Finite Group Databasebeta

← #29709291

Subgroups — #29709291

Order 15616.

868 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×#16839869C211yesmaximal, characteristic, complemented
2C2×#16839865C211yesmaximal, characteristic
3#21448362C211yesmaximal, characteristic
4#21448294C211yesmaximal, characteristic, complemented
5#21447305C211yesmaximal, characteristic, complemented
6#21447240C211yesmaximal, characteristic, complemented
7#21447214C211yesmaximal, characteristic, complemented
8#21447192C211yesmaximal, characteristic
9#21447134C211yesmaximal, characteristic
10#21447056C211yesmaximal, characteristic
11#21446910C211yesmaximal, characteristic
12#21445674C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14#21445643C211yesmaximal, characteristic
15C61×#2056C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21#16839869—24no—
22#16839865—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34#16839669C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839654—21no—
39#16839654—21no—
40#16839647—21no—
41#16839645—21no—
42#16839627C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44#16839613—21no—
45#16839609—21no—
46#16839609—21no—
47#16839607—21no—
48C42×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758201—22no—
64C2×#11758201—22no—
65C2×#11758201—22no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519766—22no—
69#14519766—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73Q8×Dic61D414yes—
74#14519739—22no—
75#14519739—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61D411yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97#14519710—22no—
98#14519709—22no—
99C4⋊#92418—22no—
100#14519701—22no—
101#14519701—22no—
102#14519701—22no—
103#14519699—22no—
104#14519699—22no—
105#14519697—22no—
106#14519687—22no—
107#14519687—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115C22×Q8×C61C2311yescharacteristic
116C22×Q8×C61C2311yescharacteristic
117C2×Q16×C61—22no—
118C2×Q16×C61—22no—
119C2×Q16×C61—22no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C4⋊C8×C61—22no—
126#103×C61—22no—
127#103×C61—22no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C23×D61—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×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C2×D412yes—
144C2×Q8×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244#4112yes—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168Q16×C61—28no—
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—21no—
179C42×C61—21no—
180C42×C61—21no—
181#11758201—28no—
182#11758198—24no—
183#11758198—24no—
184#11758198—42no—
185#11758198—42no—
186#11758198—42no—
187C4⋊Dic61—21no—
188C4⋊Dic61—24no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61C2×D412yes—
195#11758196—42no—
196#11758196—42no—
197C4×Dic61—22no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—24no—
201C4×Dic61—41no—
202C4×Dic61—41no—
203C4×Dic61—41no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C4×Dic61C2×D412yes—
207C4×Dic61#4112yes—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418#4112yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222Q8×C61—22no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229Q8×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—41no—
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×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61#13712yes—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
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#12112yes—
274C2×Dic61#12112yes—
275D244—22no—
276D244—22no—
277D244—41no—
278D244—41no—
279C4×D61—84no—
280C8×C61—22no—
281C8×C61—41no—
282#92418—22no—
283#92418—41no—
284#23687—611nomaximal, Sylow
285C22×C61C2×#12111yescharacteristic
286C22×C61#52511yescharacteristic
287C22×C61#53211yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57511yescharacteristic
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C4×C61—22no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61#52511yescharacteristic
306C4×C61#52511yescharacteristic
307C4×C61#52911yescharacteristic
308C4×C61#52911yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—41no—
312Dic61—41no—
313Dic61—42no—
314Dic61—81no—
315Dic61—81no—
316C2×#441—611no—
317C2×#440—611no—
318C4×#124—611no—
319#2295—611no—
320#2085—611no—
321#2074—611no—
322#2068—611no—
323#2065—611no—
324#2056D6111yescharacteristic, complemented
325#2055—611no—
326#2042—611no—
327#2015—611no—
328#1767—611no—
329#1763—611no—
330#1762—611no—
331C2×C61#239211yescharacteristic
332C2×C61#303711yescharacteristic
333C2×C61#304011yescharacteristic
334C2×C61#311111yescharacteristic
335C2×C61#311611yescharacteristic
336C2×C61#318111yescharacteristic
337C2×C61#318311yescharacteristic
338D61—82no—
339C22×Q16C2×D6111yescharacteristic
340C22×SD16—611no—
341C2×#124—611no—
342C2×C4×Q8—611no—
343C2×C4×Q8—611no—
344C2×C4×D4—611no—
345#441—1224no—
346#440—1224no—
347C2×#107—611no—
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8C2×D6111yescharacteristic
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103C2×D6111yescharacteristic
355C2×#103C2×D6111yescharacteristic
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#97—611no—
360C2×#97—611no—
361#391—611no—
362#391C2×D6111yescharacteristic
363#391C2×D6111yescharacteristic
364#390—611no—
365#389—611no—
366#389—611no—
367#388—1221no—
368#388—1221no—
369#387—1221no—
370#387—1221no—
371#382—611no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—1221no—
374C4×C22⋊C4—1221no—
375C4×C22⋊C4—1221no—
376#376—1221no—
377C43—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340C2×D6111yescharacteristic
382#336—611no—
383#336—611no—
384C61#2368711yescharacteristic, complemented, Sylow
385C22×Q8C22×D6111yescharacteristic
386C22×Q8C22×D6111yescharacteristic
387C22×D4—611no—
388C23×C4—1221no—
389C2×Q16—22no—
390C2×Q16—22no—
391C2×Q16—22no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—1222no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399#124—1222no—
400#124—1222no—
401C4×Q8—614no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#106—1222no—
433C4⋊C8—22no—
434C4⋊C8—1222no—
435#103—22no—
436#103—22no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#102—1222no—
441#102—1222no—
442#102—1222no—
443#97—1222no—
444#97—1222no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456C24—1221no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8D4×D6112yes—
468C2×Q8#1175822412yes—
469C2×D4—612no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C22×C4—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—2442no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4D4×D6111yescharacteristic
490C22×C4D4×D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492Q16—28no—
493SD16—1228no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—612no—
500C2×C8—1221no—
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—612no—
511C4⋊C4—1224no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C22⋊C4—1224no—
521C22⋊C4—1224no—
522C22⋊C4—2442no—
523C22⋊C4—2442no—
524C22⋊C4—2442no—
525C42—21no—
526C42—21no—
527C42—612no—
528C42—612no—
529C42—1224no—
530C42—1222no—
531C42—1221no—
532C42—1221no—
533C42—2441no—
534C42—2441no—
535C42—2441no—
536C42—2442no—
537C42—2442no—
538C23—1222no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23C2×D4×D6111yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554D4—1222no—
555D4—1222no—
556D4—2441no—
557D4—2441no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1222no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—4884no—
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#1451980912yes—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C8—22no—
605C8—41no—
606C8—1222no—
607C8—2441no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—4881no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22#121×D6111yescharacteristic
616C22#1684027211yescharacteristic
617C22#1684030711yescharacteristic
618C22C2×#1451980911yescharacteristic
619C22#1684055911yescharacteristic
620C22#1684056111yescharacteristic
621C22#1684056111yescharacteristic
622C4—22no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—1222no—
629C4—1222no—
630C4—4881no—
631C4—4881no—
632C4—2441no—
633C4—2441no—
634C4—2442no—
635C4#1684026811yescharacteristic
636C4#1684027011yescharacteristic
637C4#1684028911yescharacteristic
638C4#1684029111yescharacteristic
639C2—4882no—
640C2#2144906211yescharacteristic
641C2#2145529211yescharacteristic
642C2#2145531011yescharacteristic
643C2#2145592711yescharacteristic
644C2#2145598011yescharacteristic
645C2#2145652311yescharacteristic
646C2#2145654911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2056C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2056D61
hypocenterC61#23687
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
1. C2×#16839869C2semidirect
4. #21448294C2semidirect
5. #21447305C2semidirect
6. #21447240C2semidirect
7. #21447214C2semidirect
12. #21445674C2semidirect
13. #21445651C2semidirect
15. C61×#2056C2semidirect
324. #2056D61semidirect
384. C61#23687semidirect

Maximal subgroups

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

subgroupindex
C2×#168398692
C2×#168398652
#214483622
#214482942
#214473052
#214472402
#214472142
#214471922
#214471342
#214470562
#214469102
#214456742
#214456512
#214456432
C61×#20562
#2368761