Finite Group Databasebeta

← #29743388

Subgroups — #29743388

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840052C211yesmaximal, characteristic
2C2×#16839863C211yesmaximal, characteristic
3#21449660C211yesmaximal, characteristic, complemented
4#21448724C211yesmaximal, characteristic
5#21447365C211yesmaximal, characteristic, complemented
6#21447306C211yesmaximal, characteristic, complemented
7#21447214C211yesmaximal, characteristic, complemented
8#21447148C211yesmaximal, characteristic, complemented
9#21446556C211yesmaximal, characteristic, complemented
10#21446386C211yesmaximal, characteristic, complemented
11#21446155C211yesmaximal, characteristic
12#21446085C211yesmaximal, characteristic, complemented
13#21446068C211yesmaximal, characteristic
14#21446064C211yesmaximal, characteristic
15C61×#1877C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16840052—24no—
21#16839863—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33#16839694—21no—
34#16839683—21no—
35#16839669C2211yescharacteristic
36#16839666—21no—
37#16839661C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839649—21no—
42#16839635—21no—
43#16839634—21no—
44#16839627C2211yescharacteristic
45#16839620—21no—
46C4×#11758196C2211yescharacteristic
47C2×#14519684—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519777—22no—
67#14519777—41no—
68Q8×Dic61—22no—
69Q8×Dic61—41no—
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—22no—
73D4×Dic61—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196D411yescharacteristic
85C2×#11758196D411yescharacteristic
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519711—22no—
92#14519711—41no—
93#14519710—22no—
94#14519708—22no—
95#14519708—22no—
96C4⋊#92418—22no—
97#14519701—22no—
98#14519701—22no—
99#14519698—22no—
100#14519698D414yes—
101#14519687—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C23×C4×C61—21no—
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C2×SD16×C61Q814yescomplemented
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—41no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#103×C61—41no—
127#102×C61—22no—
128#95×C61—21no—
129#95×C61—21no—
130C24×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61C2×Q812yes—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61C2×D412yes—
141C2×D4×C61C2×Q812yes—
142C2×D4×C61#4112yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—81no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—24no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—81no—
160C22×Dic61—81no—
161C22×Dic61—81no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C2×Dic122—42no—
167C2×Dic122—42no—
168C2×Dic122—81no—
169SD16×C61—28no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×Q812yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C22⋊C4×C61—42no—
181C42×C61—21no—
182C42×C61—21no—
183C22⋊Dic61—24no—
184C22⋊Dic61—42no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—42no—
192C4⋊Dic61—42no—
193C4⋊Dic61—81no—
194C4⋊Dic61C2×D412yes—
195C4⋊Dic61#4112yes—
196#11758196—22no—
197#11758196—41no—
198#11758196—81no—
199#11758196—81no—
200#11758196—81no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D412yes—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220Q8×C61—22no—
221Q8×C61—22no—
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—22no—
227D4×C61C22⋊Q814yes—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
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—84no—
243C2×C4×C61#13712yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×Q811yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
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—84no—
275C2×Dic61—84no—
276Dic122—84no—
277C8×C61—22no—
278C8×C61—41no—
279#92418—22no—
280#92418—41no—
281#26396—611nomaximal, Sylow
282C22×C61—24no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61#52511yescharacteristic
290C22×C61D4211yescharacteristic
291C22×C61D4×Q811yescharacteristic
292C22×C61C2×#13711yescharacteristic
293C22×C61#57511yescharacteristic
294C22×C61#57611yescharacteristic
295C22×C61#57711yescharacteristic
296C4×C61—22no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61C2×C22⋊Q811yescharacteristic
302C4×C61#53911yescharacteristic
303C4×C61#54711yescharacteristic
304C4×C61#55411yescharacteristic
305Dic61—41no—
306Dic61—41no—
307Dic61—42no—
308Dic61—41no—
309Dic61—42no—
310Dic61—41no—
311Dic61—81no—
312Dic61—81no—
313C2×#476—611no—
314C2×#440—611no—
315#2451—611no—
316#2349—611no—
317#2097—611no—
318#2085—611no—
319#2068—611no—
320#2057—611no—
321#1943—611no—
322#1917—611no—
323#1877—611no—
324#1877D6111yescharacteristic
325#1865—611no—
326#1862—611no—
327#1862—611no—
328C2×C61—44no—
329C2×C61#244911yescharacteristic
330C2×C61#311611yescharacteristic
331C2×C61#311811yescharacteristic
332C2×C61#326611yescharacteristic
333C2×C61#327311yescharacteristic
334C2×C61#338511yescharacteristic
335C2×C61#338811yescharacteristic
336C22×SD16C2×D6111yescharacteristic
337C2×C4⋊Q8—611no—
338C2×C4×Q8—611no—
339C2×C4×D4—611no—
340C2×C4×D4—611no—
341C2×C4×D4C2×D6111yescharacteristic
342#476—1224no—
343#440—1224no—
344C2×#107—611no—
345C2×#107—611no—
346C2×#107—611no—
347C2×#106—611no—
348C2×#106—611no—
349C2×C4⋊C8—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103C2×D6111yescharacteristic
353C2×#102—611no—
354C2×#102—611no—
355C2×#102C2×D6111yescharacteristic
356C2×#97—611no—
357#398—1221no—
358#395—1221no—
359#391—611no—
360#391—1221no—
361#391C2×D6111yescharacteristic
362#389—611no—
363#389—611no—
364#389—611no—
365#389—611no—
366#387—1221no—
367#385—1221no—
368#385—1221no—
369#382—611no—
370#380—1221no—
371C4×C4⋊C4—611no—
372C2×#95—1221no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340C2×D6111yescharacteristic
377#336—611no—
378#336—611no—
379#336—611no—
380#336C2×D6111yescharacteristic
381C61#2639611yescharacteristic, complemented, Sylow
382C22×Q8—1221no—
383C22×Q8C22×D6111yescharacteristic
384C22×D4C22×D6111yescharacteristic
385C23×C4—21no—
386C23×C4—1221no—
387C23×C4—1221no—
388C2×SD16—22no—
389C2×SD16—22no—
390C2×SD16Dic12214yescomplemented
391C22×C8—611no—
392C22×C8—611no—
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395C4⋊Q8—1222no—
396C4⋊Q8—2441no—
397C4×Q8—1222no—
398C4×Q8—2441no—
399C4×D4—22no—
400C4×D4—41no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—2441no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—1221no—
425C2×C42C22×D6111yescharacteristic
426#107—1222no—
427#107—1222no—
428#107—2441no—
429#106—1222no—
430#106—1222no—
431C4⋊C8—1222no—
432#103—41no—
433#103—1222no—
434#103—1222no—
435#102—22no—
436#102—614no—
437#102—1222no—
438#97—1222no—
439#95—21no—
440#95—21no—
441#95—611no—
442#95—611no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451C24—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—4881no—
458C2×Q8—2442no—
459C2×Q8—2442no—
460C2×Q8C2×Dic12212yes—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4C2×Dic12212yes—
465C2×D4D4×D6112yes—
466C2×D4#1175822412yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1224no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—4881no—
483C22×C4—4881no—
484C22×C4—4881no—
485C22×C4—2442no—
486C22×C4—2442no—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490SD16—28no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—612no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8C2×Dic12212yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—612no—
508C4⋊C4—612no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1222no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—4881no—
519C4⋊C4—4881no—
520C4⋊C4—2441no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C22⋊C4—42no—
524C22⋊C4—1224no—
525C22⋊C4—2442no—
526C42—21no—
527C42—21no—
528C42—1221no—
529C42—1221no—
530C42—1221no—
531C42—1221no—
532C42—2442no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C2×D4×D6111yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—41no—
544Q8—41no—
545Q8—4884no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4#1451975414yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—1221no—
569C2×C4—1221no—
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—1222no—
579C2×C4—4884no—
580C2×C4—4884no—
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#1451980612yes—
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C22×Dic12211yescharacteristic
594C2×C4#41×D6111yescharacteristic
595C2×C4C2×#1175822411yescharacteristic
596C2×C4#1451982211yescharacteristic
597C2×C4#1451982411yescharacteristic
598C8—22no—
599C8—41no—
600C8—1222no—
601C8—2441no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#1684027211yescharacteristic
610C22#1684039411yescharacteristic
611C22D4×Dic12211yescharacteristic
612C22C2×#1451980611yescharacteristic
613C22#1684056111yescharacteristic
614C22#1684056411yescharacteristic
615C22#1684057211yescharacteristic
616C4—22no—
617C4—41no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—4881no—
622C4—4881no—
623C4—2441no—
624C4—2442no—
625C4—2441no—
626C4—2441no—
627C4—2442no—
628C4—2441no—
629C4C2×#1451975411yescharacteristic
630C4#1684035711yescharacteristic
631C4#1684041011yescharacteristic
632C4#1684047111yescharacteristic
633C2—44no—
634C2#2144963211yescharacteristic
635C2#2145597811yescharacteristic
636C2#2145600011yescharacteristic
637C2#2145733311yescharacteristic
638C2#2145742311yescharacteristic
639C2#2145872711yescharacteristic
640C2#2145876511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1877C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1877D61
hypocenterC61#26396
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
3. #21449660C2semidirect
5. #21447365C2semidirect
6. #21447306C2semidirect
7. #21447214C2semidirect
8. #21447148C2semidirect
9. #21446556C2semidirect
10. #21446386C2semidirect
12. #21446085C2semidirect
117. C2×SD16×C61Q8semidirect
381. C61#26396semidirect
390. C2×SD16Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168400522
C2×#168398632
#214496602
#214487242
#214473652
#214473062
#214472142
#214471482
#214465562
#214463862
#214461552
#214460852
#214460682
#214460642
C61×#18772
#2639661