Finite Group Databasebeta

← #30610477

Subgroups — #30610477

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462738C211yesmaximal, characteristic, complemented
2#21462628C211yesmaximal, characteristic, complemented
3#21461158C212yesmaximal, complemented
4#21461154C212yesmaximal, complemented
5#21461150C212yesmaximal, complemented
6#21461144C212yesmaximal, complemented
7#21461118C211yesmaximal, characteristic, complemented
8#21461114C211yesmaximal, characteristic, complemented
9#21461110C211yesmaximal, characteristic, complemented
10C2×D4⋊#92420C211yesmaximal, characteristic, complemented
11H(5,𝔽2)×Dic61C211yesmaximal, characteristic
12#16840813—21no—
13#16840813—21no—
14#16840797—21no—
15#16840797—21no—
16#16840793—22no—
17#16840793—22no—
18#16840783C2212yescomplemented
19#16840779C2212yescomplemented
20#16840778C2211yescharacteristic, complemented
21#16840778C2211yescharacteristic, complemented
22#16840765C2212yescomplemented
23C2×C4⋊#92420C2211yescharacteristic, complemented
24C4×#11758217C2212yescomplemented
25C2×C4×#92420C2211yescharacteristic, complemented
26Q8⋊#92420C2211yescharacteristic, complemented
27Q8⋊#92420C2211yescharacteristic, complemented
28Q8⋊#92420C2212yescomplemented
29D4⋊#92420C2212yescomplemented
30D4⋊#92420C2212yescomplemented
31D4⋊#92420C2211yescharacteristic, complemented
32D4⋊#92420C2211yescharacteristic, complemented
33D4⋊#92420C2212yescomplemented
34D4⋊#92420C2212yescomplemented
35D4⋊#92420C2212yescomplemented
36#16840245—21no—
37#16840245C412yes—
38#16840245C411yescharacteristic
39#16840245C411yescharacteristic
40#16840237—22no—
41#16840237—21no—
42#16840237C412yes—
43#16840237C411yescharacteristic
44#41×Dic61—21no—
45#41×Dic61C2211yescharacteristic
46#41×Dic61C2212yes—
47#41×Dic61C2211yescharacteristic
48C2×D4×Dic61—22no—
49C2×D4×Dic61—21no—
50C2×D4×Dic61C2211yescharacteristic
51C2×D4×Dic61C2212yes—
52C2×H(5,𝔽2)×C61C411yescharacteristic
53C2×#11758217—22no—
54C2×#11758217—22no—
55C22×#92420—21no—
56C22×#92420—21no—
57#14519844—22no—
58#14519844—22no—
59C4⋊#92420—21no—
60C4⋊#92420—21no—
61C4⋊#92420C2312yes—
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2311yescharacteristic
64C22⋊#92420—21no—
65C22⋊#92420—21no—
66C22⋊#92420—22no—
67C22⋊#92420—22no—
68#14519832—22no—
69#14519832—22no—
70C4×#92420—21no—
71C4×#92420—21no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C23×Dic61—21no—
76C23×Dic61—22no—
77Q8×Dic61—21no—
78Q8×Dic61C2×C411yescharacteristic
79Q8×Dic61C2×C412yes—
80Q8×Dic61D411yescharacteristic
81Q8×Dic61D411yescharacteristic
82Q8×Dic61C2311yescharacteristic
83D4×Dic61—22no—
84D4×Dic61—21no—
85D4×Dic61—21no—
86D4×Dic61—22no—
87D4×Dic61—21no—
88D4×Dic61—22no—
89D4×Dic61—22no—
90D4×Dic61—22no—
91D4×Dic61—22no—
92D4×Dic61—22no—
93D4×Dic61—21no—
94D4×Dic61—22no—
95D4×Dic61—21no—
96D4×Dic61—21no—
97D4×Dic61—22no—
98D4×Dic61—21no—
99D4×Dic61—22no—
100D4×Dic61C2×C412yes—
101D4×Dic61C2×C412yes—
102D4×Dic61C2×C412yes—
103D4×Dic61C2×C411yescharacteristic
104D4×Dic61C2×C412yes—
105D4×Dic61D412yes—
106D4×Dic61D411yescharacteristic
107D4×Dic61D412yes—
108D4×Dic61D411yescharacteristic
109D4×Dic61C2312yes—
110D4×Dic61C2311yescharacteristic
111#14519736—22no—
112#14519736—21no—
113#14519736—21no—
114#14519736—22no—
115#14519736C2×C412yes—
116#14519736C2×C411yescharacteristic
117#14519736C2×C412yes—
118#14519736C2×C411yescharacteristic
119C2×C4⋊Dic61—22no—
120C2×C4⋊Dic61—21no—
121C2×C4⋊Dic61C2×C412yes—
122C2×C4⋊Dic61C2×C411yescharacteristic
123C2×C22⋊Dic61—21no—
124C2×C22⋊Dic61—22no—
125C2×C22⋊Dic61—22no—
126C2×C22⋊Dic61—22no—
127C2×C22⋊Dic61—22no—
128C2×C4×Dic61—21no—
129C2×C4×Dic61—22no—
130C2×C4×Dic61C2311yescharacteristic
131C2×C4×Dic61C2312yes—
132H(5,𝔽2)×C61—24no—
133H(5,𝔽2)×C61—24no—
134C2×#41×C61—21no—
135C2×#41×C61C2×C412yes—
136C2×#41×C61C2×C411yescharacteristic
137C2×#41×C61C2×C411yescharacteristic
138C22×D4×C61—22no—
139C22×D4×C61—21no—
140C22×D4×C61C2×C412yes—
141C22×D4×C61C2×C411yescharacteristic
142C24×C61—22no—
143C24×C61—21no—
144#41×C61—28no—
145#41×C61—24no—
146#41×C61—24no—
147#41×C61—22no—
148#41×C61—22no—
149#41×C61—22no—
150#41×C61M1614yescomplemented
151C2×Q8×C61C22⋊C411yescharacteristic
152C2×Q8×C61C22×C411yescharacteristic
153C2×D4×C61—216no—
154C2×D4×C61—28no—
155C2×D4×C61—22no—
156C2×D4×C61—21no—
157C2×D4×C61—24no—
158C2×D4×C61—24no—
159C2×D4×C61—22no—
160C2×D4×C61—21no—
161C2×D4×C61—22no—
162C2×D4×C61C22⋊C412yes—
163C2×D4×C61C22⋊C411yescharacteristic
164C2×D4×C61M1614yescomplemented
165C2×D4×C61C22×C411yescharacteristic
166C2×D4×C61C22×C412yes—
167C22×C4×C61—22no—
168C22×C4×C61—21no—
169C22×C4×C61C22×C412yes—
170C22×C4×C61C22×C411yescharacteristic
171C22×Dic61—22no—
172C22×Dic61—21no—
173C22×Dic61—22no—
174C22×Dic61—21no—
175C22×Dic61—22no—
176C22×Dic61—22no—
177C22×Dic61—22no—
178C22×Dic61—21no—
179C22×Dic61—22no—
180C22×Dic61—81no—
181C22×Dic61—82no—
182C22×Dic61—81no—
183C22×Dic61—82no—
184C22×Dic61C2×D411yescharacteristic
185C22×Dic61C2×D412yes—
186C22×Dic61C2×D411yescharacteristic
187C22×Dic61C2×D412yes—
188#11758217—44no—
189#11758217—82no—
190C2×#92420—21no—
191C2×#92420—21no—
192C2×#92420—21no—
193C2×#92420—21no—
194C2×#92420—22no—
195C2×#92420—22no—
196C2×#92420—42no—
197C2×#92420—81no—
198C22⋊Dic61—22no—
199C22⋊Dic61—21no—
200C22⋊Dic61—21no—
201C22⋊Dic61—22no—
202C22⋊Dic61—22no—
203C22⋊Dic61—22no—
204C22⋊Dic61—22no—
205C22⋊Dic61—22no—
206C22⋊Dic61—21no—
207C22⋊Dic61C22⋊C412yes—
208C22⋊Dic61C22⋊C412yes—
209C22⋊Dic61C22⋊C412yes—
210C4⋊Dic61—22no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—22no—
214C4⋊Dic61—21no—
215C4⋊Dic61C22⋊C412yes—
216C4⋊Dic61C22⋊C411yescharacteristic
217C4⋊Dic61C22⋊C412yes—
218C4⋊Dic61C22⋊C411yescharacteristic
219C4⋊Dic61C22×C411yescharacteristic
220C4⋊Dic61C22×C412yes—
221C4⋊Dic61C22×C411yescharacteristic
222C4×Dic61—22no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—22no—
226C4×Dic61—21no—
227C4×Dic61C22×C411yescharacteristic
228C4×Dic61C22×C412yes—
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231C4×Dic61C2×D412yes—
232C4×Dic61C2×D412yes—
233C4×Dic61C2411yescharacteristic
234C23×C61—28no—
235C23×C61—22no—
236C23×C61—21no—
237C23×C61—24no—
238C23×C61—81no—
239C23×C61—82no—
240C23×C61—81no—
241C23×C61—82no—
242C23×C61C2×C22⋊C412yes—
243C23×C61C2×C22⋊C411yescharacteristic
244Q8×C61—21no—
245Q8×C61—22no—
246Q8×C61C2×M1612yes—
247D4×C61—28no—
248D4×C61—28no—
249D4×C61—24no—
250D4×C61—24no—
251D4×C61—22no—
252D4×C61—21no—
253D4×C61—22no—
254D4×C61—24no—
255D4×C61C2×M1612yes—
256D4×C61C2×M1614yes—
257C2×C4×C61—28no—
258C2×C4×C61—21no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—24no—
262C2×C4×C61C2×C22⋊C412yes—
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×M1612yes—
265C2×C4×C61C2×M1612yes—
266C2×C4×C61C23×C411yescharacteristic
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—41no—
278C2×Dic61—42no—
279C2×Dic61—41no—
280C2×Dic61—81no—
281C2×Dic61—82no—
282C2×Dic61—82no—
283C2×Dic61—82no—
284C2×Dic61—82no—
285C2×Dic61—81no—
286C2×Dic61—82no—
287C2×Dic61C2×C22⋊C411yescharacteristic
288C2×Dic61C2×C22⋊C412yes—
289C2×Dic61C2×C22⋊C411yescharacteristic
290C2×Dic61C2×C22⋊C412yes—
291C2×Dic61C22×D411yescharacteristic
292C2×Dic61C22×D411yescharacteristic
293#92420—22no—
294#92420—41no—
295#92420—42no—
296#92420—42no—
297#92420—41no—
298#92420—41no—
299#17453—611nomaximal, Sylow
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—22no—
303C22×C61—24no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—42no—
308C22×C61—41no—
309C22×C61—42no—
310C22×C61—41no—
311C22×C61—84no—
312C22×C61—88no—
313C22×C61#40712yes—
314C22×C61C22×C22⋊C411yescharacteristic
315C4×C61—22no—
316C4×C61—22no—
317C4×C61#40712yes—
318C4×C61C22×M1611yescharacteristic
319C4×C61C22×M1611yescharacteristic
320Dic61—21no—
321Dic61—22no—
322Dic61—21no—
323Dic61—21no—
324Dic61—22no—
325Dic61—42no—
326Dic61—41no—
327Dic61—41no—
328Dic61#57611yescharacteristic
329Dic61#57711yescharacteristic
330C4×H(5,𝔽2)—611no—
331#2989—611no—
332#2965—611no—
333#1490—612no—
334#1489—612no—
335#1488—612no—
336#1486—612no—
337#1478—611no—
338#1477—611no—
339#1476—611no—
340C2×D4⋊C8—611no—
341C2×C61—22no—
342C2×C61—42no—
343C2×C61—44no—
344C2×C61—42no—
345C2×C61C2×#40711yescharacteristic
346C2×C61#289711yescharacteristic
347C2×C61#290111yescharacteristic
348C2×H(5,𝔽2)#692311yescharacteristic
349#520—611no—
350#520—612no—
351#520—611no—
352#520—1221no—
353#518—612no—
354#518—611no—
355#518—1222no—
356#518—1221no—
357C4×#41—611no—
358C4×#41—611no—
359C4×#41—612no—
360C4×#41—1221no—
361C2×C4×D4—611no—
362C2×C4×D4—612no—
363C2×C4×D4—1222no—
364C2×C4×D4—1221no—
365#446—1221no—
366#446—1221no—
367#436—1221no—
368#436—1221no—
369#435—1222no—
370#435—1222no—
371#432—612no—
372#431—611no—
373#431—612no—
374#431—611no—
375#423—612no—
376C2×C4⋊C8—611no—
377C4×M16—612no—
378C2×C4×C8—611no—
379Q8⋊C8—611no—
380Q8⋊C8—611no—
381Q8⋊C8—612no—
382D4⋊C8—612no—
383D4⋊C8—612no—
384D4⋊C8—611no—
385D4⋊C8—611no—
386D4⋊C8—612no—
387D4⋊C8—612no—
388D4⋊C8—612no—
389C61#1745311yescharacteristic, complemented, Sylow
390H(5,𝔽2)—24no—
391H(5,𝔽2)—24no—
392C2×#41—21no—
393C2×#41C2×#692312yes—
394C2×#41C2×#692311yescharacteristic
395C2×#41C2×#692311yescharacteristic
396C22×D4—22no—
397C22×D4—21no—
398C22×D4C2×#692312yes—
399C22×D4C2×#692311yescharacteristic
400C23×C4—1221no—
401C23×C4—1222no—
402C2×M16—1222no—
403C2×M16—1222no—
404C22×C8—1221no—
405C22×C8—1221no—
406C4×Q8—611no—
407C4×Q8—611no—
408C4×Q8—611no—
409C4×Q8—611no—
410C4×Q8—612no—
411C4×Q8—1221no—
412C4×D4—612no—
413C4×D4—611no—
414C4×D4—612no—
415C4×D4—611no—
416C4×D4—612no—
417C4×D4—611no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—612no—
421C4×D4—611no—
422C4×D4—612no—
423C4×D4—1221no—
424C4×D4—1222no—
425C4×D4—1221no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1221no—
432C4×D4—1222no—
433C4×D4—1221no—
434C4×D4—1221no—
435C4×D4—1222no—
436C4×D4—1221no—
437C4×D4—1222no—
438C4×D4—1222no—
439C4×D4—1221no—
440#117—612no—
441#117—611no—
442#117—612no—
443#117—611no—
444#117—1221no—
445#117—1222no—
446#117—1222no—
447#117—1221no—
448C2×C4⋊C4—612no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4—1221no—
452C2×C22⋊C4—1222no—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1222no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1222no—
457C2×C42—611no—
458C2×C42—612no—
459C2×C42—1221no—
460C2×C42—1222no—
461C4⋊C8—612no—
462C4⋊C8—611no—
463C4⋊C8—611no—
464C4⋊C8—1221no—
465C4⋊C8—1221no—
466C4⋊C8—1222no—
467C4⋊C8—1222no—
468C22⋊C8—1221no—
469C22⋊C8—1221no—
470C22⋊C8—1222no—
471C22⋊C8—1222no—
472#97—1222no—
473#97—1222no—
474C4×C8—611no—
475C4×C8—611no—
476C4×C8—612no—
477C4×C8—1221no—
478C4×C8—1221no—
479C24—22no—
480C24—21no—
481#41—28no—
482#41—24no—
483#41—24no—
484#41—22no—
485#41—22no—
486#41—22no—
487#41#1175821714yescomplemented
488C2×Q8C22⋊#692311yescharacteristic
489C2×Q8C22×#692311yescharacteristic
490C2×D4—216no—
491C2×D4—28no—
492C2×D4—22no—
493C2×D4—21no—
494C2×D4—24no—
495C2×D4—24no—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—22no—
499C2×D4#1175821714yescomplemented
500C2×D4C22⋊#692312yes—
501C2×D4C22⋊#692311yescharacteristic
502C2×D4C22×#692311yescharacteristic
503C2×D4C22×#692312yes—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—611no—
507C22×C4—612no—
508C22×C4—611no—
509C22×C4—612no—
510C22×C4—1221no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1221no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1221no—
519C22×C4—4881no—
520C22×C4—4882no—
521C22×C4—4881no—
522C22×C4—4882no—
523C22×C4C22×#692312yes—
524C22×C4C22×#692311yescharacteristic
525M16—4882no—
526M16—2444no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1222no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1222no—
533C2×C8—4881no—
534C2×C8—2442no—
535C4⋊C4—612no—
536C4⋊C4—611no—
537C4⋊C4—612no—
538C4⋊C4—611no—
539C4⋊C4—611no—
540C4⋊C4—612no—
541C4⋊C4—611no—
542C4⋊C4—1221no—
543C4⋊C4—1222no—
544C4⋊C4—1221no—
545C4⋊C4—1222no—
546C4⋊C4—1221no—
547C22⋊C4—612no—
548C22⋊C4—612no—
549C22⋊C4—612no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1221no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1221no—
557C22⋊C4—1222no—
558C22⋊C4—1221no—
559C42—611no—
560C42—611no—
561C42—612no—
562C42—612no—
563C42—611no—
564C42—611no—
565C42—612no—
566C42—1221no—
567C42—1221no—
568C42—1222no—
569C42—1222no—
570C42—1221no—
571C23—28no—
572C23—22no—
573C23—21no—
574C23—24no—
575C23—81no—
576C23—82no—
577C23—81no—
578C23—82no—
579C23C2×C22⋊#692312yes—
580C23C2×C22⋊#692311yescharacteristic
581Q8—22no—
582Q8—21no—
583Q8C2×#1175821712yes—
584D4—28no—
585D4—28no—
586D4—24no—
587D4—24no—
588D4—24no—
589D4—22no—
590D4—22no—
591D4—21no—
592D4C2×#1175821712yes—
593D4C2×#1175821714yes—
594C2×C4—28no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—611no—
600C2×C4—611no—
601C2×C4—612no—
602C2×C4—611no—
603C2×C4—612no—
604C2×C4—611no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—1222no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—1221no—
612C2×C4—4881no—
613C2×C4—4882no—
614C2×C4—4882no—
615C2×C4—4882no—
616C2×C4—4882no—
617C2×C4—4881no—
618C2×C4—4882no—
619C2×C4—2442no—
620C2×C4—2441no—
621C2×C4—2442no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2442no—
625C2×C4C2×C22⋊#692312yes—
626C2×C4C2×C22⋊#692311yescharacteristic
627C2×C4C2×#1175821712yes—
628C2×C4C2×#1175821712yes—
629C2×C4C23×#692311yescharacteristic
630C8—1222no—
631C8—2441no—
632C8—2442no—
633C8—2441no—
634C8—2441no—
635C8—2442no—
636C22—21no—
637C22—21no—
638C22—22no—
639C22—24no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—84no—
648C22—88no—
649C22#1684073212yes—
650C22C22×C22⋊#692311yescharacteristic
651C4—22no—
652C4—22no—
653C4—611no—
654C4—611no—
655C4—1221no—
656C4—1222no—
657C4—1221no—
658C4—1221no—
659C4—1222no—
660C4—2442no—
661C4—2441no—
662C4—2441no—
663C4#1684073212yes—
664C4C22×#1175821711yescharacteristic
665C4C22×#1175821711yescharacteristic
666C2—22no—
667C2—42no—
668C2—44no—
669C2—42no—
670C2C2×#1684073211yescharacteristic
671C2#2146244011yescharacteristic
672C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
Fitting, prime radicalC2×H(5,𝔽2)×C61C4
socleC22×C61C22×C22⋊C4
cosocleC4×Dic61C24
hypercenterC2×H(5,𝔽2)#6923
hypocenterC61#17453
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #21462738C2semidirect
2. #21462628C2semidirect
3. #21461158C2semidirect
4. #21461154C2semidirect
5. #21461150C2semidirect
6. #21461144C2semidirect
7. #21461118C2semidirect
8. #21461114C2semidirect
9. #21461110C2semidirect
10. C2×D4⋊#92420C2semidirect
18. #16840783C22semidirect
19. #16840779C22semidirect
20. #16840778C22semidirect
21. #16840778C22semidirect
22. #16840765C22semidirect
23. C2×C4⋊#92420C22semidirect
24. C4×#11758217C22semidirect
25. C2×C4×#92420C22semidirect
26. Q8⋊#92420C22semidirect
27. Q8⋊#92420C22semidirect
28. Q8⋊#92420C22semidirect
29. D4⋊#92420C22semidirect
30. D4⋊#92420C22semidirect
31. D4⋊#92420C22semidirect
32. D4⋊#92420C22semidirect
33. D4⋊#92420C22semidirect
34. D4⋊#92420C22semidirect
35. D4⋊#92420C22semidirect
150. #41×C61M16semidirect
164. C2×D4×C61M16semidirect
389. C61#17453semidirect
487. #41#11758217semidirect
499. C2×D4#11758217semidirect

Maximal subgroups

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

subgroupindex
#214627382
#214626282
#214611582
#214611542
#214611502
#214611442
#214611182
#214611142
#214611102
C2×D4⋊#924202
H(5,𝔽2)×Dic612
#1745361