Finite Group Databasebeta

← #29732005

Subgroups — #29732005

Order 15616.

770 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840042C211yesmaximal, characteristic
2C2×#16839890C211yesmaximal, characteristic, complemented
3#21451173C211yesmaximal, characteristic
4#21451027C211yesmaximal, characteristic, complemented
5#21447316C211yesmaximal, characteristic, complemented
6#21447311C211yesmaximal, characteristic, complemented
7#21447203C211yesmaximal, characteristic
8#21447189C211yesmaximal, characteristic
9#21446792C211yesmaximal, characteristic, complemented
10#21446587C211yesmaximal, characteristic, complemented
11#21446561C211yesmaximal, characteristic, complemented
12#21446551C211yesmaximal, characteristic, complemented
13#21446518C211yesmaximal, characteristic
14#21446504C211yesmaximal, characteristic
15C61×#1762C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18#16840042—24no—
19#16839890—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839698—21no—
32#16839683—21no—
33#16839683—21no—
34#16839674—21no—
35#16839668—21no—
36#16839665C2211yescharacteristic
37#16839665C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839661C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839658C2211yescharacteristic
44#16839635—21no—
45#16839635—21no—
46#16839632—21no—
47#16839628C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122D411yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519777—22no—
66#14519777—22no—
67C4×Dic122—22no—
68C4×Dic122—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×#11758196D411yescharacteristic
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519710—22no—
90#14519710—22no—
91#14519710—22no—
92#14519708—22no—
93C4⋊#92418D414yes—
94#14519706—22no—
95#14519706—22no—
96#14519698—22no—
97#14519698—22no—
98#14519687—22no—
99C8×Dic61—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108C22×D4×C61C2311yescharacteristic
109C23×C4×C61—21no—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C42×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#102×C61—22no—
121#102×C61Q814yescomplemented
122#97×C61—22no—
123#95×C61—21no—
124C24×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61C2×Q812yes—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—81no—
136C22×C4×C61—81no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—41no—
151C2×Dic122—42no—
152C2×Dic122—81no—
153C2×Dic122—81no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C2×Q812yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61C2×Q812yes—
165C4⋊C4×C61#4112yes—
166C22⋊C4×C61—42no—
167C42×C61—42no—
168C42×C61C2×D412yes—
169C42×C61#4112yes—
170C22⋊Dic61—42no—
171C22⋊Dic61—42no—
172C22⋊Dic61—81no—
173C22⋊Dic61—81no—
174C4⋊Dic61—22no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—42no—
184C4⋊Dic61—81no—
185C4⋊Dic61—81no—
186C4⋊Dic61—81no—
187C4⋊Dic61—81no—
188#11758196—22no—
189#11758196—22no—
190#11758196—81no—
191#11758196—81no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418C2×D412yes—
201C2×#92418C2×D412yes—
202C23×C61—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61C22×D411yescharacteristic
210D4×C61—22no—
211D4×C61—22no—
212D4×C61—41no—
213D4×C61—41no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—84no—
228C2×C4×C61#12112yes—
229C2×C4×C61#12112yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×Q811yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61#13712yes—
260Dic122—44no—
261Dic122—84no—
262C8×C61—22no—
263C8×C61—41no—
264#92418—22no—
265#92418—22no—
266#25481—611nomaximal, Sylow
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61C2×#12111yescharacteristic
275C22×C61#54811yescharacteristic
276C22×C61D4×Q811yescharacteristic
277C22×C61C2×#13711yescharacteristic
278C22×C61#57511yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57611yescharacteristic
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61—81no—
286C4×C61#53311yescharacteristic
287C4×C61#54111yescharacteristic
288C4×C61#54711yescharacteristic
289C4×C61D4×Q811yescharacteristic
290Dic61—22no—
291Dic61—41no—
292Dic61—41no—
293Dic61—41no—
294Dic61—81no—
295Dic61—81no—
296Dic61—81no—
297Dic61—81no—
298C2×#474—611no—
299C2×#446—611no—
300#2580—611no—
301#2567—611no—
302#2087—611no—
303#2086—611no—
304#2067—611no—
305#2065—611no—
306#1993—611no—
307#1949—611no—
308#1944—611no—
309#1942—611no—
310#1936—611no—
311#1934—611no—
312#1762D6111yescharacteristic
313C2×C61—82no—
314C2×C61#242911yescharacteristic
315C2×C61#305611yescharacteristic
316C2×C61#305811yescharacteristic
317C2×C61#337011yescharacteristic
318C2×C61#337511yescharacteristic
319C2×C61#338511yescharacteristic
320C2×C61#338911yescharacteristic
321C2×C4⋊Q8—611no—
322C2×C4×Q8—611no—
323C2×C4×D4C2×D6111yescharacteristic
324#474—1224no—
325#446—1224no—
326C2×#107—611no—
327C2×#107—611no—
328C2×#107—611no—
329C2×#106—611no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8—611no—
332C2×C4⋊C8—611no—
333C2×#102—611no—
334C2×#102—611no—
335C2×#102C2×D6111yescharacteristic
336C2×#102C2×D6111yescharacteristic
337C2×#97—611no—
338C2×#97C2×D6111yescharacteristic
339C2×C4×C8—611no—
340#399—1221no—
341#395—1221no—
342#395—1221no—
343#393—1221no—
344#391—1221no—
345#390—611no—
346#390—611no—
347#389—611no—
348#389—611no—
349#389—611no—
350#389—611no—
351#389—611no—
352#389—611no—
353#385—1221no—
354#385—1221no—
355#384—1221no—
356#382—611no—
357C4×C4⋊C4C2×D6111yescharacteristic
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340C2×D6111yescharacteristic
365#340C2×D6111yescharacteristic
366C61#2548111yescharacteristic, complemented, Sylow
367C22×Q8—611no—
368C22×Q8—1221no—
369C22×D4C22×D6111yescharacteristic
370C23×C4—21no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C4⋊Q8—1222no—
376C4⋊Q8—1222no—
377C4×Q8—1222no—
378C4×Q8—1222no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C42—21no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42C22×D6111yescharacteristic
407#107—1222no—
408#107—1222no—
409#107—1222no—
410#106—1222no—
411C4⋊C8—614no—
412C4⋊C8—1222no—
413C4⋊C8—1222no—
414#102—22no—
415#102—1222no—
416#102—1222no—
417#102Dic12214yescomplemented
418#97—22no—
419#97—1222no—
420C4×C8—1222no—
421#95—21no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430C24—21no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—4881no—
434C2×Q8—4881no—
435C2×Q8—2441no—
436C2×Q8—2442no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×Dic12212yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—612no—
466C2×C8—612no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8C2×Dic12212yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—4881no—
490C4⋊C4—4881no—
491C4⋊C4—4881no—
492C4⋊C4—4881no—
493C4⋊C4—2442no—
494C4⋊C4C2×Dic12212yes—
495C4⋊C4#1175822212yes—
496C22⋊C4—42no—
497C22⋊C4—4881no—
498C22⋊C4—4881no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—42no—
502C42—1221no—
503C42—1221no—
504C42—1221no—
505C42—1221no—
506C42#1175822212yes—
507C42D4×D6112yes—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D6111yescharacteristic
516Q8—4884no—
517Q8—2444no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—612no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4#1451975012yes—
561C2×C4#1451975012yes—
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C22×Dic12211yescharacteristic
564C2×C4C2×#1175822211yescharacteristic
565C2×C4C2×#1175822211yescharacteristic
566C2×C4#1451982411yescharacteristic
567C2×C4#1451982411yescharacteristic
568C8—22no—
569C8—41no—
570C8—1222no—
571C8—1222no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22C2×#1451975011yescharacteristic
580C22#1684042111yescharacteristic
581C22D4×Dic12211yescharacteristic
582C22#137×D6111yescharacteristic
583C22#1684055811yescharacteristic
584C22#1684056611yescharacteristic
585C22#1684056611yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—1222no—
592C4—4881no—
593C4—4881no—
594C4—4881no—
595C4—4881no—
596C4—2441no—
597C4—2441no—
598C4—2441no—
599C4#1684031911yescharacteristic
600C4#1684037411yescharacteristic
601C4#1684041711yescharacteristic
602C4D4×Dic12211yescharacteristic
603C2—82no—
604C2#2144941811yescharacteristic
605C2#2145545211yescharacteristic
606C2#2145546611yescharacteristic
607C2#2145857711yescharacteristic
608C2#2145863411yescharacteristic
609C2#2145873011yescharacteristic
610C2#2145877811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1762D61
hypocenterC61#25481
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16839890C2semidirect
4. #21451027C2semidirect
5. #21447316C2semidirect
6. #21447311C2semidirect
9. #21446792C2semidirect
10. #21446587C2semidirect
11. #21446561C2semidirect
12. #21446551C2semidirect
121. #102×C61Q8semidirect
366. C61#25481semidirect
417. #102Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168400422
C2×#168398902
#214511732
#214510272
#214473162
#214473112
#214472032
#214471892
#214467922
#214465872
#214465612
#214465512
#214465182
#214465042
C61×#17622
#2548161