Finite Group Databasebeta

← #29783265

Subgroups — #29783265

Order 15616.

788 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454183C211yesmaximal, characteristic, complemented
2#21448281C211yesmaximal, characteristic, complemented
3#21447317C211yesmaximal, characteristic, complemented
4#21447207C211yesmaximal, characteristic, complemented
5#21447193C211yesmaximal, characteristic, complemented
6#21446696C211yesmaximal, characteristic
7#21446556C211yesmaximal, characteristic, complemented
8#21446518C211yesmaximal, characteristic, complemented
9#21446513C211yesmaximal, characteristic, complemented
10#21446449C211yesmaximal, characteristic, complemented
11#21446396C211yesmaximal, characteristic, complemented
12#21446169C211yesmaximal, characteristic
13#21446064C211yesmaximal, characteristic, complemented
14#21445674C211yesmaximal, characteristic, complemented
15C61×#2054C211yesmaximal, characteristic
16C2×#14519770—21no—
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519733—21no—
19C2×#14519710C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29#16839665C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839665C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839656—21no—
38#16839656—21no—
39#16839655—21no—
40#16839650—21no—
41#16839650—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C22⋊C4×Dic61—21no—
44C42×Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×D8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519770—44no—
63D4×Dic61—22no—
64D4×Dic61—22no—
65#14519733—44no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
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×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519710—22no—
90#14519709—22no—
91#14519708—41no—
92C4⋊#92418—22no—
93#14519698—22no—
94#14519698—22no—
95#14519698—22no—
96#14519698—41no—
97#14519687—41no—
98#14519687—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×D4×C61C2311yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C2×D8×C61—22no—
108C2×D8×C61—22no—
109C2×D8×C61—41no—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
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×C42×C61C2311yescharacteristic
119C4⋊C8×C61—22no—
120#102×C61—22no—
121#102×C61—41no—
122C24×C61—21no—
123C24×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61#4112yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—24no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152D8×C61—44no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61#4112yes—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C22⋊C4×C61—42no—
164C22⋊C4×C61—42no—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—42no—
167C42×C61—21no—
168C42×C61—21no—
169C22⋊Dic61—24no—
170C22⋊Dic61—42no—
171C22⋊Dic61—42no—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—42no—
175C22⋊Dic61—42no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—42no—
184C4⋊Dic61#4112yes—
185#11758196—42no—
186#11758196—42no—
187#11758196—42no—
188#11758196—42no—
189C4×Dic61—22no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—22no—
195C4×Dic61—41no—
196C4×Dic61—41no—
197C4×Dic61—42no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418#4112yes—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61C22×D411yescharacteristic
219D4×C61—22no—
220D4×C61—22no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61AGL(1,ℤ/8)12yes—
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C612−1+411yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—48no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61AGL(1,ℤ/8)12yes—
270C8×C61—22no—
271C8×C61—41no—
272#92418—22no—
273#92418—41no—
274#29444—611nomaximal, Sylow
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61#52511yescharacteristic
290C22×C61#53211yescharacteristic
291C22×C61#53711yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57511yescharacteristic
296C4×C61—22no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61—81no—
300C4×C61#55111yescharacteristic
301C4×C61#55211yescharacteristic
302C4×C61#55311yescharacteristic
303C4×C61#55611yescharacteristic
304Dic61—22no—
305Dic61—42no—
306Dic61—41no—
307Dic61—41no—
308Dic61—41no—
309Dic61—81no—
310Dic61—81no—
311Dic61—81no—
312#2860—611no—
313#2293—611no—
314#2088—611no—
315#2067—611no—
316#2065—611no—
317#2054D6111yescharacteristic
318#1972—611no—
319#1943—611no—
320#1936—611no—
321#1936—611no—
322#1926—611no—
323#1919—611no—
324#1879—611no—
325#1862—611no—
326#1767—611no—
327C2×C61—82no—
328C2×C61—161no—
329C2×C61#260111yescharacteristic
330C2×C61#310911yescharacteristic
331C2×C61#311111yescharacteristic
332C2×C61#317111yescharacteristic
333C2×C61#318011yescharacteristic
334C2×C61#325111yescharacteristic
335C2×C61#325111yescharacteristic
336C22×D8C2×D6111yescharacteristic
337C2×#126—1221no—
338C2×C4×D4—611no—
339C2×#117—1221no—
340C2×#107—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×C4⋊C8—611no—
344C2×C4⋊C8C2×D6111yescharacteristic
345C2×#102—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×#102—611no—
349C2×#102C2×D6111yescharacteristic
350C2×#102C2×D6111yescharacteristic
351C2×#97—611no—
352C2×#97—611no—
353#390—611no—
354#390—611no—
355#390—611no—
356#390C2×D6111yescharacteristic
357#390C2×D6111yescharacteristic
358#389—611no—
359#389—611no—
360#389—611no—
361#389—611no—
362#389—611no—
363#388—1221no—
364#388—1221no—
365#388—1221no—
366#388—1221no—
367#388—1221no—
368C4×C4⋊C4—611no—
369C4×C22⋊C4—1221no—
370C43—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340C2×D6111yescharacteristic
377#336—611no—
378#336—611no—
379C61#2944411yescharacteristic, complemented, Sylow
380C22×D4C22×D6111yescharacteristic
381C22×D4C22×D6111yescharacteristic
382C23×C4—1221no—
383C2×D8—22no—
384C2×D8—22no—
385C2×D8—41no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390#126—2444no—
391C4×D4—1222no—
392C4×D4—1222no—
393#117—2444no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—1221no—
423C2×C42C22×D6111yescharacteristic
424#107—1222no—
425#106—1222no—
426#106—2441no—
427C4⋊C8—22no—
428C4⋊C8—1222no—
429#102—22no—
430#102—41no—
431#102—1222no—
432#102—1222no—
433#102—1222no—
434#102—2441no—
435#97—2441no—
436#97—2441no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443C24—21no—
444C24—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4#1175822412yes—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1224no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4C23×D6111yescharacteristic
473D8—44no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—612no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8#1175822412yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—612no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—1224no—
508C22⋊C4—2442no—
509C22⋊C4—2442no—
510C22⋊C4—2442no—
511C22⋊C4—2442no—
512C22⋊C4—2442no—
513C22⋊C4—2442no—
514C42—21no—
515C42—21no—
516C42—1222no—
517C42—1221no—
518C42—1221no—
519C42—1221no—
520C42—1221no—
521C42—1222no—
522C42—2441no—
523C42—2441no—
524C42—2442no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C2×D4×D6111yescharacteristic
540D4—22no—
541D4—22no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
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—1222no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—2448no—
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—2441no—
584C2×C4#1451980112yes—
585C2×C4#41×D6111yescharacteristic
586C2×C4#41×D6111yescharacteristic
587C2×C4C2×#1175822411yescharacteristic
588C2×C4C2×#1175822411yescharacteristic
589C2×C4#1451982211yescharacteristic
590C2×C4#1451982411yescharacteristic
591C8—22no—
592C8—41no—
593C8—1222no—
594C8—2441no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22#1684027211yescharacteristic
610C22#1684030711yescharacteristic
611C22#1684034111yescharacteristic
612C22AGL(1,ℤ/8)×D6111yescharacteristic
613C22C2×#1451980111yescharacteristic
614C22#1684055811yescharacteristic
615C22#1684055911yescharacteristic
616C4—22no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—1222no—
621C4—4881no—
622C4—4881no—
623C4—4881no—
624C4—2441no—
625C4—2441no—
626C4—2442no—
627C4—2441no—
628C4#1684044311yescharacteristic
629C4#1684044911yescharacteristic
630C4#1684045711yescharacteristic
631C4#1684048111yescharacteristic
632C2—82no—
633C2—161no—
634C2#2145140911yescharacteristic
635C2#2145590911yescharacteristic
636C2#2145592511yescharacteristic
637C2#2145640911yescharacteristic
638C2#2145651211yescharacteristic
639C2#2145717711yescharacteristic
640C2#2145717811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2054C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenter#2054D61
hypocenterC61#29444
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
1. #21454183C2semidirect
2. #21448281C2semidirect
3. #21447317C2semidirect
4. #21447207C2semidirect
5. #21447193C2semidirect
7. #21446556C2semidirect
8. #21446518C2semidirect
9. #21446513C2semidirect
10. #21446449C2semidirect
11. #21446396C2semidirect
13. #21446064C2semidirect
14. #21445674C2semidirect
379. C61#29444semidirect

Maximal subgroups

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

subgroupindex
#214541832
#214482812
#214473172
#214472072
#214471932
#214466962
#214465562
#214465182
#214465132
#214464492
#214463962
#214461692
#214460642
#214456742
C61×#20542
#2944461