Finite Group Databasebeta

← #29743078

Subgroups — #29743078

Order 15616.

900 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×#16840050C211yesmaximal, characteristic
2#21449652C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic
4#21447360C211yesmaximal, characteristic, complemented
5#21447298C211yesmaximal, characteristic, complemented
6#21447198C211yesmaximal, characteristic, complemented
7#21447150C211yesmaximal, characteristic, complemented
8#21446547C211yesmaximal, characteristic, complemented
9#21446376C211yesmaximal, characteristic, complemented
10#21446155C211yesmaximal, characteristic
11#21446155C211yesmaximal, characteristic
12#21446087C211yesmaximal, characteristic, complemented
13#21446082C211yesmaximal, characteristic
14#21446080C211yesmaximal, characteristic
15C2×C4×SD16×C61C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C22×#11758196—21no—
20#16840050—24no—
21C2×#14519709C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30#16839695—21no—
31#16839682—21no—
32#16839669C2211yescharacteristic
33#16839669C2211yescharacteristic
34#16839668—21no—
35#16839661C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839646—21no—
40#16839635—21no—
41#16839634—21no—
42#16839631C2211yescharacteristic
43#16839620—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C4×SD16×C61—24no—
57C2×#106×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519776—22no—
67#14519776—41no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—22no—
71D4×Dic61—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—44no—
84C2×#11758196D411yescharacteristic
85C2×#11758196D411yescharacteristic
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519709—22no—
91#14519709—22no—
92#14519708—22no—
93#14519708—41no—
94#14519706—22no—
95#14519701—22no—
96#14519701—41no—
97#14519698—22no—
98#14519698—22no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C2×SD16×C61—22no—
113C2×SD16×C61—22no—
114C2×SD16×C61Q814yescomplemented
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C4×Q8×C61—22no—
118C4×Q8×C61—41no—
119C4×D4×C61—22no—
120C4×D4×C61—41no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#106×C61—22no—
128#103×C61—22no—
129#102×C61D414yes—
130C4×C8×C61—22no—
131C24×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61C2×Q812yes—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×D412yes—
142C2×D4×C61C2×Q812yes—
143C2×D4×C61#4112yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61—81no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—24no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—81no—
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122—81no—
170SD16×C61—28no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D412yes—
176C2×C8×C61C2×Q812yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—42no—
181C4⋊C4×C61C2×D412yes—
182C22⋊C4×C61—42no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—42no—
186C22⋊Dic61—24no—
187C22⋊Dic61—42no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—81no—
196C4⋊Dic61#4112yes—
197#11758196—41no—
198#11758196—42no—
199#11758196—42no—
200#11758196—42no—
201#11758196—42no—
202#11758196—81no—
203#11758196—81no—
204#11758196SD1614yescomplemented
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61C22⋊Q814yes—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
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×C61—84no—
246C2×C4×C61C2×SD1612yes—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×Q811yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C612−1+411yescharacteristic
253C2×Dic61—21no—
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—48no—
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—41no—
275C2×Dic61—41no—
276C2×Dic61—84no—
277C2×Dic61C2×SD1612yes—
278C2×Dic61C2×SD1612yes—
279Dic122—84no—
280C8×C61—22no—
281C8×C61—22no—
282#92418—41no—
283#92418—41no—
284#26376—611nomaximal, Sylow
285C22×C61—24no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61#52511yescharacteristic
293C22×C61D4211yescharacteristic
294C22×C61D4×Q811yescharacteristic
295C22×C61C22×SD1611yescharacteristic
296C22×C61#57511yescharacteristic
297C22×C61#57711yescharacteristic
298C22×C61#57711yescharacteristic
299C4×C61—22no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—81no—
304C4×C61C2×C22⋊Q811yescharacteristic
305C4×C61#53911yescharacteristic
306C4×C61#54711yescharacteristic
307C4×C61#55411yescharacteristic
308Dic61—22no—
309Dic61—22no—
310Dic61—42no—
311Dic61—41no—
312Dic61—41no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316C2×#476—611no—
317C2×C4×SD16D6111yescharacteristic
318#2451—611no—
319D4×C4⋊C4—611no—
320#2096—611no—
321#2084—611no—
322#2066—611no—
323#2058—611no—
324#1941—611no—
325#1916—611no—
326#1877—611no—
327#1877—611no—
328#1865—611no—
329#1864—611no—
330#1864—611no—
331C2×C61—44no—
332C2×C61#244911yescharacteristic
333C2×C61#310111yescharacteristic
334C2×C61#312511yescharacteristic
335C2×C61#326711yescharacteristic
336C2×C61D4×SD1611yescharacteristic
337C2×C61Q8×SD1611yescharacteristic
338C2×C61#338811yescharacteristic
339C22×SD16C2×D6111yescharacteristic
340C2×C4⋊Q8—611no—
341C2×C4×Q8C2×D6111yescharacteristic
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C2×C4×D4C2×D6111yescharacteristic
345C22×C4⋊C4—1221no—
346#476—1224no—
347C4×SD16—24no—
348C2×#106—611no—
349C2×#106—611no—
350C2×#106—611no—
351C2×#106—611no—
352C2×#106C2×D6111yescharacteristic
353C2×C4⋊C8—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103C2×D6111yescharacteristic
357C2×#102—611no—
358C2×#102—611no—
359C2×#102C2×D6111yescharacteristic
360C2×C4×C8C2×D6111yescharacteristic
361#398—1221no—
362#395—1221no—
363#391—611no—
364#391—611no—
365#391—1221no—
366#389—611no—
367#389—611no—
368#389—611no—
369#389—611no—
370#387—1221no—
371#385—1221no—
372#385—1221no—
373#384—611no—
374#380—1221no—
375C4×C4⋊C4—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#336—611no—
381#336—611no—
382#336—611no—
383#336—611no—
384C61#2637611yescharacteristic, complemented, Sylow
385C22×Q8—1221no—
386C22×Q8C22×D6111yescharacteristic
387C22×D4C22×D6111yescharacteristic
388C23×C4—21no—
389C23×C4—1221no—
390C23×C4—1221no—
391C2×SD16—22no—
392C2×SD16—22no—
393C2×SD16Dic12214yescomplemented
394C22×C8—611no—
395C22×C8—611no—
396C22×C8C22×D6111yescharacteristic
397C22×C8C22×D6111yescharacteristic
398C4⋊Q8—1222no—
399C4⋊Q8—2441no—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×D4—22no—
403C4×D4—41no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C4×D4—2441no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—2444no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C42—21no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42C22×D6111yescharacteristic
432#106—22no—
433#106—1222no—
434#106—1222no—
435#106—1222no—
436#106—2441no—
437C4⋊C8—1222no—
438#103—22no—
439#103—1222no—
440#103—2441no—
441#102—1222no—
442#102—1222no—
443#102C22⋊D6114yes—
444C4×C8—22no—
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—
455C24—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—4881no—
462C2×Q8—2442no—
463C2×Q8—2442no—
464C2×Q8C2×Dic12212yes—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×Dic12212yes—
469C2×D4#1175822212yes—
470C2×D4C2×C22⋊D6112yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—81no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—1224no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—4881no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494SD16—28no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8C2×Dic12212yes—
506C2×C8C2×C22⋊D6112yes—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—614no—
512C4⋊C4—612no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—4881no—
521C4⋊C4—4881no—
522C4⋊C4—4881no—
523C4⋊C4—2441no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4C2×C22⋊D6112yes—
529C22⋊C4—42no—
530C22⋊C4—1224no—
531C22⋊C4—2442no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—1221no—
536C42—1221no—
537C42—1221no—
538C42—1221no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C2×D4×D6111yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—4884no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4#1451975614yes—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1222no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—4884no—
585C2×C4—2448no—
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—2441no—
597C2×C4SD16×D6112yes—
598C2×C4C22×C22⋊D6111yescharacteristic
599C2×C4C22×Dic12211yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4C2×#1175822211yescharacteristic
602C2×C4#1451982211yescharacteristic
603C2×C4#1451982411yescharacteristic
604C8—22no—
605C8—22no—
606C8—2441no—
607C8—2441no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1684027111yescharacteristic
616C22D4×C22⋊D6111yescharacteristic
617C22D4×Dic12211yescharacteristic
618C22C2×SD16×D6111yescharacteristic
619C22#1684055911yescharacteristic
620C22#1684056811yescharacteristic
621C22#1684056911yescharacteristic
622C4—22no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—1222no—
628C4—1222no—
629C4—4881no—
630C4—4881no—
631C4—4881no—
632C4—2441no—
633C4—2442no—
634C4—2441no—
635C4C2×#1451975611yescharacteristic
636C4#1684035411yescharacteristic
637C4#1684041111yescharacteristic
638C4#1684047111yescharacteristic
639C2—44no—
640C2#2144962211yescharacteristic
641C2#2145582511yescharacteristic
642C2#2145609211yescharacteristic
643C2#2145734411yescharacteristic
644C2SD16×C22⋊D6111yescharacteristic
645C2SD16×Dic12211yescharacteristic
646C2#2145876011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×SD16×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×C4×SD16D61
hypocenterC61#26376
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. #21449652C2semidirect
4. #21447360C2semidirect
5. #21447298C2semidirect
6. #21447198C2semidirect
7. #21447150C2semidirect
8. #21446547C2semidirect
9. #21446376C2semidirect
12. #21446087C2semidirect
114. C2×SD16×C61Q8semidirect
204. #11758196SD16semidirect
384. C61#26376semidirect
393. C2×SD16Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168400502
#214496522
D4×#117581962
#214473602
#214472982
#214471982
#214471502
#214465472
#214463762
#214461552
#214460872
#214460822
#214460802
C2×C4×SD16×C612
#2637661