Finite Group Databasebeta

← #29749939

Subgroups — #29749939

Order 15616.

800 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840120C211yesmaximal, characteristic
2C2×#16839867C211yesmaximal, characteristic, complemented
3#21449652C211yesmaximal, characteristic, complemented
4#21449618C211yesmaximal, characteristic
5#21447189C211yesmaximal, characteristic
6#21447151C211yesmaximal, characteristic, complemented
7#21447140C211yesmaximal, characteristic, complemented
8#21446911C211yesmaximal, characteristic
9#21446675C211yesmaximal, characteristic
10#21446519C211yesmaximal, characteristic
11#21446511C211yesmaximal, characteristic, complemented
12#21446496C211yesmaximal, characteristic, complemented
13#21446090C211yesmaximal, characteristic, complemented
14#21445685C211yesmaximal, characteristic, complemented
15C61×#1880C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16840120—24no—
21#16839867—24no—
22C2×#14519709C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839698—21no—
35#16839695—21no—
36#16839690—21no—
37#16839682—21no—
38#16839669C2211yescharacteristic
39#16839669C2211yescharacteristic
40#16839668—21no—
41#16839667C2211yescharacteristic
42#16839665C2211yescharacteristic
43#16839661C2211yescharacteristic
44#16839661C2211yescharacteristic
45#16839660C2211yescharacteristic
46#16839656—21no—
47#16839655—21no—
48#16839646—21no—
49C4×#11758196C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519776—22no—
69#14519776—41no—
70#14519764—22no—
71#14519764—41no—
72C4×Dic122—22no—
73C4×Dic122—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519709—41no—
92#14519708—22no—
93#14519706—22no—
94#14519702—22no—
95#14519702—22no—
96#14519701—22no—
97#14519700D414yes—
98#14519698—22no—
99#14519698—22no—
100#14519688—22no—
101#14519687—22no—
102C8×Dic61—22no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C22×Q8×C61C2311yescharacteristic
115C22×D4×C61C2311yescharacteristic
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61—41no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×Q8×C61—22no—
122C4×Q8×C61—41no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130#103×C61—22no—
131#102×C61—22no—
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×C61#4112yes—
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×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—42no—
165C2×Dic122—42no—
166C2×Dic122—81no—
167C2×Dic122C2×D412yes—
168SD16×C61—44no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61#4112yes—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61C2×D412yes—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—81no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—42no—
185C22⋊Dic61—42no—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
188C22⋊Dic61—81no—
189#11758202—28no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—81no—
197C4⋊Dic61—81no—
198#11758196—22no—
199#11758196—41no—
200#11758196—42no—
201#11758196—42no—
202#11758196—81no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×D412yes—
213C2×#92418#4112yes—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C22×D411yescharacteristic
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—41no—
225Q8×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×C4×C612−1+411yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
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#13712yes—
275Dic122—22no—
276Dic122—22no—
277Dic122—41no—
278Dic122—41no—
279Dic122—84no—
280C8×C61—22no—
281C8×C61—41no—
282#92418—22no—
283#92418—22no—
284#26846—611nomaximal, Sylow
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61#53211yescharacteristic
293C22×C61D4211yescharacteristic
294C22×C61#54811yescharacteristic
295C22×C61C2×#13711yescharacteristic
296C22×C61#57511yescharacteristic
297C22×C61#57711yescharacteristic
298C22×C61#57711yescharacteristic
299C4×C61—42no—
300C4×C61—42no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61#52511yescharacteristic
305C4×C61#54211yescharacteristic
306C4×C61#54711yescharacteristic
307C4×C61#55111yescharacteristic
308Dic61—22no—
309Dic61—41no—
310Dic61—42no—
311Dic61—41no—
312Dic61—41no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316C2×#489—611no—
317C2×#441—611no—
318#2451—611no—
319#2448—611no—
320#2065—611no—
321#2058—611no—
322#2056—611no—
323#2015—611no—
324#1968—611no—
325#1936—611no—
326#1935—611no—
327#1933—611no—
328#1880D6111yescharacteristic
329#1865—611no—
330#1770—611no—
331C2×C61—161no—
332C2×C61#246111yescharacteristic
333C2×C61#317311yescharacteristic
334C2×C61#318111yescharacteristic
335C2×C61#327511yescharacteristic
336C2×C61#327811yescharacteristic
337C2×C61#336911yescharacteristic
338C2×C61#337111yescharacteristic
339C22×Q16—611no—
340C22×SD16C2×D6111yescharacteristic
341C2×C4⋊Q8—611no—
342C2×#124—611no—
343C2×C4×Q8—611no—
344C2×C4×Q8C2×D6111yescharacteristic
345#489—1224no—
346#441—1224no—
347C2×#106—611no—
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103C2×D6111yescharacteristic
355C2×#102—611no—
356C2×#102—611no—
357C2×#102C2×D6111yescharacteristic
358C2×#97—611no—
359C2×#97—611no—
360C2×C4×C8—611no—
361#399—1221no—
362#398—1221no—
363#397—1221no—
364#395—1221no—
365#391—611no—
366#391—611no—
367#391—611no—
368#391—1221no—
369#390—611no—
370#390C2×D6111yescharacteristic
371#389—611no—
372#389—611no—
373#389—611no—
374#388—1221no—
375#388—1221no—
376#387—1221no—
377C4×C4⋊C4—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340C2×D6111yescharacteristic
382#336—611no—
383#336C2×D6111yescharacteristic
384C61#2684611yescharacteristic, complemented, Sylow
385C22×Q8—611no—
386C22×Q8—1221no—
387C22×Q8C22×D6111yescharacteristic
388C22×D4C22×D6111yescharacteristic
389C2×Q16—1222no—
390C2×Q16—1222no—
391C2×Q16—1222no—
392C2×SD16—22no—
393C2×SD16—22no—
394C2×SD16—41no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399C4⋊Q8—1222no—
400C4⋊Q8—2441no—
401#124—1222no—
402#124—2441no—
403C4×Q8—22no—
404C4×Q8—41no—
405C4×Q8—1222no—
406C4×Q8—2441no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—21no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—1221no—
430C2×C42C22×D6111yescharacteristic
431#106—1222no—
432#106—2441no—
433C4⋊C8—1222no—
434#103—22no—
435#103—614no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#102—22no—
440#102—1222no—
441#102—1222no—
442#97—1222no—
443#97—1222no—
444C4×C8—1222no—
445#95—611no—
446#95—611no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456C24—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—612no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—4881no—
469C2×Q8—2442no—
470C2×Q8—2442no—
471C2×Q8#1175822412yes—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4D4×D6111yescharacteristic
490C22×C4D4×D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492Q16—1228no—
493SD16—44no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—612no—
500C2×C8—612no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8#1175822212yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—1222no—
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—2441no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4D4×D6112yes—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—4881no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—1221no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C42—2442no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C2×D4×D6111yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—1222no—
551Q8—1222no—
552Q8—4884no—
553Q8—2441no—
554Q8—2441no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—612no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1222no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4C2×#1175822411yescharacteristic
601C2×C4C2×#1175822211yescharacteristic
602C2×C4#1451982211yescharacteristic
603C2×C4#1451982411yescharacteristic
604C8—22no—
605C8—41no—
606C8—1222no—
607C8—1222no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#1684030911yescharacteristic
616C22#1684039411yescharacteristic
617C22#1684041811yescharacteristic
618C22#137×D6111yescharacteristic
619C22#1684055811yescharacteristic
620C22#1684056811yescharacteristic
621C22#1684056911yescharacteristic
622C4—41no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—81no—
627C4—1222no—
628C4—4881no—
629C4—4881no—
630C4—4881no—
631C4—2441no—
632C4—2441no—
633C4—2442no—
634C4—2441no—
635C4#1684027211yescharacteristic
636C4#1684038011yescharacteristic
637C4#1684041711yescharacteristic
638C4#1684043811yescharacteristic
639C2—161no—
640C2#2144979111yescharacteristic
641C2#2145643211yescharacteristic
642C2#2145652411yescharacteristic
643C2#2145744811yescharacteristic
644C2#2145748311yescharacteristic
645C2#2145855411yescharacteristic
646C2#2145858011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1880C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1880D61
hypocenterC61#26846
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
2. C2×#16839867C2semidirect
3. #21449652C2semidirect
6. #21447151C2semidirect
7. #21447140C2semidirect
11. #21446511C2semidirect
12. #21446496C2semidirect
13. #21446090C2semidirect
14. #21445685C2semidirect
384. C61#26846semidirect

Maximal subgroups

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

subgroupindex
C2×#168401202
C2×#168398672
#214496522
#214496182
#214471892
#214471512
#214471402
#214469112
#214466752
#214465192
#214465112
#214464962
#214460902
#214456852
C61×#18802
#2684661