Finite Group Databasebeta

← #29681281

Subgroups — #29681281

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×#16840329C211yesmaximal, characteristic
2C2×#16840242C211yesmaximal, characteristic, complemented
3C22×#14519698C211yesmaximal, characteristic, complemented
4C2×#16839730C211yesmaximal, characteristic, complemented
5#21446138C214yesmaximal, complemented
6#21446133C214yesmaximal, complemented
7#21445539C211yesmaximal, characteristic, complemented
8#21445535C211yesmaximal, characteristic, complemented
9C2×C61×#418C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×#92418C2211yescharacteristic
12#16840329—24no—
13#16840329C418yescomplemented
14C2×#14519773—21no—
15C2×#14519764—21no—
16C2×#14519748C2214yes—
17C2×#14519746—22no—
18#16840242—24no—
19#16840242—24no—
20C2×C4×Dic122—21no—
21C2×C4×D244—21no—
22C2×#14519737—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519735C2212yescomplemented
25C22×C4⋊Dic61C2211yescharacteristic
26C2×C4⋊C4×D61C2212yescomplemented
27C2×#14519698—24no—
28C2×#14519698C2214yes—
29#16839730—24no—
30C2×#14519689C2214yescomplemented
31#16839525C2214yescomplemented
32#16839524C2212yescomplemented
33#16839524C2212yescomplemented
34C23×D4×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C61×#418—24no—
38C2×#102×C61C2212yes—
39C2×#102×C61C2212yes—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222C2×C414yes—
45C2×#11758222D414yes—
46C22×Dic122D411yescharacteristic
47C22×D244D411yescharacteristic
48C22×C22⋊D61—24no—
49C22×C22⋊D61D412yes—
50C23×Dic61—21no—
51C22×C4×D61—22no—
52C22×C4×D61C2312yes—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418—22no—
56C22×#92418C2312yes—
57#14519773—28no—
58#14519764—28no—
59#14519748—28no—
60#14519748C2×C4116yescomplemented
61#14519746—216no—
62C4×Dic122—28no—
63C4×D244—28no—
64#14519737—216no—
65#14519736—22no—
66#14519736C2×C414yescomplemented
67#14519735—24no—
68#14519735—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61D414yescomplemented
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—22no—
75C4⋊C4×D61—24no—
76C4⋊C4×D61—24no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—24no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—22no—
82#14519698—28no—
83#14519689—28no—
84C25×C61—21no—
85C22×D4×C61—22no—
86C22×D4×C61—22no—
87C22×D4×C61—21no—
88C22×D4×C61—22no—
89C22×D4×C61—22no—
90C22×D4×C61—22no—
91C22×D4×C61C2×C414yescomplemented
92C22×D4×C61C2312yes—
93C23×C4×C61C2311yescharacteristic
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C2×M16×C61—22no—
97C22×C8×C61C2312yes—
98#117×C61—22no—
99#117×C61D414yescomplemented
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61—21no—
104#102×C61—24no—
105#102×C61—24no—
106C24×C61—22no—
107C24×C61—24no—
108C24×C61—42no—
109C24×C61—42no—
110C24×C61—82no—
111C24×C61—82no—
112C24×C61C2×D411yescharacteristic
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—42no—
123C2×D4×C61—44no—
124C2×D4×C61—44no—
125C2×D4×C61—42no—
126C2×D4×C61C22×C414yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61#4112yes—
135C22×C4×C61#4112yes—
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—24no—
138C2×C22⋊D61—24no—
139C2×C22⋊D61—216no—
140C2×C22⋊D61—84no—
141C2×C22⋊D61—84no—
142C2×C22⋊D61C22⋊C418yes—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—24no—
146C22×Dic61—82no—
147C22×Dic61C2×D412yes—
148#11758222—28no—
149#11758222—28no—
150#11758222—28no—
151#11758222—28no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244C22⋊C414yes—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—28no—
160C2×C4×D61—84no—
161C2×C4×D61C22×C414yes—
162C2×C4×D61C2×D414yes—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122C22⋊C414yes—
166M16×C61—44no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C2×D412yes—
173C4⋊C4×C61C2×D412yes—
174C22⋊C4×C61—24no—
175C42×C61—24no—
176C22⋊Dic61—24no—
177C22⋊Dic61—216no—
178#11758198—28no—
179C4⋊Dic61—24no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61C22×C414yes—
183C4⋊Dic61C2×D414yes—
184#11758196—28no—
185C4×Dic61—28no—
186C4×Dic61—24no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418—44no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—24no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—44no—
197C23×C61—44no—
198C23×C61—48no—
199C23×C61—88no—
200C23×C61—88no—
201C23×C61—84no—
202C23×C61—84no—
203C23×C61C2×C22⋊C412yes—
204C23×C61#12012yes—
205C23×C61#12312yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61C2×C22⋊C414yes—
212D4×C61—24no—
213D4×C61—24no—
214D4×C61—42no—
215D4×C61—42no—
216D4×C61—48no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C4×D414yes—
223C2×C4×C61C4×D414yes—
224C2×C4×C61#12012yes—
225C2×C4×C61#12114yes—
226C2×C4×C61#12114yes—
227C2×C4×C61#12312yes—
228C2×C4×C61C23×C411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C22⋊D61—216no—
235C22⋊D61—816no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—44no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—84no—
244C2×Dic61C2×C22⋊C414yes—
245D244—28no—
246C4×D61—24no—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—24no—
250C4×D61—88no—
251Dic122—28no—
252C8×C61—42no—
253#92418—42no—
254#20914—611nomaximal, Sylow
255C22×C61—24no—
256C22×C61—24no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—48no—
260C22×C61—44no—
261C22×C61—816no—
262C22×C61—84no—
263C22×C61—82no—
264C22×C61—82no—
265C22×C61#38614yes—
266C22×C61C22×C22⋊C411yescharacteristic
267C22×C61C2×#12011yescharacteristic
268C22×C61C2×#12311yescharacteristic
269C22×C61#57612yes—
270C22×C61#57712yes—
271C2×D61—28no—
272C2×D61—24no—
273C2×D61—44no—
274C2×D61—42no—
275C2×D61—42no—
276C4×C61—42no—
277C4×C61—42no—
278C4×C61#38614yes—
279C4×C61C2×C4×D411yescharacteristic
280C4×C61C2×C4×D411yescharacteristic
281C4×C61C2×#12111yescharacteristic
282C4×C61C2×#12111yescharacteristic
283Dic61—24no—
284Dic61—44no—
285Dic61—44no—
286C2×#535—611no—
287C2×#520—611no—
288C2×#418D6111yescharacteristic, complemented
289C22×#102—611no—
290C2×#408—611no—
291#1875—614no—
292#1874—614no—
293#1736—611no—
294#1735—611no—
295C2×C61—24no—
296C2×C61—84no—
297C2×C61C2×#38611yescharacteristic
298C2×C61#289712yes—
299C2×C61#300811yescharacteristic
300C2×C61#301011yescharacteristic
301C2×C61#309612yes—
302D61—44no—
303C22×#41—611no—
304C23×D4C2×D6111yescharacteristic
305C22×M16C2×D6111yescharacteristic
306C23×C8—611no—
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—1221no—
317C2×C4×D4—1222no—
318C2×#117—611no—
319C2×#117—612no—
320C2×#117C2×D6111yescharacteristic
321C22×C4⋊C4—611no—
322C22×C4⋊C4—612no—
323#418—24no—
324C2×#102—614no—
325C2×#102—1224no—
326C2×#102C2×D6112yes—
327C2×#102C2×D6112yes—
328#408—1224no—
329C2×C22⋊C8—614no—
330#340—614no—
331#340—612no—
332#340—612no—
333C61#2091411yescharacteristic, complemented, Sylow
334C25—21no—
335C2×#41—614no—
336C2×#41—614no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C2×#41—1222no—
341C22×Q8—611no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—21no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—612no—
349C22×D4—611no—
350C22×D4—1224no—
351C22×D4C4×D6114yescomplemented
352C22×D4C22×D6112yes—
353C23×C4—612no—
354C23×C4—1222no—
355C23×C4—1221no—
356C23×C4C22×D6111yescharacteristic
357C2×M16—22no—
358C2×M16—22no—
359C2×M16—22no—
360C22×C8—612no—
361C22×C8—1222no—
362C22×C8—1222no—
363C22×C8—1222no—
364C22×C8C22×D6112yes—
365#127—1228no—
366#124—1228no—
367#121—6116no—
368#121—1228no—
369#120—12216no—
370C4×Q8—1228no—
371C4×D4—1228no—
372C4×D4—12216no—
373#117—22no—
374#117—614no—
375#117—1224no—
376#117—1222no—
377#117—1224no—
378#117C22⋊D6114yescomplemented
379C2×C4⋊C4—614no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—1224no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1224no—
392C2×C22⋊C4—1221no—
393C2×C42—21no—
394C2×C42—1222no—
395C2×C42—1221no—
396#102—24no—
397#102—24no—
398#102—1228no—
399C22⋊C8—1228no—
400C24—22no—
401C24—24no—
402C24—42no—
403C24—42no—
404C24—82no—
405C24—82no—
406C24—612no—
407C24D4×D6111yescharacteristic
408#41—1228no—
409#41—1228no—
410#41—1228no—
411#41—1228no—
412C2×Q8—614no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—618no—
428C2×D4—614no—
429C2×D4—1224no—
430C2×D4—1224no—
431C2×D4—12216no—
432C2×D4—1222no—
433C2×D4—1222no—
434C2×D4—4884no—
435C2×D4—4884no—
436C2×D4C2×C4×D6114yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—612no—
440C22×C4—614no—
441C22×C4—614no—
442C22×C4—1228no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1224no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—4884no—
451C22×C4—4882no—
452C22×C4C2×C4×D6112yes—
453C22×C4#1175822212yes—
454C22×C4D4×D6112yes—
455C22×C4D4×D6112yes—
456C22×C4D4×D6111yescharacteristic
457C22×C4#1175822412yes—
458C22×C4C2×C22⋊D6112yes—
459C22×C4C23×D6111yescharacteristic
460M16—44no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—2444no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—614no—
471C4⋊C4—614no—
472C4⋊C4—1228no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1224no—
476C4⋊C4C2×C22⋊D6112yes—
477C4⋊C4C2×C22⋊D6112yes—
478C22⋊C4—24no—
479C22⋊C4—1228no—
480C22⋊C4—12216no—
481C22⋊C4—1224no—
482C42—24no—
483C42—1228no—
484C42—1224no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—48no—
493C23—88no—
494C23—84no—
495C23—88no—
496C23—84no—
497C23—614no—
498C23—1222no—
499C23—1222no—
500C23—1224no—
501C23—1224no—
502C23C22⋊C4×D6112yes—
503C23#1451974512yes—
504C23#1451975912yes—
505C23C2×D4×D6111yescharacteristic
506Q8—1228no—
507D4—24no—
508D4—24no—
509D4—42no—
510D4—42no—
511D4—48no—
512D4—12216no—
513D4—1228no—
514D4—48816no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—614no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—4888no—
529C2×C4—4884no—
530C2×C4—2444no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4C22⋊C4×D6112yes—
534C2×C4#1451973714yes—
535C2×C4C4×C22⋊D6114yes—
536C2×C4#1451974512yes—
537C2×C4#1451974814yes—
538C2×C4#1451975014yes—
539C2×C4#1451975912yes—
540C2×C4C22×C4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C2×#1175822411yescharacteristic
545C2×C4C2×#1175822211yescharacteristic
546C8—42no—
547C8—2442no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—48no—
553C22—44no—
554C22—816no—
555C22—82no—
556C22—84no—
557C22—82no—
558C22—1228no—
559C22—1224no—
560C22—2442no—
561C22—2444no—
562C22—2442no—
563C22#1683963814yes—
564C22C2×C22⋊C4×D6111yescharacteristic
565C22C2×#1451974511yescharacteristic
566C22C2×#1451975911yescharacteristic
567C22#1684056612yes—
568C22#1684057212yes—
569C4—42no—
570C4—42no—
571C4—1224no—
572C4—2444no—
573C4—2444no—
574C4#1683963814yes—
575C4C2×#1451973711yescharacteristic
576C4C2×C4×C22⋊D6111yescharacteristic
577C4C2×#1451974811yescharacteristic
578C4C2×#1451975011yescharacteristic
579C2—24no—
580C2—84no—
581C2—2444no—
582C2C2×#1683963811yescharacteristic
583C2#2145434412yes—
584C2#2145508611yescharacteristic
585C2#2145510411yescharacteristic
586C2#2145577512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#418C2
socleC23×C61C22×D4
prime radicalC2×#16840329C2
cosocleC22×C4×C61C24
hypercenterC2×#418D61
hypocenterC61#20914
LCSG ▷ C2×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×#16840242C2semidirect
3. C22×#14519698C2semidirect
4. C2×#16839730C2semidirect
5. #21446138C2semidirect
6. #21446133C2semidirect
7. #21445539C2semidirect
8. #21445535C2semidirect
9. C2×C61×#418C2semidirect
13. #16840329C4semidirect
24. C2×#14519735C22semidirect
26. C2×C4⋊C4×D61C22semidirect
30. C2×#14519689C22semidirect
31. #16839525C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
60. #14519748C2×C4semidirect
66. #14519736C2×C4semidirect
71. C2×C4⋊Dic61D4semidirect
91. C22×D4×C61C2×C4semidirect
99. #117×C61D4semidirect
288. C2×#418D61semidirect
333. C61#20914semidirect
351. C22×D4C4×D61semidirect
378. #117C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403292
C2×#168402422
C22×#145196982
C2×#168397302
#214461382
#214461332
#214455392
#214455352
C2×C61×#4182
#2091461