Finite Group Databasebeta

← #29707697

Subgroups — #29707697

Order 15616.

1522 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×#16840310C211yesmaximal, characteristic
2C2×#16839755C212yesmaximal, complemented
3C2×#16839755C212yesmaximal, complemented
4#21448300C211yesmaximal, characteristic, complemented
5#21447066C211yesmaximal, characteristic, complemented
6#21447065C211yesmaximal, characteristic, complemented
7#21447059C212yesmaximal, complemented
8#21447058C212yesmaximal, complemented
9#21447050C211yesmaximal, characteristic, complemented
10#336×D61C211yesmaximal, characteristic, complemented
11C61×#2041C211yesmaximal, characteristic, complemented
12C22×#11758224C2212yes—
13C22×C8×D61C2212yes—
14#16840310—28no—
15C2×#14519777C2211yescharacteristic
16C2×#14519764C2212yes—
17C2×#14519748—21no—
18C2×#14519748—21no—
19C2×D4×Dic61—21no—
20C2×D4×Dic61—21no—
21C2×#14519735C2211yescharacteristic
22C2×#14519735C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16839755—28no—
25#16839755—28no—
26C2×#14519702C2212yes—
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519698C2212yes—
31C2×#14519698C2212yes—
32#16839661C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839636—21no—
36#16839633—21no—
37C4⋊C4×Dic61—21no—
38#16839613—21no—
39#16839517C2211yescharacteristic
40#16839516C2212yes—
41C2×#127×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×#102×C61C2212yes—
44C61×#389C2211yescharacteristic
45C61×#336C2211yescharacteristic
46C2×#11758224—24no—
47C2×#11758224—24no—
48C2×#11758224—24no—
49C2×#11758224—24no—
50C2×#11758224—24no—
51C2×#11758224D418yes—
52C22×Dic122C2312yes—
53C22×C22⋊D61—22no—
54C23×Dic61—22no—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C22×#92418C2312yes—
59C2×C8×D61—24no—
60C2×C8×D61—24no—
61C2×C8×D61—24no—
62#14519777—24no—
63#14519764—24no—
64#14519764—24no—
65#14519748—28no—
66#14519748—28no—
67D4×Dic61—28no—
68D4×Dic61—28no—
69#14519735—22no—
70#14519735—24no—
71#14519735D414yescomplemented
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—22no—
80C2×C22⋊Dic61—22no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61—24no—
89#14519702—24no—
90#14519702—24no—
91#14519700—24no—
92#14519700—24no—
93#14519698—24no—
94#14519698—24no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98C22×D4×C61—21no—
99C22×D4×C61—21no—
100C22×D4×C61C2312yes—
101C22×C8×C61C2312yes—
102#127×C61—24no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61C2311yescharacteristic
107#102×C61—24no—
108#102×C61—24no—
109#95×C61—21no—
110C24×C61—22no—
111C23×D61C2×D411yescharacteristic
112C2×D4×C61—24no—
113C2×D4×C61—24no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61C2×D414yes—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C2×C22⋊D61—28no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—44no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—28no—
135C22×Dic61—21no—
136C22×Dic61—44no—
137C22×Dic61—44no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141#11758224—216no—
142#11758224—216no—
143#11758224—48no—
144#11758224—48no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×C4×D61C2×D412yes—
152C2×C4×D61C2×D414yes—
153C2×C4×D61#4114yes—
154C2×C4×D61#4112yes—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122C2×D414yes—
161C2×C8×C61—22no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—42no—
167C4⋊C4×C61C2×D412yes—
168C42×C61—22no—
169C22⋊Dic61—28no—
170C22⋊Dic61—28no—
171#11758198—24no—
172#11758198—42no—
173C4⋊Dic61—24no—
174C4⋊Dic61—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61—22no—
177C4⋊Dic61—42no—
178C4⋊Dic61C2×D412yes—
179#11758196—42no—
180#11758196—42no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61—24no—
185C4×Dic61—42no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C8×D61—48no—
190C23×C61—24no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61C22×D411yescharacteristic
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61#12012yes—
203C22×D61#12112yes—
204C22×D61#12412yes—
205D4×C61—24no—
206D4×C61—24no—
207D4×C61—48no—
208D4×C61—48no—
209D4×C61—42no—
210D4×C61—42no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61#13512yes—
222C2×C4×C61#13512yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D412yes—
225C2×C4×C61C2×#4112yes—
226C2×C4×C61C2×#4111yescharacteristic
227C22⋊D61—416no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—24no—
231C2×Dic61—22no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—416no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61#12012yes—
245C2×Dic61#12112yes—
246C2×Dic61#12412yes—
247C4×D61—44no—
248C4×D61—44no—
249C4×D61#12118yes—
250C4×D61#12318yes—
251Dic122—24no—
252Dic122—24no—
253Dic122—42no—
254Dic122—42no—
255C8×C61—42no—
256#92418—42no—
257#23504—611nomaximal, Sylow
258C22×C61—44no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—82no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61C2×#12011yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×#12411yescharacteristic
268C22×C61C2×#13511yescharacteristic
269C22×C61C2×#13511yescharacteristic
270C22×C61#57511yescharacteristic
271C22×C61#57511yescharacteristic
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61#39414yes—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—42no—
282C4×C61—81no—
283C4×C61C2×#12112yes—
284C4×C61C2×#12312yes—
285Dic61—44no—
286Dic61—41no—
287Dic61—41no—
288Dic61—42no—
289Dic61—81no—
290Dic61#39414yes—
291C2×#532—611no—
292C2×#416—612no—
293C2×#416—612no—
294#2296—611no—
295#2043—611no—
296#2043—611no—
297#2042—612no—
298#2042—612no—
299#2041—611no—
300#2041D6111yescharacteristic, complemented
301C2×#336—611no—
302C2×C61—84no—
303C2×C61C2×#39411yescharacteristic
304C2×C61#300211yescharacteristic
305C2×C61#300211yescharacteristic
306C2×C61#303511yescharacteristic
307C2×C61#303511yescharacteristic
308C2×C61#313111yescharacteristic
309C2×C61#313511yescharacteristic
310D61—24no—
311C22×#41—612no—
312C23×C8—612no—
313#532—1228no—
314C2×C4⋊Q8—611no—
315C2×#127C2×D6111yescharacteristic
316C2×#124—612no—
317C2×#121—1221no—
318C2×#121—1221no—
319C2×C4×D4—1221no—
320C2×C4×D4—1221no—
321C2×#117—611no—
322C2×#117—611no—
323C22×C42—611no—
324#416—1228no—
325#416—1228no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×#102—612no—
331C2×#102—612no—
332C2×#102C2×D6112yes—
333C2×#102C2×D6112yes—
334#389—611no—
335#389—611no—
336#389—611no—
337#389C2×D6111yescharacteristic
338#385—1221no—
339#385—1221no—
340C4×C4⋊C4—1221no—
341C4×C4⋊C4—1221no—
342#336—612no—
343#336—611no—
344#336C2×D6111yescharacteristic
345C61#2350411yescharacteristic, complemented, Sylow
346C2×#41—618no—
347C2×#41—1224no—
348C2×#41—1224no—
349C2×#41—1224no—
350C2×#41—1224no—
351C2×#41—1224no—
352C22×Q8—612no—
353C22×D4—21no—
354C22×D4—21no—
355C22×D4—1222no—
356C22×D4C22×D6112yes—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—1222no—
361C22×C8—612no—
362C22×C8—1224no—
363C22×C8—1224no—
364C22×C8—1224no—
365C22×C8C22×D6112yes—
366C4⋊Q8—1224no—
367#127—24no—
368#124—1224no—
369#124—1224no—
370#121—1228no—
371#121—1228no—
372C4×D4—1228no—
373C4×D4—1228no—
374#117—614no—
375#117—1222no—
376#117—1224no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1221no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—1224no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398#103—1224no—
399#103—1224no—
400#103—1224no—
401#103—1224no—
402#102—24no—
403#102—24no—
404#102—1224no—
405#102—1224no—
406#95—21no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410C24—22no—
411C24—611no—
412#41—12216no—
413#41—12216no—
414#41—2448no—
415#41—2448no—
416C2×Q8—614no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—1228no—
434C2×D4—2444no—
435C2×D4—2444no—
436C2×D4D4×D6114yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—611no—
440C22×C4—614no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—614no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1228no—
449C22×C4—1221no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1221no—
454C22×C4—2441no—
455C22×C4—2441no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4D4×D6111yescharacteristic
459C22×C4D4×D6111yescharacteristic
460C22×C4C23×D6111yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—2448no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—612no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C4⋊C4—1221no—
475C4⋊C4—1222no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4D4×D6112yes—
480C22⋊C4—1224no—
481C22⋊C4—1228no—
482C22⋊C4—1228no—
483C22⋊C4—2442no—
484C42—22no—
485C42—1222no—
486C42—1222no—
487C42—1224no—
488C42—1222no—
489C42—2442no—
490C23—24no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—612no—
498C23—612no—
499C23—612no—
500C23—1221no—
501C23—1221no—
502C23—1221no—
503C23—1221no—
504C23C2×D4×D6111yescharacteristic
505Q8—1224no—
506Q8—1224no—
507Q8—2442no—
508Q8—2442no—
509D4—24no—
510D4—24no—
511D4—48no—
512D4—48no—
513D4—42no—
514D4—42no—
515D4—24416no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—618no—
530C2×C4—618no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1224no—
534C2×C4—1222no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—2444no—
540C2×C4—2444no—
541C2×C4—24416no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4#1451979612yes—
550C2×C4#1451979812yes—
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C2×D4×D6112yes—
553C2×C4#41×D6112yes—
554C2×C4#41×D6111yescharacteristic
555C8—42no—
556C8—2442no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—614no—
565C22—1222no—
566C22—1222no—
567C22—1222no—
568C22—1222no—
569C22—1222no—
570C22—1222no—
571C22#120×D6111yescharacteristic
572C22#121×D6111yescharacteristic
573C22#124×D6111yescharacteristic
574C22C2×#1451979611yescharacteristic
575C22C2×#1451979811yescharacteristic
576C22#1684055811yescharacteristic
577C22#1684055811yescharacteristic
578C4—22no—
579C4—22no—
580C4—42no—
581C4—81no—
582C4—614no—
583C4—4881no—
584C4—2444no—
585C4—2442no—
586C4—2441no—
587C4—2441no—
588C4#121×D6112yes—
589C4#123×D6112yes—
590C2—84no—
591C2—1224no—
592C2#394×D6111yescharacteristic
593C2#2145503211yescharacteristic
594C2#2145503811yescharacteristic
595C2#2145526711yescharacteristic
596C2#2145526911yescharacteristic
597C2#2145613811yescharacteristic
598C2#2145616311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2041C2
socleC23×C61C22×D4
prime radicalC2×#16840310C2
hypercenter#2041D61
hypocenterC61#23504
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
2. C2×#16839755C2semidirect
3. C2×#16839755C2semidirect
4. #21448300C2semidirect
5. #21447066C2semidirect
6. #21447065C2semidirect
7. #21447059C2semidirect
8. #21447058C2semidirect
9. #21447050C2semidirect
10. #336×D61C2semidirect
11. C61×#2041C2semidirect
71. #14519735D4semidirect
300. #2041D61semidirect
345. C61#23504semidirect

Maximal subgroups

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

subgroupindex
C2×#168403102
C2×#168397552
#214483002
#214470662
#214470652
#214470592
#214470582
#214470502
#336×D612
C61×#20412
#2350461