Finite Group Databasebeta

← #29709192

Subgroups — #29709192

Order 15616.

956 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic61C211yesmaximal, characteristic, complemented
2C2×D8×Dic61C211yesmaximal, characteristic
3#124×Dic61C211yesmaximal, characteristic, complemented
4#21448281C211yesmaximal, characteristic, complemented
5#21447302C211yesmaximal, characteristic, complemented
6#21447232C211yesmaximal, characteristic, complemented
7#21447212C211yesmaximal, characteristic, complemented
8#21447191C211yesmaximal, characteristic, complemented
9#21447132C211yesmaximal, characteristic, complemented
10#21447127C211yesmaximal, characteristic
11#21447055C211yesmaximal, characteristic, complemented
12#21445667C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2012C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519733—21no—
20SD16×Dic61—24no—
21D8×Dic61—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839669C2211yescharacteristic
35#16839665C2211yescharacteristic
36#16839665C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839656—21no—
40#16839655—21no—
41#16839655—21no—
42#16839650—21no—
43#16839628C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C22⋊C4×Dic61—21no—
46C22⋊C4×Dic61—21no—
47C22⋊C4×Dic61—21no—
48C42×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C22×D8×C61C2211yescharacteristic
56C2×#124×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61D414yes—
71#14519733—44no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519711—22no—
94#14519709—22no—
95#14519706—22no—
96#14519706—22no—
97#14519701—22no—
98#14519701—22no—
99#14519698—22no—
100#14519698—22no—
101#14519698—22no—
102#14519698—22no—
103C8×Dic61—22no—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C22×Q8×C61C2311yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×D8×C61—22no—
119C2×D8×C61—22no—
120C2×D8×C61—22no—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123#124×C61—22no—
124#124×C61—22no—
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#103×C61—22no—
132#102×C61—22no—
133C24×C61—21no—
134C24×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61#4112yes—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61C2×D412yes—
152C2×D4×C61#4112yes—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—24no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—42no—
164C22×Dic61—42no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172SD16×C61—28no—
173D8×C61—28no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61#4112yes—
179C2×C8×C61#4112yes—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C22⋊C4×C61—24no—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—42no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—24no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—42no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—42no—
202C4⋊Dic61C2×D412yes—
203#11758196—42no—
204#11758196—42no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—41no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61C2×D412yes—
214C4×Dic61#4112yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61C22×D411yescharacteristic
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61—41no—
247D4×C61—41no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—48no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—84no—
287C2×Dic61#12112yes—
288C2×Dic61#12112yes—
289C2×Dic61#13512yes—
290C2×Dic61#13512yes—
291C2×Dic61AGL(1,ℤ/8)12yes—
292C8×C61—22no—
293C8×C61—22no—
294#92418—41no—
295#92418—41no—
296#23680—611nomaximal, Sylow
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61C2×#12111yescharacteristic
312C22×C61#52511yescharacteristic
313C22×C61#53211yescharacteristic
314C22×C61C2×#13511yescharacteristic
315C22×C61C2×#13511yescharacteristic
316C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
317C22×C61#57511yescharacteristic
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—42no—
321C4×C61—81no—
322C4×C61#52511yescharacteristic
323C4×C61#52511yescharacteristic
324C4×C61#52911yescharacteristic
325C4×C61#52911yescharacteristic
326Dic61—22no—
327Dic61—22no—
328Dic61—41no—
329Dic61—42no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333Dic61#46314yes—
334C2×C4×SD16—611no—
335C2×C4×D8—611no—
336C4×#124—611no—
337#2293—611no—
338#2085—611no—
339#2073—611no—
340#2068—611no—
341#2065—611no—
342#2055—611no—
343#2054—611no—
344#2042—611no—
345#2012D6111yescharacteristic
346C4×C4⋊C8—611no—
347C4×#103—611no—
348C4×#102—611no—
349C2×C61—82no—
350C2×C61—82no—
351C2×C61#239211yescharacteristic
352C2×C61C2×#46311yescharacteristic
353C2×C61#303511yescharacteristic
354C2×C61#310211yescharacteristic
355C2×C61#310611yescharacteristic
356C2×C61#316811yescharacteristic
357C2×C61#317111yescharacteristic
358C22×SD16C2×D6111yescharacteristic
359C22×D8C2×D6111yescharacteristic
360C2×#124C2×D6111yescharacteristic
361C2×C4×Q8—611no—
362C2×C4×D4—611no—
363C2×C4×D4—611no—
364C2×#117—1221no—
365C4×SD16—1224no—
366C4×D8—1224no—
367C2×#107—611no—
368C2×#106—611no—
369C2×C4⋊C8—611no—
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103C2×D6111yescharacteristic
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102C2×D6111yescharacteristic
379C2×C4×C8—611no—
380C2×C4×C8—611no—
381#391—611no—
382#390—611no—
383#390—611no—
384#390C2×D6111yescharacteristic
385#389—611no—
386#389—611no—
387#388—1221no—
388#388—1221no—
389#388—1221no—
390#388—1221no—
391#382—611no—
392C4×C4⋊C4—611no—
393C4×C22⋊C4—1221no—
394C4×C22⋊C4—1221no—
395C4×C22⋊C4—1221no—
396C43—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401#336—611no—
402#336C2×D6111yescharacteristic
403C61#2368011yescharacteristic, complemented, Sylow
404C22×Q8C22×D6111yescharacteristic
405C22×D4C22×D6111yescharacteristic
406C22×D4C22×D6111yescharacteristic
407C23×C4—1221no—
408C23×C4—1221no—
409C2×SD16—22no—
410C2×SD16—22no—
411C2×SD16—22no—
412C2×D8—22no—
413C2×D8—22no—
414C2×D8—22no—
415C22×C8—611no—
416C22×C8—611no—
417C22×C8C22×D6111yescharacteristic
418C22×C8C22×D6111yescharacteristic
419#124—22no—
420#124—22no—
421C4×Q8—1222no—
422C4×Q8—1222no—
423C4×D4—614no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427#117—2444no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42C22×D6111yescharacteristic
455#107—1222no—
456#106—1222no—
457C4⋊C8—1222no—
458C4⋊C8—1222no—
459#103—22no—
460#103—1222no—
461#103—1222no—
462#102—22no—
463#102—1222no—
464#102—1222no—
465#102—1222no—
466#102—1222no—
467C4×C8—1222no—
468C4×C8—1222no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—21no—
477C24—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8#1175822412yes—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4D4×D6112yes—
495C2×D4#1175822412yes—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—1224no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4D4×D6111yescharacteristic
513C22×C4D4×D6111yescharacteristic
514C22×C4C23×D6111yescharacteristic
515SD16—28no—
516D8—28no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8#1175822412yes—
528C2×C8#1175822412yes—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—612no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—1224no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C42—21no—
553C42—21no—
554C42—612no—
555C42—612no—
556C42—1222no—
557C42—1222no—
558C42—1222no—
559C42—1221no—
560C42—1221no—
561C42—2441no—
562C42—2442no—
563C42—2442no—
564C23—22no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23C2×D4×D6111yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—41no—
582Q8—41no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—22no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—41no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—612no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1222no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—4884no—
617C2×C4—2448no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4#41×D6111yescharacteristic
631C2×C4C2×#1175822411yescharacteristic
632C2×C4C2×#1175822411yescharacteristic
633C2×C4C2×#1175822411yescharacteristic
634C2×C4C2×#1175822411yescharacteristic
635C8—22no—
636C8—22no—
637C8—2441no—
638C8—2441no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22#121×D6111yescharacteristic
654C22#1684026811yescharacteristic
655C22#1684031011yescharacteristic
656C22#135×D6111yescharacteristic
657C22#135×D6111yescharacteristic
658C22AGL(1,ℤ/8)×D6111yescharacteristic
659C22#1684055811yescharacteristic
660C4—41no—
661C4—42no—
662C4—41no—
663C4—81no—
664C4—614no—
665C4—1222no—
666C4—1222no—
667C4—4881no—
668C4—4881no—
669C4—4881no—
670C4—2441no—
671C4—2442no—
672C4#1684026811yescharacteristic
673C4#1684026811yescharacteristic
674C4#1684029311yescharacteristic
675C4#1684029311yescharacteristic
676C2—82no—
677C2—82no—
678C2#2144905411yescharacteristic
679C2#463×D6111yescharacteristic
680C2#2145527111yescharacteristic
681C2#2145583011yescharacteristic
682C2#2145587211yescharacteristic
683C2#2145638611yescharacteristic
684C2#2145641111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2012C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenter#2012D61
hypocenterC61#23680
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2012

Complements

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

NH ≅ G/Ntype
1. C2×SD16×Dic61C2semidirect
3. #124×Dic61C2semidirect
4. #21448281C2semidirect
5. #21447302C2semidirect
6. #21447232C2semidirect
7. #21447212C2semidirect
8. #21447191C2semidirect
9. #21447132C2semidirect
11. #21447055C2semidirect
12. #21445667C2semidirect
13. #103×Dic61C2semidirect
14. #102×Dic61C2semidirect
403. C61#23680semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic612
C2×D8×Dic612
#124×Dic612
#214482812
#214473022
#214472322
#214472122
#214471912
#214471322
#214471272
#214470552
#214456672
#103×Dic612
#102×Dic612
C61×#20122
#2368061