Finite Group Databasebeta

← #29707503

Subgroups — #29707503

Order 15616.

1902 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
1C22×#14519748C211yesmaximal, characteristic
2C2×#16839805C211yesmaximal, characteristic, complemented
3C2×#16839767C211yesmaximal, characteristic, complemented
4C2×#16839725C211yesmaximal, characteristic, complemented
5#21448198C211yesmaximal, characteristic, complemented
6#21447006C214yesmaximal, complemented
7#21446999C214yesmaximal, complemented
8#21445539C211yesmaximal, characteristic, complemented
9C2×C61×#418C211yesmaximal, characteristic, complemented
10C23×C22⋊D61—21no—
11C23×C4×D61C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13C2×#14519748—24no—
14C2×#14519748—24no—
15C2×#14519748—22no—
16C2×#14519748C2214yes—
17C2×#14519736C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C22×C22⋊Dic61—21no—
20#16839805—24no—
21C2×#14519710C2212yescomplemented
22C2×#14519708C2212yescomplemented
23#16839767—24no—
24C2×#14519698C2214yescomplemented
25#16839725—24no—
26C2×#14519689C2214yescomplemented
27#16839623—21no—
28#16839619C2212yescomplemented
29#16839619C2212yescomplemented
30#16839617—21no—
31C4×#11758198—22no—
32#16839525C2214yescomplemented
33C23×D4×C61C2211yescharacteristic
34C22×M16×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#418—24no—
37C2×#102×C61C2212yescomplemented
38C2×#102×C61C2212yescomplemented
39C24×D61D411yescharacteristic
40C22×C22⋊D61—28no—
41C22×C22⋊D61—24no—
42C22×C22⋊D61—48no—
43C23×Dic61—21no—
44C23×Dic61D411yescharacteristic
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61D412yes—
52C22×C4×D61C2312yes—
53C2×#11758194—22no—
54C2×#11758194—22no—
55C2×#11758194D414yescomplemented
56C22×#92418C2312yes—
57#14519748—216no—
58#14519748—28no—
59#14519748—28no—
60#14519736—22no—
61#14519736—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61D414yes—
65C2×C4⋊Dic61C2312yes—
66C2×C4⋊Dic61C2312yes—
67C2×C22⋊Dic61—28no—
68C2×C22⋊Dic61—24no—
69C2×C22⋊Dic61—21no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×C4×Dic61—21no—
73#14519710—24no—
74#14519708—24no—
75#14519698—28no—
76#14519689—28no—
77#14519684—22no—
78#14519684—22no—
79C25×C61—21no—
80C22×D4×C61—22no—
81C22×D4×C61—21no—
82C22×D4×C61—22no—
83C22×D4×C61—22no—
84C22×D4×C61—22no—
85C22×D4×C61—22no—
86C22×D4×C61D414yes—
87C22×D4×C61C2312yes—
88C23×C4×C61C2311yescharacteristic
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C2×M16×C61—22no—
92C22×C8×C61C2312yes—
93#117×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61C2311yescharacteristic
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C42×C61—21no—
99#102×C61—24no—
100#102×C61—24no—
101C24×C61—24no—
102C24×C61—22no—
103C24×C61—44no—
104C24×C61—44no—
105C24×C61—42no—
106C24×C61—42no—
107C24×C61C2×D411yescharacteristic
108C23×D61—22no—
109C23×D61—24no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—42no—
113C23×D61C2×D412yes—
114C2×D4×C61—24no—
115C2×D4×C61—28no—
116C2×D4×C61—28no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—44no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—44no—
126C2×D4×C61C2×D414yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—216no—
138C2×C22⋊D61—432no—
139C2×C22⋊D61—48no—
140C2×C22⋊D61—48no—
141C22×Dic61—22no—
142C22×Dic61—24no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—44no—
148C22×Dic61C2×D412yes—
149C2×C4×D61—24no—
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162M16×C61—44no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C22⋊C4×C61—42no—
171C42×C61—42no—
172C22⋊Dic61—216no—
173C22⋊Dic61—42no—
174#11758198—44no—
175#11758198—44no—
176C4⋊Dic61—24no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61C2×D414yes—
181C4×Dic61—42no—
182#11758194—28no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418C2×D414yescomplemented
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—24no—
189C23×C61—48no—
190C23×C61—48no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—44no—
194C23×C61—44no—
195C23×C61—48no—
196C23×C61—88no—
197C23×C61—88no—
198C23×C61#12012yes—
199C23×C61#12112yes—
200C23×C61#12312yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—28no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—48no—
210C22×D61—48no—
211C22×D61—44no—
212C22×D61—44no—
213C22×D61AGL(1,ℤ/8)14yes—
214D4×C61—24no—
215D4×C61—24no—
216D4×C61—42no—
217D4×C61—42no—
218D4×C61—48no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61#12012yes—
224C2×C4×C61#12114yes—
225C2×C4×C61#12114yes—
226C2×C4×C61#12112yes—
227C2×C4×C61#12314yes—
228C2×C4×C61#12312yes—
229C2×C4×C61#12414yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C22⋊D61—432no—
237C2×Dic61—24no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—48no—
242C2×Dic61—44no—
243C2×Dic61—44no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—42no—
251C4×D61—48no—
252C8×C61—42no—
253#92418—24no—
254#23480—611nomaximal, Sylow
255C22×C61—24no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—24no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—44no—
262C22×C61—816no—
263C22×C61—84no—
264C22×C61—84no—
265C22×C61#39414yes—
266C22×C61C2×#12011yescharacteristic
267C22×C61C2×#12111yescharacteristic
268C22×C61C2×#12311yescharacteristic
269C22×C61C2×AGL(1,ℤ/8)12yes—
270C22×C61#57512yes—
271C2×D61—24no—
272C2×D61—24no—
273C2×D61—24no—
274C2×D61—416no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61#39414yes—
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×#12111yescharacteristic
284C4×C61C2×#12311yescharacteristic
285C4×C61C2×#12411yescharacteristic
286Dic61—42no—
287Dic61—44no—
288Dic61—82no—
289C22×#121—611no—
290C2×#428—611no—
291C2×#418—611no—
292C2×#418D6111yescharacteristic, complemented
293C2×#407—611no—
294#2280—611no—
295#2034—614no—
296#2033—614no—
297#1736—611no—
298C2×C61—24no—
299C2×C61—84no—
300C2×C61C2×#39411yescharacteristic
301C2×C61#300311yescharacteristic
302C2×C61#300511yescharacteristic
303C2×C61#305512yes—
304C2×C61#309412yes—
305D61—44no—
306C23×D4—1221no—
307C23×D4C2×D6111yescharacteristic
308C24×C4—611no—
309C22×M16—611no—
310C22×M16C2×D6111yescharacteristic
311C2×#121—614no—
312C2×#121—1224no—
313C2×#121—1222no—
314C2×#121—1224no—
315C2×#117—611no—
316C2×#117C2×D6111yescharacteristic
317C22×C4⋊C4—611no—
318C22×C22⋊C4—1221no—
319#428—1224no—
320C2×#107—612no—
321C2×#106—612no—
322#418—24no—
323#418—1224no—
324C2×#102—614no—
325C2×#102C2×D6112yescomplemented
326C2×#102C2×D6112yescomplemented
327#407—1224no—
328C2×C22⋊C8—614no—
329#381—1221no—
330#380—612no—
331#380—612no—
332#379—1221no—
333C4×C22⋊C4—1222no—
334#340—614no—
335C61#2348011yescharacteristic, complemented, Sylow
336C25—21no—
337C25—611no—
338C22×D4—22no—
339C22×D4—22no—
340C22×D4—22no—
341C22×D4—22no—
342C22×D4—21no—
343C22×D4—22no—
344C22×D4—1224no—
345C22×D4—1228no—
346C22×D4—2448no—
347C22×D4C22⋊D6114yes—
348C22×D4C22×D6112yes—
349C23×C4—611no—
350C23×C4—612no—
351C23×C4—612no—
352C23×C4—1221no—
353C23×C4—1222no—
354C23×C4—1222no—
355C23×C4—1222no—
356C23×C4—1222no—
357C23×C4—1222no—
358C23×C4—1222no—
359C23×C4C22×D6111yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—22no—
363C2×M16—614no—
364C2×M16—1222no—
365C2×M16—1222no—
366C22×C8—612no—
367C22×C8C22×D6112yes—
368#121—1228no—
369#121—12216no—
370#121—1228no—
371#117—22no—
372#117—22no—
373#117—1222no—
374#117—1222no—
375C2×C4⋊C4—614no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1224no—
384C2×C22⋊C4—1228no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4—1221no—
388C2×C42—21no—
389C2×C42—1221no—
390#107—1224no—
391#106—1224no—
392#102—24no—
393#102—24no—
394#102—1228no—
395C22⋊C8—1228no—
396#95—1222no—
397#95—1222no—
398C24—22no—
399C24—24no—
400C24—44no—
401C24—44no—
402C24—42no—
403C24—42no—
404C24—612no—
405C24—1224no—
406C24—1222no—
407C24—1222no—
408C24—1222no—
409C24—2442no—
410C24D4×D6111yescharacteristic
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—12216no—
424C2×D4—2448no—
425C2×D4—24432no—
426C2×D4—2448no—
427C2×D4C2×C22⋊D6114yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—612no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1224no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1222no—
446C22×C4—2444no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4—2444no—
450C22×C4—2444no—
451C22×C4#1175822212yes—
452C22×C4#1175822212yes—
453C22×C4D4×D6112yes—
454C22×C4D4×D6112yes—
455C22×C4D4×D6111yescharacteristic
456C22×C4#1175822412yes—
457C22×C4C2×C22⋊D6112yes—
458C22×C4C23×D6111yescharacteristic
459M16—44no—
460M16—1228no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—614no—
465C2×C8—1222no—
466C2×C8—1222no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—614no—
472C4⋊C4—1222no—
473C4⋊C4—1224no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C22⋊C4—42no—
477C22⋊C4—12216no—
478C22⋊C4—2444no—
479C22⋊C4—2444no—
480C22⋊C4—2442no—
481C42—42no—
482C42—2442no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—48no—
487C23—48no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—48no—
493C23—88no—
494C23—88no—
495C23—614no—
496C23—1228no—
497C23—1222no—
498C23—1222no—
499C23—1222no—
500C23—1222no—
501C23—1222no—
502C23—1222no—
503C23—2448no—
504C23—2448no—
505C23—2444no—
506C23—2444no—
507C23#1451974512yes—
508C23#1451975012yes—
509C23#1451975912yes—
510C23C2×D4×D6111yescharacteristic
511D4—24no—
512D4—24no—
513D4—42no—
514D4—42no—
515D4—48no—
516D4—24432no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—1222no—
522C2×C4—1224no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2448no—
529C2×C4—2444no—
530C2×C4—2444no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2448no—
536C2×C4#1451974512yes—
537C2×C4#1451974914yes—
538C2×C4#1451975014yes—
539C2×C4#1451975012yes—
540C2×C4#1451975912yes—
541C2×C4#1451976014yes—
542C2×C4#1451976314yes—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4C2×#1175822411yescharacteristic
547C2×C4C2×#1175822211yescharacteristic
548C2×C4C2×#1175822211yescharacteristic
549C8—42no—
550C8—1224no—
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—816no—
559C22—84no—
560C22—84no—
561C22—1224no—
562C22—1224no—
563C22—1224no—
564C22—24416no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22#1683967714yes—
570C22C2×#1451974511yescharacteristic
571C22C2×#1451975011yescharacteristic
572C22C2×#1451975911yescharacteristic
573C22AGL(1,ℤ/8)×D6112yes—
574C22#1684055812yes—
575C4—81no—
576C4—81no—
577C4—4882no—
578C4—2442no—
579C4—2444no—
580C4#1683967714yes—
581C4C2×#1451974911yescharacteristic
582C4C2×#1451975011yescharacteristic
583C4C2×#1451976011yescharacteristic
584C4C2×#1451976311yescharacteristic
585C2—24no—
586C2—84no—
587C2—2444no—
588C2C2×#1683967711yescharacteristic
589C2#2145504411yescharacteristic
590C2#2145505811yescharacteristic
591C2#2145543612yes—
592C2#2145575712yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#418C2
socleC23×C61C22×D4
prime radicalC22×#14519748C2
hypercenterC2×#418D61
hypocenterC61#23480
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#16839805C2semidirect
3. C2×#16839767C2semidirect
4. C2×#16839725C2semidirect
5. #21448198C2semidirect
6. #21447006C2semidirect
7. #21446999C2semidirect
8. #21445539C2semidirect
9. C2×C61×#418C2semidirect
21. C2×#14519710C22semidirect
22. C2×#14519708C22semidirect
24. C2×#14519698C22semidirect
26. C2×#14519689C22semidirect
28. #16839619C22semidirect
29. #16839619C22semidirect
32. #16839525C22semidirect
37. C2×#102×C61C22semidirect
38. C2×#102×C61C22semidirect
55. C2×#11758194D4semidirect
185. C2×#92418C2×D4semidirect
292. C2×#418D61semidirect
325. C2×#102C2×D61semidirect
326. C2×#102C2×D61semidirect
335. C61#23480semidirect

Maximal subgroups

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

subgroupindex
C22×#145197482
C2×#168398052
C2×#168397672
C2×#168397252
#214481982
#214470062
#214469992
#214455392
C2×C61×#4182
#2348061