Finite Group Databasebeta

← #29709812

Subgroups — #29709812

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839799C211yesmaximal, characteristic, complemented
2C2×#16839797C211yesmaximal, characteristic, complemented
3C2×#16839757C211yesmaximal, characteristic
4C2×#16839757C211yesmaximal, characteristic
5#21448308C211yesmaximal, characteristic
6#21448225C211yesmaximal, characteristic, complemented
7#21447367C211yesmaximal, characteristic, complemented
8#21447360C211yesmaximal, characteristic, complemented
9#21447355C211yesmaximal, characteristic, complemented
10#21447349C211yesmaximal, characteristic, complemented
11#21447344C211yesmaximal, characteristic, complemented
12#21447343C211yesmaximal, characteristic, complemented
13#21447339C211yesmaximal, characteristic, complemented
14#336×D61C211yesmaximal, characteristic, complemented
15C61×#2097C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519735C2211yescharacteristic
22C2×C42×D61C2211yescharacteristic
23#16839799—24no—
24#16839797—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31#16839757—24no—
32#16839757—24no—
33C2×#14519701C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37C2×#14519697C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39#16839669C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839663C2211yescharacteristic
42#16839640—22no—
43#16839631C2211yescharacteristic
44#16839631C2211yescharacteristic
45#16839628C2211yescharacteristic
46#16839621—22no—
47#16839613—22no—
48#16839609—22no—
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#107×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C2×#11758226—22no—
60C2×#11758226—22no—
61C2×#11758226—22no—
62C2×#11758226—22no—
63C2×#11758226—22no—
64C2×#11758226Q814yes—
65C22×D244—21no—
66C22×D244C2311yescharacteristic
67C22×C4×D61—21no—
68C22×C4×D61D412yes—
69C22×C4×D61C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C2×C8×D61—22no—
73C2×C8×D61—22no—
74C2×C8×D61—22no—
75C2×C8×D61—22no—
76C2×C8×D61D414yes—
77C2×C8×D61D414yes—
78#14519735—22no—
79#14519735—22no—
80#14519735—22no—
81#14519735—22no—
82#14519735—22no—
83#14519735—22no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—22no—
88C2×#11758196—22no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—22no—
93C2×#11758198—22no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61D412yes—
98C2×C4×Dic61C2311yescharacteristic
99C42×D61—22no—
100C42×D61—22no—
101#14519711—22no—
102#14519710—22no—
103#14519710—22no—
104#14519709—22no—
105#14519708—22no—
106#14519708—22no—
107#14519701—22no—
108#14519701—22no—
109#14519699—22no—
110#14519699—22no—
111#14519697—22no—
112#14519697—22no—
113#14519684—22no—
114#14519684—22no—
115C22×Q8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122#107×C61—22no—
123#106×C61—22no—
124#103×C61—22no—
125#103×C61—22no—
126#95×C61—22no—
127#95×C61—22no—
128C23×D61—21no—
129C23×D61C2×D411yescharacteristic
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61C2×Q812yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D411yescharacteristic
147#11758226—28no—
148#11758226—28no—
149#11758226—44no—
150#11758226—44no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—24no—
156C2×D244—21no—
157C2×D244—42no—
158C2×D244—42no—
159C2×D244C2×Q812yes—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61C2×D412yes—
167C2×C4×D61C2×D412yes—
168C2×C4×D61C2×Q812yes—
169C2×C4×D61#4112yes—
170C2×C4×D61#4112yes—
171C2×C4×D61#4112yes—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D412yes—
177C2×C8×C61C2×D412yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186#11758198—44no—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190#11758198—44no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197#11758196—44no—
198#11758196—44no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—24no—
202C4×Dic61—42no—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D412yes—
210C2×#92418C2×D412yes—
211C8×D61—28no—
212C8×D61—28no—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61C22⋊Q812yes—
222C22×D61#12312yes—
223C22×D61#12712yes—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—41no—
227Q8×C61—41no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×Q811yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61C22⋊Q812yes—
255C2×Dic61#12312yes—
256C2×Dic61#12712yes—
257C2×Dic61#13514yes—
258D244—22no—
259D244—22no—
260D244—48no—
261D244—41no—
262D244—41no—
263C4×D61—48no—
264C4×D61—48no—
265C4×D61C22⋊Q814yes—
266C4×D61C22⋊Q814yes—
267C4×D61#12314yes—
268C4×D61#12314yes—
269C8×C61—22no—
270C8×C61—22no—
271#92418—22no—
272#92418—22no—
273#23730—611nomaximal, Sylow
274C22×C61C2×C22⋊Q811yescharacteristic
275C22×C61C2×#12311yescharacteristic
276C22×C61C2×#12711yescharacteristic
277C22×C61C2×#13512yes—
278C22×C61#57512yes—
279C2×D61—24no—
280C2×D61—24no—
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—44no—
284C2×D61—42no—
285C2×D61—82no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61#39914yes—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61—81no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61C2×C22⋊Q811yescharacteristic
295C4×C61C2×C22⋊Q811yescharacteristic
296C4×C61C2×#12311yescharacteristic
297C4×C61C2×#12311yescharacteristic
298Dic61—24no—
299Dic61—81no—
300Dic61—81no—
301Dic61—81no—
302Dic61#39914yes—
303C2×#427—611no—
304C2×#427—611no—
305C2×#416—611no—
306C2×#416—611no—
307#2297—611no—
308#2285—611no—
309#2097—611no—
310#2097D6111yescharacteristic, complemented
311#2096—611no—
312#2095—611no—
313#2094—611no—
314#2093—611no—
315#2092—611no—
316#2092—611no—
317C2×#336—611no—
318C2×C61C2×#39911yescharacteristic
319C2×C61#307612yes—
320C2×C61#309112yes—
321C2×C61#314711yescharacteristic
322C2×C61#315011yescharacteristic
323D61—24no—
324D61—82no—
325C22×#41—611no—
326C23×C8—611no—
327C23×C8—611no—
328C2×#117—611no—
329C2×#117—611no—
330C2×#117—611no—
331C22×C42—611no—
332#427—1224no—
333#427—1224no—
334C2×#107—611no—
335C2×#107—611no—
336C2×#107—611no—
337C2×#107C2×D6111yescharacteristic
338C2×#106—611no—
339C2×#106—611no—
340C2×#106—611no—
341C2×#106C2×D6111yescharacteristic
342#416—1224no—
343#416—1224no—
344C2×#103—611no—
345C2×#103—611no—
346C2×#103C2×D6111yescharacteristic
347C2×#103C2×D6111yescharacteristic
348C2×#102—611no—
349C2×#102—611no—
350C2×#102—611no—
351C2×#102—611no—
352#391—611no—
353#391C2×D6111yescharacteristic
354#390—611no—
355#390—611no—
356#386—1222no—
357#384—611no—
358#384—611no—
359#382—611no—
360#382C2×D6111yescharacteristic
361#380—1222no—
362C4×C4⋊C4—1222no—
363C4×C22⋊C4—1222no—
364#336—611no—
365#336—611no—
366#336—611no—
367#336C2×D6111yescharacteristic
368C61#2373011yescharacteristic, complemented, Sylow
369C2×#41—614no—
370C2×#41—1222no—
371C2×#41—1222no—
372C2×#41—1222no—
373C2×#41—1222no—
374C2×#41—1222no—
375C22×Q8C22×D6111yescharacteristic
376C22×D4—611no—
377C22×D4—1221no—
378C23×C4—611no—
379C23×C4—612no—
380C23×C4—1221no—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8—614no—
384C22×C8—614no—
385C22×C8—1222no—
386C22×C8—1222no—
387C22×C8—1222no—
388C22×C8—1222no—
389C22×C8C22×D6111yescharacteristic
390C22×C8C22×D6111yescharacteristic
391#117—1222no—
392#117—1222no—
393#117—1222no—
394#117—1222no—
395#117—1222no—
396#117—1222no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1222no—
410C2×C42—612no—
411C2×C42—611no—
412C2×C42—1222no—
413C2×C42—1222no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42—1221no—
417C2×C42C22×D6111yescharacteristic
418#107—22no—
419#107—1222no—
420#107—1222no—
421#107—1222no—
422#106—22no—
423#106—1222no—
424#106—1222no—
425#106—1222no—
426#103—22no—
427#103—22no—
428#103—1222no—
429#103—1222no—
430#102—1222no—
431#102—1222no—
432#102—1222no—
433#102—1222no—
434#95—22no—
435#95—22no—
436#95—1222no—
437#95—1222no—
438C24—611no—
439C24—1221no—
440#41—1228no—
441#41—1228no—
442#41—2444no—
443#41—2444no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8Q8×D6112yes—
450C2×D4—612no—
451C2×D4—1224no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—2442no—
458C2×D4—2442no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—611no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—1224no—
472C22×C4—1224no—
473C22×C4—1224no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4D4×D6112yes—
481C22×C4C23×D6111yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—612no—
487C2×C8—612no—
488C2×C8—1228no—
489C2×C8—1228no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8D4×D6112yes—
495C2×C8D4×D6112yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—2444no—
509C4⋊C4—2444no—
510C22⋊C4—2444no—
511C22⋊C4—2444no—
512C22⋊C4—2442no—
513C22⋊C4—2442no—
514C22⋊C4—2442no—
515C42—21no—
516C42—21no—
517C42—1224no—
518C42—1221no—
519C42—1221no—
520C42—2442no—
521C42—2442no—
522C42—2442no—
523C23—612no—
524C23—612no—
525C23—612no—
526C23—1222no—
527C23—1222no—
528C23—1222no—
529C23—2441no—
530C23—2442no—
531C23—2441no—
532C23—2442no—
533C23C2×D4×D6111yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538D4—1222no—
539D4—1222no—
540D4—2448no—
541D4—2441no—
542D4—2441no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—614no—
559C2×C4—614no—
560C2×C4—614no—
561C2×C4—614no—
562C2×C4—614no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—2448no—
566C2×C4—2448no—
567C2×C4—2442no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C2×Q8×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C2×C4#41×D6111yescharacteristic
579C8—22no—
580C8—22no—
581C8—1222no—
582C8—1222no—
583C22—614no—
584C22—1222no—
585C22—1224no—
586C22—1222no—
587C22—1224no—
588C22—4881no—
589C22—4882no—
590C22—4881no—
591C22—2442no—
592C22—2444no—
593C22C22⋊Q8×D6111yescharacteristic
594C22#123×D6111yescharacteristic
595C22#127×D6111yescharacteristic
596C22#135×D6112yes—
597C22#1684056112yes—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—614no—
604C4—1224no—
605C4—4881no—
606C4—4881no—
607C4—4881no—
608C4C22⋊Q8×D6111yescharacteristic
609C4C22⋊Q8×D6111yescharacteristic
610C4#123×D6111yescharacteristic
611C4#123×D6111yescharacteristic
612C2—1224no—
613C2—4882no—
614C2#399×D6111yescharacteristic
615C2#2145560612yes—
616C2#2145573712yes—
617C2#2145624411yescharacteristic
618C2#2145626311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2097C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
hypercenter#2097D61
hypocenterC61#23730
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2097

Complements

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

NH ≅ G/Ntype
1. C2×#16839799C2semidirect
2. C2×#16839797C2semidirect
6. #21448225C2semidirect
7. #21447367C2semidirect
8. #21447360C2semidirect
9. #21447355C2semidirect
10. #21447349C2semidirect
11. #21447344C2semidirect
12. #21447343C2semidirect
13. #21447339C2semidirect
14. #336×D61C2semidirect
15. C61×#2097C2semidirect
310. #2097D61semidirect
368. C61#23730semidirect

Maximal subgroups

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

subgroupindex
C2×#168397992
C2×#168397972
C2×#168397572
#214483082
#214482252
#214473672
#214473602
#214473552
#214473492
#214473442
#214473432
#214473392
#336×D612
C61×#20972
#2373061