Finite Group Databasebeta

← #29681276

Subgroups — #29681276

Order 15616.

1970 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840326C211yesmaximal, characteristic
2C2×#16840245C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C22×#14519699C211yesmaximal, characteristic, complemented
5#21446139C214yesmaximal, complemented
6#21446134C214yesmaximal, complemented
7#21445539C211yesmaximal, characteristic, complemented
8#21445536C211yesmaximal, characteristic, complemented
9C2×C61×#408C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11#16840326—24no—
12#16840326C418yescomplemented
13C2×#14519773—21no—
14C2×#14519766—21no—
15C2×#14519751C2214yes—
16C2×#14519745—22no—
17#16840245—24no—
18#16840245—24no—
19C2×Q8×Dic61—21no—
20C2×D4×Dic61—22no—
21C2×D4×Dic61—21no—
22C2×#14519736C2211yescharacteristic
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2212yescomplemented
25C22×C4⋊Dic61C2211yescharacteristic
26C22×C4⋊Dic61C2212yescomplemented
27#16839768—24no—
28C2×#14519699—24no—
29C2×#14519699C2214yes—
30C2×#14519699C2212yes—
31C2×#14519699C2212yes—
32#16839525C2212yescomplemented
33#16839525C2212yescomplemented
34#16839525C2214yescomplemented
35C22×#41×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C23×C8×C61C2211yescharacteristic
38C61×#408—24no—
39C2×C22⋊C8×C61C2214yescomplemented
40C24×D61—21no—
41C22×D244—22no—
42C22×D244—22no—
43C22×D244—22no—
44C22×D244—22no—
45C22×D244—21no—
46C22×D244—22no—
47C22×D244C2×C414yescomplemented
48C22×D244C2312yes—
49C22×C22⋊D61—24no—
50C23×Dic61—21no—
51C23×Dic61—22no—
52#14519773—28no—
53#14519766—28no—
54#14519751—28no—
55#14519751C2×C4116yescomplemented
56#14519745—216no—
57Q8×Dic61—28no—
58D4×Dic61—216no—
59D4×Dic61—28no—
60#14519736—22no—
61#14519736—22no—
62#14519736—24no—
63#14519736—24no—
64#14519736C2×C414yescomplemented
65#14519736D414yescomplemented
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—24no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61D414yescomplemented
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—22no—
79C2×#11758198—24no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—22no—
83#14519699—28no—
84#14519699—24no—
85#14519699—24no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61C2×C414yes—
91C2×#41×C61D414yes—
92C22×Q8×C61D411yescharacteristic
93C22×D4×C61D412yes—
94C22×D4×C61D411yescharacteristic
95C23×C4×C61C2312yes—
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61—22no—
98C2×M16×C61—22no—
99C2×M16×C61—22no—
100C22×C8×C61—22no—
101C22×C8×C61—22no—
102C22×C8×C61—22no—
103C22×C8×C61C2312yes—
104C22×C8×C61C2312yes—
105C22⋊C8×C61—28no—
106C24×C61C2×D412yes—
107C24×C61C2×D411yescharacteristic
108C23×D61—22no—
109C23×D61—24no—
110C23×D61—42no—
111C23×D61—42no—
112C23×D61—82no—
113C23×D61—82no—
114#41×C61—28no—
115#41×C61—28no—
116#41×C61—28no—
117#41×C61—28no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61C22⋊C414yes—
121C2×D4×C61—24no—
122C2×D4×C61—24no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61C22⋊C418yes—
126C2×D4×C61C22⋊C414yes—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61C22×C414yes—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D414yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61#4112yes—
140C22×C4×C61#4112yes—
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—216no—
143C2×C22⋊D61—84no—
144C2×C22⋊D61—84no—
145C22×Dic61—22no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—28no—
150C22×Dic61—24no—
151C22×Dic61—82no—
152C22×Dic61—84no—
153C2×D244—28no—
154C2×D244—28no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—24no—
160C2×D244—22no—
161C2×D244—42no—
162C2×D244—44no—
163C2×D244—44no—
164C2×D244—42no—
165C2×D244C22×C414yes—
166M16×C61—44no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—44no—
174C22⋊Dic61—28no—
175C22⋊Dic61—24no—
176C22⋊Dic61—24no—
177#11758198—216no—
178C4⋊Dic61—28no—
179C4⋊Dic61—24no—
180C4⋊Dic61—22no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—22no—
184C4⋊Dic61C22×C414yes—
185C4⋊Dic61C2×D412yes—
186C4⋊Dic61C2×D414yes—
187C4⋊Dic61C2×D412yes—
188C4×Dic61—28no—
189C4×Dic61—24no—
190C4×Dic61—24no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—24no—
196C23×C61—24no—
197C23×C61C2×C22⋊C414yes—
198C23×C61C2×C22⋊C412yes—
199C23×C61#12012yes—
200C23×C61#12312yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—24no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—44no—
206C22×D61—44no—
207C22×D61—48no—
208C22×D61—88no—
209C22×D61—88no—
210C22×D61—84no—
211C22×D61—84no—
212Q8×C61—28no—
213D4×C61—216no—
214D4×C61—28no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61C2×C22⋊C414yes—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C4×D414yes—
224C2×C4×C61C4×D414yes—
225C2×C4×C61#12012yes—
226C2×C4×C61#12114yes—
227C2×C4×C61#12114yes—
228C2×C4×C61#12312yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—816no—
236C2×Dic61—24no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—22no—
240C2×Dic61—44no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—84no—
246C2×Dic61—88no—
247D244—24no—
248D244—24no—
249D244—42no—
250D244—42no—
251D244—48no—
252C8×C61—42no—
253C8×C61—42no—
254#20914—611nomaximal, Sylow
255C22×C61—24no—
256C22×C61—24no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—28no—
260C22×C61—24no—
261C22×C61—44no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61#38614yes—
265C22×C61C22×C22⋊C411yescharacteristic
266C22×C61C2×#12011yescharacteristic
267C22×C61C2×#12311yescharacteristic
268C22×C61#57612yes—
269C22×C61#57712yes—
270C2×D61—48no—
271C2×D61—44no—
272C2×D61—816no—
273C2×D61—84no—
274C2×D61—82no—
275C2×D61—82no—
276C4×C61—24no—
277C4×C61#38614yes—
278C4×C61C2×C4×D411yescharacteristic
279C4×C61C2×C4×D411yescharacteristic
280C4×C61C2×#12111yescharacteristic
281C4×C61C2×#12111yescharacteristic
282Dic61—44no—
283Dic61—44no—
284Dic61—42no—
285Dic61—42no—
286C2×#535—611no—
287C2×#520—611no—
288C2×#418—611no—
289C22×#102—611no—
290C2×#408D6111yescharacteristic, complemented
291#1875—614no—
292#1874—614no—
293#1736—611no—
294#1735—611no—
295C2×C61—24no—
296C2×C61—44no—
297C2×C61C2×#38611yescharacteristic
298C2×C61#289712yes—
299C2×C61#300811yescharacteristic
300C2×C61#301011yescharacteristic
301C2×C61#309612yes—
302D61—84no—
303C22×#41C2×D6111yescharacteristic
304C23×D4—611no—
305C22×M16C2×D6111yescharacteristic
306C23×C8C2×D6111yescharacteristic
307#535—618no—
308#535—1224no—
309C2×#127—1221no—
310C2×#124—1221no—
311C2×#121—614no—
312C2×#120—1222no—
313#520—1224no—
314#520—1224no—
315C2×C4×Q8—1221no—
316C2×C4×D4—1222no—
317C2×C4×D4—1221no—
318C2×#117—611no—
319C2×#117—611no—
320C2×#117—612no—
321C22×C4⋊C4—611no—
322C22×C4⋊C4—612no—
323#418—1224no—
324C2×#102—614no—
325C2×#102—612no—
326C2×#102—612no—
327C2×#102—1224no—
328#408—24no—
329C2×C22⋊C8C2×D6114yescomplemented
330#340—612no—
331#340—614no—
332#340—612no—
333C61#2091411yescharacteristic, complemented, Sylow
334C25—1221no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41C4×D6114yes—
340C2×#41C22⋊D6114yes—
341C22×Q8D24411yescharacteristic
342C22×D4—614no—
343C22×D4—612no—
344C22×D4—1224no—
345C22×D4—1221no—
346C22×D4—1222no—
347C22×D4—1222no—
348C22×D4—1222no—
349C22×D4—1222no—
350C22×D4—1222no—
351C22×D4D24412yes—
352C22×D4D24411yescharacteristic
353C23×C4—1222no—
354C23×C4—1221no—
355C23×C4C22×D6112yes—
356C23×C4C22×D6111yescharacteristic
357C2×M16—22no—
358C2×M16—22no—
359C2×M16—22no—
360C22×C8—22no—
361C22×C8—22no—
362C22×C8—22no—
363C22×C8C22×D6112yes—
364C22×C8C22×D6112yes—
365#127—1228no—
366#124—1228no—
367#121—6116no—
368#121—1228no—
369#120—12216no—
370C4×Q8—1228no—
371C4×D4—12216no—
372C4×D4—1228no—
373#117—614no—
374#117—614no—
375#117—1224no—
376#117—1224no—
377#117—1222no—
378#117—1222no—
379C2×C4⋊C4—614no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—1224no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1224no—
393C2×C42—1222no—
394C2×C42—1221no—
395C2×C42—1221no—
396#102—1228no—
397#102—1224no—
398#102—1224no—
399C22⋊C8—28no—
400C24—1224no—
401C24—1222no—
402C24—4882no—
403C24—4882no—
404C24—2442no—
405C24—2442no—
406C24C2×D24412yes—
407C24C2×D24411yescharacteristic
408#41—28no—
409#41—28no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8#1175819814yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—614no—
420C2×D4—12216no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1224no—
427C2×D4—1228no—
428C2×D4—1228no—
429C2×D4—4884no—
430C2×D4—4884no—
431C2×D4—2442no—
432C2×D4—2444no—
433C2×D4—2444no—
434C2×D4—2442no—
435C2×D4#1175819818yes—
436C2×D4#1175819814yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—1228no—
442C22×C4—1224no—
443C22×C4—1221no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1221no—
447C22×C4—4882no—
448C22×C4—4884no—
449C22×C4C2×C4×D6114yes—
450C22×C4C2×C4×D6112yes—
451C22×C4C2×D24412yes—
452C22×C4C2×D24411yescharacteristic
453C22×C4D4×D6112yes—
454C22×C4D4×D6112yes—
455C22×C4#1175822412yes—
456C22×C4#1175822412yes—
457C22×C4C2×C22⋊D6114yes—
458C22×C4C2×C22⋊D6112yes—
459C22×C4C23×D6111yescharacteristic
460M16—44no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—44no—
468C4⋊C4—614no—
469C4⋊C4—614no—
470C4⋊C4—612no—
471C4⋊C4—612no—
472C4⋊C4—1228no—
473C4⋊C4—1221no—
474C4⋊C4—1222no—
475C4⋊C4—1221no—
476C4⋊C4—1222no—
477C4⋊C4—1224no—
478C22⋊C4—1228no—
479C22⋊C4—1224no—
480C22⋊C4—1224no—
481C22⋊C4—12216no—
482C42—1228no—
483C42—1224no—
484C42—1224no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—1224no—
492C23—4884no—
493C23—4888no—
494C23—4884no—
495C23—4888no—
496C23—2442no—
497C23—2442no—
498C23—2444no—
499C23—2444no—
500C23—2448no—
501C23C2×#1175819814yes—
502C23C2×#1175819812yes—
503C23#1451974712yes—
504C23#1451976112yes—
505C23C22×D24411yescharacteristic
506Q8—28no—
507D4—216no—
508D4—28no—
509D4—1224no—
510D4—1224no—
511D4—48816no—
512D4—2442no—
513D4—2442no—
514D4—2448no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—1224no—
522C2×C4—1222no—
523C2×C4—1224no—
524C2×C4—1222no—
525C2×C4—4884no—
526C2×C4—4888no—
527C2×C4—2444no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4C2×#1175819814yes—
533C2×C4C2×#1175819812yes—
534C2×C4#1451973714yes—
535C2×C4#1451973714yes—
536C2×C4#1451974712yes—
537C2×C4#1451974914yes—
538C2×C4#1451974914yes—
539C2×C4#1451976112yes—
540C2×C4C22×C4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C2×#1175822411yescharacteristic
545C2×C4C2×#1175822411yescharacteristic
546C8—42no—
547C8—42no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—28no—
553C22—24no—
554C22—42no—
555C22—44no—
556C22—42no—
557C22—48816no—
558C22—4882no—
559C22—4884no—
560C22—4882no—
561C22—2448no—
562C22—2444no—
563C22#1683963914yes—
564C22C22×#1175819811yescharacteristic
565C22C2×#1451974711yescharacteristic
566C22C2×#1451976111yescharacteristic
567C22#1684056712yes—
568C22#1684057412yes—
569C4—24no—
570C4—2442no—
571C4—2444no—
572C4—2444no—
573C4—2442no—
574C4#1683963914yes—
575C4C2×#1451973711yescharacteristic
576C4C2×#1451973711yescharacteristic
577C4C2×#1451974911yescharacteristic
578C4C2×#1451974911yescharacteristic
579C2—24no—
580C2—44no—
581C2—4884no—
582C2C2×#1683963911yescharacteristic
583C2#2145434612yes—
584C2#2145508911yescharacteristic
585C2#2145510711yescharacteristic
586C2#2145577812yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#408C2
socleC23×C61C22×D4
prime radicalC2×#16840326C2
cosocleC22×C4×C61C24
hypercenterC2×#408D61
hypocenterC61#20914
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C2×#16840245C2semidirect
3. C2×#16839768C2semidirect
4. C22×#14519699C2semidirect
5. #21446139C2semidirect
6. #21446134C2semidirect
7. #21445539C2semidirect
8. #21445536C2semidirect
9. C2×C61×#408C2semidirect
12. #16840326C4semidirect
24. C2×#14519736C22semidirect
26. C22×C4⋊Dic61C22semidirect
32. #16839525C22semidirect
33. #16839525C22semidirect
34. #16839525C22semidirect
39. C2×C22⋊C8×C61C22semidirect
47. C22×D244C2×C4semidirect
55. #14519751C2×C4semidirect
64. #14519736C2×C4semidirect
65. #14519736D4semidirect
71. C2×C4⋊Dic61D4semidirect
290. C2×#408D61semidirect
329. C2×C22⋊C8C2×D61semidirect
333. C61#20914semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
C2×#168402452
C2×#168397682
C22×#145196992
#214461392
#214461342
#214455392
#214455362
C2×C61×#4082
#2091461