Finite Group Databasebeta

← #29732069

Subgroups — #29732069

Order 15616.

790 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840052C211yesmaximal, characteristic
2C2×#16839893C211yesmaximal, characteristic, complemented
3#21450804C211yesmaximal, characteristic, complemented
4#21447321C211yesmaximal, characteristic, complemented
5#21447320C211yesmaximal, characteristic, complemented
6#21447207C211yesmaximal, characteristic
7#21447193C211yesmaximal, characteristic
8#21446791C211yesmaximal, characteristic, complemented
9#21446585C211yesmaximal, characteristic, complemented
10#21446554C211yesmaximal, characteristic, complemented
11#21446544C211yesmaximal, characteristic, complemented
12#21446519C211yesmaximal, characteristic
13#21446505C211yesmaximal, characteristic
14#21445644C211yesmaximal, characteristic
15C61×#2580C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519775—21no—
18#16840052—24no—
19#16839893—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519688C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34#16839692—21no—
35#16839692—21no—
36#16839667—21no—
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C61×#399—21no—
53C61×#393—21no—
54C61×#390C2211yescharacteristic
55C61×#390C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#385—21no—
59C61×#385—21no—
60C61×#382C2211yescharacteristic
61C22×Dic122—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519777—22no—
67#14519777—22no—
68#14519775—44no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C4×Dic61—21no—
79#14519710—22no—
80#14519708—22no—
81#14519708—22no—
82#14519708—22no—
83C4⋊#92418—22no—
84C4⋊#92418—22no—
85C4⋊#92418D414yes—
86#14519698—22no—
87#14519698—22no—
88#14519698—22no—
89#14519698Q814yescomplemented
90#14519688—22no—
91#14519688—22no—
92C4×#92418—22no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98C22×Q8×C61D411yescharacteristic
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C4×Q8×C61—22no—
102C4×Q8×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125C24×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—41no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C2×Q812yes—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—42no—
139C22×C4×C61—81no—
140C22×C4×C61—81no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—42no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—81no—
165C4⋊C4×C61—81no—
166C22⋊C4×C61—42no—
167C22⋊C4×C61—42no—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—81no—
170C22⋊C4×C61—81no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61C2×D412yes—
176C42×C61#4112yes—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—42no—
184C4⋊Dic61C2×Q812yes—
185C4⋊Dic61#4112yes—
186#11758196—42no—
187#11758196—42no—
188#11758196—42no—
189#11758196—42no—
190C4×Dic61—42no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418C2×D412yes—
201C2×#92418C2×D412yes—
202C2×#92418C2×Q812yes—
203C23×C61—22no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61C22×D411yescharacteristic
211Q8×C61—44no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—41no—
215D4×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—84no—
241C2×C4×C61#12112yes—
242C2×C4×C61#12112yes—
243C2×C4×C61AGL(1,ℤ/8)12yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×Q811yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—22no—
251C2×Dic61—22no—
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—41no—
261C2×Dic61—41no—
262Dic122—84no—
263#92418—22no—
264#92418—22no—
265#92418—22no—
266#92418—41no—
267#25485—611nomaximal, Sylow
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61C2×#12111yescharacteristic
276C22×C61#54811yescharacteristic
277C22×C61D4×Q811yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61#57511yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57711yescharacteristic
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61#53311yescharacteristic
292C4×C61#54111yescharacteristic
293C4×C61#54711yescharacteristic
294C4×C61D4×Q811yescharacteristic
295Dic61—42no—
296Dic61—42no—
297Dic61—81no—
298Dic61—81no—
299C2×#476—611no—
300C2×#446—611no—
301#2580D6111yescharacteristic
302#2546—611no—
303#2088—611no—
304#2088—611no—
305#2067—611no—
306#2065—611no—
307#1993—611no—
308#1949—611no—
309#1943—611no—
310#1941—611no—
311#1936—611no—
312#1934—611no—
313#1762—611no—
314C2×C61—82no—
315C2×C61#242911yescharacteristic
316C2×C61#305511yescharacteristic
317C2×C61#305711yescharacteristic
318C2×C61#337111yescharacteristic
319C2×C61#337311yescharacteristic
320C2×C61#338411yescharacteristic
321C2×C61#338811yescharacteristic
322C2×C4⋊Q8—611no—
323C2×C4⋊Q8—1221no—
324C2×C4×Q8C2×D6111yescharacteristic
325C2×C4×D4C2×D6111yescharacteristic
326#476—1224no—
327#446—1224no—
328C2×#107—611no—
329C2×#106—611no—
330C2×#106—611no—
331C2×#106—611no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8—611no—
334C2×C4⋊C8—611no—
335C2×#102—611no—
336C2×#102—611no—
337C2×#102—611no—
338C2×#102—611no—
339C2×#97—611no—
340C2×#97—611no—
341C2×C4×C8—611no—
342#399—21no—
343#398—1221no—
344#398—1221no—
345#393—21no—
346#391—1221no—
347#390C2×D6111yescharacteristic
348#390C2×D6111yescharacteristic
349#389—611no—
350#389—611no—
351#389—611no—
352#389—611no—
353#389C2×D6111yescharacteristic
354#389C2×D6111yescharacteristic
355#385—21no—
356#385—21no—
357#382C2×D6111yescharacteristic
358C4×C4⋊C4—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367C61#2548511yescharacteristic, complemented, Sylow
368C22×Q8—1221no—
369C22×Q8C22⋊D6111yescharacteristic
370C22×D4C22×D6111yescharacteristic
371C23×C4—21no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8—611no—
376C4⋊Q8—1222no—
377C4⋊Q8—1222no—
378C4⋊Q8—2444no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—22no—
382C4×D4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
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⋊C4C22⋊D6111yescharacteristic
395C2×C4⋊C4C22⋊D6111yescharacteristic
396C2×C4⋊C4C22⋊D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408C2×C42C22×D6111yescharacteristic
409#107—1222no—
410#106—1222no—
411#106—1222no—
412#106—1222no—
413C4⋊C8—614no—
414C4⋊C8—1222no—
415C4⋊C8—1222no—
416#102—614no—
417#102—1222no—
418#102—1222no—
419#102—1222no—
420#97—1222no—
421#97—1222no—
422C4×C8—1222no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432C24—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—41no—
436C2×Q8—2442no—
437C2×Q8—2442no—
438C2×Q8—2442no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4Q8×D6112yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C2×C22⋊D6111yescharacteristic
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C23×D6111yescharacteristic
462C2×C8—612no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—612no—
486C4⋊C4—612no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4—2442no—
496C4⋊C4—2442no—
497C4⋊C4—2442no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—2442no—
508C42D4×D6112yes—
509C42#1175822412yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C22×C22⋊D6111yescharacteristic
518Q8—44no—
519Q8—4884no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—84no—
549C2×C4—1222no—
550C2×C4—1222no—
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—2441no—
561C2×C4#1451974912yes—
562C2×C4#1451974912yes—
563C2×C4#1451980212yes—
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C2×Q8×D6111yescharacteristic
566C2×C4#41×D6111yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C2×C4#1451982411yescharacteristic
569C2×C4#1451982511yescharacteristic
570C8—1222no—
571C8—1222no—
572C8—1222no—
573C8—2441no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#1451974911yescharacteristic
582C22#1684041911yescharacteristic
583C22Q8×C22⋊D6111yescharacteristic
584C22C2×#1451980211yescharacteristic
585C22#1684056011yescharacteristic
586C22#1684057311yescharacteristic
587C22#1684057311yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—4881no—
598C4—4881no—
599C4—2442no—
600C4—2442no—
601C4#1684031411yescharacteristic
602C4#1684037211yescharacteristic
603C4#1684041411yescharacteristic
604C4#1684042511yescharacteristic
605C2—82no—
606C2#2144942011yescharacteristic
607C2#2145544011yescharacteristic
608C2#2145545511yescharacteristic
609C2#2145858111yescharacteristic
610C2#2145860711yescharacteristic
611C2#2145871811yescharacteristic
612C2#2145876411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2580C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2580D61
hypocenterC61#25485
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. C2×#16839893C2semidirect
3. #21450804C2semidirect
4. #21447321C2semidirect
5. #21447320C2semidirect
8. #21446791C2semidirect
9. #21446585C2semidirect
10. #21446554C2semidirect
11. #21446544C2semidirect
89. #14519698Q8semidirect
367. C61#25485semidirect

Maximal subgroups

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

subgroupindex
C2×#168400522
C2×#168398932
#214508042
#214473212
#214473202
#214472072
#214471932
#214467912
#214465852
#214465542
#214465442
#214465192
#214465052
#214456442
C61×#25802
#2548561