C2#2145888211yescharacteristic
678C2#2145890211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2054C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenter#2054D61
hypocenterC61#25298
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2054

Complements

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

NH ≅ G/Ntype
2. C2×#16839891C2semidirect
3. #21451304C2semidirect
4. #21451175C2semidirect
5. #21447203C2semidirect
6. #21447199C2semidirect
7. #21447135C2semidirect
8. #21446699C2semidirect
10. #21446629C2semidirect
11. #21446586C2semidirect
12. #21446519C2semidirect
13. #21446510C2semidirect
14. #102×Dic61C2semidirect
400. C61#25298semidirect

Maximal subgroups

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

subgroupindex
C2×#168399752
C2×#168398912
#214513042
#214511752
#214472032
#214471992
#214471352
#214466992
#214466962
#214466292
#214465862
#214465192
#214465102
#102×Dic612
C61×#20542
#2529861