Finite Group Databasebeta

← #29707506

Subgroups — #29707506

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×#14519751C211yesmaximal, characteristic
2C2×#16839804C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C2×#16839768C211yesmaximal, characteristic, complemented
5#21448202C211yesmaximal, characteristic, complemented
6#21447009C214yesmaximal, complemented
7#21447002C214yesmaximal, complemented
8#21445539C211yesmaximal, characteristic, complemented
9C2×C61×#407C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C23×C22⋊D61—21no—
12C2×#14519751—24no—
13C2×#14519751—22no—
14C2×#14519751—24no—
15C2×#14519751C2214yes—
16C2×#14519736C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C22×#11758198—21no—
20#16839804—24no—
21C2×#14519711C2212yescomplemented
22C2×#14519709C2212yescomplemented
23#16839768—24no—
24#16839768—24no—
25C2×#14519699C2212yescomplemented
26C2×#14519699C2214yescomplemented
27C2×#14519699C2212yescomplemented
28#16839625—21no—
29#16839622C2212yescomplemented
30#16839622C2212yescomplemented
31#16839618—21no—
32C4×C22⋊Dic61—22no—
33#16839525C2214yescomplemented
34C24×C4×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C61×#407—24no—
38C2×C22⋊C8×C61C2214yescomplemented
39C24×D61—21no—
40C22×D244—22no—
41C22×D244—22no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—21no—
45C22×D244—22no—
46C22×D244D414yes—
47C22×D244C2312yes—
48C22×C22⋊D61—28no—
49C22×C22⋊D61—24no—
50C22×C22⋊D61—48no—
51C23×Dic61—21no—
52#14519751—216no—
53#14519751—28no—
54#14519751—28no—
55#14519736—22no—
56#14519736—22no—
57#14519736—22no—
58#14519736—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61D414yes—
62C2×C4⋊Dic61C2312yes—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×C4⋊Dic61C2312yes—
66C2×C22⋊Dic61—21no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—22no—
69C2×C22⋊Dic61—22no—
70C2×#11758198—28no—
71C2×#11758198—24no—
72C2×C4×Dic61—21no—
73C2×C4×Dic61—21no—
74#14519711—24no—
75#14519709—24no—
76#14519699—24no—
77#14519699—24no—
78#14519699—28no—
79#14519684—22no—
80#14519684—22no—
81C25×C61D411yescharacteristic
82C23×C4×C61—22no—
83C23×C4×C61—22no—
84C23×C4×C61—22no—
85C23×C4×C61—22no—
86C23×C4×C61—22no—
87C23×C4×C61—22no—
88C23×C4×C61D411yescharacteristic
89C23×C4×C61D412yes—
90C23×C4×C61C2312yes—
91C23×C4×C61C2311yescharacteristic
92C2×M16×C61—22no—
93C2×M16×C61—22no—
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C2×M16×C61—22no—
97C2×M16×C61D414yescomplemented
98C22×C8×C61C2312yes—
99C22×C8×C61C2312yes—
100C22⋊C8×C61—28no—
101C24×C61—22no—
102C24×C61—24no—
103C24×C61—22no—
104C24×C61—22no—
105C24×C61—42no—
106C24×C61C2×D412yes—
107C24×C61C2×D411yescharacteristic
108C23×D61—24no—
109C23×D61—22no—
110C23×D61—44no—
111C23×D61—44no—
112C23×D61—42no—
113C23×D61—42no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—24no—
118C22×C4×C61—24no—
119C22×C4×C61—24no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—44no—
127C22×C4×C61—44no—
128C22×C4×C61—44no—
129C22×C4×C61—44no—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61#4112yes—
136C22×C4×C61#4112yes—
137C22×C4×C61#4112yes—
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—216no—
140C2×C22⋊D61—432no—
141C2×C22⋊D61—48no—
142C2×C22⋊D61—48no—
143C22×Dic61—24no—
144C22×Dic61—22no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—44no—
149C2×D244—28no—
150C2×D244—28no—
151C2×D244—22no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—22no—
155C2×D244—24no—
156C2×D244—22no—
157C2×D244—44no—
158C2×D244—42no—
159C2×D244—42no—
160C2×D244—44no—
161C2×D244C2×D414yes—
162M16×C61—28no—
163M16×C61—44no—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61C2×D414yescomplemented
170C22⋊Dic61—44no—
171C22⋊Dic61—42no—
172C22⋊Dic61—42no—
173C22⋊Dic61—44no—
174#11758198—216no—
175C4⋊Dic61—24no—
176C4⋊Dic61—22no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—22no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—22no—
183C4⋊Dic61C2×D414yes—
184C4×Dic61—42no—
185C4×Dic61—42no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—28no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—48no—
196C23×C61—48no—
197C23×C61—44no—
198C23×C61—44no—
199C23×C61#12012yes—
200C23×C61#12112yes—
201C23×C61#12312yes—
202C23×C61AGL(1,ℤ/8)14yes—
203C23×C61C22×D411yescharacteristic
204C22×D61—24no—
205C22×D61—48no—
206C22×D61—48no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—44no—
210C22×D61—44no—
211C22×D61—48no—
212C22×D61—88no—
213C22×D61—88no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—44no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—48no—
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—48no—
239C2×Dic61—44no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247D244—24no—
248D244—24no—
249D244—42no—
250D244—42no—
251D244—48no—
252C8×C61—24no—
253C8×C61—42no—
254#23480—611nomaximal, Sylow
255C22×C61—24no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—24no—
259C22×C61—24no—
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—416no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61#39414yes—
268C22×C61C2×#12011yescharacteristic
269C22×C61C2×#12111yescharacteristic
270C22×C61C2×#12311yescharacteristic
271C22×C61C2×AGL(1,ℤ/8)12yes—
272C22×C61#57512yes—
273C2×D61—44no—
274C2×D61—44no—
275C2×D61—44no—
276C2×D61—816no—
277C2×D61—84no—
278C2×D61—84no—
279C4×C61—42no—
280C4×C61#39414yes—
281C4×C61C2×#12111yescharacteristic
282C4×C61C2×#12111yescharacteristic
283C4×C61C2×#12311yescharacteristic
284C4×C61C2×#12411yescharacteristic
285Dic61—44no—
286Dic61—81no—
287Dic61—81no—
288Dic61—82no—
289C22×#121—611no—
290C2×#428—611no—
291C2×#418—611no—
292C2×#418—611no—
293C2×#407D6111yescharacteristic, complemented
294#2280—611no—
295#2034—614no—
296#2033—614no—
297#1736—611no—
298C2×C61—24no—
299C2×C61—44no—
300C2×C61C2×#39411yescharacteristic
301C2×C61#300311yescharacteristic
302C2×C61#300511yescharacteristic
303C2×C61#305512yes—
304C2×C61#309412yes—
305D61—84no—
306C23×D4—611no—
307C23×D4—1221no—
308C24×C4C2×D6111yescharacteristic
309C22×M16C2×D6111yescharacteristic
310C22×M16C2×D6111yescharacteristic
311C2×#121—614no—
312C2×#121—1224no—
313C2×#121—1224no—
314C2×#121—1222no—
315C2×#117—611no—
316C2×#117—611no—
317C22×C4⋊C4—611no—
318C22×C22⋊C4—1221no—
319#428—1224no—
320C2×#107—612no—
321C2×#106—612no—
322#418—1224no—
323#418—1224no—
324C2×#102—612no—
325C2×#102—614no—
326C2×#102—612no—
327#407—24no—
328C2×C22⋊C8C2×D6114yescomplemented
329#381—1221no—
330#380—612no—
331#380—612no—
332#379—1221no—
333C4×C22⋊C4—1222no—
334#340—614no—
335C61#2348011yescharacteristic, complemented, Sylow
336C25—1221no—
337C25D24411yescharacteristic
338C22×D4—614no—
339C22×D4—612no—
340C22×D4—1228no—
341C22×D4—1224no—
342C22×D4—1221no—
343C22×D4—1222no—
344C22×D4—1222no—
345C22×D4—1222no—
346C22×D4—1222no—
347C22×D4—1222no—
348C22×D4—2448no—
349C23×C4—22no—
350C23×C4—22no—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—22no—
354C23×C4—22no—
355C23×C4—1221no—
356C23×C4D24411yescharacteristic
357C23×C4D24412yes—
358C23×C4C22×D6112yes—
359C23×C4C22×D6111yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—22no—
363C2×M16—22no—
364C2×M16—22no—
365C2×M16C22⋊D6114yescomplemented
366C22×C8C22×D6112yes—
367C22×C8C22×D6112yes—
368#121—1228no—
369#121—1228no—
370#121—12216no—
371#117—1222no—
372#117—1222no—
373#117—1222no—
374#117—1222no—
375C2×C4⋊C4—614no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1222no—
382C2×C22⋊C4—1228no—
383C2×C22⋊C4—1224no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4—1222no—
388C2×C42—1221no—
389C2×C42—1221no—
390#107—1224no—
391#106—1224no—
392#102—1224no—
393#102—1224no—
394#102—1228no—
395C22⋊C8—28no—
396#95—1222no—
397#95—1222no—
398C24—22no—
399C24—24no—
400C24—22no—
401C24—22no—
402C24—42no—
403C24—1222no—
404C24—1224no—
405C24—2444no—
406C24—2444no—
407C24—2442no—
408C24—2442no—
409C24C2×D24412yes—
410C24C2×D24411yescharacteristic
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—24432no—
422C2×D4—2448no—
423C2×D4—2448no—
424C2×D4—2444no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4—2444no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—1222no—
445C22×C4—1224no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1222no—
449C22×C4—2444no—
450C22×C4C2×D24412yes—
451C22×C4C2×D24411yescharacteristic
452C22×C4#1175822212yes—
453C22×C4D4×D6112yes—
454C22×C4D4×D6112yes—
455C22×C4#1175822412yes—
456C22×C4#1175822412yes—
457C22×C4C2×C22⋊D6112yes—
458C22×C4C23×D6111yescharacteristic
459M16—28no—
460M16—44no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8C2×C22⋊D6114yescomplemented
467C4⋊C4—614no—
468C4⋊C4—1222no—
469C4⋊C4—1224no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C22⋊C4—12216no—
477C22⋊C4—2444no—
478C22⋊C4—2442no—
479C22⋊C4—2442no—
480C22⋊C4—2444no—
481C42—2442no—
482C42—2442no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—48no—
493C23—48no—
494C23—44no—
495C23—44no—
496C23—1224no—
497C23—4888no—
498C23—4888no—
499C23—2448no—
500C23—2448no—
501C23—2442no—
502C23—2442no—
503C23—2444no—
504C23—2444no—
505C23—2448no—
506C23#1451974712yes—
507C23#1451975112yes—
508C23#1451976112yes—
509C23#1451980514yes—
510C23C22×D24411yescharacteristic
511D4—1224no—
512D4—1224no—
513D4—24432no—
514D4—2442no—
515D4—2442no—
516D4—2448no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—1224no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2448no—
535C2×C4—2444no—
536C2×C4#1451974712yes—
537C2×C4#1451974814yes—
538C2×C4#1451974914yes—
539C2×C4#1451975112yes—
540C2×C4#1451975914yes—
541C2×C4#1451976112yes—
542C2×C4#1451976314yes—
543C2×C4C2×D4×D6111yescharacteristic
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4C2×#1175822411yescharacteristic
547C2×C4C2×#1175822411yescharacteristic
548C2×C4C2×#1175822211yescharacteristic
549C8—24no—
550C8—42no—
551C22—24no—
552C22—24no—
553C22—22no—
554C22—22no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—416no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—48816no—
564C22—4884no—
565C22—4884no—
566C22—2444no—
567C22—2444no—
568C22—2444no—
569C22#1683967814yes—
570C22C2×#1451974711yescharacteristic
571C22C2×#1451975111yescharacteristic
572C22C2×#1451976111yescharacteristic
573C22C2×#1451980512yes—
574C22#1684056212yes—
575C4—42no—
576C4—4882no—
577C4—4881no—
578C4—4881no—
579C4—2444no—
580C4#1683967814yes—
581C4C2×#1451974811yescharacteristic
582C4C2×#1451974911yescharacteristic
583C4C2×#1451975911yescharacteristic
584C4C2×#1451976311yescharacteristic
585C2—24no—
586C2—44no—
587C2—4884no—
588C2C2×#1683967811yescharacteristic
589C2#2145504911yescharacteristic
590C2#2145506411yescharacteristic
591C2#2145543812yes—
592C2#2145576012yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#407C2
socleC23×C61C22×D4
prime radicalC22×#14519751C2
hypercenterC2×#407D61
hypocenterC61#23480
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×#16839804C2semidirect
3. C2×#16839768C2semidirect
4. C2×#16839768C2semidirect
5. #21448202C2semidirect
6. #21447009C2semidirect
7. #21447002C2semidirect
8. #21445539C2semidirect
9. C2×C61×#407C2semidirect
21. C2×#14519711C22semidirect
22. C2×#14519709C22semidirect
25. C2×#14519699C22semidirect
26. C2×#14519699C22semidirect
27. C2×#14519699C22semidirect
29. #16839622C22semidirect
30. #16839622C22semidirect
33. #16839525C22semidirect
38. C2×C22⋊C8×C61C22semidirect
97. C2×M16×C61D4semidirect
169. C2×C8×C61C2×D4semidirect
293. C2×#407D61semidirect
328. C2×C22⋊C8C2×D61semidirect
335. C61#23480semidirect
365. C2×M16C22⋊D61semidirect
466. C2×C8C2×C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197512
C2×#168398042
C2×#168397682
#214482022
#214470092
#214470022
#214455392
C2×C61×#4072
#2348061