Finite Group Databasebeta

← #29740016

Subgroups — #29740016

Order 15616.

908 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×Dic61C211yesmaximal, characteristic
2#21449602C211yesmaximal, characteristic, complemented
3#21448724C211yesmaximal, characteristic, complemented
4#21447365C211yesmaximal, characteristic, complemented
5#21447307C211yesmaximal, characteristic, complemented
6#21447235C211yesmaximal, characteristic, complemented
7#21447055C211yesmaximal, characteristic, complemented
8#21446492C211yesmaximal, characteristic, complemented
9#21446373C211yesmaximal, characteristic, complemented
10#21446169C211yesmaximal, characteristic
11#21446155C211yesmaximal, characteristic, complemented
12#21446155C211yesmaximal, characteristic, complemented
13#21446080C211yesmaximal, characteristic, complemented
14#21446078C211yesmaximal, characteristic, complemented
15C2×C61×#463C211yesmaximal, characteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19D8×Dic61—24no—
20C2×#14519711C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839690—21no—
30#16839690—21no—
31#16839669C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839635—21no—
38#16839634—21no—
39#16839634—21no—
40#16839627C2211yescharacteristic
41#16839623—21no—
42#16839620—21no—
43C4×#11758196C2211yescharacteristic
44C2×#14519684—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C22×D8×C61C2211yescharacteristic
55C2×#124×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C61×#463—24no—
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C23×Dic61—21no—
62C23×Dic61—21no—
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66D4×Dic61—22no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—41no—
71D4×Dic61—41no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196D411yescharacteristic
80C2×#11758196D411yescharacteristic
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519711—22no—
89#14519708—41no—
90#14519701—22no—
91#14519701—41no—
92#14519698—22no—
93#14519698—22no—
94#14519698—22no—
95#14519698—22no—
96C8×Dic61—22no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×Q8×C61C2311yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C2×SD16×C61—22no—
113C2×SD16×C61—22no—
114C2×SD16×C61—22no—
115C2×D8×C61—22no—
116C2×D8×C61—22no—
117C2×D8×C61D414yes—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120#124×C61—22no—
121#124×C61—41no—
122C4×D4×C61—22no—
123C4×D4×C61—41no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—22no—
130#103×C61—22no—
131#102×C61—22no—
132C24×C61—21no—
133C24×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×D4×C61—21no—
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×C61C2×D412yes—
149C2×D4×C61C2×D412yes—
150C2×D4×C61#4112yes—
151C2×D4×C61#4112yes—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61—81no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C22×Dic61—21no—
161C22×Dic61—24no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61—42no—
168C22×Dic61—42no—
169C22×Dic61—81no—
170C22×Dic61—81no—
171C22×Dic61—81no—
172C22×Dic61—81no—
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175C22×Dic61C2×D411yescharacteristic
176C22×Dic61C2×D411yescharacteristic
177SD16×C61—44no—
178D8×C61—28no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C2×D412yes—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C22⋊C4×C61—42no—
188C22⋊C4×C61—42no—
189C22⋊C4×C61—42no—
190C42×C61—21no—
191C42×C61—21no—
192C22⋊Dic61—24no—
193C22⋊Dic61—42no—
194C22⋊Dic61—42no—
195C22⋊Dic61—42no—
196C22⋊Dic61—81no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61#4112yes—
205#11758196—22no—
206#11758196—41no—
207#11758196—81no—
208#11758196—81no—
209#11758196—81no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
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×C61C22×D411yescharacteristic
235Q8×C61—41no—
236Q8×C61—41no—
237Q8×C61—41no—
238Q8×C61—41no—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—41no—
245D4×C61—41no—
246D4×C61#12314yes—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—84no—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61H(5,𝔽2)11yescharacteristic
265C2×C4×C612−1+411yescharacteristic
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—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
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—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—84no—
290C2×Dic61—84no—
291C2×Dic61—84no—
292C2×Dic61#13512yes—
293C8×C61—22no—
294C8×C61—41no—
295#92418—41no—
296#92418—41no—
297#26160—611nomaximal, Sylow
298C22×C61—24no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61#52511yescharacteristic
313C22×C61D4211yescharacteristic
314C22×C61#54811yescharacteristic
315C22×C61C2×#13511yescharacteristic
316C22×C61#57511yescharacteristic
317C22×C61#57611yescharacteristic
318C22×C61#57711yescharacteristic
319C4×C61—41no—
320C4×C61—41no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61C2×#12311yescharacteristic
324C4×C61#53911yescharacteristic
325C4×C61#54711yescharacteristic
326C4×C61#55311yescharacteristic
327Dic61—22no—
328Dic61—41no—
329Dic61—41no—
330Dic61—42no—
331Dic61—41no—
332Dic61—81no—
333Dic61—81no—
334Dic61—81no—
335C2×#463D6111yescharacteristic
336C2×C4×D8—611no—
337#2447—611no—
338#2349—611no—
339#2097—611no—
340#2086—611no—
341#2073—611no—
342#2042—611no—
343#1933—611no—
344#1916—611no—
345#1879—611no—
346#1877—611no—
347#1877—611no—
348#1864—611no—
349#1864—611no—
350C2×C61—44no—
351C2×C61—82no—
352C2×C61#244311yescharacteristic
353C2×C61#310211yescharacteristic
354C2×C61#311811yescharacteristic
355C2×C61#326711yescharacteristic
356C2×C61#328011yescharacteristic
357C2×C61#336011yescharacteristic
358C2×C61#337011yescharacteristic
359C22×SD16C2×D6111yescharacteristic
360C22×D8C2×D6111yescharacteristic
361C2×#124C2×D6111yescharacteristic
362C2×C4×D4—611no—
363C2×C4×D4—611no—
364C2×C4×D4—611no—
365C2×C4×D4C2×D6111yescharacteristic
366#463—24no—
367C4×D8—1224no—
368C2×#107—611no—
369C2×#106—611no—
370C2×C4⋊C8C2×D6111yescharacteristic
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—
380#397—1221no—
381#397—1221no—
382#391—611no—
383#391—611no—
384#390—611no—
385#389—611no—
386#389—611no—
387#389—611no—
388#385—1221no—
389#385—1221no—
390#385—1221no—
391#382—611no—
392#381—1221no—
393#380—1221no—
394C4×C4⋊C4—611no—
395C2×#95—1221no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#336—611no—
401#336—611no—
402#336—611no—
403#336—611no—
404C61#2616011yescharacteristic, complemented, Sylow
405C22×Q8C22×D6111yescharacteristic
406C22×D4C22×D6111yescharacteristic
407C22×D4C22×D6111yescharacteristic
408C23×C4—21no—
409C23×C4—1221no—
410C23×C4—1221no—
411C23×C4—1221no—
412C2×SD16—22no—
413C2×SD16—22no—
414C2×SD16—22no—
415C2×D8—22no—
416C2×D8—22no—
417C2×D8C22⋊D6114yes—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8C22×D6111yescharacteristic
421C22×C8C22×D6111yescharacteristic
422#124—22no—
423#124—41no—
424C4×D4—22no—
425C4×D4—41no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—2441no—
431C4×D4—2441no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42C22×D6111yescharacteristic
453#107—1222no—
454#106—2441no—
455C4⋊C8—22no—
456#103—22no—
457#103—1222no—
458#103—2441no—
459#102—22no—
460#102—1222no—
461#102—1222no—
462#102—1222no—
463#102—1222no—
464C4×C8—1222no—
465#95—611no—
466#95—611no—
467#95—1221no—
468#95—1221no—
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—21no—
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#1175822412yes—
493C2×D4#1175822412yes—
494C2×D4C2×C22⋊D6112yes—
495C2×D4C2×C22⋊D6112yes—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—81no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—1224no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—4881no—
511C22×C4—4881no—
512C22×C4—4881no—
513C22×C4—4881no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4C23×D6111yescharacteristic
521SD16—44no—
522D8—28no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8C2×C22⋊D6112yes—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—612no—
538C4⋊C4—1222no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—4881no—
547C4⋊C4—4881no—
548C4⋊C4—4881no—
549C4⋊C4—2441no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—1224no—
554C22⋊C4—4881no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C42—21no—
559C42—21no—
560C42—1221no—
561C42—1221no—
562C42—1221no—
563C42—1221no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23C2×D4×D6111yescharacteristic
579Q8—41no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—22no—
587D4—22no—
588D4—41no—
589D4—41no—
590D4#1451975814yes—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—84no—
604C2×C4—612no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1222no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—4884no—
618C2×C4—4884no—
619C2×C4—4884no—
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×C4—2441no—
630C2×C4—2441no—
631C2×C4C22×C22⋊D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4C2×#1175822411yescharacteristic
634C2×C4C2×#1175822411yescharacteristic
635C2×C4#1451982211yescharacteristic
636C2×C4#1451982411yescharacteristic
637C8—22no—
638C8—41no—
639C8—2441no—
640C8—2441no—
641C22—24no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22#1684027211yescharacteristic
656C22D4×C22⋊D6111yescharacteristic
657C22#1684041811yescharacteristic
658C22#135×D6111yescharacteristic
659C22#1684055811yescharacteristic
660C22#1684056411yescharacteristic
661C22#1684057211yescharacteristic
662C4—41no—
663C4—41no—
664C4—81no—
665C4—81no—
666C4—1222no—
667C4—4881no—
668C4—4881no—
669C4—4881no—
670C4—2441no—
671C4—2442no—
672C4—2441no—
673C4—2441no—
674C4C2×#1451975811yescharacteristic
675C4#1684035711yescharacteristic
676C4#1684041111yescharacteristic
677C4#1684045511yescharacteristic
678C2—44no—
679C2—82no—
680C2#2144955011yescharacteristic
681C2#2145583411yescharacteristic
682C2#2145600011yescharacteristic
683C2#2145733811yescharacteristic
684C2#2145749811yescharacteristic
685C2#2145846311yescharacteristic
686C2#2145856911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#463C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenterC2×#463D61
hypocenterC61#26160
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
2. #21449602C2semidirect
3. #21448724C2semidirect
4. #21447365C2semidirect
5. #21447307C2semidirect
6. #21447235C2semidirect
7. #21447055C2semidirect
8. #21446492C2semidirect
9. #21446373C2semidirect
11. #21446155C2semidirect
12. #21446155C2semidirect
13. #21446080C2semidirect
14. #21446078C2semidirect
404. C61#26160semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic612
#214496022
#214487242
#214473652
#214473072
#214472352
#214470552
#214464922
#214463732
#214461692
#214461552
#214460802
#214460782
C2×C61×#4632
#2616061