Finite Group Databasebeta

← #29753159

Subgroups — #29753159

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450427C211yesmaximal, characteristic, complemented
2#21447340C211yesmaximal, characteristic, complemented
3#21447136C211yesmaximal, characteristic, complemented
4#21447050C211yesmaximal, characteristic
5#21446907C211yesmaximal, characteristic
6#21446672C211yesmaximal, characteristic, complemented
7#21446460C211yesmaximal, characteristic, complemented
8#21446436C211yesmaximal, characteristic, complemented
9#21446397C211yesmaximal, characteristic, complemented
10#21446378C211yesmaximal, characteristic, complemented
11#21446153C211yesmaximal, characteristic, complemented
12#21446153C211yesmaximal, characteristic, complemented
13#21445829C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic, complemented
15C61×#2366C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519700C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30C61⋊#401—21no—
31#16839700—21no—
32#16839668C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839668C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839643—21no—
37#16839629—21no—
38#16839628C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839517C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×#127×C61C2211yescharacteristic
49C2×C4×D4×C61—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C61×#390—21no—
53C61×#390C2211yescharacteristic
54C61×#390C2211yescharacteristic
55C61×#386—21no—
56C61×#386—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C2×#11758200—22no—
61C2×#11758200—22no—
62C2×#11758200—22no—
63C2×#11758200—41no—
64C2×#11758200—41no—
65C2×#11758200—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×C4×Dic61—21no—
75#14519710—41no—
76#14519708—22no—
77C4⋊#92418—41no—
78#14519700—22no—
79#14519700—41no—
80#14519698—22no—
81#14519698—22no—
82#14519698—22no—
83#14519698—41no—
84#14519698—41no—
85#14519698—41no—
86C4×#92418—41no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×D4×C61—21no—
98C22×D4×C61—21no—
99C22×D4×C61C2311yescharacteristic
100C22×D4×C61C2311yescharacteristic
101C23×C4×C61—21no—
102C23×C4×C61—21no—
103#127×C61—22no—
104#127×C61—22no—
105C4×D4×C61—22no—
106C4×D4×C61—22no—
107C4×D4×C61—41no—
108C4×D4×C61—41no—
109C4×D4×C61—44no—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124#95×C61—21no—
125#95×C61D411yescharacteristic
126#95×C61D411yescharacteristic
127C24×C61—21no—
128C24×C61—21no—
129C2×D4×C61—24no—
130C2×D4×C61—24no—
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#4112yes—
146C2×D4×C61#4112yes—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61—42no—
152C22×C4×C61—42no—
153C22×C4×C61—81no—
154C22×C4×C61—81no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61#4112yes—
178C22⋊C4×C61—42no—
179C22⋊C4×C61—42no—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—81no—
183C22⋊C4×C61—81no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—42no—
191#11758200—44no—
192#11758200—44no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199#11758196—42no—
200C4×Dic61—42no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418#4112yes—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
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×C61C22×D411yescharacteristic
228D4×C61—22no—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—48no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—84no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—84no—
261C2×C4×C61—84no—
262C2×C4×C61#13512yes—
263C2×C4×C61AGL(1,ℤ/8)12yes—
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
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—41no—
282Dic122—41no—
283Dic122—41no—
284Dic122—41no—
285Dic122—41no—
286#92418—22no—
287#92418—41no—
288#92418—41no—
289#92418—41no—
290#27063—611nomaximal, Sylow
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
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#52511yescharacteristic
306C22×C61#54611yescharacteristic
307C22×C61#55011yescharacteristic
308C22×C61C2×#13511yescharacteristic
309C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C61#57711yescharacteristic
311C22×C61#57711yescharacteristic
312C4×C61—22no—
313C4×C61—22no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—42no—
319C4×C61—81no—
320C4×C61#52911yescharacteristic
321C4×C61#53811yescharacteristic
322C4×C61#55111yescharacteristic
323C4×C61#55311yescharacteristic
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327Dic61—81no—
328#2514—611no—
329#2366D6111yescharacteristic
330#2092—611no—
331#2055—611no—
332#2041—611no—
333#2014—611no—
334#1968—611no—
335#1928—611no—
336#1925—611no—
337#1919—611no—
338#1916—611no—
339#1877—611no—
340#1877—611no—
341#1804—611no—
342C4×#102—611no—
343C2×C61—82no—
344C2×C61—82no—
345C2×C61#246711yescharacteristic
346C2×C61#310511yescharacteristic
347C2×C61#312311yescharacteristic
348C2×C61#330411yescharacteristic
349C2×C61#331011yescharacteristic
350C2×C61#339411yescharacteristic
351C2×C61#339711yescharacteristic
352C22×SD16—611no—
353C22×SD16—611no—
354C2×#127C2×D6111yescharacteristic
355C2×C4×D4—21no—
356C2×C4×D4C2×D6111yescharacteristic
357C2×C4×D4C2×D6111yescharacteristic
358C2×#107—611no—
359C2×#106—611no—
360C2×C4⋊C8—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×C4×C8—611no—
370#401—1221no—
371#400—1221no—
372#391—611no—
373#391—611no—
374#391—611no—
375#390—21no—
376#390C2×D6111yescharacteristic
377#390C2×D6111yescharacteristic
378#389—611no—
379#387—1221no—
380#386—21no—
381#386—21no—
382#383—1221no—
383#382—611no—
384C4×C4⋊C4—611no—
385C4×C4⋊C4C2×D6111yescharacteristic
386C4×C4⋊C4C2×D6111yescharacteristic
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#336—611no—
392#336—611no—
393#336—611no—
394#336—611no—
395C61#2706311yescharacteristic, complemented, Sylow
396C22×Q8—611no—
397C22×D4—21no—
398C22×D4—21no—
399C22×D4C22×D6111yescharacteristic
400C22×D4C22×D6111yescharacteristic
401C23×C4—21no—
402C23×C4—21no—
403C2×SD16—1222no—
404C2×SD16—1222no—
405C2×SD16—1222no—
406C2×SD16—2441no—
407C2×SD16—2441no—
408C2×SD16—2441no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8—611no—
413#127—22no—
414#127—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—41no—
418C4×D4—41no—
419C4×D4—44no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4C22⋊D6111yescharacteristic
425C2×C4⋊C4C22⋊D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C42—21no—
435C2×C42—1221no—
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439#107—2441no—
440#106—1222no—
441C4⋊C8—2441no—
442#103—1222no—
443#103—2441no—
444#102—1222no—
445#102—1222no—
446#102—1222no—
447#102—2441no—
448#102—2441no—
449#102—2441no—
450C4×C8—2441no—
451#95—21no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95C22⋊D6111yescharacteristic
463#95C22⋊D6111yescharacteristic
464C24—21no—
465C24—21no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—81no—
487C2×D4—81no—
488C2×D4#1175822212yes—
489C2×D4#1175822212yes—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C2×C22⋊D6111yescharacteristic
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C2×C22⋊D6111yescharacteristic
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C23×D6111yescharacteristic
509SD16—2444no—
510SD16—2444no—
511C2×C8—612no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—2442no—
535C4⋊C4#1175822412yes—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—42no—
549C42—2442no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C22×C22⋊D6111yescharacteristic
565Q8—2441no—
566Q8—2441no—
567Q8—2441no—
568Q8—2441no—
569D4—22no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4—48no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—84no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4#1451979712yes—
616C2×C4#1451980312yes—
617C2×C4#41×D6111yescharacteristic
618C2×C4C2×#1175822411yescharacteristic
619C2×C4C2×#1175822211yescharacteristic
620C2×C4C2×#1175822211yescharacteristic
621C2×C4#1451982211yescharacteristic
622C2×C4#1451982311yescharacteristic
623C8—1222no—
624C8—2441no—
625C8—2441no—
626C8—2441no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22#1684027311yescharacteristic
642C22#1684040611yescharacteristic
643C22#1684043111yescharacteristic
644C22C2×#1451979711yescharacteristic
645C22C2×#1451980311yescharacteristic
646C22#1684057311yescharacteristic
647C22#1684057311yescharacteristic
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—42no—
654C4—41no—
655C4—81no—
656C4—4881no—
657C4—4881no—
658C4—4881no—
659C4—4881no—
660C4#1684029211yescharacteristic
661C4#1684034511yescharacteristic
662C4#1684044111yescharacteristic
663C4#1684045811yescharacteristic
664C2—82no—
665C2—82no—
666C2#2144986411yescharacteristic
667C2#2145586611yescharacteristic
668C2#2145607111yescharacteristic
669C2#2145775811yescharacteristic
670C2#2145784011yescharacteristic
671C2#2145881111yescharacteristic
672C2#2145883511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2366C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenter#2366D61
hypocenterC61#27063
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. #21450427C2semidirect
2. #21447340C2semidirect
3. #21447136C2semidirect
6. #21446672C2semidirect
7. #21446460C2semidirect
8. #21446436C2semidirect
9. #21446397C2semidirect
10. #21446378C2semidirect
11. #21446153C2semidirect
12. #21446153C2semidirect
13. #21445829C2semidirect
14. C4×#14519698C2semidirect
395. C61#27063semidirect

Maximal subgroups

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

subgroupindex
#214504272
#214473402
#214471362
#214470502
#214469072
#214466722
#214464602
#214464362
#214463972
#214463782
#214461532
#214458292
C4×#145196982
C61×#23662
#2706361