Finite Group Databasebeta

← #29729575

Subgroups — #29729575

Order 15616.

800 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839981C211yesmaximal, characteristic
2C2×C8×Dic122C211yesmaximal, characteristic, complemented
3#21451308C211yesmaximal, characteristic
4#21451186C211yesmaximal, characteristic, complemented
5#21447203C211yesmaximal, characteristic
6#21447196C211yesmaximal, characteristic, complemented
7#21447140C211yesmaximal, characteristic, complemented
8#21447135C211yesmaximal, characteristic
9#21446677C211yesmaximal, characteristic, complemented
10#21446674C211yesmaximal, characteristic
11#21446585C211yesmaximal, characteristic, complemented
12#21446504C211yesmaximal, characteristic
13#21446497C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C61×#1958C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16839981—24no—
21C8×Dic122—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839702—21no—
34#16839687—21no—
35#16839683—21no—
36#16839674—21no—
37#16839669C2211yescharacteristic
38#16839669C2211yescharacteristic
39#16839667C2211yescharacteristic
40#16839667C2211yescharacteristic
41#16839665C2211yescharacteristic
42#16839665C2211yescharacteristic
43#16839661C2211yescharacteristic
44#16839661C2211yescharacteristic
45#16839656—21no—
46#16839656—21no—
47#16839647—21no—
48#16839647—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×C4×C8×C61C2211yescharacteristic
60C61×#382C2211yescharacteristic
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#14519710—41no—
93#14519708—41no—
94C4⋊#92418D414yes—
95#14519706—22no—
96#14519706—22no—
97#14519702—22no—
98#14519702—22no—
99#14519701—22no—
100#14519698—22no—
101C8×Dic61—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×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61C2311yescharacteristic
125#103×C61—22no—
126#103×C61—41no—
127#102×C61—22no—
128#102×C61—41no—
129C4×C8×C61—22no—
130#95×C61—21no—
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×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—22no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
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—
178C22⋊C4×C61—42no—
179C22⋊C4×C61—42no—
180C42×C61C2×D412yes—
181C42×C61#4112yes—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C22⋊Dic61—81no—
185C22⋊Dic61—81no—
186#11758202—44no—
187#11758200—44no—
188C4⋊Dic61—22no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61—42no—
197#11758196—22no—
198#11758196—22no—
199#11758196—42no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—42no—
205C4×Dic61—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C2×D412yes—
211C2×#92418C2×D412yes—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
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#12112yes—
242C2×C4×C61#12112yes—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
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#13512yes—
272C2×Dic61#13512yes—
273Dic122—28no—
274Dic122—22no—
275Dic122—22no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—22no—
279C8×C61—41no—
280#92418—22no—
281#92418—22no—
282#25297—611nomaximal, Sylow
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61C2×#12111yescharacteristic
291C22×C61#54811yescharacteristic
292C22×C61#55011yescharacteristic
293C22×C61C2×#13511yescharacteristic
294C22×C61C2×#13511yescharacteristic
295C22×C61#57711yescharacteristic
296C22×C61#57811yescharacteristic
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61#52911yescharacteristic
303C4×C61#54211yescharacteristic
304C4×C61#54611yescharacteristic
305C4×C61#54811yescharacteristic
306Dic61—22no—
307Dic61—22no—
308Dic61—42no—
309Dic61—41no—
310Dic61—41no—
311Dic61—81no—
312Dic61—81no—
313Dic61—81no—
314C2×#464—611no—
315C2×C8×Q8—611no—
316#2592—611no—
317#2581—611no—
318#2067—611no—
319#2066—611no—
320#2056—611no—
321#2055—611no—
322#1969—611no—
323#1968—611no—
324#1958D6111yescharacteristic
325#1949—611no—
326#1934—611no—
327#1933—611no—
328C4×#103—611no—
329C2×C61—161no—
330C2×C61#242611yescharacteristic
331C2×C61#305011yescharacteristic
332C2×C61#306311yescharacteristic
333C2×C61#336411yescharacteristic
334C2×C61#336511yescharacteristic
335C2×C61#339311yescharacteristic
336C2×C61#339411yescharacteristic
337C22×Q16—611no—
338C22×SD16—611no—
339C2×#124C2×D6111yescharacteristic
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342#464—1224no—
343C8×Q8—1224no—
344C2×#107—611no—
345C2×#106—611no—
346C2×C4⋊C8—611no—
347C2×C4⋊C8—611no—
348C2×C4⋊C8—611no—
349C2×#103—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103C2×D6111yescharacteristic
353C2×#103C2×D6111yescharacteristic
354C2×#102—611no—
355C2×#102C2×D6111yescharacteristic
356C2×#102C2×D6111yescharacteristic
357C2×C4×C8—611no—
358C2×C4×C8—611no—
359C2×C4×C8C2×D6111yescharacteristic
360#400—1221no—
361#397—1221no—
362#395—1221no—
363#393—1221no—
364#391—611no—
365#391—611no—
366#391—611no—
367#391—611no—
368#390—611no—
369#390—611no—
370#389—611no—
371#389—611no—
372#388—1221no—
373#388—1221no—
374#387—1221no—
375#387—1221no—
376#382C2×D6111yescharacteristic
377C4×C4⋊C4—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382C61#2529711yescharacteristic, complemented, Sylow
383C22×Q8—611no—
384C22×Q8—611no—
385C22×Q8C22×D6111yescharacteristic
386C22×D4C22×D6111yescharacteristic
387C2×Q16—1222no—
388C2×Q16—1222no—
389C2×Q16—2441no—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—2441no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397#124—22no—
398#124—22no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—1221no—
423C2×C42—1221no—
424C2×C42C22×D6111yescharacteristic
425#107—2441no—
426#106—2441no—
427C4⋊C8—614no—
428C4⋊C8—1222no—
429C4⋊C8—1222no—
430#103—22no—
431#103—41no—
432#103—1222no—
433#103—1222no—
434#103—1222no—
435#102—22no—
436#102—41no—
437#102—1222no—
438C4×C8—22no—
439C4×C8—1222no—
440C4×C8—1222no—
441#95—21no—
442#95—21no—
443#95—1221no—
444#95—1221no—
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—
454C24—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—612no—
462C2×Q8—1222no—
463C2×Q8—1222no—
464C2×Q8—1221no—
465C2×Q8—1222no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
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—2444no—
491SD16—2444no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—612no—
498C2×C8—612no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8#1175822412yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1221no—
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—42no—
521C22⋊C4—42no—
522C22⋊C4—4881no—
523C22⋊C4—4881no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—4881no—
531C42—2442no—
532C42#1175822212yes—
533C42D4×D6112yes—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×D4×D6111yescharacteristic
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—1228no—
547Q8—1222no—
548Q8—1222no—
549Q8—2441no—
550Q8—2441no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1222no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4#1451975012yes—
593C2×C4#1451975012yes—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4C2×#1175822411yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4#1451982211yescharacteristic
599C2×C4#1451982411yescharacteristic
600C8—22no—
601C8—41no—
602C8—1222no—
603C8—1222no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#1451975011yescharacteristic
612C22#1684042011yescharacteristic
613C22#1684043211yescharacteristic
614C22#135×D6111yescharacteristic
615C22#135×D6111yescharacteristic
616C22#1684056911yescharacteristic
617C22#1684057611yescharacteristic
618C4—22no—
619C4—22no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—1222no—
624C4—1222no—
625C4—4881no—
626C4—4881no—
627C4—4881no—
628C4—2441no—
629C4—2441no—
630C4—2442no—
631C4#1684029411yescharacteristic
632C4#1684037711yescharacteristic
633C4#1684040911yescharacteristic
634C4#1684041811yescharacteristic
635C2—161no—
636C2#2144938611yescharacteristic
637C2#2145539911yescharacteristic
638C2#2145551311yescharacteristic
639C2#2145850111yescharacteristic
640C2#2145851211yescharacteristic
641C2#2145880411yescharacteristic
642C2#2145881211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1958C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1958D61
hypocenterC61#25297
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×C8×Dic122C2semidirect
4. #21451186C2semidirect
6. #21447196C2semidirect
7. #21447140C2semidirect
9. #21446677C2semidirect
11. #21446585C2semidirect
13. #21446497C2semidirect
14. C4×#14519702C2semidirect
382. C61#25297semidirect

Maximal subgroups

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

subgroupindex
C2×#168399812
C2×C8×Dic1222
#214513082
#214511862
#214472032
#214471962
#214471402
#214471352
#214466772
#214466742
#214465852
#214465042
#214464972
C4×#145197022
C61×#19582
#2529761