Finite Group Databasebeta

← #29743090

Subgroups — #29743090

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
2C2×SD16×Dic61C211yesmaximal, characteristic
3#21449661C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic
5#21447362C211yesmaximal, characteristic, complemented
6#21447297C211yesmaximal, characteristic, complemented
7#21447196C211yesmaximal, characteristic, complemented
8#21447151C211yesmaximal, characteristic, complemented
9#21446546C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446155C211yesmaximal, characteristic
12#21446089C211yesmaximal, characteristic, complemented
13#21446080C211yesmaximal, characteristic
14#21446078C211yesmaximal, characteristic
15C61×#1877C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C22×C4⋊Dic61—21no—
21#16840050—24no—
22SD16×Dic61—24no—
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839696—21no—
35#16839683—21no—
36#16839669C2211yescharacteristic
37#16839666—21no—
38#16839661C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839645—21no—
43#16839637—21no—
44#16839637—21no—
45#16839632C2211yescharacteristic
46#16839622—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
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#14519776—22no—
67#14519776—41no—
68Q8×Dic61—22no—
69Q8×Dic61—41no—
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—41no—
73D4×Dic61—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—44no—
80C2×C4⋊Dic61D411yescharacteristic
81C2×C4⋊Dic61D411yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519709—22no—
95#14519709—22no—
96#14519708—22no—
97#14519708—22no—
98#14519708—41no—
99#14519706—22no—
100#14519701—22no—
101#14519701—41no—
102#14519698—22no—
103#14519698D414yes—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C22×Q8×C61C2311yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C23×C4×C61—21no—
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61Q814yescomplemented
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×D4×C61—22no—
122C4×D4×C61—22no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#103×C61—22no—
128#102×C61—22no—
129#95×C61—21no—
130#95×C61—21no—
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—24no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—81no—
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×C61—21no—
176C2×C8×C61C2×Q812yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61#4112yes—
181C22⋊C4×C61—24no—
182C42×C61—21no—
183C42×C61—21no—
184C22⋊Dic61—42no—
185C22⋊Dic61—42no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—41no—
194C4⋊Dic61—42no—
195C4⋊Dic61—42no—
196C4⋊Dic61—42no—
197C4⋊Dic61—42no—
198C4⋊Dic61—81no—
199C4⋊Dic61SD1614yescomplemented
200C4⋊Dic61C2×D412yes—
201#11758196—42no—
202#11758196—81no—
203#11758196—81no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D412yes—
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—21no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—48no—
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×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—21no—
259C2×Dic61—22no—
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—41no—
275C2×Dic61—84no—
276C2×Dic61—84no—
277C2×Dic61C2×SD1612yes—
278C2×Dic61C2×SD1612yes—
279Dic122—84no—
280C8×C61—22no—
281C8×C61—41no—
282#92418—22no—
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—41no—
302C4×C61—42no—
303C4×C61—81no—
304C4×C61C2×C22⋊Q811yescharacteristic
305C4×C61#53911yescharacteristic
306C4×C61#54711yescharacteristic
307C4×C61#55411yescharacteristic
308Dic61—22no—
309Dic61—22no—
310Dic61—41no—
311Dic61—42no—
312Dic61—41no—
313Dic61—81no—
314Dic61—81no—
315Dic61—81no—
316C2×#476—611no—
317C2×C4×SD16—611no—
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#1877D6111yescharacteristic
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×Q8—611no—
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C2×C4×D4C2×D6111yescharacteristic
345C22×C4⋊C4—1221no—
346#476—1224no—
347C4×SD16—1224no—
348C2×#106—611no—
349C2×#106—611no—
350C2×#106—611no—
351C2×#106—611no—
352C2×#106—611no—
353C2×C4⋊C8—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103C2×D6111yescharacteristic
357C2×#102—611no—
358C2×#102—611no—
359C2×#102C2×D6111yescharacteristic
360C2×C4×C8—611no—
361#398—1221no—
362#395—1221no—
363#391—611no—
364#391—1221no—
365#391C2×D6111yescharacteristic
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#340C2×D6111yescharacteristic
380#336—611no—
381#336—611no—
382#336—611no—
383#336C2×D6111yescharacteristic
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—1222no—
401C4×Q8—2441no—
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—2441no—
407C4×D4—2441no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
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—611no—
429C2×C42—611no—
430C2×C42—1221no—
431C2×C42C22×D6111yescharacteristic
432#106—1222no—
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—22no—
442#102—614no—
443#102—1222no—
444C4×C8—1222no—
445#95—21no—
446#95—21no—
447#95—611no—
448#95—611no—
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×D4D4×D6112yes—
470C2×D4#1175822412yes—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—4881no—
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—21no—
500C2×C8—612no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8C2×Dic12212yes—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—614no—
511C4⋊C4—612no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—4881no—
520C4⋊C4—4881no—
521C4⋊C4—4881no—
522C4⋊C4—2441no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4#1175822412yes—
529C22⋊C4—24no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C42—21no—
533C42—21no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C42—1221no—
538C42—2442no—
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#1451975414yes—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—48no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—612no—
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—1222no—
584C2×C4—4884no—
585C2×C4—4884no—
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×C4C2×D4×D6111yescharacteristic
599C2×C4C22×Dic12211yescharacteristic
600C2×C4C2×#1175822411yescharacteristic
601C2×C4C2×#1175822411yescharacteristic
602C2×C4#1451982311yescharacteristic
603C2×C4#1451982411yescharacteristic
604C8—22no—
605C8—41no—
606C8—1222no—
607C8—2441no—
608C22—24no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1684026811yescharacteristic
616C22#1684039611yescharacteristic
617C22D4×Dic12211yescharacteristic
618C22C2×SD16×D6111yescharacteristic
619C22#1684055811yescharacteristic
620C22#1684056911yescharacteristic
621C22#1684057111yescharacteristic
622C4—22no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—81no—
627C4—1222no—
628C4—1222no—
629C4—4881no—
630C4—4881no—
631C4—4881no—
632C4—2441no—
633C4—2441no—
634C4—2442no—
635C4C2×#1451975411yescharacteristic
636C4#1684035611yescharacteristic
637C4#1684041211yescharacteristic
638C4#1684046911yescharacteristic
639C2—44no—
640C2#2144963411yescharacteristic
641C2#2145582211yescharacteristic
642C2#2145608911yescharacteristic
643C2#2145733911yescharacteristic
644C2#2145751811yescharacteristic
645C2SD16×Dic12211yescharacteristic
646C2#2145876911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1877C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1877D61
hypocenterC61#26376
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. #21449661C2semidirect
5. #21447362C2semidirect
6. #21447297C2semidirect
7. #21447196C2semidirect
8. #21447151C2semidirect
9. #21446546C2semidirect
10. #21446375C2semidirect
12. #21446089C2semidirect
118. C2×SD16×C61Q8semidirect
199. C4⋊Dic61SD16semidirect
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
C2×SD16×Dic612
#214496612
D4×C4⋊Dic612
#214473622
#214472972
#214471962
#214471512
#214465462
#214463752
#214461552
#214460892
#214460802
#214460782
C61×#18772
#2637661