Finite Group Databasebeta

← #29713929

Subgroups — #29713929

Order 15616.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic, complemented
2C2×#16840053C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic, complemented
4#21450735C211yesmaximal, characteristic
5#21447014C212yesmaximal, complemented
6#21446821C212yesmaximal, complemented
7#21446779C211yesmaximal, characteristic, complemented
8#21446612C211yesmaximal, characteristic, complemented
9#21446135C212yesmaximal, complemented
10#21446013C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13#16840472—24no—
14#16840472—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519769—21no—
17C2×#14519755—22no—
18C2×#14519753C2212yes—
19C2×#14519752—21no—
20C2×#14519752C2212yes—
21C2×C4×D244—21no—
22C2×C4×D244C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×C4⋊C4×D61C2212yes—
25#16840053—24no—
26C4×#11758201—24no—
27C2×#14519709C2212yes—
28C2×#14519708C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519699C2212yes—
31C2×#14519697C2211yescharacteristic
32C2×#14519689C2212yes—
33C2×#14519689C2212yes—
34C2×C4×#92418C2211yescharacteristic
35#16839679—22no—
36#16839619—21no—
37#16839619C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—21no—
46C22×D244—22no—
47C22×D244D411yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C22×C4×D61C2312yes—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C22×#92418C2311yescharacteristic
56C22×#92418C2311yescharacteristic
57#14519777—22no—
58#14519777D414yescomplemented
59#14519769—28no—
60#14519755—216no—
61#14519753—24no—
62#14519753—24no—
63#14519752—24no—
64#14519752—42no—
65#14519752—44no—
66C4×D244—28no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70C4×D244—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—22no—
76C2×#11758196—22no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61—24no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×#11758198D412yes—
84#14519709—24no—
85#14519708—22no—
86#14519701—22no—
87#14519699—24no—
88#14519697—22no—
89#14519689—24no—
90#14519689—42no—
91C4×#92418—22no—
92#14519684—22no—
93C22×Q8×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C4×Q8×C61—22no—
96C4×Q8×C61D414yescomplemented
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2312yes—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—22no—
105#103×C61—42no—
106C23×D61—21no—
107C23×D61C2×D412yes—
108C2×Q8×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C22×Dic61—22no—
121C22×Dic61—21no—
122C22×Dic61C2×D412yes—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—44no—
131C2×D244—44no—
132C2×D244—82no—
133C2×D244SD1614yescomplemented
134C2×D244#4112yes—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—28no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C4×D61—82no—
146C2×C4×D61—82no—
147C2×Dic122—24no—
148C2×Dic122—81no—
149C2×Dic122—81no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—42no—
158C4⋊C4×C61—81no—
159C4⋊C4×C61C2×D412yes—
160C42×C61—24no—
161C42×C61C2×D412yes—
162C42×C61C2×D412yes—
163#11758201—28no—
164#11758198—28no—
165#11758198—24no—
166#11758198—44no—
167#11758198—44no—
168#11758198—42no—
169C4⋊Dic61—24no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61C2×D414yes—
175#11758196—28no—
176#11758196—28no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418#4112yes—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61C2×SD1614yes—
198Q8×C61—22no—
199Q8×C61—22no—
200Q8×C61—41no—
201Q8×C61—41no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61#12012yes—
214C2×C4×C61#12012yes—
215C2×C4×C61C2×SD1612yes—
216C2×C4×C61#13712yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×Dic61—22no—
224C2×Dic61—24no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232D244—22no—
233D244—22no—
234D244—28no—
235D244—41no—
236D244—41no—
237D244—88no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—42no—
241C4×D61—42no—
242C4×D61—88no—
243C4×D61—84no—
244Dic122—84no—
245C8×C61—42no—
246#92418—22no—
247#92418—41no—
248#24118—611nomaximal, Sylow
249C22×C61C2×#12011yescharacteristic
250C22×C61#54611yescharacteristic
251C22×C61#54711yescharacteristic
252C22×C61C22×SD1611yescharacteristic
253C22×C61C2×#13711yescharacteristic
254C22×C61#57611yescharacteristic
255C22×C61#57711yescharacteristic
256C2×D61—24no—
257C2×D61—24no—
258C2×D61—24no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C4×C61—22no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—82no—
274C4×C61#44814yes—
275C4×C61#52511yescharacteristic
276C4×C61#53711yescharacteristic
277C4×C61#54811yescharacteristic
278C4×C61#55011yescharacteristic
279Dic61—42no—
280Dic61—44no—
281Dic61—81no—
282C2×#554—611no—
283C2×#476—611no—
284C2×C4×SD16—611no—
285#2537—611no—
286#2035—612no—
287#2000—612no—
288#1990—611no—
289#1954—611no—
290#1874—612no—
291#1853—612no—
292#1803D6111yescharacteristic, complemented
293C2×C61#240711yescharacteristic
294C2×C61C2×#44811yescharacteristic
295C2×C61#301211yescharacteristic
296C2×C61#329511yescharacteristic
297C2×C61#333111yescharacteristic
298C2×C61#333511yescharacteristic
299C2×C61#335711yescharacteristic
300D61—44no—
301D61—82no—
302C22×SD16—611no—
303#554—1224no—
304#554—1224no—
305C2×C4⋊Q8—611no—
306C2×#125—1221no—
307C2×C22⋊Q8—612no—
308C2×C22⋊Q8—1222no—
309C2×#121—612no—
310C2×#121—1221no—
311C2×C4×Q8C2×D6111yescharacteristic
312C2×C4×D4—611no—
313C2×C4×D4—611no—
314C2×C4×D4—1221no—
315C22×C4⋊C4—612no—
316#476—1224no—
317C4×SD16—1224no—
318C2×#106—612no—
319C2×#106—611no—
320C2×C4⋊C8C2×D6111yescharacteristic
321C2×#103—611no—
322C2×#103C2×D6112yes—
323C2×#102—612no—
324C2×#102—611no—
325C2×C22⋊C8—612no—
326C2×C22⋊C8—612no—
327C2×C4×C8—611no—
328#394—1222no—
329#384C2×D6111yescharacteristic
330#380—612no—
331#380—1221no—
332#340—612no—
333#340—612no—
334#340C2×D6112yes—
335C61#2411811yescharacteristic, complemented, Sylow
336C22×Q8—1221no—
337C22×Q8C22×D6111yescharacteristic
338C22×D4—611no—
339C22×D4—611no—
340C22×D4—1222no—
341C23×C4—612no—
342C23×C4—1222no—
343C23×C4—1221no—
344C2×SD16—1222no—
345C2×SD16—1222no—
346C2×SD16—1222no—
347C22×C8—611no—
348C22×C8—611no—
349C22×C8C22×D6112yes—
350C4⋊Q8—614no—
351C4⋊Q8—1222no—
352#125—1228no—
353C22⋊Q8—12216no—
354C22⋊Q8—1224no—
355C22⋊Q8—1224no—
356#121—1224no—
357#121—2442no—
358#121—2444no—
359C4×Q8—22no—
360C4×Q8D24414yescomplemented
361C4×D4—1228no—
362C4×D4—1222no—
363C4×D4—1222no—
364C4×D4—1222no—
365C4×D4—1222no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1224no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4C22×D6112yes—
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—612no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1221no—
384C2×C42—21no—
385C2×C42C22×D6111yescharacteristic
386#106—1224no—
387#106—1222no—
388C4⋊C8—22no—
389#103—42no—
390#103—1222no—
391#102—1224no—
392#102—1222no—
393C22⋊C8—1224no—
394C22⋊C8—2442no—
395C4×C8—1222no—
396#95—1222no—
397C24—612no—
398C24—1221no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—1224no—
405C2×Q8—4881no—
406C2×Q8—4881no—
407C2×Q8C2×D24412yes—
408C2×D4—612no—
409C2×D4—614no—
410C2×D4—1222no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—1222no—
417C2×D4—4882no—
418C2×D4—2444no—
419C2×D4—2444no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—612no—
424C22×C4—1228no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1221no—
433C22×C4—4882no—
434C22×C4—4882no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4D4×D6111yescharacteristic
439C22×C4D4×D6111yescharacteristic
440C22×C4C23×D6111yescharacteristic
441SD16—1228no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—612no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C4⋊C4—21no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—614no—
458C4⋊C4—1228no—
459C4⋊C4—1228no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1224no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4C2×D24412yes—
466C22⋊C4—1228no—
467C22⋊C4—1224no—
468C22⋊C4—2444no—
469C22⋊C4—2444no—
470C22⋊C4—2442no—
471C42—24no—
472C42C2×D24412yes—
473C42D4×D6112yes—
474C23—614no—
475C23—1222no—
476C23—1222no—
477C23—1222no—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—2441no—
483C23—2441no—
484C23—2441no—
485C23—2441no—
486C23—2441no—
487C23—2441no—
488C23C2×D4×D6111yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—41no—
492Q8—41no—
493Q8—4884no—
494D4—1222no—
495D4—1228no—
496D4—1222no—
497D4—4888no—
498D4—2441no—
499D4—2441no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—1224no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—4888no—
516C2×C4—4884no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974712yes—
527C2×C4#1451974712yes—
528C2×C4SD16×D6112yes—
529C2×C4#1451980912yes—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×D24411yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C2×C4#41×D6111yescharacteristic
534C2×C4#1451982211yescharacteristic
535C2×C4#1451982511yescharacteristic
536C8—42no—
537C8—1222no—
538C8—2441no—
539C22—1224no—
540C22—1224no—
541C22—1224no—
542C22—4881no—
543C22—4881no—
544C22—4881no—
545C22—4881no—
546C22—2442no—
547C22—2442no—
548C22—2442no—
549C22—2442no—
550C22—2442no—
551C22—2442no—
552C22—2442no—
553C22C2×#1451974711yescharacteristic
554C22#1684040911yescharacteristic
555C22#1684041411yescharacteristic
556C22C2×SD16×D6111yescharacteristic
557C22C2×#1451980911yescharacteristic
558C22#1684056611yescharacteristic
559C22#1684057111yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—82no—
564C4—4881no—
565C4—2444no—
566C4—2442no—
567C4#1683990214yes—
568C4#1684027411yescharacteristic
569C4#1684034311yescharacteristic
570C4#1684042211yescharacteristic
571C4#1684043011yescharacteristic
572C2—4882no—
573C2—2444no—
574C2#2144916511yescharacteristic
575C2C2×#1683990211yescharacteristic
576C2#2145512511yescharacteristic
577C2#2145764611yescharacteristic
578C2#2145813811yescharacteristic
579C2#2145817911yescharacteristic
580C2#2145844211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radical#21450735C2
hypercenter#1803D61
hypocenterC61#24118
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×#16840472C2semidirect
2. C2×#16840053C2semidirect
3. C2×C4×#11758201C2semidirect
5. #21447014C2semidirect
6. #21446821C2semidirect
7. #21446779C2semidirect
8. #21446612C2semidirect
9. #21446135C2semidirect
10. #21446013C2semidirect
11. C61×#1803C2semidirect
58. #14519777D4semidirect
96. C4×Q8×C61D4semidirect
133. C2×D244SD16semidirect
292. #1803D61semidirect
335. C61#24118semidirect
360. C4×Q8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
C2×#168400532
C2×C4×#117582012
#214507352
#214470142
#214468212
#214467792
#214466122
#214461352
#214460132
C61×#18032
#2411861