Finite Group Databasebeta

← #29686266

Subgroups — #29686266

Order 15616.

1260 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839954C211yesmaximal, characteristic
2C2×#16839871C212yesmaximal, complemented
3#127×Dic61C211yesmaximal, characteristic, complemented
4C4×#14519739C211yesmaximal, characteristic, complemented
5#21446582C211yesmaximal, characteristic, complemented
6#21446449C212yesmaximal, complemented
7#21446164C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9#21445674C211yesmaximal, characteristic, complemented
10#21445646C212yesmaximal, complemented
11C61×#1982C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×C4×D244C2211yescharacteristic
14C2×D4×Dic61—21no—
15C2×D4×Dic61—21no—
16C2×D4×Dic61C2212yes—
17C2×#14519739C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×C42×D61—21no—
20#16839954C418yescomplemented
21#16839871—28no—
22C2×#14519708C2212yes—
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519699C2212yes—
25C2×#14519698C2212yes—
26C2×#14519687C2212yes—
27#16839665—21no—
28#16839665—21no—
29#16839665C2212yes—
30#16839632C2211yescharacteristic
31C4⋊C4×Dic61C2211yescharacteristic
32C4⋊C4×Dic61C2211yescharacteristic
33C4×#11758198—21no—
34#16839609—21no—
35#16839609—21no—
36C42×Dic61C2211yescharacteristic
37#16839525C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C2×#102×C61C2212yes—
44C4×C4⋊C4×C61C2211yescharacteristic
45C22×D244C2311yescharacteristic
46C23×Dic61—22no—
47C22×C4×D61—21no—
48C22×C4×D61—21no—
49C22×C4×D61—21no—
50C2×#11758199—24no—
51C2×#11758199—24no—
52C2×#11758199C2×C418yescomplemented
53C22×#92418C2312yes—
54C4×D244—22no—
55C4×D244C2×C414yescomplemented
56D4×Dic61—24no—
57D4×Dic61—24no—
58D4×Dic61—44no—
59D4×Dic61—44no—
60#14519739—22no—
61#14519739—22no—
62#14519739—22no—
63#14519739D414yescomplemented
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198—21no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61D411yescharacteristic
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C42×D61—44no—
83#14519708—24no—
84C4⋊#92418C2×C414yescomplemented
85#14519699—24no—
86#14519698C2×C418yescomplemented
87#14519687—24no—
88#14519684—21no—
89#14519684—21no—
90C22×D4×C61—21no—
91C22×D4×C61—21no—
92C22×D4×C61C2312yes—
93C22×C8×C61C2312yes—
94#127×C61—22no—
95#127×C61C2×C414yescomplemented
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C42×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61—22no—
101#102×C61—24no—
102#102×C61—24no—
103#95×C61—21no—
104C24×C61—22no—
105C23×D61—21no—
106C2×D4×C61—24no—
107C2×D4×C61—24no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C2×D4×C61—42no—
117C2×D4×C61C22×C414yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—44no—
126C22×Dic61—44no—
127C22×Dic61—44no—
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244C22×C412yes—
136C2×D244C2×D412yes—
137C2×D244#4112yes—
138C2×C4×D61—24no—
139C2×C4×D61—24no—
140C2×C4×D61—24no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61#4112yes—
154C42×C61—42no—
155C42×C61C22×C412yes—
156C42×C61#4112yes—
157C22⋊Dic61—44no—
158C22⋊Dic61—44no—
159C22⋊Dic61—44no—
160#11758199—216no—
161#11758198—24no—
162#11758198—24no—
163#11758198—24no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—42no—
168C4⋊Dic61—42no—
169C4⋊Dic61C22×C412yes—
170C4×Dic61—22no—
171C4×Dic61—22no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C4×Dic61—22no—
175C4×Dic61—41no—
176C4×Dic61—42no—
177C4×Dic61—42no—
178C4×Dic61C2×D412yes—
179C4×Dic61#4112yes—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418C22×C414yes—
183C23×C61—24no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61C22×D411yescharacteristic
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198D4×C61—24no—
199D4×C61—24no—
200D4×C61—48no—
201D4×C61—42no—
202D4×C61—42no—
203D4×C61—48no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61C4×D412yes—
213C2×C4×C61C4×D412yes—
214C2×C4×C61AGL(1,ℤ/8)12yes—
215C2×C4×C61AGL(1,ℤ/8)12yes—
216C2×C4×C61C23×C411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—88no—
238C2×Dic61#12112yes—
239C2×Dic61#12112yes—
240C2×Dic61AGL(1,ℤ/8)12yes—
241D244—22no—
242D244—22no—
243D244—22no—
244D244C4×D414yescomplemented
245C4×D61—48no—
246C4×D61—48no—
247C4×D61—48no—
248C8×C61—42no—
249#92418—24no—
250#21503—611nomaximal, Sylow
251C22×C61—44no—
252C22×C61—44no—
253C22×C61—44no—
254C22×C61—82no—
255C22×C61—82no—
256C22×C61—82no—
257C22×C61—82no—
258C22×C61C2×C4×D411yescharacteristic
259C22×C61C2×#12111yescharacteristic
260C22×C61#52511yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C61#57511yescharacteristic
265C2×D61—24no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61#44214yescomplemented
276C4×C61C2×C4×D411yescharacteristic
277C4×C61C4×#4111yescharacteristic
278C4×C61#52911yescharacteristic
279C4×C61#53211yescharacteristic
280Dic61—22no—
281Dic61—22no—
282Dic61—22no—
283Dic61—41no—
284Dic61—42no—
285Dic61—82no—
286C2×#459—611no—
287C2×#442—612no—
288C4×#127—611no—
289C42×D4—611no—
290#1982D6111yescharacteristic, complemented
291#1948—611no—
292#1926—612no—
293#1879—612no—
294#1802—611no—
295#1767—611no—
296#1762—612no—
297C2×C61—84no—
298C2×C61C4×#12111yescharacteristic
299C2×C61C2×#44211yescharacteristic
300C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C61#303611yescharacteristic
302C2×C61#303811yescharacteristic
303C2×C61#310911yescharacteristic
304C2×C61#311511yescharacteristic
305D61—44no—
306C22×D8—612no—
307C2×#127C2×D6111yescharacteristic
308C2×C4×D4—612no—
309C2×C4×D4—611no—
310C2×C4×D4—611no—
311C2×C4×D4—611no—
312C2×C4×D4—1221no—
313C2×C4×D4—1221no—
314C22×C42—1221no—
315#459—618no—
316#442—1228no—
317C2×#106—612no—
318C2×C4⋊C8—611no—
319C2×C4⋊C8C2×D6111yescharacteristic
320C2×#102—612no—
321C2×#102—612no—
322C2×#102C2×D6112yes—
323C2×#102C2×D6112yes—
324C2×#97—612no—
325#390—612no—
326#390—1221no—
327#390—1221no—
328#384—611no—
329C4×C4⋊C4—611no—
330C4×C4⋊C4—611no—
331C4×C4⋊C4C2×D6111yescharacteristic
332C4×C22⋊C4—1221no—
333C4×C22⋊C4—1221no—
334C4×C22⋊C4—1221no—
335C43—611no—
336#340—612no—
337#340—612no—
338#336—612no—
339C61#2150311yescharacteristic, complemented, Sylow
340C22×D4—21no—
341C22×D4—21no—
342C22×D4—611no—
343C22×D4C22×D6112yes—
344C23×C4—1221no—
345C23×C4—1221no—
346C23×C4—1222no—
347C23×C4—1221no—
348C2×D8—618no—
349C2×D8—1224no—
350C2×D8—1224no—
351C22×C8—612no—
352C22×C8C22×D6112yes—
353#127—22no—
354#127C4×D6114yescomplemented
355C4×D4—614no—
356C4×D4—614no—
357C4×D4—1222no—
358C4×D4—1224no—
359C4×D4—1224no—
360C4×D4—1222no—
361C4×D4—1222no—
362C4×D4—1222no—
363C4×D4—2444no—
364C4×D4—2444no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C4⋊C4C22×D6111yescharacteristic
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C42—21no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—1221no—
385C2×C42—1221no—
386C2×C42—2444no—
387C2×C42C22×D6111yescharacteristic
388#106—1224no—
389C4⋊C8—22no—
390C4⋊C8—614no—
391#102—24no—
392#102—24no—
393#102—618no—
394#102—1224no—
395#97—1224no—
396#95—21no—
397#95—1221no—
398#95—1221no—
399C24—22no—
400C24—1221no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—612no—
413C2×D4—612no—
414C2×D4—612no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4—1221no—
418C2×D4C2×C4×D6114yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—1224no—
426C22×C4—1224no—
427C22×C4—1221no—
428C22×C4—1224no—
429C22×C4—1222no—
430C22×C4—2442no—
431C22×C4—2442no—
432C22×C4—2442no—
433C22×C4—2442no—
434C22×C4—2444no—
435C22×C4—2444no—
436C22×C4—2444no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442D8—12216no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—614no—
447C2×C8—1222no—
448C2×C8—1222no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—612no—
452C4⋊C4—1221no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—2442no—
456C4⋊C4—2442no—
457C4⋊C4D4×D6112yes—
458C4⋊C4#1175822412yes—
459C22⋊C4—1224no—
460C22⋊C4—1224no—
461C22⋊C4—1224no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C22⋊C4—2444no—
465C42—42no—
466C42—612no—
467C42—612no—
468C42—1222no—
469C42—1221no—
470C42—1221no—
471C42—1222no—
472C42—1222no—
473C42—2441no—
474C42—2442no—
475C42—2442no—
476C42C2×C4×D6112yes—
477C42#1175822412yes—
478C23—24no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—1222no—
486C23—1222no—
487C23—1222no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23C2×D4×D6111yescharacteristic
493D4—24no—
494D4—24no—
495D4—48no—
496D4—42no—
497D4—42no—
498D4—48no—
499D4—614no—
500D4—1222no—
501D4—1222no—
502D4—1222no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1222no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—4888no—
525C2×C4—2448no—
526C2×C4—2448no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2448no—
533C2×C4#1451973712yes—
534C2×C4#1451973712yes—
535C2×C4#1451980112yes—
536C2×C4#1451980412yes—
537C2×C4C22×C4×D6111yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4#41×D6111yescharacteristic
540C2×C4#41×D6111yescharacteristic
541C2×C4C2×#1175822411yescharacteristic
542C2×C4C2×#1175822411yescharacteristic
543C8—42no—
544C8—1224no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—1224no—
553C22—2442no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22—2442no—
558C22—2442no—
559C22C2×#1451973711yescharacteristic
560C22#121×D6111yescharacteristic
561C22#1684027211yescharacteristic
562C22AGL(1,ℤ/8)×D6111yescharacteristic
563C22C2×#1451980111yescharacteristic
564C22C2×#1451980411yescharacteristic
565C22#1684055811yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—1222no—
570C4—1222no—
571C4—1222no—
572C4—4882no—
573C4—2442no—
574C4—2441no—
575C4#1683987114yescomplemented
576C4C4×D4×D6111yescharacteristic
577C4C4×#1175822411yescharacteristic
578C4#1684029111yescharacteristic
579C4#1684031011yescharacteristic
580C2—84no—
581C2—2444no—
582C2#2144833511yescharacteristic
583C2C2×#1683987111yescharacteristic
584C2#2145462311yescharacteristic
585C2#2145527811yescharacteristic
586C2#2145529611yescharacteristic
587C2#2145590911yescharacteristic
588C2#2145596911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1982C2
socleC23×C61C22×D4
prime radicalC2×#16839954C2
cosocleC22×C4×C61C24
hypercenter#1982D61
hypocenterC61#21503
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. C2×#16839871C2semidirect
3. #127×Dic61C2semidirect
4. C4×#14519739C2semidirect
5. #21446582C2semidirect
6. #21446449C2semidirect
7. #21446164C2semidirect
8. #21445820C2semidirect
9. #21445674C2semidirect
10. #21445646C2semidirect
11. C61×#1982C2semidirect
20. #16839954C4semidirect
52. C2×#11758199C2×C4semidirect
55. C4×D244C2×C4semidirect
63. #14519739D4semidirect
84. C4⋊#92418C2×C4semidirect
86. #14519698C2×C4semidirect
95. #127×C61C2×C4semidirect
244. D244C4×D4semidirect
275. C4×C61#442semidirect
290. #1982D61semidirect
339. C61#21503semidirect
354. #127C4×D61semidirect
575. C4#16839871semidirect

Maximal subgroups

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

subgroupindex
C2×#168399542
C2×#168398712
#127×Dic612
C4×#145197392
#214465822
#214464492
#214461642
#214458202
#214456742
#214456462
C61×#19822
#2150361