Finite Group Databasebeta

← #29779540

Subgroups — #29779540

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454172C211yesmaximal, characteristic, complemented
2#390×D61C211yesmaximal, characteristic
3#21447223C211yesmaximal, characteristic, complemented
4#21446956C211yesmaximal, characteristic, complemented
5#21446955C211yesmaximal, characteristic, complemented
6#21446518C212yesmaximal, complemented
7#21446517C212yesmaximal, complemented
8#21445908C212yesmaximal, complemented
9#21445748C212yesmaximal
10#21445580C211yesmaximal, characteristic, complemented
11C61×#2071C211yesmaximal, characteristic, complemented
12C22×D4×D61C2211yescharacteristic
13C2×#14519777C2211yescharacteristic, complemented
14C2×#14519757—21no—
15C2×C4⋊C4×D61C2211yescharacteristic
16C2×C22⋊C4×D61—21no—
17C2×C22⋊C4×D61—21no—
18C2×C42×D61C2211yescharacteristic
19C2×#14519699C2212yes—
20C2×#14519698C2212yes—
21C2×#14519697C2212yes—
22C2×#14519689C2212yes—
23C2×#14519687C2212yescomplemented
24#16839667—21no—
25#16839665C2211yescharacteristic
26#16839664C2211yescharacteristic
27#16839663C2211yescharacteristic
28#16839662—21no—
29#16839662—21no—
30#16839661—21no—
31#16839661C2212yescomplemented
32#16839619—21no—
33#16839619C2212yes—
34#16839525C2212yescomplemented
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839517C2211yescharacteristic, complemented
38C2×#102×C61C2212yes—
39C61×#390C2211yescharacteristic
40C61×#384C2211yescharacteristic, complemented
41C61×#340C2212yescomplemented
42C61×#336C2211yescharacteristic, complemented
43C24×D61—21no—
44C22×Dic122—21no—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C2×D4×D61—22no—
48C2×D4×D61—22no—
49C2×D4×D61—24no—
50C2×D4×D61—42no—
51C22×C4×D61—21no—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C22×#92418C2312yes—
56#14519777—42no—
57#14519757—44no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—21no—
63C4⋊C4×D61—24no—
64C2×C22⋊Dic61—21no—
65C2×C22⋊Dic61—21no—
66C2×#11758198—22no—
67C2×#11758198—21no—
68C2×#11758198—21no—
69C2×#11758198—21no—
70C2×#11758198—21no—
71C22⋊C4×D61—44no—
72C22⋊C4×D61—44no—
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61D411yescharacteristic
75C2×C4×Dic61C2311yescharacteristic
76C42×D61—42no—
77#14519699—42no—
78#14519698—42no—
79#14519697—24no—
80#14519689—24no—
81#14519687—42no—
82#14519684—22no—
83#14519684—22no—
84C22×D4×C61C2311yescharacteristic
85C22×C8×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2312yes—
90C2×C22⋊C4×C61—21no—
91C2×C22⋊C4×C61—21no—
92C2×C42×C61C2311yescharacteristic
93#102×C61—24no—
94C24×C61—21no—
95C23×D61—24no—
96C23×D61—21no—
97C23×D61—42no—
98C23×D61—42no—
99C23×D61—42no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61C2×D411yescharacteristic
104C2×D4×C61—22no—
105C2×D4×C61—21no—
106C2×D4×C61—22no—
107C2×D4×C61#4112yes—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C2×C22⋊D61—24no—
114C2×C22⋊D61—42no—
115C2×C22⋊D61—42no—
116C22×Dic61—21no—
117C22×Dic61—22no—
118C22×Dic61C2×D411yescharacteristic
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121D4×D61—48no—
122D4×D61—48no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—21no—
126C2×D244#4112yes—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—41no—
135C2×C4×D61—41no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61#4114yes—
140C2×Dic122—42no—
141C2×Dic122—42no—
142C2×Dic122—42no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—42no—
150C4⋊C4×C61—42no—
151C22⋊C4×C61—42no—
152C22⋊C4×C61—42no—
153C42×C61—22no—
154C22⋊Dic61—42no—
155C22⋊Dic61—42no—
156#11758198—42no—
157#11758198—42no—
158#11758198—44no—
159#11758198—42no—
160#11758198—42no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—44no—
165C4⋊Dic61—42no—
166#11758196—42no—
167C4×Dic61—22no—
168C4×Dic61—41no—
169C4×Dic61—41no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C23×C61—22no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61C22×D411yescharacteristic
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—21no—
185C22×D61—22no—
186C22×D61—48no—
187C22×D61—48no—
188C22×D61—48no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—84no—
196C22×D61—84no—
197C22×D61—84no—
198C22×D61—84no—
199C22×D61#12412yes—
200C22×D61AGL(1,ℤ/8)12yes—
201C22×D61#13712yes—
202D4×C61—24no—
203D4×C61—42no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61AGL(1,ℤ/8)12yes—
215C2×C4×C61AGL(1,ℤ/8)12yes—
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4112yes—
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61H(5,𝔽2)12yes—
220C22⋊D61—48no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61#12412yes—
236D244—24no—
237D244—42no—
238C4×D61—24no—
239C4×D61—24no—
240C4×D61—44no—
241C4×D61—44no—
242C4×D61—84no—
243Dic122—84no—
244C8×C61—42no—
245#92418—42no—
246#29109—611nomaximal, Sylow
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61#52511yescharacteristic
255C22×C61C2×#12411yescharacteristic
256C22×C61#53411yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61C2×#13711yescharacteristic
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—816no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61#48914yes—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—82no—
279C4×C61—81no—
280C4×C61#55112yes—
281C4×C61#55112yes—
282Dic61—22no—
283Dic61—41no—
284Dic61—41no—
285Dic61—81no—
286Dic61—82no—
287#2857—611no—
288C2×#390—611no—
289#2071—611no—
290#2071D6111yescharacteristic, complemented
291#2023—611no—
292#2023—611no—
293#1936—612no—
294#1936—612no—
295#1824—612no—
296#1786—612no—
297#1749—611no—
298C2×C61—82no—
299C2×C61#256111yescharacteristic
300C2×C61#310911yescharacteristic
301C2×C61#311011yescharacteristic
302C2×C61C2×#48911yescharacteristic
303C2×C61#313811yescharacteristic
304C2×C61#319211yescharacteristic
305C2×C61#319311yescharacteristic
306D61—24no—
307D61—82no—
308C23×D4—611no—
309C2×C4⋊Q8—611no—
310C2×C22⋊Q8—1221no—
311C22×C4⋊C4—611no—
312C22×C22⋊C4—1221no—
313C22×C22⋊C4—1221no—
314C22×C42—611no—
315C2×#102—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102C2×D6112yes—
319C2×C22⋊C8—612no—
320C2×#97—612no—
321#391—1221no—
322#390—611no—
323#390—611no—
324#390—611no—
325#390—1221no—
326#390—1221no—
327#390C2×D6111yescharacteristic
328#389—612no—
329#389—1221no—
330#384C2×D6111yescharacteristic, complemented
331#380—612no—
332#380—1221no—
333#340—612no—
334#340—612no—
335#340—612no—
336#340C2×D6112yescomplemented
337#336—611no—
338#336C2×D6111yescharacteristic, complemented
339C61#2910911yescharacteristic, complemented, Sylow
340C25—1221no—
341C22×Q8—1221no—
342C22×D4—611no—
343C22×D4—1224no—
344C22×D4—1221no—
345C22×D4—1222no—
346C22×D4—1222no—
347C22×D4—2442no—
348C22×D4C22×D6111yescharacteristic
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—1221no—
353C22×C8—612no—
354C22×C8C22×D6112yes—
355C4⋊Q8—2442no—
356C22⋊Q8—2444no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—612no—
361C2×C4⋊C4—1224no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4C22×D6111yescharacteristic
366C2×C4⋊C4C22×D6112yes—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—1221no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1222no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—2444no—
377C2×C22⋊C4—2444no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—2442no—
382C2×C42C22×D6111yescharacteristic
383#102—24no—
384#102—1224no—
385#102—2442no—
386#102—2442no—
387C22⋊C8—1224no—
388#97—2442no—
389#95—1222no—
390#95—1222no—
391C24—21no—
392C24—611no—
393C24—1224no—
394C24—1221no—
395C24—2442no—
396C24—2442no—
397C24—2442no—
398C24—2442no—
399C24—2442no—
400C24—2442no—
401C2×Q8—2442no—
402C2×Q8—2442no—
403C2×Q8—2442no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—612no—
408C2×D4—1224no—
409C2×D4—1222no—
410C2×D4—1221no—
411C2×D4—1222no—
412C2×D4—2442no—
413C2×D4—2448no—
414C2×D4—2448no—
415C2×D4—2442no—
416C2×D4#1175822412yes—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—611no—
420C22×C4—614no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1221no—
432C22×C4—2441no—
433C22×C4—2441no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—2442no—
455C4⋊C4—2442no—
456C4⋊C4—2444no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—2442no—
460C22⋊C4—2442no—
461C22⋊C4—2444no—
462C22⋊C4—2442no—
463C22⋊C4—2442no—
464C22⋊C4—2442no—
465C22⋊C4—2442no—
466C42—22no—
467C42—1222no—
468C42—2441no—
469C42—2441no—
470C23—22no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—612no—
478C23—612no—
479C23—612no—
480C23—1221no—
481C23—1221no—
482C23—1221no—
483C23—1221no—
484C23—1222no—
485C23—4884no—
486C23—4884no—
487C23—4884no—
488C23—4884no—
489C23—2448no—
490C23—2448no—
491C23—2448no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23—2441no—
498C23C2×D4×D6111yescharacteristic
499Q8—4884no—
500D4—24no—
501D4—42no—
502D4—1224no—
503D4—2448no—
504D4—2442no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—612no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—4884no—
527C2×C4—2444no—
528C2×C4—2444no—
529C2×C4—2441no—
530C2×C4—2442no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4#1451980112yes—
536C2×C4#1451980412yes—
537C2×C4#41×D6112yes—
538C2×C4#41×D6111yescharacteristic
539C2×C4C2×#1175822411yescharacteristic
540C2×C4#1451982312yes—
541C8—42no—
542C8—2442no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—614no—
551C22—1222no—
552C22—1222no—
553C22—1222no—
554C22—1222no—
555C22—1222no—
556C22—1222no—
557C22—48816no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22#1684027211yescharacteristic
566C22#124×D6111yescharacteristic
567C22#1684032311yescharacteristic
568C22AGL(1,ℤ/8)×D6111yescharacteristic
569C22C2×#1451980111yescharacteristic
570C22C2×#1451980411yescharacteristic
571C22#137×D6111yescharacteristic
572C4—22no—
573C4—22no—
574C4—82no—
575C4—81no—
576C4—1222no—
577C4—4881no—
578C4—4882no—
579C4—2441no—
580C4—2441no—
581C4#1684043512yes—
582C4#1684043712yes—
583C2—82no—
584C2—1224no—
585C2—4882no—
586C2#2145097111yescharacteristic
587C2#2145590911yescharacteristic
588C2#2145591711yescharacteristic
589C2#489×D6111yescharacteristic
590C2#2145619011yescharacteristic
591C2#2145661811yescharacteristic
592C2#2145662711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2071C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
hypercenter#2071D61
hypocenterC61#29109
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
1. #21454172C2semidirect
3. #21447223C2semidirect
4. #21446956C2semidirect
5. #21446955C2semidirect
6. #21446518C2semidirect
7. #21446517C2semidirect
8. #21445908C2semidirect
10. #21445580C2semidirect
11. C61×#2071C2semidirect
13. C2×#14519777C22semidirect
23. C2×#14519687C22semidirect
31. #16839661C22semidirect
34. #16839525C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839517C22semidirect
40. C61×#384C22semidirect
41. C61×#340C22semidirect
42. C61×#336C22semidirect
290. #2071D61semidirect
330. #384C2×D61semidirect
336. #340C2×D61semidirect
338. #336C2×D61semidirect
339. C61#29109semidirect

Maximal subgroups

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

subgroupindex
#214541722
#390×D612
#214472232
#214469562
#214469552
#214465182
#214465172
#214459082
#214457482
#214455802
C61×#20712
#2910961