Finite Group Databasebeta

← #29681279

Subgroups — #29681279

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×#16840331C211yesmaximal, characteristic
2C2×#16840243C211yesmaximal, characteristic, complemented
3C2×#16839768C211yesmaximal, characteristic, complemented
4C22×#14519697C211yesmaximal, characteristic, complemented
5C2×#16839730C211yesmaximal, characteristic, complemented
6#21446140C214yesmaximal, complemented
7#21446135C214yesmaximal, complemented
8#21445535C211yesmaximal, characteristic, complemented
9C61×#1736C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C23×D244C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13#16840331—24no—
14#16840331C418yescomplemented
15C2×#14519774—21no—
16C2×#14519765—21no—
17C2×#14519752C2214yes—
18C2×#14519747—22no—
19#16840243—24no—
20#16840243—24no—
21C2×#14519742—21no—
22C2×#14519739—21no—
23C2×C4×C22⋊D61—22no—
24C2×#14519736C2211yescharacteristic
25C2×#14519735C2212yescomplemented
26C2×C4⋊C4×D61C2212yescomplemented
27#16839768—24no—
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30C2×#14519697—24no—
31C2×#14519697C2214yes—
32#16839730—24no—
33C2×#14519689C2214yescomplemented
34#16839524C2212yescomplemented
35#16839524C2212yescomplemented
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C22×C4⋊C4×C61C2211yescharacteristic
39C61×#340C2214yescomplemented
40C24×D61—21no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222C2×C414yes—
46C2×#11758222D414yes—
47C22×Dic122D411yescharacteristic
48C22×D244—22no—
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—22no—
52C22×D244—21no—
53C22×D244—22no—
54C22×D244—24no—
55C22×D244C2×C414yescomplemented
56C22×D244D411yescharacteristic
57C22×D244C2312yes—
58C22×C22⋊D61D412yes—
59C22×C4×D61—22no—
60C22×C4×D61C2312yes—
61C22×#92418—22no—
62C22×#92418—22no—
63C22×#92418—22no—
64C22×#92418C2312yes—
65#14519774—28no—
66#14519765—28no—
67#14519752—28no—
68#14519752C2×C4116yescomplemented
69#14519747—216no—
70#14519742—28no—
71#14519739—28no—
72C4×C22⋊D61—216no—
73#14519736—22no—
74#14519736D414yescomplemented
75#14519735—24no—
76#14519735—24no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—22no—
80C4⋊C4×D61—24no—
81C4⋊C4×D61—24no—
82C2×C22⋊Dic61—21no—
83C2×#11758198—22no—
84C2×#11758198—24no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—22no—
87#14519699—24no—
88#14519699—24no—
89#14519697—28no—
90#14519689—28no—
91C23×C4×C61—21no—
92C23×C4×C61C2311yescharacteristic
93C2×M16×C61—22no—
94C2×M16×C61—22no—
95C2×M16×C61—22no—
96C22×C8×C61C2312yes—
97#117×C61—22no—
98#117×C61C2×C414yescomplemented
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61D414yescomplemented
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2312yes—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61—21no—
106C24×C61C2×D411yescharacteristic
107C23×D61—22no—
108C23×D61—24no—
109C23×D61—42no—
110C23×D61—42no—
111C23×D61—82no—
112C23×D61—82no—
113C23×D61C2×D412yes—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—82no—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×C4×C61C2411yescharacteristic
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61C22⋊C418yes—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D412yes—
132#11758222—28no—
133#11758222—28no—
134#11758222—28no—
135#11758222—28no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—28no—
139C2×D244—28no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—24no—
145C2×D244—22no—
146C2×D244—216no—
147C2×D244—42no—
148C2×D244—44no—
149C2×D244—44no—
150C2×D244—42no—
151C2×D244—84no—
152C2×D244—84no—
153C2×D244C22⋊C414yes—
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—24no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61C22×C414yes—
174C4⋊C4×C61C2×D414yes—
175C22⋊C4×C61—24no—
176C42×C61—24no—
177C22⋊Dic61—24no—
178#11758198—28no—
179#11758198—216no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61C2×D412yes—
183C4⋊Dic61C2×D412yes—
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×C61C2×C22⋊C412yes—
194C23×C61#12012yes—
195C23×C61#12312yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—24no—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—44no—
205C22×D61—44no—
206C22×D61—48no—
207C22×D61—88no—
208C22×D61—88no—
209C22×D61—84no—
210C22×D61—84no—
211C22×D61C2×C22⋊C414yes—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—44no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—84no—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C4×D414yes—
220C2×C4×C61C4×D414yes—
221C2×C4×C61#12012yes—
222C2×C4×C61#12114yes—
223C2×C4×C61#12114yes—
224C2×C4×C61#12312yes—
225C2×C4×C61C23×C411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C22⋊D61—216no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61C2×C22⋊C414yes—
239D244—24no—
240D244—24no—
241D244—28no—
242D244—42no—
243D244—42no—
244D244—48no—
245D244—816no—
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#38614yes—
260C22×C61C22×C22⋊C411yescharacteristic
261C22×C61C2×#12011yescharacteristic
262C22×C61C2×#12311yescharacteristic
263C22×C61#57612yes—
264C22×C61#57712yes—
265C2×D61—28no—
266C2×D61—24no—
267C2×D61—44no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—48no—
271C2×D61—44no—
272C2×D61—816no—
273C2×D61—84no—
274C2×D61—82no—
275C2×D61—82no—
276C4×C61—44no—
277C4×C61—44no—
278C4×C61#38614yes—
279C4×C61C2×C4×D411yescharacteristic
280C4×C61C2×C4×D411yescharacteristic
281C4×C61C2×#12111yescharacteristic
282C4×C61C2×#12111yescharacteristic
283Dic61—24no—
284Dic61—42no—
285Dic61—42no—
286C2×#535—611no—
287C2×#520—611no—
288C2×#418—611no—
289C22×#102—611no—
290C2×#408—611no—
291#1875—614no—
292#1874—614no—
293#1736D6111yescharacteristic, complemented
294#1735—611no—
295C2×C61—24no—
296C2×C61C2×#38611yescharacteristic
297C2×C61#289712yes—
298C2×C61#300811yescharacteristic
299C2×C61#301011yescharacteristic
300C2×C61#309612yes—
301D61—44no—
302D61—84no—
303C22×#41—611no—
304C23×D4—611no—
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—612no—
322C22×C4⋊C4C2×D6111yescharacteristic
323#418—1224no—
324C2×#102—612no—
325C2×#102—612no—
326C2×#102—614no—
327C2×#102—1224no—
328#408—1224no—
329C2×C22⋊C8—614no—
330#340—612no—
331#340—612no—
332#340C2×D6114yescomplemented
333C61#2091411yescharacteristic, complemented, Sylow
334C25—1221no—
335C2×#41—614no—
336C2×#41—614no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C2×#41—1222no—
341C22×Q8—611no—
342C22×D4—612no—
343C22×D4—614no—
344C22×D4—611no—
345C22×D4—612no—
346C22×D4—1224no—
347C22×D4—1222no—
348C22×D4—1222no—
349C22×D4—1222no—
350C22×D4—1221no—
351C22×D4—1222no—
352C22×D4—1222no—
353C23×C4—21no—
354C23×C4—612no—
355C23×C4—1222no—
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#117C4×D6114yescomplemented
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1224no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4D24414yescomplemented
387C2×C4⋊C4C22×D6112yes—
388C2×C4⋊C4C22×D6112yes—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1221no—
393C2×C42—21no—
394C2×C42—1222no—
395C2×C42—1221no—
396#102—1224no—
397#102—1224no—
398#102—1228no—
399C22⋊C8—1228no—
400C24—612no—
401C24—1224no—
402C24—1222no—
403C24—4882no—
404C24—4882no—
405C24—2442no—
406C24—2442no—
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—618no—
416C2×D4—614no—
417C2×D4—614no—
418C2×D4—12216no—
419C2×D4—1224no—
420C2×D4—1224no—
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—1222no—
430C2×D4—1222no—
431C2×D4—4884no—
432C2×D4—4884no—
433C2×D4—2442no—
434C2×D4—2444no—
435C2×D4—2444no—
436C2×D4—2442no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—82no—
441C22×C4—612no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—1228no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—4884no—
452C22×C4C2×C4×D6112yes—
453C22×C4C2×D24412yes—
454C22×C4#1175822212yes—
455C22×C4D4×D6112yes—
456C22×C4D4×D6112yes—
457C22×C4D4×D6111yescharacteristic
458C22×C4#1175822612yes—
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—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—612no—
472C4⋊C4—612no—
473C4⋊C4—1228no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4C2×C4×D6114yes—
477C4⋊C4C2×D24414yes—
478C22⋊C4—24no—
479C22⋊C4—12216no—
480C22⋊C4—1228no—
481C22⋊C4—1224no—
482C42—24no—
483C42—1228no—
484C42—1224no—
485C23—22no—
486C23—22no—
487C23—614no—
488C23—1222no—
489C23—1222no—
490C23—1224no—
491C23—1224no—
492C23—1224no—
493C23—4884no—
494C23—4888no—
495C23—4884no—
496C23—4888no—
497C23—2442no—
498C23—2442no—
499C23—2444no—
500C23—2444no—
501C23—2448no—
502C23C22⋊C4×D6112yes—
503C23#1451974712yes—
504C23#1451976212yes—
505C23C2×D4×D6111yescharacteristic
506Q8—1228no—
507D4—12216no—
508D4—1224no—
509D4—1224no—
510D4—1228no—
511D4—48816no—
512D4—2442no—
513D4—2442no—
514D4—2448no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—614no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1224no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—4888no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4C22⋊C4×D6112yes—
534C2×C4#1451973914yes—
535C2×C4C4×D24414yes—
536C2×C4#1451974712yes—
537C2×C4#1451975014yes—
538C2×C4#1451975214yes—
539C2×C4#1451976212yes—
540C2×C4C22×C4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×D24411yescharacteristic
544C2×C4C2×#1175822211yescharacteristic
545C2×C4C2×#1175822611yescharacteristic
546C8—42no—
547C8—2442no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—1228no—
553C22—1224no—
554C22—48816no—
555C22—4882no—
556C22—4884no—
557C22—4882no—
558C22—2448no—
559C22—2444no—
560C22—2442no—
561C22—2444no—
562C22—2442no—
563C22#1683964014yes—
564C22C2×C22⋊C4×D6111yescharacteristic
565C22C2×#1451974711yescharacteristic
566C22C2×#1451976211yescharacteristic
567C22#1684056612yes—
568C22#1684057112yes—
569C4—44no—
570C4—44no—
571C4—1224no—
572C4—2442no—
573C4—2442no—
574C4#1683964014yes—
575C4C2×#1451973911yescharacteristic
576C4C2×C4×D24411yescharacteristic
577C4C2×#1451975011yescharacteristic
578C4C2×#1451975211yescharacteristic
579C2—24no—
580C2—4884no—
581C2—2444no—
582C2C2×#1683964011yescharacteristic
583C2#2145434312yes—
584C2#2145508811yescharacteristic
585C2#2145510611yescharacteristic
586C2#2145577912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC2×#16840331C2
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20914
LCSG ▷ C2×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×#16840243C2semidirect
3. C2×#16839768C2semidirect
4. C22×#14519697C2semidirect
5. C2×#16839730C2semidirect
6. #21446140C2semidirect
7. #21446135C2semidirect
8. #21445535C2semidirect
9. C61×#1736C2semidirect
14. #16840331C4semidirect
25. C2×#14519735C22semidirect
26. C2×C4⋊C4×D61C22semidirect
33. C2×#14519689C22semidirect
34. #16839524C22semidirect
35. #16839524C22semidirect
39. C61×#340C22semidirect
55. C22×D244C2×C4semidirect
68. #14519752C2×C4semidirect
74. #14519736D4semidirect
98. #117×C61C2×C4semidirect
101. C2×C4⋊C4×C61D4semidirect
293. #1736D61semidirect
332. #340C2×D61semidirect
333. C61#20914semidirect
378. #117C4×D61semidirect
386. C2×C4⋊C4D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168403312
C2×#168402432
C2×#168397682
C22×#145196972
C2×#168397302
#214461402
#214461352
#214455352
C61×#17362
#2091461