Finite Group Databasebeta

← #30610406

Subgroups — #30610406

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#21461120C211yesmaximal, characteristic, complemented
4#21461119C212yesmaximal, complemented
5#21461117C212yesmaximal, complemented
6#21461114C211yesmaximal, characteristic, complemented
7#21461113C212yesmaximal, complemented
8#21461110C211yesmaximal, characteristic, complemented
9C2×#16840610C211yesmaximal, characteristic, complemented
10D61⋊(D4⋊C8)C212yesmaximal, complemented
11#21459025C211yesmaximal, 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
26Q8⋊#92420C2211yescharacteristic, complemented
27Q8⋊#92420C2211yescharacteristic, complemented
28Q8⋊#92420C2212yescomplemented
29#16840610C2212yescomplemented
30#16840610C2212yescomplemented
31#16840610C2212yescomplemented
32#16840610C2212yescomplemented
33D4⋊#92420C2211yescharacteristic, complemented
34D4⋊#92420C2212yescomplemented
35D4⋊#92420C2211yescharacteristic, complemented
36C2×#14519823C411yescharacteristic
37#16840244C412yes—
38#16840243—21no—
39#16840243C411yescharacteristic
40#16840236—21no—
41#16840236C411yescharacteristic
42#16840234—22no—
43#16840234C412yes—
44#41×Dic61C2211yescharacteristic
45C4×#11758226C2212yes—
46#16840230—21no—
47#16840230C2211yescharacteristic
48C2×#14519739—21no—
49C2×#14519739C2211yescharacteristic
50C4×D4×D61—22no—
51C4×D4×D61C2212yes—
52C61×#520C411yescharacteristic
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#14519823—24no—
76#14519823C818yescomplemented
77C2×#11758226C2×C412yes—
78C2×#11758222—21no—
79C2×#11758222C2×C411yescharacteristic
80C22×D244—21no—
81C22×D244C2×C411yescharacteristic
82C2×D4×D61—22no—
83C2×D4×D61C2×C412yes—
84C22×C4×D61—21no—
85C22×C4×D61—22no—
86Q8×Dic61C2×C411yescharacteristic
87Q8×Dic61C2311yescharacteristic
88#14519742—21no—
89#14519742D411yescharacteristic
90#14519742D411yescharacteristic
91C4×D244—22no—
92C4×D244—22no—
93C4×D244C2×C412yes—
94C4×D244C2×C412yes—
95D4×Dic61—21no—
96D4×Dic61—21no—
97D4×Dic61C2×C411yescharacteristic
98D4×Dic61C2311yescharacteristic
99#14519739—21no—
100#14519739—22no—
101#14519739—21no—
102#14519739—21no—
103#14519739—22no—
104#14519739—21no—
105#14519739—21no—
106#14519739C2×C412yes—
107#14519739D411yescharacteristic
108#14519739D411yescharacteristic
109#14519739C2312yes—
110C4×C22⋊D61—22no—
111C4×C22⋊D61—22no—
112C4×C22⋊D61—22no—
113#14519737—22no—
114#14519737—22no—
115#14519737D412yes—
116#14519737D412yes—
117#14519736—21no—
118#14519736C2×C411yescharacteristic
119#14519735—22no—
120#14519735C2×C412yes—
121#14519734—22no—
122#14519734C2×C412yes—
123C4⋊C4×D61—22no—
124C4⋊C4×D61C2×C412yes—
125C2×#11758198—21no—
126C2×#11758198—22no—
127C22⋊C4×D61—22no—
128C22⋊C4×D61—22no—
129C22⋊C4×D61—22no—
130C2×C4×Dic61—21no—
131C2×C4×Dic61C2311yescharacteristic
132C42×D61—22no—
133C42×D61C2312yes—
134C2×#41×C61C2×C411yescharacteristic
135C4×Q8×C61C2×C412yes—
136C4×D4×C61—22no—
137C4×D4×C61C2×C412yes—
138#117×C61—21no—
139#117×C61C2×C411yescharacteristic
140C2×C4⋊C4×C61—21no—
141C2×C4⋊C4×C61C2×C411yescharacteristic
142C23×D61—21no—
143C23×D61—22no—
144#41×C61—22no—
145#41×C61C2×C814yescomplemented
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61C22×C411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—22no—
152C22×C4×C61C22×C411yescharacteristic
153C2×C22⋊D61—22no—
154C2×C22⋊D61C22⋊C412yes—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159#11758226—24no—
160#11758226C2×C818yescomplemented
161D4×D61—216no—
162D4×D61—24no—
163D4×D61C2×C818yescomplemented
164#11758222—28no—
165#11758222C2×C814yes—
166#11758222M1614yes—
167C2×D244—22no—
168C2×D244—21no—
169C2×D244—22no—
170C2×D244—28no—
171C2×D244C22⋊C411yescharacteristic
172C2×D244C2×C814yescomplemented
173C2×D244C22×C412yes—
174C2×C4×D61—22no—
175C2×C4×D61—22no—
176C2×C4×D61—22no—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—22no—
180C2×C4×D61—82no—
181C2×C4×D61—82no—
182C2×C4×D61—81no—
183C2×C4×D61—81no—
184C2×C4×D61C22×C412yes—
185C2×C4×D61C2×D412yes—
186C2×C4×D61C2×D412yes—
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×C61—21no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61C22⋊C411yescharacteristic
201C4⋊C4×C61C22⋊C411yescharacteristic
202C4⋊C4×C61C22×C412yes—
203C22⋊C4×C61—22no—
204C22⋊C4×C61C22⋊C412yes—
205C42×C61—22no—
206C42×C61C22×C412yes—
207C22⋊Dic61—21no—
208C22⋊Dic61—21no—
209C22⋊Dic61—21no—
210#11758198—22no—
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215#11758198C22⋊C412yes—
216#11758198C22⋊C412yes—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61C22×C411yescharacteristic
220C4⋊Dic61C22×C411yescharacteristic
221#11758196—22no—
222#11758196C22⋊C412yes—
223#11758196C22⋊C412yes—
224C4×Dic61—21no—
225C4×Dic61—22no—
226C4×Dic61—21no—
227C4×Dic61—21no—
228C4×Dic61C22×C411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231C4×Dic61C2×D412yes—
232C4×Dic61C2×D412yes—
233C4×Dic61C2411yescharacteristic
234C23×C61—21no—
235C23×C61C2×C22⋊C411yescharacteristic
236C22×D61—22no—
237C22×D61—24no—
238C22×D61—28no—
239C22×D61—82no—
240C22×D61—82no—
241C22×D61—81no—
242C22×D61—81no—
243C22×D61C2×C22⋊C412yes—
244Q8×C61—21no—
245Q8×C61C22×C812yes—
246D4×C61—24no—
247D4×C61—21no—
248D4×C61C22×C812yes—
249C2×C4×C61—21no—
250C2×C4×C61—24no—
251C2×C4×C61—22no—
252C2×C4×C61—82no—
253C2×C4×C61—81no—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C22×C812yes—
258C2×C4×C61C2×M1612yes—
259C2×C4×C61C23×C411yescharacteristic
260C22⋊D61—28no—
261C22⋊D61C22⋊C818yes—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—81no—
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C412yes—
273C2×Dic61C22×D411yescharacteristic
274C2×Dic61C22×D411yescharacteristic
275D244—28no—
276D244—22no—
277D244—24no—
278D244C22⋊C814yes—
279D244C22×C814yes—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—28no—
283C4×D61—42no—
284C4×D61—42no—
285C4×D61—42no—
286C4×D61—82no—
287C4×D61—82no—
288C4×D61—82no—
289C4×D61—82no—
290C4×D61C22×C814yes—
291C4×D61C2×M1614yes—
292Dic122C22⋊C814yes—
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—22no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61C2×C22⋊C812yes—
306C22×C61C22×C22⋊C411yescharacteristic
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—24no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—88no—
313C2×D61—84no—
314C2×D61C2×C22⋊C814yes—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—22no—
318C4×C61—42no—
319C4×C61C2×C22⋊C812yes—
320C4×C61C23×C811yescharacteristic
321C4×C61C22×M1611yescharacteristic
322Dic61—21no—
323Dic61—21no—
324Dic61—21no—
325Dic61—41no—
326Dic61—41no—
327Dic61C2×C22⋊C814yes—
328Dic61#57611yescharacteristic
329Dic61#57711yescharacteristic
330C4×H(5,𝔽2)—611no—
331#2988—611no—
332C8×#41—611no—
333#1478—612no—
334#1478—611no—
335#1477—611no—
336#1477—612no—
337#1476—611no—
338#1476—612no—
339C2×D4⋊C8—612no—
340C2×D4⋊C8—611no—
341C2×C61—22no—
342C2×C61—42no—
343C2×C61C22×C22⋊C811yescharacteristic
344C2×C61#289711yescharacteristic
345C2×C61#290111yescharacteristic
346D61—24no—
347D61—44no—
348C2×H(5,𝔽2)—611no—
349#520—611no—
350#520—612no—
351#520—1221no—
352#520#692311yescharacteristic
353#518—612no—
354#518—611no—
355#518—1221no—
356#518—1222no—
357C4×#41—611no—
358C4×#41—611no—
359C4×#41—612no—
360C4×#41—1221no—
361C2×C4×D4—612no—
362C2×C4×D4—611no—
363C2×C4×D4—1221no—
364C2×C4×D4—1222no—
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—611no—
381Q8⋊C8—612no—
382D4⋊C8—611no—
383D4⋊C8—612no—
384D4⋊C8—611no—
385D4⋊C8—612no—
386D4⋊C8—612no—
387D4⋊C8—612no—
388D4⋊C8—612no—
389C61#1743811yescharacteristic, complemented, Sylow
390H(5,𝔽2)—618no—
391H(5,𝔽2)—1224no—
392C2×#41—611no—
393C2×#41—612no—
394C2×#41—1221no—
395C2×#41C2×#692311yescharacteristic
396C22×D4—612no—
397C22×D4—611no—
398C22×D4—1221no—
399C22×D4—1222no—
400C23×C4—1221no—
401C23×C4—1222no—
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—1221no—
411C4×Q8C2×#692312yes—
412C4×D4—22no—
413C4×D4—611no—
414C4×D4—611no—
415C4×D4—612no—
416C4×D4—612no—
417C4×D4—611no—
418C4×D4—611no—
419C4×D4—612no—
420C4×D4—612no—
421C4×D4—612no—
422C4×D4—612no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1221no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1221no—
436C4×D4—1221no—
437C4×D4—1221no—
438C4×D4—1221no—
439C4×D4C2×#692312yes—
440#117—21no—
441#117—612no—
442#117—611no—
443#117—612no—
444#117—1222no—
445#117—1222no—
446#117—1221no—
447#117C2×#692311yescharacteristic
448C2×C4⋊C4—21no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—1222no—
451C2×C4⋊C4C2×#692311yescharacteristic
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—1222no—
460C2×C42—1221no—
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—1222no—
480C24—1221no—
481#41—22no—
482#41—614no—
483#41—614no—
484#41—618no—
485#41—1228no—
486#41—1224no—
487#41D61⋊C814yescomplemented
488C2×Q8—611no—
489C2×Q8C22×#692311yescharacteristic
490C2×D4—21no—
491C2×D4—612no—
492C2×D4—618no—
493C2×D4—611no—
494C2×D4—614no—
495C2×D4—612no—
496C2×D4—1228no—
497C2×D4—1222no—
498C2×D4—1224no—
499C2×D4—1222no—
500C2×D4—12216no—
501C2×D4—1221no—
502C2×D4—1222no—
503C2×D4C22×#692311yescharacteristic
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—612no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1222no—
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—22no—
536C4⋊C4—21no—
537C4⋊C4—612no—
538C4⋊C4—612no—
539C4⋊C4—611no—
540C4⋊C4—611no—
541C4⋊C4—1222no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4C22⋊#692311yescharacteristic
545C4⋊C4C22⋊#692311yescharacteristic
546C4⋊C4C22×#692312yes—
547C22⋊C4—22no—
548C22⋊C4—612no—
549C22⋊C4—612no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1221no—
554C22⋊C4—1221no—
555C22⋊C4—1221no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4C22⋊#692312yes—
559C42—22no—
560C42—612no—
561C42—611no—
562C42—612no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—1221no—
567C42—1222no—
568C42—1221no—
569C42—1221no—
570C42C22×#692312yes—
571C23—21no—
572C23—612no—
573C23—1228no—
574C23—1224no—
575C23—1222no—
576C23—4881no—
577C23—4881no—
578C23—4882no—
579C23—4882no—
580C23C2×C22⋊#692311yescharacteristic
581Q8—21no—
582Q8—614no—
583Q8C2×D61⋊C812yes—
584D4—24no—
585D4—21no—
586D4—618no—
587D4—614no—
588D4—614no—
589D4—1228no—
590D4—1228no—
591D4—1224no—
592D4—1222no—
593D4C2×D61⋊C812yes—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—612no—
600C2×C4—611no—
601C2×C4—614no—
602C2×C4—614no—
603C2×C4—611no—
604C2×C4—611no—
605C2×C4—611no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1228no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—4882no—
614C2×C4—4882no—
615C2×C4—4882no—
616C2×C4—4882no—
617C2×C4—4881no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2441no—
622C2×C4—2441no—
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—22no—
639C22—41no—
640C22—41no—
641C22—614no—
642C22—1222no—
643C22—1222no—
644C22—1224no—
645C22—4884no—
646C22—4888no—
647C22—2442no—
648C22—2442no—
649C22D61⋊(C22⋊C8)12yes—
650C22C22×C22⋊#692311yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—614no—
656C4—611no—
657C4—611no—
658C4—1221no—
659C4—1221no—
660C4—1221no—
661C4—2441no—
662C4—2441no—
663C4D61⋊(C22⋊C8)12yes—
664C4C22×D61⋊C811yescharacteristic
665C4C2×D61⋊M1611yescharacteristic
666C2—22no—
667C2—42no—
668C2—1224no—
669C2—2444no—
670C2C2×D61⋊(C22⋊C8)11yescharacteristic
671C2#2146244011yescharacteristic
672C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC4×C61C23×C8
FrattiniC2×C4C23×#6923
FittingC61×#520C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519823C4
cosocleC4×Dic61C24
hypercenter#520#6923
hypocenterC61#17438
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #21462736C2semidirect
2. #21462623C2semidirect
3. #21461120C2semidirect
4. #21461119C2semidirect
5. #21461117C2semidirect
6. #21461114C2semidirect
7. #21461113C2semidirect
8. #21461110C2semidirect
9. C2×#16840610C2semidirect
10. D61⋊(D4⋊C8)C2semidirect
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. Q8⋊#92420C22semidirect
27. Q8⋊#92420C22semidirect
28. Q8⋊#92420C22semidirect
29. #16840610C22semidirect
30. #16840610C22semidirect
31. #16840610C22semidirect
32. #16840610C22semidirect
33. D4⋊#92420C22semidirect
34. D4⋊#92420C22semidirect
35. D4⋊#92420C22semidirect
76. #14519823C8semidirect
145. #41×C61C2×C8semidirect
160. #11758226C2×C8semidirect
163. D4×D61C2×C8semidirect
172. C2×D244C2×C8semidirect
389. C61#17438semidirect
487. #41D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
#214627362
#214626232
#214611202
#214611192
#214611172
#214611142
#214611132
#214611102
C2×#168406102
D61⋊(D4⋊C8)2
#214590252
#1743861