Finite Group Databasebeta

← #29740148

Subgroups — #29740148

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839981C211yesmaximal, characteristic
2C2×#16839854C211yesmaximal, characteristic, complemented
3#21449613C211yesmaximal, characteristic
4#21448733C211yesmaximal, characteristic, complemented
5#21447345C211yesmaximal, characteristic
6#21447310C211yesmaximal, characteristic, complemented
7#21447230C211yesmaximal, characteristic, complemented
8#21446504C211yesmaximal, characteristic
9#21446379C211yesmaximal, characteristic, complemented
10#21446171C211yesmaximal, characteristic
11#21446171C211yesmaximal, characteristic
12#21446157C211yesmaximal, characteristic, complemented
13#21446076C211yesmaximal, characteristic, complemented
14#21446075C211yesmaximal, characteristic, complemented
15C61×#2042C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22#16839981—24no—
23#16839854—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839698—21no—
33#16839687—21no—
34#16839669C2211yescharacteristic
35#16839665C2211yescharacteristic
36#16839665C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839659—21no—
40#16839657—21no—
41#16839650—21no—
42#16839650—21no—
43#16839631—21no—
44#16839631C2211yescharacteristic
45C4×#11758196C2211yescharacteristic
46#16839613—21no—
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#124×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202D414yes—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C2×#11758200—22no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70Q8×Dic61—22no—
71Q8×Dic61—41no—
72C4×Dic122—22no—
73C4×Dic122—41no—
74#14519742—22no—
75#14519742—22no—
76#14519742—22no—
77#14519742—41no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196D411yescharacteristic
90C2×#11758196D411yescharacteristic
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519711—41no—
102#14519710—22no—
103C4⋊#92418—22no—
104#14519702—22no—
105#14519702—22no—
106#14519701—22no—
107#14519698—22no—
108C8×Dic61—22no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×Q8×C61C2311yescharacteristic
116C22×D4×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119#124×C61—22no—
120#124×C61—41no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#103×C61—22no—
128#103×C61—22no—
129#102×C61—22no—
130#102×C61—41no—
131#95×C61—21no—
132C24×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61C2×D412yes—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122C2×D412yes—
163C2×Dic122#4112yes—
164C2×Dic122#4112yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—81no—
175C4⋊C4×C61#4112yes—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—42no—
178C42×C61—21no—
179C42×C61—21no—
180C22⋊Dic61—42no—
181C22⋊Dic61—42no—
182C22⋊Dic61—81no—
183C22⋊Dic61—81no—
184#11758202—28no—
185#11758200—44no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—42no—
193C4⋊Dic61—81no—
194#11758196—24no—
195#11758196—22no—
196#11758196—22no—
197#11758196—42no—
198#11758196—42no—
199#11758196—42no—
200#11758196—81no—
201#11758196—81no—
202C4×Dic61—24no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C4×Dic61—42no—
209C4×Dic61—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61C22×D411yescharacteristic
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C612−1+411yescharacteristic
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61#13512yes—
276C2×Dic61#13712yes—
277Dic122—22no—
278Dic122—22no—
279Dic122—22no—
280Dic122#12314yes—
281C8×C61—41no—
282C8×C61—41no—
283#92418—22no—
284#92418—41no—
285#26169—611nomaximal, Sylow
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61#52511yescharacteristic
294C22×C61D4211yescharacteristic
295C22×C61#54811yescharacteristic
296C22×C61C2×#13511yescharacteristic
297C22×C61C2×#13711yescharacteristic
298C22×C61#57611yescharacteristic
299C22×C61#57811yescharacteristic
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—81no—
305C4×C61C2×#12311yescharacteristic
306C4×C61#53911yescharacteristic
307C4×C61#54711yescharacteristic
308C4×C61#55311yescharacteristic
309Dic61—22no—
310Dic61—24no—
311Dic61—22no—
312Dic61—41no—
313Dic61—41no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317C2×#464—611no—
318C2×C4×Q16—611no—
319#2448—611no—
320Q8×C4⋊C4—611no—
321#2093—611no—
322#2086—611no—
323#2072—611no—
324#2042D6111yescharacteristic
325#1934—611no—
326#1916—611no—
327#1880—611no—
328#1880—611no—
329#1878—611no—
330#1863—611no—
331#1863—611no—
332C2×C61—161no—
333C2×C61#244311yescharacteristic
334C2×C61#310811yescharacteristic
335C2×C61#312411yescharacteristic
336C2×C61#327011yescharacteristic
337C2×C61#328711yescharacteristic
338C2×C61#336411yescharacteristic
339C2×C61#337511yescharacteristic
340C22×Q16—611no—
341C22×SD16—611no—
342C2×#124C2×D6111yescharacteristic
343C2×C4×Q8—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
347#464—1224no—
348C4×Q16—1224no—
349C2×#107—611no—
350C2×#107—611no—
351C2×C4⋊C8—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103C2×D6111yescharacteristic
356C2×#103C2×D6111yescharacteristic
357C2×#102—611no—
358C2×#102C2×D6111yescharacteristic
359C2×#102C2×D6111yescharacteristic
360C2×C4×C8—611no—
361#399—1221no—
362#397—1221no—
363#391—611no—
364#390—611no—
365#390—611no—
366#389—611no—
367#389—611no—
368#389—1221no—
369#389—1221no—
370#389C2×D6111yescharacteristic
371#388—1221no—
372#388—1221no—
373#384—611no—
374#384—1221no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—1221no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#336—611no—
382#336—611no—
383#336—611no—
384#336C2×D6111yescharacteristic
385C61#2616911yescharacteristic, complemented, Sylow
386C22×Q8—611no—
387C22×Q8C22×D6111yescharacteristic
388C22×D4C22×D6111yescharacteristic
389C2×Q16—614no—
390C2×Q16—1222no—
391C2×Q16—1222no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—1222no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399#124—22no—
400#124—41no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—2441no—
407C4×Q8—2441no—
408C4×Q8—2441no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42C22×D6111yescharacteristic
438#107—1222no—
439#107—2441no—
440C4⋊C8—1222no—
441#103—22no—
442#103—22no—
443#103—1222no—
444#103—1222no—
445#103—1222no—
446#102—22no—
447#102—41no—
448#102—1222no—
449C4×C8—1222no—
450#95—21no—
451#95—611no—
452#95—611no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457C24—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—612no—
464C2×Q8—612no—
465C2×Q8—612no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8D4×D6112yes—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490Q16—1228no—
491SD16—2444no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—612no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—81no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1224no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—4881no—
518C4⋊C4—4881no—
519C4⋊C4—4881no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C4⋊C4#1175822412yes—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—4881no—
528C22⋊C4—4881no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—21no—
532C42—21no—
533C42—1224no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C42—1221no—
538C42—2442no—
539C42—2442no—
540C42—2442no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×D4×D6111yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—41no—
552Q8—41no—
553Q8—614no—
554Q8—1222no—
555Q8—1222no—
556Q8—1222no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—41no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1221no—
579C2×C4—1222no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4#41×D6111yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982411yescharacteristic
606C8—41no—
607C8—41no—
608C8—1222no—
609C8—2441no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22#1684027211yescharacteristic
618C22#1684039411yescharacteristic
619C22#1684042011yescharacteristic
620C22#135×D6111yescharacteristic
621C22#137×D6111yescharacteristic
622C22#1684056611yescharacteristic
623C22#1684057611yescharacteristic
624C4—42no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—81no—
629C4—1222no—
630C4—1224no—
631C4—1222no—
632C4—4881no—
633C4—4881no—
634C4—4881no—
635C4—2441no—
636C4—2441no—
637C4#123×D6111yescharacteristic
638C4#1684035511yescharacteristic
639C4#1684041211yescharacteristic
640C4#1684045911yescharacteristic
641C2—161no—
642C2#2144954711yescharacteristic
643C2#2145589811yescharacteristic
644C2#2145607811yescharacteristic
645C2#2145738711yescharacteristic
646C2#2145757211yescharacteristic
647C2#2145850111yescharacteristic
648C2#2145863011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2042C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2042D61
hypocenterC61#26169
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
2. C2×#16839854C2semidirect
4. #21448733C2semidirect
6. #21447310C2semidirect
7. #21447230C2semidirect
9. #21446379C2semidirect
12. #21446157C2semidirect
13. #21446076C2semidirect
14. #21446075C2semidirect
385. C61#26169semidirect

Maximal subgroups

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

subgroupindex
C2×#168399812
C2×#168398542
#214496132
#214487332
#214473452
#214473102
#214472302
#214465042
#214463792
#214461712
#214461572
#214460762
#214460752
C61×#20422
#2616961