Finite Group Databasebeta

← #29679544

Subgroups — #29679544

Order 15616.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840245C211yesmaximal, characteristic
2C2×#16839790C211yesmaximal, characteristic, complemented
3C2×#16839758C211yesmaximal, characteristic
4#21448443C211yesmaximal, characteristic, complemented
5#21445832C212yesmaximal, complemented
6#21445827C212yesmaximal, complemented
7#21445822C212yesmaximal, complemented
8#21445817C212yesmaximal, complemented
9#21445539C211yesmaximal, characteristic, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C2×C61×#417C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13#16840245—24no—
14#16840245C418yescomplemented
15C2×Q8×Dic61C2212yes—
16C2×D4×Dic61—22no—
17C2×D4×Dic61C2212yes—
18C2×#14519736—21no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61—21no—
22C22×C4⋊Dic61C2211yescharacteristic
23#16839790—24no—
24C2×#14519706C2212yes—
25C2×#14519706C2212yes—
26#16839758—24no—
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31#16839658—21no—
32#16839658—21no—
33#16839635—22no—
34#16839631C2212yes—
35#16839627C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40C22×#41×C61C2211yescharacteristic
41C22×M16×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#417—24no—
44C2×#103×C61C2212yes—
45C2×#102×C61C2212yes—
46C23×Dic61—21no—
47C23×Dic61—22no—
48C22×#92418—22no—
49C22×#92418—22no—
50C22×#92418—22no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53Q8×Dic61—24no—
54Q8×Dic61C2×C418yescomplemented
55D4×Dic61—216no—
56D4×Dic61—24no—
57D4×Dic61C2×C418yescomplemented
58#14519736—28no—
59#14519736—22no—
60#14519736—22no—
61#14519736C2×C414yes—
62#14519736D414yes—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—28no—
67C2×C4⋊Dic61C2×C414yes—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×#11758196—22no—
75C2×C22⋊Dic61—22no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61D412yes—
78C2×C4×Dic61—22no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61C2312yes—
81#14519706—24no—
82#14519706—24no—
83#14519701—22no—
84#14519701—22no—
85#14519698—22no—
86#14519698—22no—
87#14519684—22no—
88#14519684—22no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C2×#41×C61C2×C414yes—
93C2×#41×C61D414yes—
94C2×#41×C61Q814yes—
95C22×Q8×C61C2311yescharacteristic
96C22×D4×C61—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C2×M16×C61—22no—
102C2×M16×C61—22no—
103C2×M16×C61—22no—
104C22×C8×C61C2312yes—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2312yes—
109#103×C61—24no—
110#102×C61—24no—
111C24×C61—21no—
112C24×C61C2×D411yescharacteristic
113#41×C61—28no—
114#41×C61—28no—
115#41×C61—28no—
116#41×C61C4⋊C4116yescomplemented
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61C22×C412yes—
121C2×Q8×C61C2×D412yes—
122C2×Q8×C61C2×Q812yes—
123C2×D4×C61—24no—
124C2×D4×C61—24no—
125C2×D4×C61—24no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61C22×C412yes—
130C2×D4×C61C2×D412yes—
131C2×D4×C61C2×Q812yes—
132C22×C4×C61—21no—
133C22×C4×C61—24no—
134C22×C4×C61—24no—
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—42no—
138C22×C4×C61—81no—
139C22×C4×C61—81no—
140C22×C4×C61C22×C412yes—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×Q812yes—
145C22×C4×C61#4112yes—
146C22×C4×C61#4112yes—
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—22no—
149C22×Dic61—24no—
150C22×Dic61—28no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—82no—
154C22×Dic61—82no—
155C22×Dic61—81no—
156C22×Dic61—81no—
157C22×Dic61C2×D412yes—
158C22×Dic61C2×D412yes—
159M16×C61—44no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—42no—
166C22⋊Dic61—28no—
167C22⋊Dic61—42no—
168C22⋊Dic61C22⋊C418yes—
169C4⋊Dic61—24no—
170C4⋊Dic61—22no—
171C4⋊Dic61—28no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61C22⋊C414yes—
177C4⋊Dic61C22⋊C414yes—
178C4⋊Dic61C22×C414yes—
179#11758196—82no—
180#11758196—82no—
181C4×Dic61—28no—
182C4×Dic61—42no—
183C4×Dic61C22×C414yes—
184C4×Dic61C2×D414yes—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—44no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61C2×C22⋊C412yes—
204C23×C61#12112yes—
205C23×C61C22⋊Q812yes—
206C23×C61C22×D411yescharacteristic
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61C2×C4⋊C414yes—
211D4×C61—216no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—22no—
215D4×C61C2×C4⋊C414yes—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—216no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—84no—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C2×C4⋊C414yes—
226C2×C4×C61#11714yes—
227C2×C4×C61#12114yes—
228C2×C4×C61#12112yes—
229C2×C4×C61C22⋊Q814yes—
230C2×C4×C61C22⋊Q812yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×Q811yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—88no—
251C2×Dic61—84no—
252C2×Dic61C2×C22⋊C414yes—
253C2×Dic61C2×C22⋊C414yes—
254C8×C61—42no—
255#92418—41no—
256#92418—41no—
257#20652—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—24no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61#38014yes—
272C22×C61C22×C22⋊C411yescharacteristic
273C22×C61C2×#12111yescharacteristic
274C22×C61C2×C22⋊Q811yescharacteristic
275C22×C61#57611yescharacteristic
276C22×C61#57711yescharacteristic
277C22×C61#57711yescharacteristic
278C22×C61#57811yescharacteristic
279C4×C61—24no—
280C4×C61—82no—
281C4×C61#38014yes—
282C4×C61C22×C4⋊C411yescharacteristic
283C4×C61C2×#11711yescharacteristic
284C4×C61C2×#12111yescharacteristic
285C4×C61C2×C22⋊Q811yescharacteristic
286Dic61—24no—
287Dic61—24no—
288Dic61—44no—
289Dic61—81no—
290Dic61—81no—
291C2×#520—611no—
292C2×#424—611no—
293C2×#417D6111yescharacteristic
294C2×#416—611no—
295#2317—611no—
296#1805—612no—
297#1804—612no—
298#1803—612no—
299#1802—612no—
300#1736—611no—
301C2×#340—611no—
302C2×C61—24no—
303C2×C61—44no—
304C2×C61C2×#38011yescharacteristic
305C2×C61#289711yescharacteristic
306C2×C61#289811yescharacteristic
307C2×C61#304511yescharacteristic
308C2×C61#304611yescharacteristic
309C2×C61#308311yescharacteristic
310C2×C61#308411yescharacteristic
311C22×#41C2×D6111yescharacteristic
312C22×M16C2×D6111yescharacteristic
313C23×C8—611no—
314#520—618no—
315#520—1224no—
316C2×C4×Q8—612no—
317C2×C4×D4—612no—
318C2×C4×D4—1222no—
319C2×#117—611no—
320C2×#117—611no—
321C2×#117—1221no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—1221no—
324C22×C4⋊C4C2×D6111yescharacteristic
325#424—1224no—
326C2×C4⋊C8—612no—
327C2×C4⋊C8—612no—
328#417—24no—
329#416—1224no—
330C2×#103—611no—
331C2×#103—611no—
332C2×#103C2×D6112yes—
333C2×#102—611no—
334C2×#102—611no—
335C2×#102C2×D6112yes—
336#389—1221no—
337#389—1221no—
338#385—1222no—
339#384—612no—
340#382—612no—
341#340—612no—
342#340—612no—
343#340—612no—
344#340—612no—
345C61#2065211yescharacteristic, complemented, Sylow
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41Dic12214yes—
350C2×#41C4×D6114yes—
351C2×#41C22⋊D6114yes—
352C22×Q8C22×D6111yescharacteristic
353C22×D4—21no—
354C22×D4C22×D6111yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—1222no—
358C23×C4—1221no—
359C23×C4C22×D6111yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—22no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8—1222no—
366C22×C8—1222no—
367C22×C8—1222no—
368C22×C8C22×D6112yes—
369C4×Q8—618no—
370C4×Q8—1224no—
371C4×D4—618no—
372C4×D4—1224no—
373C4×D4—12216no—
374#117—614no—
375#117—614no—
376#117—1228no—
377#117—1222no—
378#117—1222no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—614no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—1228no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4C22×D6112yes—
395C2×C22⋊C4—612no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1221no—
398C2×C42—612no—
399C2×C42—1222no—
400C2×C42—1221no—
401C4⋊C8—1224no—
402C4⋊C8—1224no—
403#103—24no—
404#103—1222no—
405#103—1222no—
406#102—24no—
407#102—1222no—
408#102—1222no—
409#95—1222no—
410#95—1222no—
411C24—21no—
412C24D4×D6111yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41#11758196116yescomplemented
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C2×Dic12212yes—
421C2×Q8C2×C4×D6112yes—
422C2×Q8C2×C22⋊D6112yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×Dic12212yes—
430C2×D4C2×C4×D6112yes—
431C2×D4C2×C22⋊D6112yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—1228no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—4882no—
448C22×C4—4882no—
449C22×C4—4881no—
450C22×C4—4881no—
451C22×C4C2×Dic12212yes—
452C22×C4C2×C4×D6112yes—
453C22×C4D4×D6112yes—
454C22×C4D4×D6111yescharacteristic
455C22×C4#1175822412yes—
456C22×C4#1175822412yes—
457C22×C4C2×C22⋊D6112yes—
458C22×C4C23×D6111yescharacteristic
459M16—44no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—2444no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4—614no—
474C4⋊C4—614no—
475C4⋊C4—614no—
476C4⋊C4—1228no—
477C4⋊C4—1222no—
478C4⋊C4—1224no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—4882no—
484C4⋊C4—4882no—
485C22⋊C4—618no—
486C22⋊C4—1228no—
487C22⋊C4—2442no—
488C42—614no—
489C42—614no—
490C42—1228no—
491C42—2442no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C22⋊C4×D6112yes—
504C23#1451974912yes—
505C23#1451975412yes—
506C23C2×D4×D6111yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8C2×#1175819614yes—
511D4—216no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C2×#1175819614yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—84no—
524C2×C4—614no—
525C2×C4—614no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1224no—
531C2×C4—4888no—
532C2×C4—4884no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4C22⋊C4×D6112yes—
542C2×C4C2×#1175819614yes—
543C2×C4#1451973314yes—
544C2×C4#1451974914yes—
545C2×C4#1451974912yes—
546C2×C4#1451975414yes—
547C2×C4#1451975412yes—
548C2×C4C22×C4×D6111yescharacteristic
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4C22×Dic12211yescharacteristic
552C2×C4C2×#1175822411yescharacteristic
553C2×C4C2×#1175822411yescharacteristic
554C8—42no—
555C8—2441no—
556C8—2441no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#1683962014yes—
571C22C2×C22⋊C4×D6111yescharacteristic
572C22C2×#1451974911yescharacteristic
573C22C2×#1451975411yescharacteristic
574C22#1684056611yescharacteristic
575C22#1684057111yescharacteristic
576C22#1684057211yescharacteristic
577C22#1684057811yescharacteristic
578C4—24no—
579C4—82no—
580C4—1224no—
581C4—1224no—
582C4—4881no—
583C4—4881no—
584C4—2444no—
585C4#1683962014yes—
586C4C22×#1175819611yescharacteristic
587C4C2×#1451973311yescharacteristic
588C4C2×#1451974911yescharacteristic
589C4C2×#1451975411yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×#1683962011yescharacteristic
593C2#2145434311yescharacteristic
594C2#2145435011yescharacteristic
595C2#2145536211yescharacteristic
596C2#2145537311yescharacteristic
597C2#2145567411yescharacteristic
598C2#2145568511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#417D61
hypocenterC61#20652
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#16839790C2semidirect
4. #21448443C2semidirect
5. #21445832C2semidirect
6. #21445827C2semidirect
7. #21445822C2semidirect
8. #21445817C2semidirect
9. #21445539C2semidirect
10. C2×#16839524C2semidirect
14. #16840245C4semidirect
54. Q8×Dic61C2×C4semidirect
57. D4×Dic61C2×C4semidirect
116. #41×C61C4⋊C4semidirect
345. C61#20652semidirect
416. #41#11758196semidirect

Maximal subgroups

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

subgroupindex
C2×#168402452
C2×#168397902
C2×#168397582
#214484432
#214458322
#214458272
#214458222
#214458172
#214455392
C2×#168395242
C2×C61×#4172
#2065261