Finite Group Databasebeta

← #29783266

Subgroups — #29783266

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#21454186C211yesmaximal, characteristic, complemented
2#21448283C211yesmaximal, characteristic, complemented
3#21447317C211yesmaximal, characteristic, complemented
4#21447206C211yesmaximal, characteristic, complemented
5#21447192C211yesmaximal, characteristic, complemented
6#21447130C211yesmaximal, characteristic
7#21446695C211yesmaximal, characteristic
8#21446557C211yesmaximal, characteristic, complemented
9#21446516C211yesmaximal, characteristic, complemented
10#21446516C211yesmaximal, characteristic, complemented
11#21446447C211yesmaximal, characteristic, complemented
12#21446398C211yesmaximal, characteristic, complemented
13#21446166C211yesmaximal, characteristic
14#21446069C211yesmaximal, characteristic, complemented
15C61×#1767C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C2×#14519771—21no—
18C2×C4×D244C2211yescharacteristic
19C2×#14519734—21no—
20C2×#14519711C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29#16839664C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839664C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839653—21no—
40#16839653—21no—
41#16839653—21no—
42#16839653—21no—
43#16839653—21no—
44C4×C4⋊Dic61C2211yescharacteristic
45C4×#11758198—21no—
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C43×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×D488—22no—
63C2×D488—22no—
64C2×D488—41no—
65#14519771—44no—
66C4×D244—22no—
67C4×D244—22no—
68#14519734—44no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×C4×Dic61—21no—
92#14519711—22no—
93#14519709—22no—
94#14519709—41no—
95#14519699—22no—
96#14519699—22no—
97#14519699—22no—
98#14519699—22no—
99#14519699—41no—
100#14519699—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C42×C61D411yescharacteristic
112C2×C42×C61D411yescharacteristic
113C2×C42×C61D411yescharacteristic
114C2×C42×C61D411yescharacteristic
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C4⋊C8×C61—22no—
119C4⋊C8×C61—22no—
120#97×C61—41no—
121#97×C61—41no—
122C23×D61—21no—
123C23×D61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244#4112yes—
149C2×C4×D61—24no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61#4112yes—
163C2×C8×C61#4112yes—
164C42×C61—22no—
165C42×C61—22no—
166C42×C61—21no—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—41no—
173C42×C61—41no—
174#11758198—24no—
175#11758198—42no—
176#11758198—42no—
177#11758198—42no—
178#11758198—42no—
179#11758198—42no—
180#11758198—42no—
181#11758198—42no—
182#11758198—42no—
183#11758198—42no—
184#11758198—42no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61#4112yes—
197#11758196—42no—
198#11758196—42no—
199#11758196—42no—
200#11758196—42no—
201#11758196—42no—
202C4×Dic61—42no—
203D488—44no—
204C23×C61C22×D411yescharacteristic
205C22×D61—22no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61AGL(1,ℤ/8)12yes—
236C2×C4×C61AGL(1,ℤ/8)12yes—
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C2×C4×C612−1+411yescharacteristic
243C2×Dic61—22no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
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×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261D244—22no—
262D244—22no—
263D244—41no—
264D244—41no—
265D244—41no—
266D244—41no—
267D244—41no—
268D244—41no—
269C4×D61—48no—
270C8×C61—22no—
271C8×C61—22no—
272C8×C61—41no—
273C8×C61—41no—
274#29444—611nomaximal, Sylow
275C22×C61#52511yescharacteristic
276C22×C61#53211yescharacteristic
277C22×C61#53711yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61#57511yescharacteristic
281C22×C61#57511yescharacteristic
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—81no—
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C4×C61—22no—
297C4×C61—22no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61#55111yescharacteristic
303C4×C61#55211yescharacteristic
304C4×C61#55311yescharacteristic
305C4×C61#55611yescharacteristic
306Dic61—42no—
307Dic61—81no—
308Dic61—81no—
309Dic61—81no—
310Dic61—81no—
311Dic61—81no—
312#2860—611no—
313#2293—611no—
314#2088—611no—
315#2067—611no—
316#2065—611no—
317#2054—611no—
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#1767D6111yescharacteristic, complemented
327C2×C61#260111yescharacteristic
328C2×C61#310911yescharacteristic
329C2×C61#311111yescharacteristic
330C2×C61#317111yescharacteristic
331C2×C61#318011yescharacteristic
332C2×C61#325111yescharacteristic
333C2×C61#325111yescharacteristic
334D61—82no—
335D61—161no—
336C22×D8—611no—
337C2×#126—1221no—
338C2×C4×D4—611no—
339C2×#117—1221no—
340C2×#107—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×C4⋊C8C2×D6111yescharacteristic
344C2×C4⋊C8C2×D6111yescharacteristic
345C2×#102—611no—
346C2×#102—611no—
347C2×#102—611no—
348C2×#102—611no—
349C2×#102—611no—
350C2×#102—611no—
351C2×#97C2×D6111yescharacteristic
352C2×#97C2×D6111yescharacteristic
353#390—611no—
354#390—611no—
355#390—611no—
356#390—611no—
357#390—611no—
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—
370C43C2×D6111yescharacteristic
371#340—611no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#336C2×D6111yescharacteristic
378#336C2×D6111yescharacteristic
379C61#2944411yescharacteristic, complemented, Sylow
380C22×D4—611no—
381C22×D4—611no—
382C23×C4—1221no—
383C2×D8—1222no—
384C2×D8—1222no—
385C2×D8—2441no—
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
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—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
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×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C42—1221no—
417C2×C42D24411yescharacteristic
418C2×C42D24411yescharacteristic
419C2×C42D24411yescharacteristic
420C2×C42D24411yescharacteristic
421C2×C42C22×D6111yescharacteristic
422C2×C42C22×D6111yescharacteristic
423C2×C42C22×D6111yescharacteristic
424#107—1222no—
425#106—1222no—
426#106—2441no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429#102—1222no—
430#102—1222no—
431#102—1222no—
432#102—1222no—
433#102—2441no—
434#102—2441no—
435#97—41no—
436#97—41no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443C24—1221no—
444C24—1221no—
445C2×D4—612no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C22×C4—1224no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4C2×D24411yescharacteristic
467C22×C4C2×D24411yescharacteristic
468C22×C4C2×D24411yescharacteristic
469C22×C4C2×D24411yescharacteristic
470C22×C4C2×D24411yescharacteristic
471C22×C4C2×D24411yescharacteristic
472C22×C4C23×D6111yescharacteristic
473D8—2444no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8#1175822212yes—
485C2×C8#1175822212yes—
486C4⋊C4—612no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
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—1224no—
504C22⋊C4—2442no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C22⋊C4—2442no—
508C22⋊C4—2442no—
509C22⋊C4—2442no—
510C22⋊C4—2442no—
511C22⋊C4—2442no—
512C22⋊C4—2442no—
513C22⋊C4—2442no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—22no—
521C42—22no—
522C42—41no—
523C42—41no—
524C42—2442no—
525C23—1222no—
526C23—2441no—
527C23—2441no—
528C23—2441no—
529C23—2441no—
530C23—2441no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537C23—2441no—
538C23—2441no—
539C23C22×D24411yescharacteristic
540D4—1222no—
541D4—1222no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—1222no—
565C2×C4—2448no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
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#1451980512yes—
584C2×C4#1451980512yes—
585C2×C4#41×D6111yescharacteristic
586C2×C4#41×D6111yescharacteristic
587C2×C4C2×#1175822211yescharacteristic
588C2×C4C2×#1175822211yescharacteristic
589C2×C4#1451982211yescharacteristic
590C2×C4#1451982511yescharacteristic
591C8—22no—
592C8—22no—
593C8—41no—
594C8—41no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22—4881no—
603C22—4881no—
604C22—4881no—
605C22—4881no—
606C22—2442no—
607C22—2442no—
608C22—2442no—
609C22#1684027411yescharacteristic
610C22#1684031211yescharacteristic
611C22#1684034311yescharacteristic
612C22C2×#1451980511yescharacteristic
613C22C2×#1451980511yescharacteristic
614C22#1684056211yescharacteristic
615C22#1684056211yescharacteristic
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—4881no—
623C4—4881no—
624C4—4881no—
625C4—4881no—
626C4—4881no—
627C4—2442no—
628C4#1684044111yescharacteristic
629C4#1684045311yescharacteristic
630C4#1684046111yescharacteristic
631C4#1684048411yescharacteristic
632C2—9761no—
633C2—4882no—
634C2#2145141011yescharacteristic
635C2#2145591111yescharacteristic
636C2#2145593011yescharacteristic
637C2#2145641411yescharacteristic
638C2#2145651711yescharacteristic
639C2#2145718211yescharacteristic
640C2#2145718211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1767C2
socleC23×C61C22×D4
prime radicalC22×D488C22
hypercenter#1767D61
hypocenterC61#29444
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
1. #21454186C2semidirect
2. #21448283C2semidirect
3. #21447317C2semidirect
4. #21447206C2semidirect
5. #21447192C2semidirect
8. #21446557C2semidirect
9. #21446516C2semidirect
10. #21446516C2semidirect
11. #21446447C2semidirect
12. #21446398C2semidirect
14. #21446069C2semidirect
15. C61×#1767C2semidirect
326. #1767D61semidirect
379. C61#29444semidirect

Maximal subgroups

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

subgroupindex
#214541862
#214482832
#214473172
#214472062
#214471922
#214471302
#214466952
#214465572
#214465162
#214464472
#214463982
#214461662
#214460692
C61×#17672
#2944461