Finite Group Databasebeta

← #29732001

Subgroups — #29732001

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×#16839891C211yesmaximal, characteristic, complemented
2#21451173C211yesmaximal, characteristic
3#21451028C211yesmaximal, characteristic, complemented
4#21447313C211yesmaximal, characteristic, complemented
5#21447308C211yesmaximal, characteristic, complemented
6#21447205C211yesmaximal, characteristic
7#21447191C211yesmaximal, characteristic
8#21446793C211yesmaximal, characteristic, complemented
9#21446589C211yesmaximal, characteristic, complemented
10#21446560C211yesmaximal, characteristic, complemented
11#21446550C211yesmaximal, characteristic, complemented
12#21446513C211yesmaximal, characteristic
13#21446499C211yesmaximal, characteristic
14#21445644C211yesmaximal, characteristic
15C2×C61×#474C211yesmaximal, characteristic
16C2×C4×Dic122C2211yescharacteristic
17#16839891—24no—
18C2×#14519711C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519688C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839698—21no—
28#16839681—21no—
29#16839681—21no—
30#16839674—21no—
31#16839667—21no—
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839658C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839635—21no—
41#16839635—21no—
42#16839631—21no—
43#16839628C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C61×#474—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C22×Dic122D411yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C4×Dic122—22no—
65C4×Dic122—22no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196D411yescharacteristic
78C2×#11758196D411yescharacteristic
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519711—22no—
87#14519709—22no—
88#14519706—22no—
89#14519706—22no—
90#14519698—22no—
91#14519698Q814yescomplemented
92#14519688—22no—
93#14519687—22no—
94C8×Dic61—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104C22×Q8×C61—21no—
105C22×D4×C61C2311yescharacteristic
106C23×C4×C61—21no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C4⋊Q8×C61—22no—
110C4⋊Q8×C61—22no—
111C4×D4×C61—22no—
112C4×D4×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#107×C61—22no—
120#107×C61—22no—
121C4⋊C8×C61D414yes—
122#102×C61—22no—
123#102×C61—22no—
124C24×C61—21no—
125C2×Q8×C61—42no—
126C2×Q8×C61—81no—
127C2×Q8×C61—81no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×Q812yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61—81no—
139C22×C4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—41no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C2×D412yes—
159C2×C8×C61C2×D412yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—42no—
167C22⋊C4×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⋊Dic61C2×Q812yes—
182C4⋊Dic61#4112yes—
183#11758196—22no—
184#11758196—22no—
185#11758196—81no—
186#11758196—81no—
187#11758196—81no—
188#11758196—81no—
189#11758196—81no—
190#11758196—81no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—42no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C2×Q812yes—
202C23×C61—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61C22×D411yescharacteristic
210Q8×C61—84no—
211D4×C61—22no—
212D4×C61—22no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—41no—
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—41no—
228C2×C4×C61—84no—
229C2×C4×C61#12112yes—
230C2×C4×C61#12112yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×Q811yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
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—41no—
260C2×Dic61#13712yes—
261Dic122—44no—
262C8×C61—22no—
263C8×C61—22no—
264#92418—22no—
265#92418—41no—
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—81no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61#53311yescharacteristic
287C4×C61#54111yescharacteristic
288C4×C61#54711yescharacteristic
289C4×C61D4×Q811yescharacteristic
290Dic61—22no—
291Dic61—42no—
292Dic61—42no—
293Dic61—41no—
294Dic61—41no—
295Dic61—41no—
296Dic61—81no—
297Dic61—81no—
298C2×#474D6111yescharacteristic
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#1762—611no—
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⋊Q8C2×D6111yescharacteristic
322C2×C4×Q8—611no—
323C2×C4×D4C2×D6111yescharacteristic
324#474—24no—
325#446—1224no—
326C2×#107—611no—
327C2×#107C2×D6111yescharacteristic
328C2×#107C2×D6111yescharacteristic
329C2×#106—611no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8—611no—
332C2×C4⋊C8C2×D6111yescharacteristic
333C2×#102—611no—
334C2×#102—611no—
335C2×#102C2×D6111yescharacteristic
336C2×#102C2×D6111yescharacteristic
337C2×#97—611no—
338C2×#97—611no—
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⋊C4—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366C61#2548111yescharacteristic, complemented, Sylow
367C22×Q8—21no—
368C22×Q8—611no—
369C22×D4C22×D6111yescharacteristic
370C23×C4—21no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C4⋊Q8—22no—
376C4⋊Q8—22no—
377C4×Q8—1222no—
378C4×Q8—1222no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—21no—
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—611no—
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—611no—
404C2×C42—611no—
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407#107—22no—
408#107—22no—
409#107—1222no—
410#106—1222no—
411C4⋊C8—1222no—
412C4⋊C8—1222no—
413C4⋊C8C22⋊D6114yes—
414#102—22no—
415#102—22no—
416#102—614no—
417#102—1222no—
418#97—1222no—
419#97—1222no—
420C4×C8—1222no—
421#95—1221no—
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—42no—
432C2×Q8—81no—
433C2×Q8—81no—
434C2×Q8—1222no—
435C2×Q8—1222no—
436C2×Q8—2441no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4Q8×D6112yes—
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—612no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8C2×C22⋊D6112yes—
471C2×C8C2×C22⋊D6112yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—612no—
480C4⋊C4—612no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—4881no—
491C4⋊C4—4881no—
492C4⋊C4—4881no—
493C4⋊C4—4881no—
494C4⋊C4—4881no—
495C4⋊C4—4881no—
496C22⋊C4—42no—
497C22⋊C4—4881no—
498C22⋊C4—4881no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—1221no—
502C42—1221no—
503C42—1221no—
504C42—1221no—
505C42—2442no—
506C42#1175822412yes—
507C42C2×C22⋊D6112yes—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D6111yescharacteristic
516Q8—84no—
517Q8—2444no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
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—1222no—
549C2×C4—1222no—
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#1451974812yes—
561C2×C4#1451974812yes—
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4C2×Q8×D6111yescharacteristic
564C2×C4#41×D6111yescharacteristic
565C2×C4C2×#1175822411yescharacteristic
566C2×C4#1451982411yescharacteristic
567C2×C4#1451982511yescharacteristic
568C8—22no—
569C8—22no—
570C8—1222no—
571C8—2441no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22C2×#1451974811yescharacteristic
580C22#1684042211yescharacteristic
581C22#1684042511yescharacteristic
582C22#137×D6111yescharacteristic
583C22#1684056111yescharacteristic
584C22#1684056311yescharacteristic
585C22#1684056311yescharacteristic
586C4—22no—
587C4—22no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—1222no—
592C4—4881no—
593C4—4881no—
594C4—2441no—
595C4—2442no—
596C4—2441no—
597C4—2441no—
598C4—2442no—
599C4#1684031411yescharacteristic
600C4#1684037211yescharacteristic
601C4#1684041111yescharacteristic
602C4Q8×C22⋊D6111yescharacteristic
603C2—82no—
604C2#2144941411yescharacteristic
605C2#2145544311yescharacteristic
606C2#2145546311yescharacteristic
607C2#2145857911yescharacteristic
608C2#2145863611yescharacteristic
609C2#2145872411yescharacteristic
610C2#2145877211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#474C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#474D61
hypocenterC61#25481
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#16839891C2semidirect
3. #21451028C2semidirect
4. #21447313C2semidirect
5. #21447308C2semidirect
8. #21446793C2semidirect
9. #21446589C2semidirect
10. #21446560C2semidirect
11. #21446550C2semidirect
91. #14519698Q8semidirect
366. C61#25481semidirect

Maximal subgroups

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

subgroupindex
C2×#168398912
#214511732
#214510282
#214473132
#214473082
#214472052
#214471912
#214467932
#214465892
#214465602
#214465502
#214465132
#214464992
#214456442
C2×C61×#4742
#2548161