Finite Group Databasebeta

← #29748189

Subgroups — #29748189

Order 15616.

824 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450173C211yesmaximal, characteristic, complemented
2#21448861C211yesmaximal, characteristic, complemented
3#21447127C211yesmaximal, characteristic, complemented
4#21446998C211yesmaximal, characteristic, complemented
5#21446963C211yesmaximal, characteristic, complemented
6#21446773C211yesmaximal, characteristic, complemented
7#21446504C211yesmaximal, characteristic, complemented
8#21446442C211yesmaximal, characteristic, complemented
9#21446422C211yesmaximal, characteristic, complemented
10#21446414C211yesmaximal, characteristic, complemented
11#21446169C211yesmaximal, characteristic, complemented
12#21446142C211yesmaximal, characteristic, complemented
13#21445915C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#1999C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839687—21no—
28#16839676—21no—
29#16839665—21no—
30#16839665—21no—
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839656—21no—
36#16839655—21no—
37#16839642—21no—
38#16839622C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C22⋊C4×Dic61—21no—
42C4×C22⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×D8×C61C2211yescharacteristic
53C2×#121×C61C2211yescharacteristic
54C2×#121×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519768—22no—
63#14519768—41no—
64D4×Dic61—22no—
65D4×Dic61—41no—
66#14519736—22no—
67#14519736—41no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61D411yescharacteristic
85C2×C22⋊Dic61D411yescharacteristic
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519710—22no—
92#14519706—41no—
93#14519698—22no—
94#14519698—22no—
95#14519698—41no—
96#14519698—41no—
97C22⋊#92418—41no—
98C8×Dic61—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C22×D4×C61—21no—
106C22×D4×C61—21no—
107C22×D4×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C23×C4×C61C2311yescharacteristic
110C2×D8×C61—22no—
111C2×D8×C61—41no—
112C2×D8×C61—41no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115#121×C61—22no—
116#121×C61—22no—
117#121×C61—22no—
118#121×C61—41no—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123#102×C61—22no—
124#102×C61—22no—
125C22⋊C8×C61—41no—
126C24×C61—21no—
127C24×C61—21no—
128C24×C61C2×D411yescharacteristic
129C2×D4×C61—24no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×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—42no—
142C2×D4×C61—42no—
143C2×D4×C61—81no—
144C2×D4×C61—81no—
145C2×D4×C61—81no—
146C2×D4×C61#4112yes—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—81no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168D8×C61—44no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61#4112yes—
178C4⋊C4×C61#4112yes—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—42no—
181C22⋊Dic61—22no—
182C22⋊Dic61—41no—
183C22⋊Dic61—42no—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187C22⋊Dic61—81no—
188C22⋊Dic61—81no—
189C22⋊Dic61—81no—
190C22⋊Dic61—81no—
191C22⋊Dic61—81no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—81no—
201C4⋊Dic61#4112yes—
202#11758196—42no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C4×Dic61—81no—
209C4×Dic61—81no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C23×C61—22no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
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×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61C22×D411yescharacteristic
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—84no—
247D4×C61—84no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—84no—
291C2×Dic61#13512yes—
292C8×C61—41no—
293C8×C61—41no—
294#92418—41no—
295#92418—41no—
296#26730—611nomaximal, Sylow
297C22×C61—41no—
298C22×C61—42no—
299C22×C61—41no—
300C22×C61—42no—
301C22×C61—41no—
302C22×C61—42no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61—81no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61—81no—
317C22×C61—81no—
318C22×C61#52511yescharacteristic
319C22×C61#54611yescharacteristic
320C22×C61#54611yescharacteristic
321C22×C61C2×#13511yescharacteristic
322C22×C61#57511yescharacteristic
323C22×C61#57611yescharacteristic
324C22×C61#57611yescharacteristic
325C4×C61—42no—
326C4×C61—41no—
327C4×C61—42no—
328C4×C61#52911yescharacteristic
329C4×C61#53811yescharacteristic
330C4×C61#55511yescharacteristic
331C4×C61#55911yescharacteristic
332Dic61—22no—
333Dic61—42no—
334Dic61—41no—
335Dic61—41no—
336Dic61—41no—
337Dic61—81no—
338Dic61—81no—
339Dic61—81no—
340#2490—611no—
341#2363—611no—
342#2054—611no—
343#2033—611no—
344#2025—611no—
345#1999D6111yescharacteristic
346#1989—611no—
347#1934—611no—
348#1925—611no—
349#1923—611no—
350#1921—611no—
351#1879—611no—
352#1875—611no—
353#1825—611no—
354C4×#102—611no—
355C2×C61—81no—
356C2×C61—82no—
357C2×C61—161no—
358C2×C61#245911yescharacteristic
359C2×C61#310211yescharacteristic
360C2×C61#312211yescharacteristic
361C2×C61#329711yescharacteristic
362C2×C61#330511yescharacteristic
363C2×C61#330911yescharacteristic
364C2×C61#332211yescharacteristic
365C22×D8C2×D6111yescharacteristic
366C2×#125—611no—
367C2×#121C2×D6111yescharacteristic
368C2×#121C2×D6111yescharacteristic
369C2×C4×D4—611no—
370C2×#117—611no—
371C2×#107—611no—
372C2×C4⋊C8—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102C2×D6111yescharacteristic
378C2×#102C2×D6111yescharacteristic
379C2×C22⋊C8—611no—
380C2×C22⋊C8C2×D6111yescharacteristic
381C2×C4×C8—611no—
382#397—1221no—
383#394—1221no—
384#390—611no—
385#390—611no—
386#390—611no—
387#390—1221no—
388#390—1221no—
389#389—611no—
390#388—1221no—
391#388—1221no—
392#386—1221no—
393#380—611no—
394C4×C4⋊C4—611no—
395C4×C4⋊C4—611no—
396C4×C22⋊C4—611no—
397C4×C22⋊C4—1221no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#340—611no—
405#340C2×D6111yescharacteristic
406#336—611no—
407#336—611no—
408C61#2673011yescharacteristic, complemented, Sylow
409C22×D4—21no—
410C22×D4—21no—
411C22×D4C22×D6111yescharacteristic
412C22×D4C22×D6111yescharacteristic
413C23×C4—1221no—
414C23×C4C22×D6111yescharacteristic
415C2×D8—22no—
416C2×D8—41no—
417C2×D8—41no—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8C22×D6111yescharacteristic
421C22×C8C22×D6111yescharacteristic
422#125—1222no—
423#125—2441no—
424#121—22no—
425#121—22no—
426#121—22no—
427#121—41no—
428C4×D4—1222no—
429C4×D4—2441no—
430#117—1222no—
431#117—2441no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—611no—
444C2×C22⋊C4—611no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—1221no—
457C2×C42—1221no—
458C2×C42—1221no—
459#107—1222no—
460C4⋊C8—2441no—
461#102—22no—
462#102—22no—
463#102—1222no—
464#102—1222no—
465#102—2441no—
466#102—2441no—
467C22⋊C8—41no—
468C22⋊C8—2441no—
469C4×C8—2441no—
470#95—611no—
471#95—611no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—21no—
477C24—21no—
478C24D4×D6111yescharacteristic
479C2×D4—24no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
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—42no—
492C2×D4—42no—
493C2×D4—81no—
494C2×D4—81no—
495C2×D4—81no—
496C2×D4#1175822412yes—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—4881no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4D4×D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518D8—44no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—612no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—4881no—
542C4⋊C4—2442no—
543C4⋊C4—2442no—
544C4⋊C4#1175822412yes—
545C4⋊C4#1175822412yes—
546C22⋊C4—24no—
547C22⋊C4—42no—
548C22⋊C4—1222no—
549C22⋊C4—4881no—
550C22⋊C4—4881no—
551C22⋊C4—4881no—
552C22⋊C4—4881no—
553C22⋊C4—4881no—
554C22⋊C4—2442no—
555C22⋊C4—2441no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C42—1221no—
560C42—1221no—
561C42—1221no—
562C42—1221no—
563C42—4881no—
564C42—4881no—
565C42—2442no—
566C23—22no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23C2×D4×D6111yescharacteristic
588D4—22no—
589D4—22no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—84no—
597D4—84no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—612no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1222no—
624C2×C4—4884no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4#41×D6111yescharacteristic
637C2×C4C2×#1175822411yescharacteristic
638C2×C4C2×#1175822411yescharacteristic
639C2×C4C2×#1175822411yescharacteristic
640C2×C4#1451982211yescharacteristic
641C2×C4#1451982211yescharacteristic
642C8—41no—
643C8—41no—
644C8—2441no—
645C8—2441no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—41no—
655C22—42no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22#1684026811yescharacteristic
668C22#1684039911yescharacteristic
669C22#1684040111yescharacteristic
670C22#135×D6111yescharacteristic
671C22#1684055811yescharacteristic
672C22#1684056311yescharacteristic
673C22#1684056611yescharacteristic
674C4—41no—
675C4—42no—
676C4—42no—
677C4—1222no—
678C4—4881no—
679C4—4881no—
680C4—4881no—
681C4—2442no—
682C4—2441no—
683C4—2441no—
684C4—2441no—
685C4#1684029311yescharacteristic
686C4#1684034711yescharacteristic
687C4#1684047611yescharacteristic
688C4#1684049811yescharacteristic
689C2—81no—
690C2—82no—
691C2—161no—
692C2#2144975111yescharacteristic
693C2#2145583011yescharacteristic
694C2#2145605611yescharacteristic
695C2#2145766411yescharacteristic
696C2#2145776211yescharacteristic
697C2#2145782011yescharacteristic
698C2#2145802411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#1999C2
socleC23×C61C22×D4
hypercenter#1999D61
hypocenterC61#26730
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. #21450173C2semidirect
2. #21448861C2semidirect
3. #21447127C2semidirect
4. #21446998C2semidirect
5. #21446963C2semidirect
6. #21446773C2semidirect
7. #21446504C2semidirect
8. #21446442C2semidirect
9. #21446422C2semidirect
10. #21446414C2semidirect
11. #21446169C2semidirect
12. #21446142C2semidirect
13. #21445915C2semidirect
14. #102×Dic61C2semidirect
408. C61#26730semidirect

Maximal subgroups

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

subgroupindex
#214501732
#214488612
#214471272
#214469982
#214469632
#214467732
#214465042
#214464422
#214464222
#214464142
#214461692
#214461422
#214459152
#102×Dic612
C61×#19992
#2673061