Finite Group Databasebeta

← #29828812

Subgroups — #29828812

Order 15616.

1576 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic61C211yesmaximal, characteristic, complemented
2C2×#16839960C212yesmaximal, complemented
3C2×#16839901C211yesmaximal, characteristic, complemented
4#21454162C211yesmaximal, characteristic
5#21446176C212yesmaximal, complemented
6#21446172C212yesmaximal, complemented
7#21446137C211yesmaximal, characteristic, complemented
8#21446134C211yesmaximal, characteristic, complemented
9#21445822C212yesmaximal, complemented
10#21445797C211yesmaximal, characteristic, complemented
11C2×C61×#448C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519773C2211yescharacteristic
14C2×#14519751C2211yescharacteristic
15C2×#14519749—21no—
16C2×#14519749—21no—
17C2×Q8×Dic61—28no—
18C2×Q8×Dic61—21no—
19C2×Q8×Dic61—21no—
20C2×Q8×Dic61C2212yes—
21C2×Q8×Dic61C2212yes—
22C22×C4⋊Dic61—21no—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×C4×Dic61—21no—
25C22×C4×Dic61C2211yescharacteristic
26#16839960—28no—
27#16839901—24no—
28C2×#14519706C2212yes—
29C2×#14519701C2212yes—
30C2×#14519699C2212yes—
31C2×#14519698C2212yes—
32C2×C22⋊#92418C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839663C2211yescharacteristic
35#16839662—21no—
36#16839662—21no—
37#16839631C2211yescharacteristic
38#16839525C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C23×Q8×C61C2211yescharacteristic
42C22×SD16×C61C2212yes—
43C2×#121×C61C2211yescharacteristic
44C61×#448—24no—
45C2×#103×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C22×C22⋊D61—21no—
50C22×C22⋊D61—21no—
51C23×Dic61—21no—
52C23×Dic61D411yescharacteristic
53C23×Dic61D411yescharacteristic
54C2×#11758201—24no—
55C2×#11758201—24no—
56C2×#11758201—24no—
57C22×#92418C2312yes—
58#14519773—24no—
59#14519751—22no—
60#14519751—22no—
61#14519749—44no—
62#14519749—44no—
63Q8×Dic61—28no—
64Q8×Dic61—28no—
65Q8×Dic61—24no—
66Q8×Dic61—24no—
67Q8×Dic61—24no—
68Q8×Dic61D418yescomplemented
69C2×C4⋊Dic61—28no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—24no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2312yes—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—28no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61—24no—
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519706—24no—
92#14519701—24no—
93#14519699—24no—
94#14519698—24no—
95C22⋊#92418—22no—
96C22×Q8×C61—24no—
97C22×Q8×C61—24no—
98C22×Q8×C61—21no—
99C22×Q8×C61—22no—
100C22×Q8×C61D414yescomplemented
101C22×Q8×C61C2312yes—
102C22×D4×C61—21no—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61—21no—
105C23×C4×C61C2311yescharacteristic
106C2×SD16×C61—24no—
107C2×SD16×C61—24no—
108C2×SD16×C61—24no—
109C22×C8×C61C2312yes—
110#121×C61—22no—
111#121×C61—22no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114#103×C61—24no—
115C22⋊C8×C61—22no—
116C24×C61—21no—
117C24×C61C2×D411yescharacteristic
118C23×D61—21no—
119C2×Q8×C61—216no—
120C2×Q8×C61—24no—
121C2×Q8×C61—24no—
122C2×Q8×C61—216no—
123C2×Q8×C61—24no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61C2×D414yes—
128C2×Q8×C61C2×D414yes—
129C2×Q8×C61#4114yes—
130C2×D4×C61—21no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61—42no—
135C2×D4×C61—42no—
136C2×D4×C61—42no—
137C22×C4×C61—24no—
138C22×C4×C61—24no—
139C22×C4×C61—24no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61#4112yes—
147C22×C4×C61C2411yescharacteristic
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C2×C22⋊D61—42no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—42no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61—42no—
156C2×C22⋊D61—42no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—24no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—21no—
164C22×Dic61—41no—
165C22×Dic61—42no—
166C22×Dic61—42no—
167C22×Dic61SD1614yescomplemented
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C2×D244—22no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244—22no—
176SD16×C61—48no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C22⋊C4×C61—42no—
183C22⋊Dic61—42no—
184C22⋊Dic61—42no—
185#11758201—48no—
186#11758198—42no—
187#11758198—42no—
188#11758198—42no—
189C4⋊Dic61—28no—
190C4⋊Dic61—24no—
191C4⋊Dic61—22no—
192C4⋊Dic61—28no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61C2×D414yes—
196#11758196—42no—
197#11758196—42no—
198C4×Dic61—28no—
199C4×Dic61—22no—
200C4×Dic61—28no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61C2×D414yes—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—22no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61#12112yes—
219C23×C61C2×SD1612yes—
220C23×C61AGL(1,ℤ/8)12yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229Q8×C61—216no—
230Q8×C61—24no—
231Q8×C61—24no—
232Q8×C61—24no—
233Q8×C61#12118yescomplemented
234D4×C61—42no—
235D4×C61—42no—
236D4×C61—84no—
237C2×C4×C61—216no—
238C2×C4×C61—24no—
239C2×C4×C61—21no—
240C2×C4×C61—24no—
241C2×C4×C61—24no—
242C2×C4×C61—21no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61#12112yes—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D412yes—
251C2×C4×C61C2×#4112yes—
252C2×C4×C61C2×#4111yescharacteristic
253C22⋊D61—84no—
254C22⋊D61—84no—
255C22⋊D61—84no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—28no—
261C2×Dic61—21no—
262C2×Dic61—24no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—44no—
266C2×Dic61—48no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61#12012yes—
273C2×Dic61#12012yes—
274C2×Dic61C2×SD1612yes—
275C2×Dic61C2×SD1612yes—
276C2×Dic61#13712yes—
277D244—42no—
278D244—42no—
279C8×C61—42no—
280#92418—42no—
281#32903—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61#46014yescomplemented
296C22×C61C2×#12011yescharacteristic
297C22×C61C2×#12111yescharacteristic
298C22×C61#52511yescharacteristic
299C22×C61C22×SD1611yescharacteristic
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61C2×#13711yescharacteristic
302C22×C61#57511yescharacteristic
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C4×C61—28no—
311C4×C61—22no—
312C4×C61—81no—
313C4×C61C2×#12112yes—
314C4×C61#52512yes—
315Dic61—22no—
316Dic61—22no—
317Dic61—41no—
318Dic61—44no—
319Dic61—81no—
320Dic61#44814yescomplemented
321C22×C4×Q8—611no—
322C2×#460—612no—
323C2×#448—611no—
324C2×#448D6111yescharacteristic, complemented
325#2856—611no—
326#1880—612no—
327#1880—612no—
328#1874—611no—
329#1874—611no—
330#1803—612no—
331#1797—611no—
332C2×C61—24no—
333C2×C61—161no—
334C2×C61#284711yescharacteristic
335C2×C61C2×#44811yescharacteristic
336C2×C61#300611yescharacteristic
337C2×C61C2×#46011yescharacteristic
338C2×C61#303711yescharacteristic
339C2×C61#310311yescharacteristic
340C2×C61#311211yescharacteristic
341D61—161no—
342C23×Q8C2×D6111yescharacteristic
343C22×SD16—612no—
344C22×SD16C2×D6112yes—
345C2×#127—611no—
346C2×#121—611no—
347C2×#121—1221no—
348C2×#121—1221no—
349C2×#121C2×D6111yescharacteristic
350C2×C4×Q8—612no—
351C2×C4×Q8—612no—
352C2×C4×Q8—1221no—
353C2×C4×Q8—1221no—
354C2×C4×Q8—1228no—
355C22×C4⋊C4—611no—
356C22×C4⋊C4—1221no—
357C22×C42—611no—
358C22×C42—1221no—
359#460—1228no—
360#448—24no—
361#448—1224no—
362C2×C4⋊C8—612no—
363C2×#103—612no—
364C2×#103C2×D6112yes—
365C2×#102—612no—
366C2×#102—612no—
367C2×C22⋊C8—611no—
368C2×C22⋊C8C2×D6111yescharacteristic
369#390—611no—
370#390—611no—
371#390—1221no—
372#390—1221no—
373#384—611no—
374#340—612no—
375#340—612no—
376#336—612no—
377C61#3290311yescharacteristic, complemented, Sylow
378C22×Q8—24no—
379C22×Q8—24no—
380C22×Q8—21no—
381C22×Q8—22no—
382C22×Q8C22⋊D6114yescomplemented
383C22×Q8C22×D6112yes—
384C22×D4—21no—
385C22×D4—611no—
386C22×D4—1221no—
387C22×D4—1221no—
388C22×D4—1221no—
389C22×D4C22×D6111yescharacteristic
390C23×C4—21no—
391C23×C4—611no—
392C23×C4—611no—
393C23×C4—1221no—
394C23×C4C22×D6111yescharacteristic
395C2×SD16—24no—
396C2×SD16—24no—
397C2×SD16—24no—
398C2×SD16—1224no—
399C2×SD16—1224no—
400C2×SD16—1224no—
401C22×C8—612no—
402C22×C8C22×D6112yes—
403#127—1224no—
404#121—22no—
405#121—22no—
406#121—1222no—
407#121—1222no—
408#121—2444no—
409#121—2444no—
410C4×Q8—618no—
411C4×Q8—1224no—
412C4×Q8—1224no—
413C4×Q8—1228no—
414C4×Q8—1228no—
415C4×Q8—1224no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1224no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—1228no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1224no—
437C2×C42—1222no—
438C2×C42—1222no—
439C2×C42—1228no—
440C4⋊C8—1224no—
441#103—24no—
442#103—1224no—
443#102—1224no—
444#102—1224no—
445C22⋊C8—22no—
446C22⋊C8—1222no—
447C24—21no—
448C24—1221no—
449C24D4×D6111yescharacteristic
450C2×Q8—24no—
451C2×Q8—216no—
452C2×Q8—24no—
453C2×Q8—216no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—24no—
458C2×Q8D4×D6114yes—
459C2×Q8#1175822414yes—
460C2×Q8C2×C22⋊D6114yes—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—1222no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1222no—
472C2×D4—2442no—
473C2×D4—2442no—
474C2×D4—2442no—
475C2×D4—2442no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4—2442no—
479C2×D4—2442no—
480C2×D4—2442no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—614no—
492C22×C4—611no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1224no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1221no—
500C22×C4—2441no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4D4×D6111yescharacteristic
504C22×C4#1175822412yes—
505C22×C4C2×C22⋊D6112yes—
506C22×C4C23×D6111yescharacteristic
507SD16—48no—
508SD16—2448no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—1222no—
513C2×C8—1222no—
514C2×C8—1222no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—614no—
518C4⋊C4—1228no—
519C4⋊C4—1228no—
520C4⋊C4—1224no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C22⋊C4—42no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C42—614no—
533C42—1222no—
534C42—1222no—
535C42—1228no—
536C42—1228no—
537C42—1222no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23#1451974912yes—
557C23SD16×D6112yes—
558C23#1451980412yes—
559C23C2×D4×D6111yescharacteristic
560Q8—216no—
561Q8—24no—
562Q8—24no—
563Q8—24no—
564Q8#1451974918yescomplemented
565D4—42no—
566D4—42no—
567D4—84no—
568D4—4884no—
569D4—4884no—
570D4—4884no—
571D4—2442no—
572D4—2442no—
573C2×C4—216no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—612no—
589C2×C4—1228no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1224no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—2444no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2448no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4#1451974912yes—
606C2×C4C2×D4×D6112yes—
607C2×C4C22×C22⋊D6111yescharacteristic
608C2×C4C2×#1175822412yes—
609C2×C4C2×#1175822411yescharacteristic
610C8—42no—
611C8—2442no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—4881no—
629C22—4881no—
630C22—4881no—
631C22—4881no—
632C22#1683996014yescomplemented
633C22#120×D6111yescharacteristic
634C22C2×#1451974911yescharacteristic
635C22#1684026811yescharacteristic
636C22C2×SD16×D6111yescharacteristic
637C22C2×#1451980411yescharacteristic
638C22#137×D6111yescharacteristic
639C22#1684056111yescharacteristic
640C4—28no—
641C4—22no—
642C4—81no—
643C4—614no—
644C4—1222no—
645C4—1222no—
646C4—4881no—
647C4—2441no—
648C4—2444no—
649C4C2×#1451974912yes—
650C4#1684026812yes—
651C2—24no—
652C2—161no—
653C2—9761no—
654C2#2145409911yescharacteristic
655C2#448×D6111yescharacteristic
656C2#2145507011yescharacteristic
657C2C2×#1683996011yescharacteristic
658C2#2145528811yescharacteristic
659C2#2145584511yescharacteristic
660C2#2145593611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#448C2
socleC23×C61C22×D4
prime radical#21454162C2
hypercenterC2×#448D61
hypocenterC61#32903
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic61C2semidirect
2. C2×#16839960C2semidirect
3. C2×#16839901C2semidirect
5. #21446176C2semidirect
6. #21446172C2semidirect
7. #21446137C2semidirect
8. #21446134C2semidirect
9. #21445822C2semidirect
10. #21445797C2semidirect
11. C2×C61×#448C2semidirect
68. Q8×Dic61D4semidirect
100. C22×Q8×C61D4semidirect
167. C22×Dic61SD16semidirect
233. Q8×C61#121semidirect
295. C22×C61#460semidirect
320. Dic61#448semidirect
324. C2×#448D61semidirect
377. C61#32903semidirect
382. C22×Q8C22⋊D61semidirect
564. Q8#14519749semidirect
632. C22#16839960semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic612
C2×#168399602
C2×#168399012
#214541622
#214461762
#214461722
#214461372
#214461342
#214458222
#214457972
C2×C61×#4482
#3290361