Finite Group Databasebeta

← #29731756

Subgroups — #29731756

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840168C211yesmaximal, characteristic, complemented
2#21451015C211yesmaximal, characteristic
3#21448909C211yesmaximal, characteristic, complemented
4#21447303C211yesmaximal, characteristic, complemented
5#21447293C211yesmaximal, characteristic, complemented
6#21447203C211yesmaximal, characteristic
7#21447189C211yesmaximal, characteristic
8#21446774C211yesmaximal, characteristic, complemented
9#21446584C211yesmaximal, characteristic, complemented
10#21446574C211yesmaximal, characteristic, complemented
11#21446540C211yesmaximal, characteristic, complemented
12#21446449C211yesmaximal, characteristic
13#21446442C211yesmaximal, characteristic
14#21445644C211yesmaximal, characteristic
15C2×C61×#474C211yesmaximal, characteristic
16C2×#14519769C2211yescharacteristic
17#16840168—24no—
18C2×#14519710C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519688C2211yescharacteristic
27#16839685—21no—
28#16839685—21no—
29#16839669—21no—
30#16839669—21no—
31#16839665C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839661—21no—
34#16839661C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839635—21no—
37#16839635—21no—
38#16839631C2211yescharacteristic
39#16839627C2211yescharacteristic
40C4⋊C4×Dic61—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
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×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519769—22no—
64#14519769—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519710—22no—
87#14519710—22no—
88#14519708—22no—
89#14519708—22no—
90#14519706—22no—
91#14519706—22no—
92#14519698—22no—
93#14519698Q814yescomplemented
94#14519688—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×Q8×C61—21no—
105C22×D4×C61C2311yescharacteristic
106C23×C4×C61—21no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C4⋊Q8×C61—22no—
110C4⋊Q8×C61D414yes—
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×C61—22no—
122#102×C61—22no—
123#102×C61—22no—
124C24×C61—21no—
125C2×Q8×C61—24no—
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×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61C2×D412yes—
163C4⋊C4×C61C2×D412yes—
164C22⋊C4×C61—42no—
165C42×C61C2×D412yes—
166C42×C61#4112yes—
167C22⋊Dic61—42no—
168C22⋊Dic61—42no—
169C22⋊Dic61—81no—
170C22⋊Dic61—81no—
171C4⋊Dic61—21no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—41no—
180C4⋊Dic61—81no—
181C4⋊Dic61—81no—
182C4⋊Dic61C2×Q812yes—
183#11758196—42no—
184#11758196—42no—
185#11758196—81no—
186#11758196—81no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—42no—
192C4×Dic61—81no—
193C4×Dic61—81no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C2×Q812yes—
199C2×#92418#4112yes—
200C23×C61—22no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61C22×D411yescharacteristic
208Q8×C61—84no—
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—41no—
212D4×C61—41no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
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—84no—
227C2×C4×C61#12112yes—
228C2×C4×C61#12112yes—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×Q811yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C612−1+411yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
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#13712yes—
259C8×C61—41no—
260C8×C61—41no—
261#92418—22no—
262#92418—22no—
263#25462—611nomaximal, Sylow
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61C2×#12111yescharacteristic
272C22×C61#54711yescharacteristic
273C22×C61D4×Q811yescharacteristic
274C22×C61C2×#13711yescharacteristic
275C22×C61#57511yescharacteristic
276C22×C61#57611yescharacteristic
277C22×C61#57611yescharacteristic
278C4×C61—22no—
279C4×C61—22no—
280C4×C61—42no—
281C4×C61—42no—
282C4×C61—81no—
283C4×C61#53311yescharacteristic
284C4×C61#54111yescharacteristic
285C4×C61#54811yescharacteristic
286C4×C61D4×Q811yescharacteristic
287Dic61—22no—
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—42no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295C2×#501—611no—
296C2×#474D6111yescharacteristic
297#2566—611no—
298#2369—611no—
299#2085—611no—
300#2083—611no—
301#2067—611no—
302#2065—611no—
303#1989—611no—
304#1949—611no—
305#1947—611no—
306#1940—611no—
307#1926—611no—
308#1925—611no—
309#1762—611no—
310C2×C61—82no—
311C2×C61#242911yescharacteristic
312C2×C61#303711yescharacteristic
313C2×C61#304111yescharacteristic
314C2×C61#334611yescharacteristic
315C2×C61#334711yescharacteristic
316C2×C61#338511yescharacteristic
317C2×C61#338911yescharacteristic
318C2×C4⋊Q8C2×D6111yescharacteristic
319C2×#125—611no—
320C2×C4×D4C2×D6111yescharacteristic
321#501—1224no—
322#474—24no—
323C2×#107—611no—
324C2×#107—611no—
325C2×#107C2×D6111yescharacteristic
326C2×#107C2×D6111yescharacteristic
327C2×#106—611no—
328C2×#106—611no—
329C2×C4⋊C8—611no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8C2×D6111yescharacteristic
332C2×#102—611no—
333C2×#102—611no—
334C2×#102C2×D6111yescharacteristic
335C2×#102C2×D6111yescharacteristic
336C2×#97—611no—
337#396—1221no—
338#396—1221no—
339#391—1221no—
340#391—1221no—
341#390—611no—
342#390—611no—
343#389—611no—
344#389—611no—
345#389—1221no—
346#385—1221no—
347#385—1221no—
348#384—611no—
349#382—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—1221no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#336—611no—
362#336—611no—
363C61#2546211yescharacteristic, complemented, Sylow
364C22×Q8—21no—
365C22×D4C22×D6111yescharacteristic
366C23×C4—21no—
367C22×C8—611no—
368C22×C8—611no—
369C22×C8C22×D6111yescharacteristic
370C22×C8C22×D6111yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8C22⋊D6114yes—
373#125—1222no—
374#125—1222no—
375C4×D4—22no—
376C4×D4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42C22×D6111yescharacteristic
404#107—22no—
405#107—22no—
406#107—1222no—
407#107—1222no—
408#106—1222no—
409#106—1222no—
410C4⋊C8—22no—
411C4⋊C8—1222no—
412C4⋊C8—1222no—
413#102—22no—
414#102—22no—
415#102—614no—
416#102—1222no—
417#97—1222no—
418#95—611no—
419#95—611no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427C24—21no—
428C2×Q8—24no—
429C2×Q8—81no—
430C2×Q8—81no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D6112yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4C23×D6111yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—612no—
461C2×C8—612no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—612no—
472C4⋊C4—1222no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—4881no—
481C4⋊C4—4881no—
482C4⋊C4—4881no—
483C4⋊C4—4881no—
484C4⋊C4—2441no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4C2×C22⋊D6112yes—
488C4⋊C4C2×C22⋊D6112yes—
489C22⋊C4—42no—
490C22⋊C4—4881no—
491C22⋊C4—4881no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C42—1221no—
495C42—1221no—
496C42—1221no—
497C42—1221no—
498C42—4881no—
499C42—4881no—
500C42—2442no—
501C42#1175822412yes—
502C42C2×C22⋊D6112yes—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D6111yescharacteristic
511Q8—84no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—612no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4#1451974912yes—
555C2×C4#1451974912yes—
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4C2×Q8×D6111yescharacteristic
558C2×C4#41×D6111yescharacteristic
559C2×C4C2×#1175822411yescharacteristic
560C2×C4#1451982411yescharacteristic
561C2×C4#1451982511yescharacteristic
562C8—41no—
563C8—41no—
564C8—1222no—
565C8—1222no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#1451974911yescharacteristic
574C22#1684041411yescharacteristic
575C22#1684042511yescharacteristic
576C22#137×D6111yescharacteristic
577C22#1684056111yescharacteristic
578C22#1684056311yescharacteristic
579C22#1684056311yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—81no—
585C4—1222no—
586C4—4881no—
587C4—4881no—
588C4—4881no—
589C4—2442no—
590C4—2441no—
591C4—2441no—
592C4—2441no—
593C4#1684031411yescharacteristic
594C4#1684037211yescharacteristic
595C4#1684041911yescharacteristic
596C4Q8×C22⋊D6111yescharacteristic
597C2—82no—
598C2#2144942011yescharacteristic
599C2#2145528811yescharacteristic
600C2#2145531911yescharacteristic
601C2#2145830111yescharacteristic
602C2#2145832611yescharacteristic
603C2#2145872411yescharacteristic
604C2#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#25462
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×#16840168C2semidirect
3. #21448909C2semidirect
4. #21447303C2semidirect
5. #21447293C2semidirect
8. #21446774C2semidirect
9. #21446584C2semidirect
10. #21446574C2semidirect
11. #21446540C2semidirect
93. #14519698Q8semidirect
363. C61#25462semidirect

Maximal subgroups

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

subgroupindex
C2×#168401682
#214510152
#214489092
#214473032
#214472932
#214472032
#214471892
#214467742
#214465842
#214465742
#214465402
#214464492
#214464422
#214456442
C2×C61×#4742
#2546261