Finite Group Databasebeta

← #29743133

Subgroups — #29743133

Order 15616.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840069C211yesmaximal, characteristic, complemented
2C2×SD16×Dic61C211yesmaximal, characteristic
3#21449693C211yesmaximal, characteristic
4Q8×#11758196C211yesmaximal, characteristic, complemented
5#21447345C211yesmaximal, characteristic
6#21447299C211yesmaximal, characteristic, complemented
7#21447207C211yesmaximal, characteristic
8#21447158C211yesmaximal, characteristic, complemented
9#21446551C211yesmaximal, characteristic, complemented
10#21446379C211yesmaximal, characteristic, complemented
11#21446176C211yesmaximal, characteristic
12#21446078C211yesmaximal, characteristic
13#21446075C211yesmaximal, characteristic, complemented
14#21446071C211yesmaximal, characteristic, complemented
15C61×#1880C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16840069—24no—
21SD16×Dic61—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839698—21no—
34#16839685—21no—
35#16839665C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839660—21no—
38#16839660C2211yescharacteristic
39#16839658—21no—
40#16839658C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839634—21no—
43#16839631—21no—
44#16839631C2211yescharacteristic
45#16839620—21no—
46C4×#11758196—21no—
47C4×#11758196C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519769—22no—
65#14519769—41no—
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68Q8×Dic61—22no—
69Q8×Dic61—41no—
70D4×Dic61—22no—
71D4×Dic61—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196D411yescharacteristic
87C2×#11758196D411yescharacteristic
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519711—41no—
96#14519710—22no—
97#14519710—22no—
98#14519709—22no—
99#14519708—22no—
100C4⋊#92418—22no—
101#14519701—22no—
102#14519701D414yes—
103#14519698—22no—
104#14519698—22no—
105C8×Dic61—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C2×SD16×C61—22no—
115C2×SD16×C61—22no—
116C2×SD16×C61Q814yes—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×Q8×C61—22no—
120C4×Q8×C61—41no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#103×C61—22no—
129#102×C61—41no—
130C24×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61C2×D412yes—
135C2×Q8×C61C2×Q812yes—
136C2×Q8×C61#4112yes—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61C2×Q812yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C22×Dic61—81no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160SD16×C61—28no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C2×Q812yes—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—42no—
172C22⋊C4×C61—81no—
173C22⋊C4×C61—81no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—42no—
177C22⋊Dic61—42no—
178C22⋊Dic61—42no—
179C22⋊Dic61—81no—
180C4⋊Dic61—24no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—42no—
188C4⋊Dic61—81no—
189C4⋊Dic61C2×D412yes—
190C4⋊Dic61#4112yes—
191#11758196—22no—
192#11758196—22no—
193#11758196—42no—
194#11758196—42no—
195#11758196—42no—
196#11758196—81no—
197#11758196—81no—
198#11758196—81no—
199#11758196—81no—
200C4×Dic61—24no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D412yes—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220Q8×C61—22no—
221Q8×C61—22no—
222Q8×C61—22no—
223Q8×C61C22⋊Q814yes—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—41no—
227D4×C61—41no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×Q811yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—84no—
272C2×Dic61#13512yes—
273C2×Dic61#13712yes—
274C8×C61—22no—
275C8×C61—41no—
276#92418—22no—
277#92418—41no—
278#26379—611nomaximal, Sylow
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61#52511yescharacteristic
287C22×C61D4211yescharacteristic
288C22×C61D4×Q811yescharacteristic
289C22×C61C2×#13511yescharacteristic
290C22×C61C2×#13711yescharacteristic
291C22×C61#57611yescharacteristic
292C22×C61#57711yescharacteristic
293C4×C61—24no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—81no—
297C4×C61—81no—
298C4×C61C2×C22⋊Q811yescharacteristic
299C4×C61#53911yescharacteristic
300C4×C61#54711yescharacteristic
301C4×C61#55411yescharacteristic
302Dic61—22no—
303Dic61—22no—
304Dic61—41no—
305Dic61—42no—
306Dic61—42no—
307Dic61—41no—
308Dic61—81no—
309Dic61—81no—
310C2×#479—611no—
311C2×C4×SD16—611no—
312#2454—611no—
313Q8×C4⋊C4—611no—
314#2093—611no—
315#2084—611no—
316#2067—611no—
317#2059—611no—
318#1942—611no—
319#1916—611no—
320#1880—611no—
321#1880D6111yescharacteristic
322#1864—611no—
323#1863—611no—
324#1863—611no—
325C2×C61—82no—
326C2×C61#244911yescharacteristic
327C2×C61#310811yescharacteristic
328C2×C61#311911yescharacteristic
329C2×C61#326611yescharacteristic
330C2×C61#328711yescharacteristic
331C2×C61#338111yescharacteristic
332C2×C61#338511yescharacteristic
333C22×SD16C2×D6111yescharacteristic
334C2×#125—611no—
335C2×C4×Q8—611no—
336C2×C4×Q8—611no—
337C2×C4×Q8C2×D6111yescharacteristic
338C2×C4×D4—611no—
339#479—1224no—
340C4×SD16—1224no—
341C2×#107—611no—
342C2×#107—611no—
343C2×#107—611no—
344C2×#106—611no—
345C2×#106—611no—
346C2×C4⋊C8—611no—
347C2×#103—611no—
348C2×#103—611no—
349C2×#103C2×D6111yescharacteristic
350C2×#102—611no—
351C2×#102—611no—
352C2×#102C2×D6111yescharacteristic
353C2×C4×C8—611no—
354#399—1221no—
355#396—1221no—
356#390—611no—
357#390C2×D6111yescharacteristic
358#389—611no—
359#389—611no—
360#389—611no—
361#389—611no—
362#389—1221no—
363#389—1221no—
364#385—1221no—
365#384—611no—
366#384—1221no—
367#380—1221no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—1221no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340C2×D6111yescharacteristic
374#336—611no—
375#336—611no—
376#336—611no—
377#336C2×D6111yescharacteristic
378C61#2637911yescharacteristic, complemented, Sylow
379C22×Q8C22×D6111yescharacteristic
380C22×D4C22×D6111yescharacteristic
381C23×C4—1221no—
382C2×SD16—22no—
383C2×SD16—22no—
384C2×SD16Dic12214yes—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8C22×D6111yescharacteristic
388C22×C8C22×D6111yescharacteristic
389#125—1222no—
390#125—2441no—
391C4×Q8—22no—
392C4×Q8—41no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—2441no—
397C4×D4—1222no—
398C4×D4—2441no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C42—21no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—1221no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429#107—1222no—
430#107—1222no—
431#107—2441no—
432#106—1222no—
433#106—1222no—
434C4⋊C8—1222no—
435#103—22no—
436#103—614no—
437#103—1222no—
438#102—41no—
439#102—1222no—
440#102—1222no—
441C4×C8—1222no—
442#95—611no—
443#95—611no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448C24—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×Dic12212yes—
453C2×Q8D4×D6112yes—
454C2×Q8#1175822412yes—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4C2×Dic12212yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—4881no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478SD16—28no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—612no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8C2×Dic12212yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—612no—
497C4⋊C4—612no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1224no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—4881no—
508C4⋊C4—4881no—
509C4⋊C4—4881no—
510C4⋊C4—4881no—
511C4⋊C4—4881no—
512C4⋊C4—2442no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—4881no—
519C22⋊C4—2442no—
520C22⋊C4—2442no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—1224no—
525C42—1221no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—2442no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C2×D4×D6111yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8#1451975414yes—
542D4—22no—
543D4—22no—
544D4—41no—
545D4—41no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1222no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1222no—
575C2×C4—4884no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4C2×D4×D6111yescharacteristic
587C2×C4C22×Dic12211yescharacteristic
588C2×C4#41×D6111yescharacteristic
589C2×C4C2×#1175822411yescharacteristic
590C2×C4#1451982211yescharacteristic
591C2×C4#1451982411yescharacteristic
592C8—22no—
593C8—41no—
594C8—1222no—
595C8—2441no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1684027211yescharacteristic
604C22#1684039411yescharacteristic
605C22D4×Dic12211yescharacteristic
606C22#135×D6111yescharacteristic
607C22#137×D6111yescharacteristic
608C22#1684056611yescharacteristic
609C22#1684056811yescharacteristic
610C4—24no—
611C4—41no—
612C4—41no—
613C4—81no—
614C4—81no—
615C4—1222no—
616C4—1222no—
617C4—4881no—
618C4—4881no—
619C4—2441no—
620C4—2442no—
621C4—2441no—
622C4—2442no—
623C4C2×#1451975411yescharacteristic
624C4#1684035711yescharacteristic
625C4#1684041011yescharacteristic
626C4#1684047111yescharacteristic
627C2—82no—
628C2#2144963211yescharacteristic
629C2#2145589811yescharacteristic
630C2#2145601711yescharacteristic
631C2#2145733211yescharacteristic
632C2#2145757211yescharacteristic
633C2#2145869511yescharacteristic
634C2#2145873011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1880C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1880D61
hypocenterC61#26379
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. C2×#16840069C2semidirect
4. Q8×#11758196C2semidirect
6. #21447299C2semidirect
8. #21447158C2semidirect
9. #21446551C2semidirect
10. #21446379C2semidirect
13. #21446075C2semidirect
14. #21446071C2semidirect
378. C61#26379semidirect

Maximal subgroups

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

subgroupindex
C2×#168400692
C2×SD16×Dic612
#214496932
Q8×#117581962
#214473452
#214472992
#214472072
#214471582
#214465512
#214463792
#214461762
#214460782
#214460752
#214460712
C61×#18802
#2637961