Finite Group Databasebeta

← #30610404

Subgroups — #30610404

Order 15616.

1290 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#21462736C211yesmaximal, characteristic, complemented
2#21462623C211yesmaximal, characteristic, complemented
3#21461119C212yesmaximal, complemented
4#21461118C211yesmaximal, characteristic, complemented
5#21461116C211yesmaximal, characteristic, complemented
6#21461115C212yesmaximal, complemented
7#21461112C211yesmaximal, characteristic, complemented
8#21461111C212yesmaximal, complemented
9D61⋊(D4⋊C8)C212yesmaximal, complemented
10C2×D4⋊#92420C211yesmaximal, characteristic, complemented
11#21459023C211yesmaximal, characteristic
12#16840811—21no—
13#16840811—21no—
14#16840790—21no—
15#16840790—21no—
16#16840789—22no—
17#16840789—22no—
18#16840781C2212yescomplemented
19#16840780C2212yescomplemented
20#16840778C2211yescharacteristic, complemented
21#16840778C2211yescharacteristic, complemented
22D61⋊(C4⋊C8)C2212yescomplemented
23C2×C4⋊#92420C2211yescharacteristic, complemented
24C4×D61⋊C8C2212yescomplemented
25C2×C4×#92420C2211yescharacteristic, complemented
26#16840612C2211yescharacteristic, complemented
27#16840612C2212yescomplemented
28#16840612C2211yescharacteristic, complemented
29#16840610C2211yescharacteristic, complemented
30#16840610C2212yescomplemented
31#16840610C2211yescharacteristic, complemented
32D4⋊#92420C2212yescomplemented
33D4⋊#92420C2212yescomplemented
34D4⋊#92420C2212yescomplemented
35D4⋊#92420C2212yescomplemented
36C2×#14519822C411yescharacteristic
37#16840242C411yescharacteristic
38#16840242C411yescharacteristic
39#16840241—21no—
40#16840241C412yes—
41#16840235—22no—
42#16840234—21no—
43#16840234C412yes—
44#16840230C2211yescharacteristic
45#16840230C2211yescharacteristic
46C4×#11758224—21no—
47C4×#11758224C2212yes—
48C2×D4×Dic61C2211yescharacteristic
49C2×#14519737—22no—
50C4×D4×D61—21no—
51C4×D4×D61C2212yes—
52C61×#518C411yescharacteristic
53C2×D61⋊C8—22no—
54C2×D61⋊C8—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
64#14519834—22no—
65#14519834—22no—
66C22⋊#92420—21no—
67C22⋊#92420—21no—
68C4×#92420—21no—
69C4×#92420—22no—
70C4×#92420—21no—
71C4×#92420—22no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75#14519822—24no—
76#14519822C818yescomplemented
77C2×#11758222C2×C411yescharacteristic
78C2×#11758222C2×C411yescharacteristic
79C2×#11758224—21no—
80C2×#11758224C2×C412yes—
81C22×C22⋊D61—22no—
82C2×D4×D61—21no—
83C2×D4×D61C2×C412yes—
84C23×Dic61—21no—
85C22×C4×D61—22no—
86C4×Dic122—21no—
87C4×Dic122C2×C412yes—
88#14519742C2×C411yescharacteristic
89#14519742D411yescharacteristic
90#14519742D411yescharacteristic
91#14519742C2311yescharacteristic
92C4×D244—21no—
93C4×D244C2×C412yes—
94D4×Dic61—21no—
95D4×Dic61—21no—
96D4×Dic61C2×C412yes—
97D4×Dic61C2312yes—
98#14519739C2×C411yescharacteristic
99#14519739D411yescharacteristic
100#14519739D411yescharacteristic
101#14519739C2311yescharacteristic
102C4×C22⋊D61—22no—
103C4×C22⋊D61—22no—
104C4×C22⋊D61—22no—
105C4×C22⋊D61—22no—
106#14519737—22no—
107#14519737—21no—
108#14519737—21no—
109#14519737—22no—
110#14519737—22no—
111#14519737—22no—
112#14519737—22no—
113#14519737—22no—
114#14519737D412yes—
115#14519737D412yes—
116#14519735—21no—
117#14519735C2×C412yes—
118#14519734—21no—
119#14519734C2×C412yes—
120#14519733—22no—
121#14519733—22no—
122C2×C4⋊Dic61C2×C411yescharacteristic
123C2×#11758196—22no—
124C4⋊C4×D61—21no—
125C4⋊C4×D61C2×C412yes—
126C2×C22⋊Dic61—21no—
127C2×#11758198—22no—
128C22⋊C4×D61—22no—
129C22⋊C4×D61—22no—
130C2×C4×Dic61—22no—
131C2×C4×Dic61C2311yescharacteristic
132C42×D61—21no—
133C42×D61C2312yes—
134C22×D4×C61C2×C411yescharacteristic
135C4×D4×C61—21no—
136C4×D4×C61—21no—
137C4×D4×C61C2×C412yes—
138C4×D4×C61C2×C412yes—
139#117×C61C2×C411yescharacteristic
140#117×C61C2×C411yescharacteristic
141C2×C22⋊C4×C61—22no—
142C24×C61—21no—
143C23×D61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61C2×C814yescomplemented
147C2×D4×C61C22×C412yes—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C22×C411yescharacteristic
151C2×C22⋊D61—21no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—216no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61C22⋊C412yes—
156C22×Dic61—22no—
157C22×Dic61—21no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—81no—
161C22×Dic61—81no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164#11758224—24no—
165#11758224—28no—
166#11758224C2×C818yescomplemented
167D4×D61—28no—
168D4×D61—24no—
169D4×D61C2×C818yescomplemented
170#11758222—22no—
171#11758222C2×C814yes—
172#11758222C2×C814yescomplemented
173#11758222M1614yes—
174C2×D244C22⋊C411yescharacteristic
175C2×D244C22×C411yescharacteristic
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—22no—
179C2×C4×D61—22no—
180C2×C4×D61—21no—
181C2×C4×D61—82no—
182C2×C4×D61—82no—
183C2×C4×D61C22×C412yes—
184C2×C4×D61C2×D412yes—
185C2×C4×D61C2×D412yes—
186C2×Dic122C22⋊C411yescharacteristic
187C2×Dic122C22×C411yescharacteristic
188C2×#92420—21no—
189C2×#92420—21no—
190C2×#92420—22no—
191C2×#92420—21no—
192C2×#92420—21no—
193C2×#92420—22no—
194C2×#92420—42no—
195C2×#92420—81no—
196D61⋊C8—44no—
197D61⋊C8—82no—
198C4⋊C4×C61C22⋊C411yescharacteristic
199C4⋊C4×C61C22⋊C411yescharacteristic
200C4⋊C4×C61C22×C411yescharacteristic
201C4⋊C4×C61C22×C411yescharacteristic
202C22⋊C4×C61—21no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61C22⋊C412yes—
206C42×C61—21no—
207C42×C61C22×C412yes—
208C22⋊Dic61—22no—
209C22⋊Dic61—22no—
210#11758198—22no—
211#11758198—22no—
212#11758198—21no—
213#11758198—21no—
214#11758198C22⋊C412yes—
215#11758198C22⋊C412yes—
216C4⋊Dic61—21no—
217C4⋊Dic61C22×C412yes—
218#11758196—22no—
219#11758196—22no—
220#11758196—21no—
221#11758196—21no—
222#11758196C22⋊C412yes—
223#11758196C22⋊C412yes—
224C4×Dic61—22no—
225C4×Dic61—21no—
226C4×Dic61—22no—
227C4×Dic61—21no—
228C4×Dic61C22×C411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231C4×Dic61C2×D412yes—
232C4×Dic61C2×D412yes—
233C4×Dic61C2411yescharacteristic
234C23×C61—22no—
235C23×C61—24no—
236C23×C61—81no—
237C23×C61—81no—
238C23×C61C2×C22⋊C411yescharacteristic
239C22×D61—21no—
240C22×D61—28no—
241C22×D61—82no—
242C22×D61—82no—
243C22×D61C2×C22⋊C412yes—
244D4×C61—22no—
245D4×C61—24no—
246D4×C61C22×C814yes—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—82no—
250C2×C4×C61—82no—
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C22×C812yes—
255C2×C4×C61C2×M1612yes—
256C2×C4×C61C23×C411yescharacteristic
257C22⋊D61—24no—
258C22⋊D61—28no—
259C22⋊D61—28no—
260C22⋊D61C22⋊C818yes—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—28no—
265C2×Dic61—22no—
266C2×Dic61—41no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—82no—
270C2×Dic61—81no—
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C22×D411yescharacteristic
275C2×Dic61C22×D411yescharacteristic
276D244—21no—
277D244C22⋊C814yes—
278D244C22×C812yes—
279C4×D61—24no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—42no—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—81no—
286C4×D61—82no—
287C4×D61—82no—
288C4×D61C22×C814yes—
289C4×D61C2×M1614yes—
290Dic122—21no—
291Dic122C22⋊C814yes—
292Dic122C22×C812yes—
293#92420—22no—
294#92420—42no—
295#92420—41no—
296#92420—41no—
297#92420—42no—
298#92420—41no—
299#17438—611nomaximal, Sylow
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—24no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—84no—
306C22×C61C2×C22⋊C812yes—
307C22×C61C22×C22⋊C411yescharacteristic
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—88no—
314C2×D61C2×C22⋊C814yes—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61C2×C22⋊C812yes—
320C4×C61C23×C811yescharacteristic
321C4×C61C22×M1611yescharacteristic
322Dic61—22no—
323Dic61—21no—
324Dic61—21no—
325Dic61—21no—
326Dic61—42no—
327Dic61C2×C22⋊C814yes—
328Dic61#57611yescharacteristic
329Dic61#57711yescharacteristic
330C4×H(5,𝔽2)—611no—
331#2988—611no—
332C8×#41—611no—
333#1478—611no—
334#1478—612no—
335#1477—612no—
336#1477—611no—
337#1476—612no—
338#1476—611no—
339C2×D4⋊C8—611no—
340C2×D4⋊C8—612no—
341C2×C61—22no—
342C2×C61—44no—
343C2×C61C22×C22⋊C811yescharacteristic
344C2×C61#289711yescharacteristic
345C2×C61#290111yescharacteristic
346D61—24no—
347D61—42no—
348C2×H(5,𝔽2)—611no—
349#520—611no—
350#520—612no—
351#520—611no—
352#520—1221no—
353#518—612no—
354#518—1222no—
355#518—1221no—
356#518#692311yescharacteristic
357C4×#41—611no—
358C4×#41—612no—
359C4×#41—611no—
360C4×#41—1221no—
361C2×C4×D4—611no—
362C2×C4×D4—612no—
363C2×C4×D4—1222no—
364C2×C4×D4—1221no—
365C8×Q8—1221no—
366C8×Q8—1221no—
367C8×D4—1222no—
368C8×D4—1222no—
369C8×D4—1221no—
370C8×D4—1221no—
371#431—611no—
372#431—612no—
373#431—611no—
374#431—612no—
375C2×C4⋊C8—611no—
376C2×C4⋊C8—612no—
377C2×C4×C8—611no—
378C2×C4×C8—612no—
379Q8⋊C8—611no—
380Q8⋊C8—612no—
381Q8⋊C8—611no—
382D4⋊C8—612no—
383D4⋊C8—612no—
384D4⋊C8—612no—
385D4⋊C8—612no—
386D4⋊C8—611no—
387D4⋊C8—612no—
388D4⋊C8—611no—
389C61#1743811yescharacteristic, complemented, Sylow
390H(5,𝔽2)—618no—
391H(5,𝔽2)—1224no—
392C2×#41—611no—
393C2×#41—612no—
394C2×#41—611no—
395C2×#41—1221no—
396C22×D4—612no—
397C22×D4—1221no—
398C22×D4—1222no—
399C22×D4C2×#692311yescharacteristic
400C23×C4—1222no—
401C23×C4—1221no—
402C22×C8—1221no—
403C22×C8—1222no—
404C22×C8—1221no—
405C22×C8—1222no—
406C4×Q8—611no—
407C4×Q8—611no—
408C4×Q8—611no—
409C4×Q8—611no—
410C4×Q8—612no—
411C4×Q8—1221no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—611no—
415C4×D4—611no—
416C4×D4—612no—
417C4×D4—612no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—611no—
421C4×D4—611no—
422C4×D4—612no—
423C4×D4—1221no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1221no—
431C4×D4—1222no—
432C4×D4—1221no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4C2×#692312yes—
439C4×D4C2×#692312yes—
440#117—612no—
441#117—612no—
442#117—1221no—
443#117—1222no—
444#117—1222no—
445#117—1221no—
446#117C2×#692311yescharacteristic
447#117C2×#692311yescharacteristic
448C2×C4⋊C4—611no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4—1221no—
452C2×C22⋊C4—22no—
453C2×C22⋊C4—1222no—
454C2×C22⋊C4—1222no—
455C2×C22⋊C4—1222no—
456C2×C22⋊C4—1221no—
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—
472C4×C8—611no—
473C4×C8—611no—
474C4×C8—612no—
475C4×C8—1221no—
476C4×C8—1222no—
477C4×C8—1221no—
478C4×C8—1222no—
479C24—21no—
480C24—1222no—
481#41—614no—
482#41—614no—
483#41—618no—
484#41—614no—
485#41—1224no—
486#41—1228no—
487#41—1222no—
488C2×Q8—611no—
489C2×Q8—611no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—612no—
493C2×D4—618no—
494C2×D4—611no—
495C2×D4—611no—
496C2×D4—1224no—
497C2×D4—1222no—
498C2×D4—1228no—
499C2×D4—12216no—
500C2×D4—1222no—
501C2×D4—1221no—
502C2×D4D61⋊C814yescomplemented
503C2×D4C22×#692312yes—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—612no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—612no—
510C22×C4—612no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1222no—
520C22×C4—4882no—
521C22×C4—4882no—
522C22×C4—4881no—
523C22×C4—4881no—
524C22×C4C22×#692311yescharacteristic
525C2×C8—1221no—
526C2×C8—1222no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1222no—
531C2×C8—4882no—
532C2×C8—4881no—
533C2×C8—2444no—
534C2×C8—2442no—
535C4⋊C4—612no—
536C4⋊C4—612no—
537C4⋊C4—612no—
538C4⋊C4—1222no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1222no—
543C4⋊C4C22⋊#692311yescharacteristic
544C4⋊C4C22⋊#692311yescharacteristic
545C4⋊C4C22×#692311yescharacteristic
546C4⋊C4C22×#692311yescharacteristic
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—612no—
551C22⋊C4—612no—
552C22⋊C4—1221no—
553C22⋊C4—1221no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4C22⋊#692312yes—
559C42—21no—
560C42—612no—
561C42—611no—
562C42—612no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—1221no—
567C42—1222no—
568C42—1221no—
569C42—1222no—
570C42C22×#692312yes—
571C23—22no—
572C23—24no—
573C23—81no—
574C23—81no—
575C23—612no—
576C23—1228no—
577C23—1221no—
578C23—4882no—
579C23—4882no—
580C23C2×C22⋊#692311yescharacteristic
581Q8—614no—
582Q8—612no—
583Q8—1221no—
584D4—24no—
585D4—22no—
586D4—618no—
587D4—614no—
588D4—612no—
589D4—1228no—
590D4—1228no—
591D4—1224no—
592D4—1221no—
593D4C2×D61⋊C814yes—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—612no—
599C2×C4—611no—
600C2×C4—614no—
601C2×C4—614no—
602C2×C4—611no—
603C2×C4—611no—
604C2×C4—611no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1228no—
610C2×C4—1221no—
611C2×C4—1224no—
612C2×C4—1222no—
613C2×C4—4882no—
614C2×C4—4881no—
615C2×C4—4882no—
616C2×C4—4882no—
617C2×C4—4881no—
618C2×C4—2442no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2441no—
624C2×C4C2×C22⋊#692312yes—
625C2×C4C2×C22⋊#692312yes—
626C2×C4C2×C22⋊#692311yescharacteristic
627C2×C4C2×D61⋊C812yes—
628C2×C4D61⋊M1612yes—
629C2×C4C23×#692311yescharacteristic
630C8—1222no—
631C8—2441no—
632C8—2441no—
633C8—2442no—
634C8—2442no—
635C8—2441no—
636C22—21no—
637C22—21no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—84no—
642C22—614no—
643C22—1222no—
644C22—1222no—
645C22—1222no—
646C22—4888no—
647C22—2441no—
648C22—2441no—
649C22D61⋊(C22⋊C8)12yes—
650C22C22×C22⋊#692311yescharacteristic
651C4—22no—
652C4—22no—
653C4—41no—
654C4—41no—
655C4—614no—
656C4—611no—
657C4—611no—
658C4—1222no—
659C4—1221no—
660C4—1221no—
661C4—1221no—
662C4—2442no—
663C4D61⋊(C22⋊C8)12yes—
664C4C22×D61⋊C811yescharacteristic
665C4C2×D61⋊M1611yescharacteristic
666C2—22no—
667C2—44no—
668C2—1224no—
669C2—2442no—
670C2C2×D61⋊(C22⋊C8)11yescharacteristic
671C2#2146244011yescharacteristic
672C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC4×C61C23×C8
FrattiniC2×C4C23×#6923
FittingC61×#518C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519822C4
cosocleC4×Dic61C24
hypercenter#518#6923
hypocenterC61#17438
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #21462736C2semidirect
2. #21462623C2semidirect
3. #21461119C2semidirect
4. #21461118C2semidirect
5. #21461116C2semidirect
6. #21461115C2semidirect
7. #21461112C2semidirect
8. #21461111C2semidirect
9. D61⋊(D4⋊C8)C2semidirect
10. C2×D4⋊#92420C2semidirect
18. #16840781C22semidirect
19. #16840780C22semidirect
20. #16840778C22semidirect
21. #16840778C22semidirect
22. D61⋊(C4⋊C8)C22semidirect
23. C2×C4⋊#92420C22semidirect
24. C4×D61⋊C8C22semidirect
25. C2×C4×#92420C22semidirect
26. #16840612C22semidirect
27. #16840612C22semidirect
28. #16840612C22semidirect
29. #16840610C22semidirect
30. #16840610C22semidirect
31. #16840610C22semidirect
32. D4⋊#92420C22semidirect
33. D4⋊#92420C22semidirect
34. D4⋊#92420C22semidirect
35. D4⋊#92420C22semidirect
76. #14519822C8semidirect
146. C2×D4×C61C2×C8semidirect
166. #11758224C2×C8semidirect
169. D4×D61C2×C8semidirect
172. #11758222C2×C8semidirect
389. C61#17438semidirect
502. C2×D4D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214627362
#214626232
#214611192
#214611182
#214611162
#214611152
#214611122
#214611112
D61⋊(D4⋊C8)2
C2×D4⋊#924202
#214590232
#1743861