Finite Group Databasebeta

← #29729501

Subgroups — #29729501

Order 15616.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840126C211yesmaximal, characteristic
2C2×#16840070C211yesmaximal, characteristic, complemented
3#21450822C211yesmaximal, characteristic
4#21448915C211yesmaximal, characteristic, complemented
5#21447205C211yesmaximal, characteristic, complemented
6#21447191C211yesmaximal, characteristic, complemented
7#21447058C211yesmaximal, characteristic, complemented
8#21447058C211yesmaximal, characteristic, complemented
9#21446750C211yesmaximal, characteristic, complemented
10#21446750C211yesmaximal, characteristic, complemented
11#21446540C211yesmaximal, characteristic, complemented
12#21446448C211yesmaximal, characteristic, complemented
13#21446441C211yesmaximal, characteristic, complemented
14#21445652C211yesmaximal, characteristic, complemented
15C61×#1982C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16840126—24no—
21#16840070—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35#16839676—21no—
36#16839676—21no—
37#16839665—21no—
38#16839665—21no—
39#16839665C2211yescharacteristic
40#16839665C2211yescharacteristic
41#16839659—21no—
42#16839658C2211yescharacteristic
43#16839658C2211yescharacteristic
44#16839649—21no—
45#16839646—21no—
46C4⋊C4×Dic61C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4×#11758196—21no—
49C4×C4⋊Dic61C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#127×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519769—22no—
65#14519769D414yes—
66#14519764—22no—
67#14519764—22no—
68#14519764—41no—
69#14519764—41no—
70C4×Dic122—22no—
71C4×Dic122—22no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519711—22no—
96#14519709—22no—
97#14519706—41no—
98#14519706—41no—
99#14519702—22no—
100#14519702—22no—
101#14519700—22no—
102#14519700—22no—
103#14519698—22no—
104#14519698—22no—
105#14519698—22no—
106#14519698—22no—
107#14519688—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C22×D4×C61—21no—
115C22×D4×C61—21no—
116C22×D4×C61C2311yescharacteristic
117C22×D4×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120#127×C61—22no—
121#127×C61—22no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C4⋊C8×C61—22no—
127#102×C61—22no—
128#102×C61—22no—
129#102×C61—41no—
130#102×C61—41no—
131#95×C61—21no—
132C24×C61—21no—
133C24×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—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—42no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61—42no—
150C2×D4×C61—81no—
151C2×D4×C61—81no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122#4112yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61#4112yes—
181C42×C61—81no—
182C42×C61C2×D412yes—
183C42×C61#4112yes—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
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—41no—
197C4⋊Dic61—41no—
198C4⋊Dic61C2×D412yes—
199C4⋊Dic61C2×D412yes—
200#11758196—24no—
201#11758196—24no—
202#11758196—42no—
203#11758196—42no—
204#11758196—81no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—42no—
210C4×Dic61—42no—
211C4×Dic61—81no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
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×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61C22×D411yescharacteristic
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—84no—
242D4×C61—84no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61#12112yes—
254C2×C4×C61#12112yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285Dic122—22no—
286Dic122—22no—
287Dic122—41no—
288Dic122—41no—
289C8×C61—41no—
290C8×C61—41no—
291#92418—41no—
292#92418—41no—
293#25291—611nomaximal, Sylow
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
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—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61C2×#12111yescharacteristic
309C22×C61#54611yescharacteristic
310C22×C61#54811yescharacteristic
311C22×C61#57511yescharacteristic
312C22×C61#57511yescharacteristic
313C22×C61#57611yescharacteristic
314C22×C61#57711yescharacteristic
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61#52911yescharacteristic
320C4×C61#54211yescharacteristic
321C4×C61#54811yescharacteristic
322C4×C61#55011yescharacteristic
323Dic61—42no—
324Dic61—42no—
325Dic61—42no—
326Dic61—41no—
327Dic61—41no—
328Dic61—41no—
329Dic61—41no—
330Dic61—81no—
331C2×#490—611no—
332C2×#479—611no—
333#2548—611no—
334#2370—611no—
335#2067—611no—
336#2065—611no—
337#2042—611no—
338#2042—611no—
339#1983—611no—
340#1983—611no—
341#1982D6111yescharacteristic
342#1940—611no—
343#1926—611no—
344#1925—611no—
345#1763—611no—
346C2×C61—161no—
347C2×C61—161no—
348C2×C61#242611yescharacteristic
349C2×C61#304011yescharacteristic
350C2×C61#304511yescharacteristic
351C2×C61#330911yescharacteristic
352C2×C61#331711yescharacteristic
353C2×C61#337011yescharacteristic
354C2×C61#337111yescharacteristic
355C2×#127C2×D6111yescharacteristic
356C2×#125—611no—
357C2×#124—611no—
358C2×#124—611no—
359C2×C4×Q8—611no—
360#490—1224no—
361#479—1224no—
362C2×#107—611no—
363C2×#106—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C4⋊C8C2×D6111yescharacteristic
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×#102C2×D6111yescharacteristic
376C2×#102C2×D6111yescharacteristic
377C2×#102C2×D6111yescharacteristic
378C2×#102C2×D6111yescharacteristic
379C2×#97—611no—
380#394—1221no—
381#394—1221no—
382#390—611no—
383#390—611no—
384#390—1221no—
385#390—1221no—
386#389—611no—
387#389—611no—
388#389—1221no—
389#387—1221no—
390#387—1221no—
391C4×C4⋊C4—611no—
392C4×C4⋊C4—611no—
393C4×C4⋊C4—611no—
394C4×C4⋊C4—1221no—
395C4×C4⋊C4C2×D6111yescharacteristic
396#340—611no—
397#340—611no—
398#336—611no—
399#336—611no—
400C61#2529111yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×D4—21no—
403C22×D4—21no—
404C22×D4C22×D6111yescharacteristic
405C22×D4C22×D6111yescharacteristic
406C22×C8—611no—
407C22×C8—611no—
408C22×C8C22×D6111yescharacteristic
409C22×C8C22×D6111yescharacteristic
410#127—22no—
411#127—22no—
412#125—614no—
413#125—1222no—
414#124—1222no—
415#124—1222no—
416#124—2441no—
417#124—2441no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C42—21no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42C22×D6111yescharacteristic
447#107—1222no—
448#106—1222no—
449C4⋊C8—22no—
450C4⋊C8—2441no—
451C4⋊C8—2441no—
452#103—1222no—
453#103—1222no—
454#103—1222no—
455#103—1222no—
456#102—22no—
457#102—22no—
458#102—41no—
459#102—41no—
460#102—1222no—
461#102—1222no—
462#102—1222no—
463#102—1222no—
464#97—1222no—
465#95—21no—
466#95—611no—
467#95—611no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—21no—
473C24—21no—
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
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—21no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—81no—
497C2×D4—81no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—612no—
527C4⋊C4—612no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1224no—
531C4⋊C4—1224no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—4881no—
535C4⋊C4—2441no—
536C4⋊C4—2441no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4#1175822412yes—
540C22⋊C4—4881no—
541C22⋊C4—4881no—
542C22⋊C4—4881no—
543C22⋊C4—4881no—
544C22⋊C4—2442no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4—2442no—
548C42—81no—
549C42—1221no—
550C42—1221no—
551C42—1221no—
552C42—1221no—
553C42—4881no—
554C42—2442no—
555C42—2442no—
556C42#1175822212yes—
557C42D4×D6112yes—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23C2×D4×D6111yescharacteristic
573Q8—1222no—
574Q8—1222no—
575Q8—2441no—
576Q8—2441no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4—84no—
586D4—84no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1222no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4#1451975012yes—
622C2×C4#1451975012yes—
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822411yescharacteristic
626C2×C4C2×#1175822211yescharacteristic
627C2×C4#1451982211yescharacteristic
628C2×C4#1451982411yescharacteristic
629C8—41no—
630C8—41no—
631C8—2441no—
632C8—2441no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22C2×#1451975011yescharacteristic
648C22#1684040911yescharacteristic
649C22#1684041811yescharacteristic
650C22#1684055811yescharacteristic
651C22#1684055811yescharacteristic
652C22#1684056311yescharacteristic
653C22#1684056811yescharacteristic
654C4—22no—
655C4—22no—
656C4—42no—
657C4—81no—
658C4—4881no—
659C4—2442no—
660C4—2441no—
661C4—2441no—
662C4—2441no—
663C4—2441no—
664C4—2442no—
665C4—2442no—
666C4#1684029411yescharacteristic
667C4#1684037711yescharacteristic
668C4#1684042011yescharacteristic
669C4#1684043211yescharacteristic
670C2—161no—
671C2—161no—
672C2#2144938611yescharacteristic
673C2#2145531211yescharacteristic
674C2#2145536311yescharacteristic
675C2#2145783211yescharacteristic
676C2#2145795211yescharacteristic
677C2#2145856511yescharacteristic
678C2#2145858011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1982C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenter#1982D61
hypocenterC61#25291
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. C2×#16840070C2semidirect
4. #21448915C2semidirect
5. #21447205C2semidirect
6. #21447191C2semidirect
7. #21447058C2semidirect
8. #21447058C2semidirect
9. #21446750C2semidirect
10. #21446750C2semidirect
11. #21446540C2semidirect
12. #21446448C2semidirect
13. #21446441C2semidirect
14. #21445652C2semidirect
400. C61#25291semidirect

Maximal subgroups

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

subgroupindex
C2×#168401262
C2×#168400702
#214508222
#214489152
#214472052
#214471912
#214470582
#214467502
#214465402
#214464482
#214464412
#214456522
C61×#19822
#2529161