Finite Group Databasebeta

← #29707508

Subgroups — #29707508

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×#14519752C211yesmaximal, characteristic
2C2×#16839802C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C2×#16839766C211yesmaximal, characteristic, complemented
5C2×#16839725C211yesmaximal, characteristic, complemented
6#21448198C211yesmaximal, characteristic, complemented
7#21447011C214yesmaximal, complemented
8#21447004C214yesmaximal, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C23×D244—21no—
11C23×D244C2211yescharacteristic
12C23×C4×D61C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C2×#14519752—24no—
15C2×#14519752—24no—
16C2×#14519752—22no—
17C2×#14519752C2214yes—
18C2×#14519736C2211yescharacteristic
19C22×#11758198—21no—
20#16839802—24no—
21C2×#14519710C2212yescomplemented
22C2×#14519708C2212yescomplemented
23#16839768—24no—
24#16839766—24no—
25C2×#14519699C2212yescomplemented
26C2×#14519699C2212yescomplemented
27C2×#14519697C2214yescomplemented
28#16839725—24no—
29C2×#14519689C2214yescomplemented
30#16839623—21no—
31#16839619C2212yescomplemented
32#16839619C2212yescomplemented
33#16839617—21no—
34C4×#11758198—22no—
35C22×M16×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C22×C4⋊C4×C61C2211yescharacteristic
38C61×#340C2214yescomplemented
39C24×D61—21no—
40C24×D61D411yescharacteristic
41C22×D244—22no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244—28no—
46C22×D244—24no—
47C22×D244—22no—
48C22×D244—21no—
49C22×D244—48no—
50C22×D244D414yes—
51C22×D244C2312yes—
52C23×Dic61D411yescharacteristic
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61—22no—
56C22×C4×D61—22no—
57C22×C4×D61—22no—
58C22×C4×D61—22no—
59C22×C4×D61D412yes—
60C22×C4×D61C2312yes—
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194D414yescomplemented
64C22×#92418C2312yes—
65#14519752—216no—
66#14519752—28no—
67#14519752—28no—
68#14519736—22no—
69#14519736—22no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C22⋊Dic61—21no—
73C2×#11758198—28no—
74C2×#11758198—24no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×C4×Dic61—21no—
78#14519710—24no—
79#14519708—24no—
80#14519699—24no—
81#14519699—24no—
82#14519697—28no—
83#14519689—28no—
84#14519684—22no—
85#14519684—22no—
86C23×C4×C61—21no—
87C23×C4×C61C2311yescharacteristic
88C2×M16×C61—22no—
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C22×C8×C61C2312yes—
92#117×C61—22no—
93#117×C61—22no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61D414yes—
97C2×C4⋊C4×C61C2312yes—
98C2×C4⋊C4×C61C2312yes—
99C2×C22⋊C4×C61—21no—
100C2×C42×C61—21no—
101C24×C61C2×D411yescharacteristic
102C23×D61—22no—
103C23×D61—24no—
104C23×D61—24no—
105C23×D61—22no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—42no—
109C23×D61—44no—
110C23×D61—44no—
111C23×D61—42no—
112C23×D61—42no—
113C23×D61C2×D412yes—
114C22×C4×C61—24no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—44no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61#4112yes—
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D412yes—
132C2×D244—28no—
133C2×D244—28no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—216no—
139C2×D244—24no—
140C2×D244—22no—
141C2×D244—432no—
142C2×D244—48no—
143C2×D244—48no—
144C2×D244—44no—
145C2×D244—42no—
146C2×D244—42no—
147C2×D244—44no—
148C2×D244C2×D414yes—
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—24no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C2×D414yes—
171C22⋊C4×C61—42no—
172C42×C61—42no—
173C22⋊Dic61—42no—
174#11758198—216no—
175#11758198—44no—
176#11758198—44no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4×Dic61—42no—
182#11758194—28no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418C2×D414yescomplemented
186C23×C61—22no—
187C23×C61—22no—
188C23×C61#12012yes—
189C23×C61#12112yes—
190C23×C61#12312yes—
191C23×C61C22×D411yescharacteristic
192C22×D61—28no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—24no—
200C22×D61—48no—
201C22×D61—48no—
202C22×D61—44no—
203C22×D61—44no—
204C22×D61—48no—
205C22×D61—48no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—44no—
209C22×D61—44no—
210C22×D61—48no—
211C22×D61—88no—
212C22×D61—88no—
213C22×D61AGL(1,ℤ/8)14yes—
214C2×C4×C61—24no—
215C2×C4×C61—48no—
216C2×C4×C61—44no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61#12012yes—
221C2×C4×C61#12114yes—
222C2×C4×C61#12114yes—
223C2×C4×C61#12112yes—
224C2×C4×C61#12314yes—
225C2×C4×C61#12312yes—
226C2×C4×C61#12414yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—44no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241D244—24no—
242D244—24no—
243D244—432no—
244D244—42no—
245D244—42no—
246D244—48no—
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#39414yes—
260C22×C61C2×#12011yescharacteristic
261C22×C61C2×#12111yescharacteristic
262C22×C61C2×#12311yescharacteristic
263C22×C61C2×AGL(1,ℤ/8)12yes—
264C22×C61#57512yes—
265C2×D61—24no—
266C2×D61—24no—
267C2×D61—24no—
268C2×D61—416no—
269C2×D61—42no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—44no—
274C2×D61—44no—
275C2×D61—44no—
276C2×D61—816no—
277C2×D61—84no—
278C2×D61—84no—
279C4×C61—44no—
280C4×C61—82no—
281C4×C61#39414yes—
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×#12111yescharacteristic
284C4×C61C2×#12311yescharacteristic
285C4×C61C2×#12411yescharacteristic
286Dic61—42no—
287Dic61—81no—
288Dic61—81no—
289C22×#121—611no—
290C2×#428—611no—
291C2×#418—611no—
292C2×#418—611no—
293C2×#407—611no—
294#2280—611no—
295#2034—614no—
296#2033—614no—
297#1736D6111yescharacteristic, complemented
298C2×C61—24no—
299C2×C61C2×#39411yescharacteristic
300C2×C61#300311yescharacteristic
301C2×C61#300511yescharacteristic
302C2×C61#305512yes—
303C2×C61#309412yes—
304D61—44no—
305D61—84no—
306C23×D4—611no—
307C23×D4—1221no—
308C24×C4—611no—
309C22×M16—611no—
310C22×M16C2×D6111yescharacteristic
311C2×#121—614no—
312C2×#121—1222no—
313C2×#121—1224no—
314C2×#121—1224no—
315C2×#117—611no—
316C2×#117C2×D6111yescharacteristic
317C22×C4⋊C4C2×D6111yescharacteristic
318C22×C22⋊C4—1221no—
319#428—1224no—
320C2×#107—612no—
321C2×#106—612no—
322#418—1224no—
323#418—1224no—
324C2×#102—612no—
325C2×#102—612no—
326C2×#102—614no—
327#407—1224no—
328C2×C22⋊C8—614no—
329#381—1221no—
330#380—612no—
331#380—612no—
332#379—1221no—
333C4×C22⋊C4—1222no—
334#340C2×D6114yescomplemented
335C61#2348011yescharacteristic, complemented, Sylow
336C25—611no—
337C25—1221no—
338C22×D4—614no—
339C22×D4—612no—
340C22×D4—1224no—
341C22×D4—1228no—
342C22×D4—1222no—
343C22×D4—1222no—
344C22×D4—1222no—
345C22×D4—1222no—
346C22×D4—1221no—
347C22×D4—1222no—
348C22×D4—2448no—
349C23×C4—21no—
350C23×C4—611no—
351C23×C4—612no—
352C23×C4—612no—
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—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4D24414yes—
380C2×C4⋊C4C22×D6112yes—
381C2×C4⋊C4C22×D6112yes—
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—1228no—
393#102—1224no—
394#102—1224no—
395C22⋊C8—1228no—
396#95—1222no—
397#95—1222no—
398C24—612no—
399C24—1224no—
400C24—1222no—
401C24—1222no—
402C24—1224no—
403C24—1222no—
404C24—1222no—
405C24—2442no—
406C24—2444no—
407C24—2444no—
408C24—2442no—
409C24—2442no—
410C24D4×D6111yescharacteristic
411C2×D4—614no—
412C2×D4—12216no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1224no—
419C2×D4—1228no—
420C2×D4—1228no—
421C2×D4—2448no—
422C2×D4—24432no—
423C2×D4—2448no—
424C2×D4—2442no—
425C2×D4—2444no—
426C2×D4—2442no—
427C2×D4—2444no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—44no—
432C22×C4—612no—
433C22×C4—1222no—
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—1221no—
446C22×C4—1221no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4—2444no—
450C22×C4—2444no—
451C22×C4C2×D24412yes—
452C22×C4#1175822212yes—
453C22×C4#1175822212yes—
454C22×C4D4×D6112yes—
455C22×C4D4×D6112yes—
456C22×C4D4×D6111yescharacteristic
457C22×C4#1175822612yes—
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—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4C2×D24414yes—
476C22⋊C4—42no—
477C22⋊C4—12216no—
478C22⋊C4—2444no—
479C22⋊C4—2444no—
480C22⋊C4—2442no—
481C42—42no—
482C42—2442no—
483C23—22no—
484C23—22no—
485C23—614no—
486C23—1228no—
487C23—1222no—
488C23—1222no—
489C23—1222no—
490C23—1222no—
491C23—1222no—
492C23—1222no—
493C23—1224no—
494C23—4888no—
495C23—4888no—
496C23—2448no—
497C23—2448no—
498C23—2448no—
499C23—2448no—
500C23—2444no—
501C23—2448no—
502C23—2442no—
503C23—2442no—
504C23—2444no—
505C23—2444no—
506C23—2444no—
507C23#1451974712yes—
508C23#1451975012yes—
509C23#1451976212yes—
510C23C2×D4×D6111yescharacteristic
511D4—1224no—
512D4—1224no—
513D4—24432no—
514D4—2448no—
515D4—2442no—
516D4—2442no—
517C2×C4—24no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2444no—
531C2×C4—2448no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4#1451974712yes—
537C2×C4#1451975012yes—
538C2×C4#1451975014yes—
539C2×C4#1451975214yes—
540C2×C4#1451976212yes—
541C2×C4#1451976214yes—
542C2×C4#1451976514yes—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C22×D24411yescharacteristic
546C2×C4C2×#1175822211yescharacteristic
547C2×C4C2×#1175822211yescharacteristic
548C2×C4C2×#1175822611yescharacteristic
549C8—42no—
550C8—1224no—
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—1224no—
556C22—1224no—
557C22—1224no—
558C22—48816no—
559C22—4884no—
560C22—4884no—
561C22—24416no—
562C22—2444no—
563C22—2444no—
564C22—2444no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22#1683968014yes—
570C22C2×#1451974711yescharacteristic
571C22C2×#1451975011yescharacteristic
572C22C2×#1451976211yescharacteristic
573C22AGL(1,ℤ/8)×D6112yes—
574C22#1684056112yes—
575C4—44no—
576C4—82no—
577C4—4881no—
578C4—4881no—
579C4—2442no—
580C4#1683968014yes—
581C4C2×#1451975011yescharacteristic
582C4C2×#1451975211yescharacteristic
583C4C2×#1451976211yescharacteristic
584C4C2×#1451976511yescharacteristic
585C2—24no—
586C2—4884no—
587C2—2444no—
588C2C2×#1683968011yescharacteristic
589C2#2145504811yescharacteristic
590C2#2145506511yescharacteristic
591C2#2145544112yes—
592C2#2145576112yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC22×#14519752C2
hypercenter#1736D61
hypocenterC61#23480
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#16839802C2semidirect
3. C2×#16839768C2semidirect
4. C2×#16839766C2semidirect
5. C2×#16839725C2semidirect
6. #21448198C2semidirect
7. #21447011C2semidirect
8. #21447004C2semidirect
9. C61×#1736C2semidirect
21. C2×#14519710C22semidirect
22. C2×#14519708C22semidirect
25. C2×#14519699C22semidirect
26. C2×#14519699C22semidirect
27. C2×#14519697C22semidirect
29. C2×#14519689C22semidirect
31. #16839619C22semidirect
32. #16839619C22semidirect
38. C61×#340C22semidirect
63. C2×#11758194D4semidirect
185. C2×#92418C2×D4semidirect
297. #1736D61semidirect
334. #340C2×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×#145197522
C2×#168398022
C2×#168397682
C2×#168397662
C2×#168397252
#214481982
#214470112
#214470042
C61×#17362
#2348061