Finite Group Databasebeta

← #29729769

Subgroups — #29729769

Order 15616.

788 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×#16839981C211yesmaximal, characteristic
2C2×#16839890C211yesmaximal, characteristic, complemented
3#21451308C211yesmaximal, characteristic
4#21451192C211yesmaximal, characteristic, complemented
5#21447196C211yesmaximal, characteristic, complemented
6#21447189C211yesmaximal, characteristic
7#21447140C211yesmaximal, characteristic, complemented
8#21447135C211yesmaximal, characteristic
9#21446687C211yesmaximal, characteristic, complemented
10#21446674C211yesmaximal, characteristic
11#21446580C211yesmaximal, characteristic, complemented
12#21446518C211yesmaximal, characteristic
13#21446497C211yesmaximal, characteristic, complemented
14#21445650C211yesmaximal, characteristic, complemented
15C61×#1959C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16839981—24no—
21#16839890—24no—
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839692—21no—
34#16839692—21no—
35#16839687—21no—
36#16839674—21no—
37#16839669C2211yescharacteristic
38#16839669C2211yescharacteristic
39#16839667—21no—
40#16839667C2211yescharacteristic
41#16839667C2211yescharacteristic
42#16839665C2211yescharacteristic
43#16839665C2211yescharacteristic
44#16839661—21no—
45#16839661C2211yescharacteristic
46#16839661C2211yescharacteristic
47#16839656—21no—
48#16839656—21no—
49C4×C4⋊Dic61C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#124×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C61×#384C2211yescharacteristic
61C22×Dic122D411yescharacteristic
62C22×Dic122C2311yescharacteristic
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—41no—
66C2×#11758200—22no—
67C2×#11758200—22no—
68C2×#11758200—41no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C4×Dic122—22no—
72C4×Dic122—22no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
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×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519708—41no—
93#14519708—41no—
94C4⋊#92418D414yes—
95#14519706—22no—
96#14519706—22no—
97#14519702—22no—
98#14519702—22no—
99#14519701—22no—
100#14519698—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#14519684—21no—
113#14519684—21no—
114C22×Q8×C61C2311yescharacteristic
115C22×D4×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118#124×C61—22no—
119#124×C61—22no—
120C2×C4⋊C4×C61—21no—
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—41no—
129#102×C61—22no—
130#102×C61—41no—
131#97×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—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—
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—22no—
164C2×Dic122—21no—
165C2×Dic122—22no—
166C2×Dic122—41no—
167C2×Dic122#4112yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61#4112yes—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61—81no—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—42no—
182C42×C61C2×D412yes—
183C42×C61#4112yes—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186C22⋊Dic61—81no—
187C22⋊Dic61—81no—
188#11758202—44no—
189#11758200—44no—
190C4⋊Dic61—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—81no—
198C4⋊Dic61—81no—
199#11758196—22no—
200#11758196—22no—
201#11758196—42no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207C4×Dic61—81no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×D412yes—
213C2×#92418C2×D412yes—
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—41no—
223Q8×C61—41no—
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—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61#12112yes—
244C2×C4×C61#12112yes—
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—44no—
278Dic122—41no—
279Dic122—41no—
280C8×C61—22no—
281C8×C61—41no—
282#92418—22no—
283#92418—22no—
284#25311—611nomaximal, Sylow
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61C2×#12111yescharacteristic
293C22×C61#54811yescharacteristic
294C22×C61#55011yescharacteristic
295C22×C61C2×#13711yescharacteristic
296C22×C61#57511yescharacteristic
297C22×C61#57711yescharacteristic
298C22×C61#57711yescharacteristic
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61—81no—
304C4×C61#52911yescharacteristic
305C4×C61#54211yescharacteristic
306C4×C61#54611yescharacteristic
307C4×C61#54811yescharacteristic
308Dic61—22no—
309Dic61—42no—
310Dic61—41no—
311Dic61—41no—
312Dic61—41no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316C2×#464—611no—
317C2×#446—611no—
318#2592—611no—
319#2582—611no—
320#2066—611no—
321#2065—611no—
322#2056—611no—
323#2055—611no—
324#1971—611no—
325#1968—611no—
326#1959D6111yescharacteristic
327#1948—611no—
328#1936—611no—
329#1933—611no—
330#1763—611no—
331C2×C61—161no—
332C2×C61#242611yescharacteristic
333C2×C61#305611yescharacteristic
334C2×C61#306011yescharacteristic
335C2×C61#336911yescharacteristic
336C2×C61#337111yescharacteristic
337C2×C61#340011yescharacteristic
338C2×C61#340111yescharacteristic
339C22×Q16—611no—
340C22×SD16—611no—
341C2×#124C2×D6111yescharacteristic
342C2×C4×Q8—611no—
343C2×C4×Q8—611no—
344#464—1224no—
345#446—1224no—
346C2×#106—611no—
347C2×#106—611no—
348C2×C4⋊C8—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103C2×D6111yescharacteristic
355C2×#103C2×D6111yescharacteristic
356C2×#102—611no—
357C2×#102C2×D6111yescharacteristic
358C2×#102C2×D6111yescharacteristic
359C2×#97—611no—
360C2×#97C2×D6111yescharacteristic
361C2×C4×C8—611no—
362#398—1221no—
363#398—1221no—
364#397—1221no—
365#393—1221no—
366#391—611no—
367#391—611no—
368#391—611no—
369#391—611no—
370#391—1221no—
371#390—611no—
372#390—611no—
373#389—611no—
374#389—611no—
375#389—1221no—
376#388—1221no—
377#388—1221no—
378#384C2×D6111yescharacteristic
379C4×C4⋊C4—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384C61#2531111yescharacteristic, complemented, Sylow
385C22×Q8—611no—
386C22×Q8—611no—
387C22×Q8C22×D6111yescharacteristic
388C22×D4C22×D6111yescharacteristic
389C2×Q16—1222no—
390C2×Q16—1222no—
391C2×Q16—2441no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—2441no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399#124—22no—
400#124—22no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—611no—
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⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—1221no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429#106—2441no—
430#106—2441no—
431C4⋊C8—614no—
432C4⋊C8—1222no—
433C4⋊C8—1222no—
434#103—22no—
435#103—41no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#102—22no—
440#102—41no—
441#102—1222no—
442#97—22no—
443#97—1222no—
444C4×C8—1222no—
445#95—1221no—
446#95—1221no—
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—21no—
463C2×Q8—612no—
464C2×Q8—1222no—
465C2×Q8—1221no—
466C2×Q8—1222no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—2441no—
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—2444no—
493SD16—2444no—
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#1175822412yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—4881no—
522C4⋊C4—4881no—
523C4⋊C4—2442no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—4881no—
527C22⋊C4—4881no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C42—1221no—
531C42—1221no—
532C42—1221no—
533C42—1221no—
534C42—4881no—
535C42—2442no—
536C42#1175822212yes—
537C42D4×D6112yes—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C2×D4×D6111yescharacteristic
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—1222no—
551Q8—1222no—
552Q8—2444no—
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—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—612no—
573C2×C4—1221no—
574C2×C4—1221no—
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—1222no—
585C2×C4—2441no—
586C2×C4—2441no—
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#1451975012yes—
597C2×C4#1451975012yes—
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—
615C22C2×#1451975011yescharacteristic
616C22#1684042011yescharacteristic
617C22#1684043211yescharacteristic
618C22#137×D6111yescharacteristic
619C22#1684055811yescharacteristic
620C22#1684057111yescharacteristic
621C22#1684057211yescharacteristic
622C4—22no—
623C4—22no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—1222no—
628C4—4881no—
629C4—4881no—
630C4—4881no—
631C4—2441no—
632C4—2441no—
633C4—2441no—
634C4—2442no—
635C4#1684029411yescharacteristic
636C4#1684037711yescharacteristic
637C4#1684040911yescharacteristic
638C4#1684041811yescharacteristic
639C2—161no—
640C2#2144938611yescharacteristic
641C2#2145545211yescharacteristic
642C2#2145548011yescharacteristic
643C2#2145855611yescharacteristic
644C2#2145858611yescharacteristic
645C2#2145886911yescharacteristic
646C2#2145888011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1959C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1959D61
hypocenterC61#25311
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
2. C2×#16839890C2semidirect
4. #21451192C2semidirect
5. #21447196C2semidirect
7. #21447140C2semidirect
9. #21446687C2semidirect
11. #21446580C2semidirect
13. #21446497C2semidirect
14. #21445650C2semidirect
384. C61#25311semidirect

Maximal subgroups

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

subgroupindex
C2×#168399812
C2×#168398902
#214513082
#214511922
#214471962
#214471892
#214471402
#214471352
#214466872
#214466742
#214465802
#214465182
#214464972
#214456502
C61×#19592
#2531161