Finite Group Databasebeta

← #29683029

Subgroups — #29683029

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839862C211yesmaximal, characteristic
2#21448296C211yesmaximal, characteristic, complemented
3#21446460C211yesmaximal, characteristic, complemented
4#21446453C211yesmaximal, characteristic, complemented
5#21446411C211yesmaximal, characteristic, complemented
6#21446390C211yesmaximal, characteristic, complemented
7#21446153C211yesmaximal, characteristic
8#21446153C211yesmaximal, characteristic
9#21446080C211yesmaximal, characteristic
10#21446066C211yesmaximal, characteristic
11#21445717C211yesmaximal, characteristic, complemented
12#21445652C211yesmaximal, characteristic, complemented
13#21445644C211yesmaximal, characteristic
14#21445568C211yesmaximal, characteristic, complemented
15C42×D4×C61C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18#16839862C418yescomplemented
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30#16839668C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839646—21no—
35#16839646—21no—
36C4×#11758196—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839608—21no—
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C4×D4×C61C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C22×C42×C61—21no—
53C4×C4⋊C4×C61C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C4×C22⋊C4×C61—21no—
57C4×C22⋊C4×C61—21no—
58C4×C22⋊C4×C61—21no—
59C43×C61C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C2×#11758200—22no—
62C2×#11758200—22no—
63C2×#11758200C2×C414yes—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C4×Dic122—22no—
69C4×Dic122C2×C414yescomplemented
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78#14519710—22no—
79#14519710C2×C414yescomplemented
80#14519708—22no—
81#14519700—22no—
82#14519700—22no—
83#14519700C2×C414yescomplemented
84#14519698—22no—
85#14519698—22no—
86#14519698C2×C414yes—
87#14519688—22no—
88#14519688C2×C414yescomplemented
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96C22×D4×C61C2311yescharacteristic
97C23×C4×C61—21no—
98C23×C4×C61—21no—
99C23×C4×C61—21no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C4×D4×C61C2×C414yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C42×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61—44no—
116C2×C42×C61D411yescharacteristic
117C2×C42×C61D411yescharacteristic
118C2×C42×C61D411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126C24×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61C22×C412yes—
131C2×D4×C61#4112yes—
132C2×D4×C61#4112yes—
133C22×C4×C61—24no—
134C22×C4×C61—24no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—24no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C22×C412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C22×C412yes—
166C4⋊C4×C61#4112yes—
167C4⋊C4×C61#4112yes—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—24no—
171C42×C61—22no—
172C42×C61—22no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—41no—
178C42×C61—41no—
179C42×C61—42no—
180C42×C61—42no—
181C42×C61—42no—
182C42×C61C22×C412yes—
183C42×C61#4112yes—
184#11758200—28no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61C22×C412yes—
191#11758196—24no—
192#11758196—42no—
193#11758196—42no—
194C4×Dic61—24no—
195C4×Dic61—42no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418C22×C412yes—
207C2×#92418C22×C412yes—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61C22×D411yescharacteristic
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—22no—
219D4×C61#11714yes—
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—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—48no—
238C2×C4×C61—48no—
239C2×C4×C61—48no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61C4×D412yes—
247C2×C4×C61C4×D412yes—
248C2×C4×C61AGL(1,ℤ/8)12yes—
249C2×C4×C61#13712yes—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272#92418—22no—
273#92418—22no—
274#92418—41no—
275#92418—41no—
276#21121—611nomaximal, Sylow
277C22×C61—24no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61C2×C4×D411yescharacteristic
285C22×C61#52511yescharacteristic
286C22×C61#52511yescharacteristic
287C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57511yescharacteristic
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—22no—
294C4×C61—22no—
295C4×C61—42no—
296C4×C61—41no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61—42no—
300C4×C61C2×#11711yescharacteristic
301C4×C61C4×#4111yescharacteristic
302C4×C61#52911yescharacteristic
303C4×C61#52911yescharacteristic
304Dic61—42no—
305Dic61—42no—
306Dic61—81no—
307Dic61—81no—
308C2×#440—611no—
309#2295—611no—
310C42×D4D6111yescharacteristic
311#1928—611no—
312#1927—611no—
313#1921—611no—
314#1918—611no—
315#1877—611no—
316#1877—611no—
317#1864—611no—
318#1862—611no—
319#1776—611no—
320#1763—611no—
321#1762—611no—
322#1745—611no—
323C2×C61—44no—
324C2×C61#229011yescharacteristic
325C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
326C2×C61C4×#13711yescharacteristic
327C2×C61#311011yescharacteristic
328C2×C61#311111yescharacteristic
329C2×C61#311411yescharacteristic
330C2×C61#311611yescharacteristic
331C22×SD16—611no—
332C2×C4×Q8—611no—
333C2×C4×D4C2×D6111yescharacteristic
334C2×C4×D4C2×D6111yescharacteristic
335C2×C4×D4C2×D6111yescharacteristic
336C22×C42—21no—
337#440—618no—
338C2×#107—611no—
339C2×#107—611no—
340C2×#106—611no—
341C2×#103—611no—
342C2×#103—611no—
343C2×#103—611no—
344C2×#102—611no—
345C2×#102—611no—
346C2×#102—611no—
347C2×#97—611no—
348C2×#97—611no—
349#391—611no—
350#391—611no—
351#389—611no—
352#389—611no—
353#387—1221no—
354#387—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—1221no—
357C4×C4⋊C4C2×D6111yescharacteristic
358C4×C4⋊C4C2×D6111yescharacteristic
359C4×C4⋊C4C2×D6111yescharacteristic
360C4×C22⋊C4—21no—
361C4×C22⋊C4—21no—
362C4×C22⋊C4—21no—
363#376—1221no—
364C43C2×D6111yescharacteristic
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#336—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375C61#2112111yescharacteristic, complemented, Sylow
376C22×Q8—611no—
377C22×D4C22×D6111yescharacteristic
378C23×C4—21no—
379C23×C4—21no—
380C23×C4—21no—
381C2×SD16—614no—
382C2×SD16—1222no—
383C2×SD16—1222no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8—611no—
388C4×Q8—614no—
389C4×Q8—1222no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4C4×D6114yes—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—44no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42C22⋊D6111yescharacteristic
415C2×C42C22⋊D6111yescharacteristic
416C2×C42C22⋊D6111yescharacteristic
417C2×C42C22⋊D6111yescharacteristic
418C2×C42C22×D6111yescharacteristic
419C2×C42C22×D6111yescharacteristic
420C2×C42C22×D6111yescharacteristic
421#107—614no—
422#107—1222no—
423#106—1222no—
424#103—614no—
425#103—1222no—
426#103—1222no—
427#102—614no—
428#102—1222no—
429#102—1222no—
430#97—614no—
431#97—1222no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442C24—21no—
443C2×Q8—612no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4C2×C4×D6112yes—
453C2×D4#1175822212yes—
454C2×D4#1175822212yes—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C2×C22⋊D6111yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C23×D6111yescharacteristic
478SD16—1228no—
479C2×C8—612no—
480C2×C8—612no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—612no—
495C4⋊C4—1221no—
496C4⋊C4—1224no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C4⋊C4C2×C4×D6112yes—
504C4⋊C4#1175822212yes—
505C4⋊C4#1175822212yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—22no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—1224no—
521C42—2442no—
522C42C2×C4×D6112yes—
523C42#1175822212yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C22×C22⋊D6111yescharacteristic
532Q8—1222no—
533Q8—1222no—
534Q8—2441no—
535Q8—2441no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4#1451973414yes—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4C4×C22⋊D6112yes—
579C2×C4C4×C22⋊D6112yes—
580C2×C4#1451980312yes—
581C2×C4#1451980712yes—
582C2×C4C22×C4×D6111yescharacteristic
583C2×C4C2×#1175822211yescharacteristic
584C2×C4C2×#1175822211yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4C2×#1175822211yescharacteristic
587C2×C4C2×#1175822211yescharacteristic
588C8—1222no—
589C8—1222no—
590C8—2441no—
591C8—2441no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22C2×C4×C22⋊D6111yescharacteristic
600C22#1684027311yescharacteristic
601C22#1684027311yescharacteristic
602C22C2×#1451980311yescharacteristic
603C22C2×#1451980711yescharacteristic
604C22#1684056011yescharacteristic
605C22#1684056011yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—4881no—
616C4—4881no—
617C4—2442no—
618C4—2442no—
619C4C2×#1451973411yescharacteristic
620C4C4×#1175822211yescharacteristic
621C4#1684029611yescharacteristic
622C4#1684029611yescharacteristic
623C2—44no—
624C2#2144826211yescharacteristic
625C2C4×#1451980311yescharacteristic
626C2C4×#1451980711yescharacteristic
627C2#2145591811yescharacteristic
628C2#2145592811yescharacteristic
629C2#2145595711yescharacteristic
630C2#2145598111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC42×D4×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC42×D4D61
hypocenterC61#21121
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #21448296C2semidirect
3. #21446460C2semidirect
4. #21446453C2semidirect
5. #21446411C2semidirect
6. #21446390C2semidirect
11. #21445717C2semidirect
12. #21445652C2semidirect
14. #21445568C2semidirect
18. #16839862C4semidirect
69. C4×Dic122C2×C4semidirect
79. #14519710C2×C4semidirect
83. #14519700C2×C4semidirect
88. #14519688C2×C4semidirect
375. C61#21121semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
#214482962
#214464602
#214464532
#214464112
#214463902
#214461532
#214460802
#214460662
#214457172
#214456522
#214456442
#214455682
C42×D4×C612
#2112161