Finite Group Databasebeta

← #29713959

Subgroups — #29713959

Order 15616.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic
2C2×#16840058C211yesmaximal, characteristic, complemented
3C2×C4×#11758190C211yesmaximal, characteristic, complemented
4#21450741C211yesmaximal, characteristic, complemented
5#21447002C212yesmaximal, complemented
6#21446824C212yesmaximal, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446129C212yesmaximal, complemented
9#21446020C212yesmaximal, complemented
10#21445820C211yesmaximal, characteristic, complemented
11C61×#1954C211yesmaximal, characteristic, complemented
12C22×#11758190C2211yescharacteristic
13#16840477—24no—
14#16840477—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519763—22no—
17C2×#14519756C2212yes—
18C2×#14519751C2212yes—
19C2×#14519748—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519736C2212yes—
23C2×#14519735—21no—
24#16840058—24no—
25C4×#11758190—24no—
26C2×#14519711C2212yes—
27C2×#14519709C2211yescharacteristic
28C2×#14519702C2212yes—
29C2×#14519702C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2212yes—
32#16839685—22no—
33#16839665—21no—
34#16839658—21no—
35#16839632C2211yescharacteristic
36#16839622C2212yes—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839525C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2212yes—
43C2×C22⋊C8×C61C2212yes—
44C2×C4×C8×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—22no—
49C22×C4×D61—21no—
50C2×#11758190—22no—
51C2×#11758190—22no—
52C2×#11758190—22no—
53#14519777—22no—
54#14519777D414yes—
55#14519763—216no—
56#14519756—24no—
57#14519756—42no—
58#14519751—24no—
59#14519751—24no—
60#14519748—28no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63C4×D244—22no—
64C4×D244D414yes—
65#14519736—24no—
66#14519736—24no—
67#14519735—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—21no—
76C2×#11758196—22no—
77C2×C22⋊Dic61—22no—
78C2×C22⋊Dic61—22no—
79C2×C22⋊Dic61—22no—
80C2×#11758198—22no—
81C2×#11758198—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—21no—
84#14519711—24no—
85#14519709—22no—
86#14519702—24no—
87#14519702—22no—
88#14519699—22no—
89#14519699—42no—
90#14519684—22no—
91C22×D4×C61D411yescharacteristic
92C23×C4×C61C2312yes—
93C22×C8×C61C2311yescharacteristic
94C22×C8×C61C2312yes—
95C22×C8×C61C2311yescharacteristic
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C22⋊C4×C61D412yes—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102C22⋊C8×C61—24no—
103C22⋊C8×C61—42no—
104C4×C8×C61—22no—
105C24×C61C2×D412yes—
106C23×D61—21no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—28no—
121C2×C22⋊D61—82no—
122C2×C22⋊D61—82no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244C2×D412yes—
134C2×C4×D61—24no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—24no—
143C2×Dic122—81no—
144C2×Dic122—81no—
145C2×Dic122#4112yes—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—21no—
152C2×C8×C61—22no—
153C2×C8×C61—21no—
154C2×C8×C61#4112yes—
155C4⋊C4×C61—22no—
156C22⋊C4×C61—24no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159C22⋊Dic61—28no—
160C22⋊Dic61—44no—
161C22⋊Dic61—82no—
162#11758198—28no—
163#11758198—24no—
164C4⋊Dic61—24no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61—81no—
173C4⋊Dic61C2×D412yes—
174C4⋊Dic61C2×D414yes—
175#11758196—44no—
176#11758196—42no—
177C4×Dic61—28no—
178C4×Dic61—42no—
179#11758190—28no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195D4×C61—44no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61#12012yes—
206C2×C4×C61#12012yes—
207C2×C4×C61#13512yes—
208C2×C4×C61#13712yes—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61H(5,𝔽2)11yescharacteristic
214C2×C4×C612−1+411yescharacteristic
215C22⋊D61—88no—
216C2×Dic61—22no—
217C2×Dic61—24no—
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231D244—22no—
232D244—22no—
233D244—41no—
234D244—41no—
235C4×D61—48no—
236Dic122—22no—
237Dic122—22no—
238Dic122—41no—
239Dic122—41no—
240Dic122—84no—
241C8×C61—22no—
242C8×C61—41no—
243C8×C61—42no—
244#24121—611nomaximal, Sylow
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61C2×#12011yescharacteristic
253C22×C61#54611yescharacteristic
254C22×C61#54711yescharacteristic
255C22×C61C2×#13511yescharacteristic
256C22×C61C2×#13711yescharacteristic
257C22×C61#57611yescharacteristic
258C22×C61#57711yescharacteristic
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61#45214yes—
269C4×C61#52511yescharacteristic
270C4×C61#53711yescharacteristic
271C4×C61#54811yescharacteristic
272C4×C61#55011yescharacteristic
273Dic61—42no—
274Dic61—42no—
275Dic61—44no—
276Dic61—81no—
277Dic61—82no—
278C2×#555—611no—
279C2×#477—611no—
280C2×C4×SD16—611no—
281#2538—611no—
282#2033—612no—
283#2000—612no—
284#1990—611no—
285#1954D6111yescharacteristic, complemented
286#1873—612no—
287#1854—612no—
288#1802—611no—
289C2×C61—82no—
290C2×C61#240711yescharacteristic
291C2×C61C2×#45211yescharacteristic
292C2×C61#301211yescharacteristic
293C2×C61#330511yescharacteristic
294C2×C61#332911yescharacteristic
295C2×C61#334011yescharacteristic
296C2×C61#335711yescharacteristic
297D61—82no—
298C22×SD16—611no—
299#555—1224no—
300#555—1224no—
301C2×C4⋊Q8—611no—
302C2×#124—1222no—
303C2×C22⋊Q8—612no—
304C2×#121—612no—
305C2×#121—1221no—
306C2×C4×Q8—611no—
307C2×C4×D4—611no—
308C2×C4×D4C2×D6111yescharacteristic
309C2×#117—612no—
310C2×#117—1221no—
311#477—1224no—
312C4×SD16—1224no—
313C2×#107—612no—
314C2×#106—611no—
315C2×C4⋊C8C2×D6111yescharacteristic
316C2×#103—612no—
317C2×#103—611no—
318C2×#102—611no—
319C2×#102—612no—
320C2×C22⋊C8C2×D6112yes—
321C2×C22⋊C8C2×D6112yes—
322C2×C4×C8C2×D6111yescharacteristic
323#396—1222no—
324#390—1221no—
325#389—1221no—
326#384—611no—
327#380—612no—
328#340—612no—
329#340—612no—
330#340—612no—
331C61#2412111yescharacteristic, complemented, Sylow
332C22×Q8—611no—
333C22×Q8—1221no—
334C22×D4—611no—
335C22×D4—1222no—
336C22×D4D24411yescharacteristic
337C23×C4—1221no—
338C23×C4C22×D6112yes—
339C2×SD16—1222no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C22×C8C22×D6111yescharacteristic
343C22×C8C22×D6112yes—
344C22×C8C22×D6111yescharacteristic
345C4⋊Q8—614no—
346C4⋊Q8—1222no—
347#124—12216no—
348C22⋊Q8—1224no—
349C22⋊Q8—2442no—
350#121—1224no—
351#121—1224no—
352#121—1228no—
353C4×Q8—1222no—
354C4×Q8—1222no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—614no—
358C4×D4—1222no—
359#117—1228no—
360#117—1224no—
361#117—1224no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4D24411yescharacteristic
372C2×C22⋊C4—1222no—
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4D24412yes—
378C2×C42—1222no—
379C2×C42—1221no—
380C2×C42C22×D6111yescharacteristic
381#107—1224no—
382#106—1222no—
383C4⋊C8—22no—
384#103—1222no—
385#103—1224no—
386#102—1222no—
387#102—2442no—
388C22⋊C8—24no—
389C22⋊C8—42no—
390C4×C8—22no—
391#95—1222no—
392C24—1221no—
393C24C2×D24412yes—
394C2×Q8—612no—
395C2×Q8—1221no—
396C2×Q8—1224no—
397C2×Q8—1221no—
398C2×Q8—1221no—
399C2×Q8—1221no—
400C2×Q8—1221no—
401C2×Q8—4881no—
402C2×Q8—4881no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—612no—
407C2×D4—1228no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—4882no—
414C2×D4—4882no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—1221no—
424C22×C4—1224no—
425C22×C4—1221no—
426C22×C4—1222no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4C2×D24412yes—
430C22×C4C2×D24411yescharacteristic
431C22×C4C2×D24411yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—1228no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—21no—
440C2×C8—22no—
441C2×C8—21no—
442C2×C8#1175822412yes—
443C4⋊C4—22no—
444C4⋊C4—612no—
445C4⋊C4—614no—
446C4⋊C4—1221no—
447C4⋊C4—1222no—
448C4⋊C4—1224no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—4881no—
454C4⋊C4—2444no—
455C4⋊C4—2442no—
456C4⋊C4—2442no—
457C22⋊C4—24no—
458C22⋊C4—1228no—
459C22⋊C4—1228no—
460C22⋊C4—1224no—
461C22⋊C4—4882no—
462C22⋊C4—2444no—
463C42—1228no—
464C42—2442no—
465C42D4×D6112yes—
466C42D4×D6112yes—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—1222no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23C22×D24411yescharacteristic
482Q8—1222no—
483Q8—1222no—
484Q8—4884no—
485Q8—2441no—
486Q8—2441no—
487D4—44no—
488D4—1222no—
489D4—1222no—
490D4—4888no—
491D4—2441no—
492D4—2441no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—1222no—
503C2×C4—1224no—
504C2×C4—1222no—
505C2×C4—2441no—
506C2×C4—2442no—
507C2×C4—2441no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4—2442no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2441no—
517C2×C4—2448no—
518C2×C4#1451974712yes—
519C2×C4#1451974712yes—
520C2×C4#1451979912yes—
521C2×C4#1451981012yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4#41×D6111yescharacteristic
525C2×C4C2×#1175822411yescharacteristic
526C2×C4#1451982211yescharacteristic
527C2×C4#1451982411yescharacteristic
528C8—22no—
529C8—41no—
530C8—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—4881no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22C2×#1451974711yescharacteristic
546C22#1684040711yescharacteristic
547C22#1684041611yescharacteristic
548C22C2×#1451979911yescharacteristic
549C22C2×#1451981011yescharacteristic
550C22#1684056711yescharacteristic
551C22#1684057411yescharacteristic
552C4—22no—
553C4—42no—
554C4—4881no—
555C4—4882no—
556C4—2444no—
557C4—2442no—
558C4—2442no—
559C4#1683992614yes—
560C4#1684027211yescharacteristic
561C4#1684034111yescharacteristic
562C4#1684042011yescharacteristic
563C4#1684043211yescharacteristic
564C2—82no—
565C2—4882no—
566C2#2144916211yescharacteristic
567C2C2×#1683992611yescharacteristic
568C2#2145512611yescharacteristic
569C2#2145777011yescharacteristic
570C2#2145811411yescharacteristic
571C2#2145823311yescharacteristic
572C2#2145844611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1954C2
socleC23×C61C22×D4
prime radicalC2×#16840477C2
hypercenter#1954D61
hypocenterC61#24121
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1954

Complements

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

NH ≅ G/Ntype
2. C2×#16840058C2semidirect
3. C2×C4×#11758190C2semidirect
4. #21450741C2semidirect
5. #21447002C2semidirect
6. #21446824C2semidirect
7. #21446780C2semidirect
8. #21446129C2semidirect
9. #21446020C2semidirect
10. #21445820C2semidirect
11. C61×#1954C2semidirect
285. #1954D61semidirect
331. C61#24121semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
C2×#168400582
C2×C4×#117581902
#214507412
#214470022
#214468242
#214467802
#214461292
#214460202
#214458202
C61×#19542
#2412161