Finite Group Databasebeta

← #29716304

Subgroups — #29716304

Order 15616.

1092 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C211yesmaximal, characteristic, complemented
2C4×#14519802C211yesmaximal, characteristic
3#21450773C211yesmaximal, characteristic, complemented
4#21447161C212yesmaximal, complemented
5#21446972C212yesmaximal, complemented
6#21446809C211yesmaximal, characteristic, complemented
7#21446764C211yesmaximal, characteristic, complemented
8#21446195C212yesmaximal
9#21446045C212yesmaximal, complemented
10#21445839C211yesmaximal, characteristic, complemented
11C61×#3503C211yesmaximal, characteristic, complemented
12C2×#14519802C2211yescharacteristic
13C2×#14519753—22no—
14C2×C4×D244C2211yescharacteristic
15#16840059—22no—
16#16840049—21no—
17#16840039—21no—
18#16839872—21no—
19#16839860—21no—
20C4×#11758201—21no—
21C4×#11758199—21no—
22#16839805C2211yescharacteristic
23#16839805C2212yes—
24#16839786C2211yescharacteristic
25C2×#14519704C2212yes—
26#16839766C2211yescharacteristic
27#16839766C2212yes—
28#16839763C2211yescharacteristic
29C4×#11758194C2211yescharacteristic
30#16839697—22no—
31#16839663C2212yes—
32#16839631C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839531C2212yes—
35#16839531C2212yes—
36#16839521C2212yes—
37#16839521C2212yes—
38#16839521C2212yes—
39Q82×C61—21no—
40C61×#557—21no—
41C61×#554—21no—
42C61×#554—22no—
43D4×Q8×C61—22no—
44D4×Q8×C61—21no—
45C61×#544—22no—
46C61×#544C2212yes—
47C61×#533—21no—
48C61×#533C2211yescharacteristic
49C2×C4⋊Q8×C61—21no—
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#520C2212yes—
52C4×#41×C61C2211yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C4×D61—21no—
55#14519802—22no—
56#14519802Q814yescomplemented
57C2×#11758201—21no—
58C2×#11758199—21no—
59C2×#11758194C2311yescharacteristic
60C2×#11758194C2312yes—
61C2×#11758194C2311yescharacteristic
62#14519753—48no—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—24no—
70C2×#11758198—21no—
71C2×#11758198—24no—
72C2×C4×Dic61C2312yes—
73#14519710—22no—
74#14519710—21no—
75#14519708—22no—
76#14519708—21no—
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79#14519704—24no—
80#14519701—21no—
81#14519698—21no—
82#14519697—22no—
83#14519697—22no—
84#14519697—21no—
85#14519697—21no—
86#14519688—21no—
87C4×#92418—21no—
88#14519684—21no—
89C2×#41×C61C2311yescharacteristic
90C22×Q8×C61—22no—
91C22×Q8×C61D412yes—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61—24no—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61—21no—
101C4⋊Q8×C61D411yescharacteristic
102C4⋊Q8×C61D411yescharacteristic
103#125×C61—24no—
104#125×C61—22no—
105#125×C61—22no—
106#125×C61D412yes—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—24no—
110C22⋊Q8×C61—24no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—21no—
113C22⋊Q8×C61—24no—
114C4×Q8×C61—22no—
115C4×Q8×C61—22no—
116C4×Q8×C61—22no—
117C4×Q8×C61—21no—
118C4×Q8×C61—21no—
119C4×Q8×C61D411yescharacteristic
120C4×Q8×C61D411yescharacteristic
121C4×D4×C61—22no—
122C4×D4×C61—21no—
123C4×D4×C61—24no—
124C4×D4×C61—21no—
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D411yescharacteristic
127#117×C61—22no—
128#117×C61—21no—
129#117×C61C2312yes—
130#117×C61C2311yescharacteristic
131C2×C4⋊C4×C61—22no—
132C2×C4⋊C4×C61—22no—
133C2×C4⋊C4×C61D412yes—
134C2×C4⋊C4×C61C2312yes—
135C2×C42×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C23×D61—21no—
138#41×C61—21no—
139#41×C61—21no—
140#41×C61C2×Q812yes—
141#41×C61#4112yes—
142C2×Q8×C61—24no—
143C2×Q8×C61—21no—
144C2×Q8×C61—22no—
145C2×Q8×C61—21no—
146C2×Q8×C61—22no—
147C2×Q8×C61—44no—
148C2×Q8×C61—84no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—24no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D414yes—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C22×Dic61—22no—
161C22×Dic61—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244C2×Q812yescomplemented
168C2×C4×D61—44no—
169C2×C4×D61—41no—
170C2×C4×D61—41no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C2×D412yes—
186C4⋊C4×C61C2×D412yes—
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—24no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C42×C61—24no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—22no—
199C42×C61C2×D411yescharacteristic
200C42×C61C2×D411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2×D411yescharacteristic
203#11758201—24no—
204#11758199—24no—
205#11758198—48no—
206#11758198—41no—
207#11758198—41no—
208C4⋊Dic61—24no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211#11758196—48no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214#11758194—21no—
215#11758194—22no—
216#11758194—21no—
217#11758194—21no—
218#11758194—22no—
219#11758194C2×Q812yescomplemented
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—22no—
223C23×C61—21no—
224C23×C61C22×D411yescharacteristic
225C22×D61—22no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—41no—
229C22×D61—41no—
230Q8×C61—21no—
231Q8×C61—42no—
232Q8×C61—44no—
233Q8×C61—42no—
234Q8×C61—82no—
235Q8×C61—84no—
236Q8×C61C22⋊Q812yescomplemented
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—21no—
240D4×C61C22⋊Q812yescomplemented
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—24no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—44no—
254C2×C4×C61—84no—
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61#12011yescharacteristic
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×Q811yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×C4×C612−1+411yescharacteristic
265C2×Dic61—24no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270D244—22no—
271D244—22no—
272D244—41no—
273D244—41no—
274C4×D61—82no—
275C4×D61—82no—
276#92418—22no—
277#92418—41no—
278#92418—42no—
279#24320—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—22no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61C2×#12011yescharacteristic
286C22×C61#54811yescharacteristic
287C22×C61D4×Q811yescharacteristic
288C2×D61—44no—
289C2×D61—42no—
290C2×D61—82no—
291C2×D61—81no—
292C2×D61—81no—
293C4×C61—22no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—41no—
298C4×C61—44no—
299C4×C61—42no—
300C4×C61—41no—
301C4×C61—42no—
302C4×C61#45312yes—
303C4×C61C2×C22⋊Q811yescharacteristic
304C4×C61#53611yescharacteristic
305C4×C61#54811yescharacteristic
306C4×C61D4×Q811yescharacteristic
307Dic61—44no—
308Dic61—81no—
309#3503D6111yescharacteristic, complemented
310#3077—611no—
311C4×AGL(1,ℤ/8)—611no—
312#2542—611no—
313#2060—612no—
314#2027—612no—
315#1997—611no—
316#1986—611no—
317#1883—612no—
318#1859—612no—
319#1806—611no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—42no—
323C2×C61#240911yescharacteristic
324C2×C61C2×#45311yescharacteristic
325C2×C61#302011yescharacteristic
326D61—82no—
327C2×AGL(1,ℤ/8)—611no—
328Q82—21no—
329#557—21no—
330#554—21no—
331#554—22no—
332D4×Q8—22no—
333D4×Q8—21no—
334#544—22no—
335#544C2×D6112yes—
336#533—21no—
337#533C2×D6111yescharacteristic
338C2×C4⋊Q8—21no—
339C2×C4⋊Q8C2×D6111yescharacteristic
340C2×C22⋊Q8—1222no—
341#520C2×D6112yes—
342C4×#41C2×D6111yescharacteristic
343C2×C4×D4—611no—
344#478—1222no—
345#476—1221no—
346#474—1221no—
347#442—1221no—
348#440—1221no—
349C4×SD16—1221no—
350C4×D8—1221no—
351#428—612no—
352#428—611no—
353#423—611no—
354C2×#104—612no—
355#418—612no—
356#418—611no—
357#417—611no—
358C4×M16—611no—
359#399—1222no—
360#390—612no—
361#384—612no—
362C4×C4⋊C4—611no—
363#343—612no—
364#343—612no—
365#339—612no—
366#339—612no—
367#339—612no—
368C61#2432011yescharacteristic, complemented, Sylow
369C2×#41C22×D6111yescharacteristic
370C22×Q8—22no—
371C22×Q8C22⋊D6112yes—
372C22×D4—611no—
373C23×C4—1221no—
374AGL(1,ℤ/8)—614no—
375AGL(1,ℤ/8)—1222no—
376C2×SD16—1221no—
377C2×D8—1221no—
378C2×M16—611no—
379C2×M16—611no—
380C2×M16—612no—
381C4⋊Q8—22no—
382C4⋊Q8—22no—
383C4⋊Q8—24no—
384C4⋊Q8—22no—
385C4⋊Q8—22no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8C22⋊D6111yescharacteristic
391C4⋊Q8C22⋊D6111yescharacteristic
392#125—24no—
393#125—22no—
394#125—22no—
395#125C22⋊D6112yes—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—24no—
399C22⋊Q8—24no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—24no—
403C22⋊Q8—2448no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8C22⋊D6111yescharacteristic
410C4×Q8C22⋊D6111yescharacteristic
411C4×D4—22no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—24no—
415C4×D4—1221no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4C22⋊D6111yescharacteristic
420C4×D4C22⋊D6111yescharacteristic
421#117—22no—
422#117—21no—
423#117C22×D6111yescharacteristic
424#117C22×D6112yes—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1224no—
430C2×C4⋊C4C22⋊D6112yes—
431C2×C4⋊C4C22×D6112yes—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1224no—
434C2×C42—21no—
435C2×C42—612no—
436C2×C42C22×D6111yescharacteristic
437#107—1221no—
438#107—1222no—
439#106—1221no—
440#106—1222no—
441C4⋊C8—1221no—
442C4⋊C8—1221no—
443#104—1224no—
444#103—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—1222no—
449#102—1222no—
450#97—1221no—
451C4×C8—1221no—
452#95—1221no—
453C24—1221no—
454#41—21no—
455#41—21no—
456#41Q8×D6112yes—
457#41#1175822612yes—
458C2×Q8—24no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—22no—
463C2×Q8—44no—
464C2×Q8—84no—
465C2×Q8C2×C22⋊D6111yescharacteristic
466C2×D4—21no—
467C2×D4—612no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4C2×C22⋊D6111yescharacteristic
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—1222no—
479C22×C4—1221no—
480C22×C4—2444no—
481C22×C4—2441no—
482C22×C4—2441no—
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6114yes—
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C23×D6111yescharacteristic
487SD16—1224no—
488D8—1224no—
489M16—612no—
490M16—1221no—
491M16—1222no—
492M16—1221no—
493M16—1221no—
494M16—1222no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1222no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—1224no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2448no—
516C4⋊C4C2×C22⋊D6112yes—
517C4⋊C4C2×C22⋊D6111yescharacteristic
518C4⋊C4C2×C22⋊D6112yes—
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C22⋊C4—24no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—2448no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C42—24no—
529C42—21no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—1222no—
534C42—1222no—
535C42C2×C22⋊D6111yescharacteristic
536C42C2×C22⋊D6111yescharacteristic
537C42C2×C22⋊D6111yescharacteristic
538C42C2×C22⋊D6111yescharacteristic
539C23—21no—
540C23—1222no—
541C23—2442no—
542C23—2442no—
543C23—2441no—
544C23—2441no—
545C23C22×C22⋊D6111yescharacteristic
546Q8—21no—
547Q8—42no—
548Q8—44no—
549Q8—42no—
550Q8—82no—
551Q8—84no—
552Q8#1451975312yescomplemented
553D4—22no—
554D4—22no—
555D4—21no—
556D4—1222no—
557D4—1222no—
558D4—2441no—
559D4—2441no—
560D4#1451975312yescomplemented
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—84no—
575C2×C4—1224no—
576C2×C4—4882no—
577C2×C4—4882no—
578C2×C4—2442no—
579C2×C4—2441no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4#1451974611yescharacteristic
583C2×C4#1451974611yescharacteristic
584C2×C4#1451974611yescharacteristic
585C2×C4#1451974611yescharacteristic
586C2×C4C22×C22⋊D6111yescharacteristic
587C2×C4C22×C22⋊D6111yescharacteristic
588C2×C4C2×Q8×D6111yescharacteristic
589C2×C4C2×#1175822611yescharacteristic
590C2×C4#1451982511yescharacteristic
591C2×C4#1451982511yescharacteristic
592C8—1222no—
593C8—2441no—
594C8—2442no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—4882no—
601C22—4881no—
602C22—4881no—
603C22—2444no—
604C22—2442no—
605C22C2×#1451974611yescharacteristic
606C22#1684042311yescharacteristic
607C22Q8×C22⋊D6111yescharacteristic
608C4—21no—
609C4—21no—
610C4—21no—
611C4—22no—
612C4—41no—
613C4—44no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—4881no—
618C4—2444no—
619C4#1683993012yes—
620C4C2×#1451975311yescharacteristic
621C4#1684033611yescharacteristic
622C4#1684042311yescharacteristic
623C4Q8×C22⋊D6111yescharacteristic
624C2—21no—
625C2—21no—
626C2—42no—
627C2—4882no—
628C2#2144917811yescharacteristic
629C2C2×#1683993011yescharacteristic
630C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3503C2
socleC22×C61C2×#120
prime radicalC2×#14519802C22
hypercenter#3503D61
hypocenterC61#24320
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3503

Complements

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

NH ≅ G/Ntype
1. #21455612C2semidirect
3. #21450773C2semidirect
4. #21447161C2semidirect
5. #21446972C2semidirect
6. #21446809C2semidirect
7. #21446764C2semidirect
9. #21446045C2semidirect
10. #21445839C2semidirect
11. C61×#3503C2semidirect
56. #14519802Q8semidirect
167. C2×D244C2×Q8semidirect
219. #11758194C2×Q8semidirect
236. Q8×C61C22⋊Q8semidirect
240. D4×C61C22⋊Q8semidirect
309. #3503D61semidirect
368. C61#24320semidirect
552. Q8#14519753semidirect
560. D4#14519753semidirect

Maximal subgroups

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

subgroupindex
#214556122
C4×#145198022
#214507732
#214471612
#214469722
#214468092
#214467642
#214461952
#214460452
#214458392
C61×#35032
#2432061