Finite Group Databasebeta

← #29752722

Subgroups — #29752722

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450217C211yesmaximal, characteristic, complemented
2#21448837C211yesmaximal, characteristic
3#21447234C211yesmaximal, characteristic, complemented
4#21447204C211yesmaximal, characteristic, complemented
5#21447011C211yesmaximal, characteristic
6#21447004C211yesmaximal, characteristic
7#21446587C211yesmaximal, characteristic, complemented
8#21446551C211yesmaximal, characteristic, complemented
9#21446426C211yesmaximal, characteristic, complemented
10#21446400C211yesmaximal, characteristic, complemented
11#21446063C211yesmaximal, characteristic, complemented
12#21446005C211yesmaximal, characteristic, complemented
13#21445908C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic, complemented
15C61×#1921C211yesmaximal, characteristic, complemented
16C2×#14519752C2211yescharacteristic
17C2×C4×D244—21no—
18C2×#14519739C2211yescharacteristic
19C2×C4⋊C4×D61C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31C61⋊#401—21no—
32#16839688—21no—
33#16839664C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839653—21no—
37#16839651—21no—
38#16839643—21no—
39#16839640—21no—
40#16839636—21no—
41#16839628C2211yescharacteristic
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×#11758198C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#107×C61C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×D244—21no—
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519752—22no—
67#14519752—22no—
68C4×D244—44no—
69#14519739—22no—
70#14519739—41no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C4⋊C4×D61—22no—
78C4⋊C4×D61—41no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198D411yescharacteristic
86C2×#11758198D411yescharacteristic
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89#14519710—22no—
90#14519710—41no—
91#14519708—22no—
92C4⋊#92418—22no—
93#14519699—22no—
94#14519699—22no—
95#14519697—22no—
96#14519697—41no—
97#14519689—22no—
98#14519689—41no—
99C8×Dic61—41no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118#107×C61—22no—
119#95×C61—21no—
120#95×C61—21no—
121C23×D61—21no—
122C23×D61C2×D411yescharacteristic
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—24no—
141C2×D244—21no—
142C2×D244—42no—
143C2×D244—42no—
144C2×D244#4112yes—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—81no—
156C2×C4×D61—81no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—81no—
170C4⋊C4×C61#4112yes—
171C4⋊C4×C61#4112yes—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—42no—
175#11758198—24no—
176#11758198—22no—
177#11758198—22no—
178#11758198—42no—
179#11758198—42no—
180#11758198—42no—
181#11758198—81no—
182#11758198—81no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—42no—
190#11758196—42no—
191#11758196—42no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—81no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418#4112yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—21no—
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—22no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61#13512yes—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61AGL(1,ℤ/8)12yes—
258D244—22no—
259D244—22no—
260D244—48no—
261D244—41no—
262D244—41no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—84no—
268C4×D61—84no—
269C8×C61—41no—
270C8×C61—41no—
271#92418—22no—
272#92418—41no—
273#27033—611nomaximal, Sylow
274C22×C61#52511yescharacteristic
275C22×C61#54611yescharacteristic
276C22×C61#54711yescharacteristic
277C22×C61C2×#13511yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57811yescharacteristic
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61—42no—
291C2×D61—81no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61#52911yescharacteristic
302C4×C61#53911yescharacteristic
303C4×C61#55311yescharacteristic
304C4×C61#55611yescharacteristic
305Dic61—22no—
306Dic61—41no—
307Dic61—41no—
308Dic61—81no—
309Dic61—81no—
310Dic61—81no—
311#2494—611no—
312#2361—611no—
313#2073—611no—
314#2067—611no—
315#2034—611no—
316#2033—611no—
317#1949—611no—
318#1942—611no—
319#1923—611no—
320#1921D6111yescharacteristic, complemented
321#1919—611no—
322#1862—611no—
323#1852—611no—
324#1824—611no—
325C4×#102—611no—
326C2×C61#246611yescharacteristic
327C2×C61#310511yescharacteristic
328C2×C61#312411yescharacteristic
329C2×C61#330011yescharacteristic
330C2×C61#331811yescharacteristic
331C2×C61#333911yescharacteristic
332C2×C61#335311yescharacteristic
333D61—42no—
334D61—82no—
335C2×#121—611no—
336C2×C4×D4—611no—
337C2×C4×D4—1221no—
338C22×C4⋊C4—611no—
339C2×#107—611no—
340C2×#107—611no—
341C2×#107C2×D6111yescharacteristic
342C2×#106—611no—
343C2×C4⋊C8—611no—
344C2×#102—611no—
345C2×#102—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×C22⋊C8—611no—
349C2×C22⋊C8—611no—
350C2×C4×C8—611no—
351#401—1221no—
352#397—1221no—
353#390—611no—
354#389—611no—
355#389—611no—
356#389C2×D6111yescharacteristic
357#388—1221no—
358#388—1221no—
359#387—1221no—
360#386—1221no—
361#385—1221no—
362#382—611no—
363#380—611no—
364#380—611no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4C2×D6111yescharacteristic
367C4×C22⋊C4—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340C2×D6111yescharacteristic
374#340C2×D6111yescharacteristic
375#340C2×D6111yescharacteristic
376#336—611no—
377#336C2×D6111yescharacteristic
378C61#2703311yescharacteristic, complemented, Sylow
379C22×D4—611no—
380C22×D4—1221no—
381C23×C4—611no—
382C23×C4—1221no—
383C23×C4—1221no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388#121—1222no—
389#121—1222no—
390C4×D4—1222no—
391C4×D4—2444no—
392C4×D4—2441no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—2441no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—611no—
407C2×C22⋊C4—611no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—21no—
415C2×C42—611no—
416C2×C42—1221no—
417C2×C42C22×D6111yescharacteristic
418#107—22no—
419#107—1222no—
420#107—2441no—
421#106—1222no—
422C4⋊C8—1222no—
423#102—1222no—
424#102—1222no—
425#102—1222no—
426#102—2441no—
427C22⋊C8—1222no—
428C22⋊C8—2441no—
429C4×C8—2441no—
430#95—21no—
431#95—21no—
432#95—611no—
433#95—611no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441C24—611no—
442C24—1221no—
443C2×D4—612no—
444C2×D4—1224no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—2442no—
451C2×D4—2442no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—4881no—
469C22×C4—4881no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4—2442no—
474C22×C4D4×D6111yescharacteristic
475C22×C4D4×D6111yescharacteristic
476C22×C4C23×D6111yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—612no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—81no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4#1175822212yes—
506C4⋊C4#1175822612yes—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1224no—
510C22⋊C4—4881no—
511C22⋊C4—4881no—
512C22⋊C4—2442no—
513C22⋊C4—2442no—
514C22⋊C4—2442no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—1221no—
519C42—1221no—
520C42—4881no—
521C23—612no—
522C23—1221no—
523C23—1221no—
524C23—1221no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23—1222no—
529C23—2441no—
530C23—2441no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23—2441no—
535C23C2×D4×D6111yescharacteristic
536D4—1222no—
537D4—1222no—
538D4—2448no—
539D4—2441no—
540D4—2441no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—612no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—4884no—
569C2×C4—4884no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4#41×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4C2×#1175822611yescharacteristic
587C2×C4#1451982311yescharacteristic
588C2×C4#1451982411yescharacteristic
589C8—41no—
590C8—41no—
591C8—1222no—
592C8—2441no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—2442no—
601C22—2442no—
602C22—2442no—
603C22—2441no—
604C22—2441no—
605C22—2441no—
606C22—2441no—
607C22#1684027511yescharacteristic
608C22#1684040811yescharacteristic
609C22#1684041011yescharacteristic
610C22#135×D6111yescharacteristic
611C22AGL(1,ℤ/8)×D6111yescharacteristic
612C22#1684056611yescharacteristic
613C22#1684057511yescharacteristic
614C4—41no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—1222no—
621C4—4881no—
622C4—4881no—
623C4—4881no—
624C4—2441no—
625C4—2441no—
626C4#1684029011yescharacteristic
627C4#1684035311yescharacteristic
628C4#1684046511yescharacteristic
629C4#1684048511yescharacteristic
630C2—4882no—
631C2—2442no—
632C2#2144986211yescharacteristic
633C2#2145587111yescharacteristic
634C2#2145608811yescharacteristic
635C2#2145771111yescharacteristic
636C2#2145796511yescharacteristic
637C2#2145821411yescharacteristic
638C2#2145839611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1921C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#1921D61
hypocenterC61#27033
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
1. #21450217C2semidirect
3. #21447234C2semidirect
4. #21447204C2semidirect
7. #21446587C2semidirect
8. #21446551C2semidirect
9. #21446426C2semidirect
10. #21446400C2semidirect
11. #21446063C2semidirect
12. #21446005C2semidirect
13. #21445908C2semidirect
14. #21445633C2semidirect
15. C61×#1921C2semidirect
320. #1921D61semidirect
378. C61#27033semidirect

Maximal subgroups

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

subgroupindex
#214502172
#214488372
#214472342
#214472042
#214470112
#214470042
#214465872
#214465512
#214464262
#214464002
#214460632
#214460052
#214459082
#214456332
C61×#19212
#2703361