Finite Group Databasebeta

← #29713937

Subgroups — #29713937

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840482C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic, complemented
3C2×C4×#11758190C211yesmaximal, characteristic, complemented
4#21450732C211yesmaximal, characteristic
5#21447023C212yesmaximal, complemented
6#21446824C212yesmaximal, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446139C212yesmaximal, complemented
9#21446020C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C61×#1954C211yesmaximal, characteristic, complemented
12C22×#11758190C2211yescharacteristic
13#16840482—24no—
14#16840482—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519770—22no—
17C2×#14519764—21no—
18C2×#14519756C2212yes—
19C2×#14519751C2212yes—
20C2×#14519748—21no—
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519742—21no—
23C2×C4×D244C2211yescharacteristic
24C2×#14519736C2212yes—
25#16840043—24no—
26C4×#11758190—24no—
27C2×#14519711C2212yes—
28C2×#14519709C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2212yes—
31C2×#14519699C2212yes—
32C2×#14519699C2211yescharacteristic
33#16839678—22no—
34#16839632C2211yescharacteristic
35#16839622C2212yes—
36#16839619—21no—
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#14519777D414yescomplemented
55#14519770—216no—
56#14519764—28no—
57#14519756—24no—
58#14519756—24no—
59#14519751—24no—
60#14519751—42no—
61#14519748—44no—
62C4×Dic122—22no—
63C4×Dic122D414yescomplemented
64#14519742—28no—
65C4×D244—22no—
66C4×D244—22no—
67#14519736—24no—
68#14519736—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2312yes—
76C2×#11758196—22no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—22no—
79C2×C22⋊Dic61—22no—
80C2×C22⋊Dic61—22no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61—21no—
85#14519711—24no—
86#14519709—22no—
87#14519702—22no—
88#14519702—42no—
89#14519699—24no—
90#14519699—22no—
91#14519684—22no—
92C22×D4×C61D411yescharacteristic
93C23×C4×C61C2312yes—
94C22×C8×C61C2311yescharacteristic
95C22×C8×C61C2312yes—
96C22×C8×C61C2311yescharacteristic
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61D412yes—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—22no—
103C22⋊C8×C61—24no—
104C22⋊C8×C61—42no—
105C4×C8×C61—22no—
106C24×C61C2×D412yes—
107C23×D61—21no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—44no—
122C2×C22⋊D61—44no—
123C2×C22⋊D61—82no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244#4112yes—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×Dic122—21no—
139C2×Dic122—21no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122—24no—
143C2×Dic122—21no—
144C2×Dic122—42no—
145C2×Dic122—42no—
146C2×Dic122C2×D412yes—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—21no—
153C2×C8×C61—22no—
154C2×C8×C61—21no—
155C2×C8×C61#4112yes—
156C4⋊C4×C61—22no—
157C22⋊C4×C61—24no—
158C42×C61C2×D412yes—
159C42×C61C2×D412yes—
160C22⋊Dic61—28no—
161C22⋊Dic61—28no—
162C22⋊Dic61—44no—
163#11758198—44no—
164#11758198—42no—
165C4⋊Dic61—24no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—42no—
173C4⋊Dic61—81no—
174C4⋊Dic61C2×D412yes—
175C4⋊Dic61C2×D414yes—
176#11758196—28no—
177#11758196—24no—
178C4×Dic61—28no—
179C4×Dic61—24no—
180#11758190—28no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61C22×D411yescharacteristic
189C22×D61—22no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196D4×C61—44no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12012yes—
208C2×C4×C61#13512yes—
209C2×C4×C61AGL(1,ℤ/8)12yes—
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61H(5,𝔽2)11yescharacteristic
215C2×C4×C612−1+411yescharacteristic
216C22⋊D61—88no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—24no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—42no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232D244—22no—
233D244—22no—
234D244—41no—
235D244—41no—
236C4×D61—84no—
237Dic122—22no—
238Dic122—22no—
239Dic122—41no—
240Dic122—41no—
241Dic122—48no—
242C8×C61—22no—
243C8×C61—41no—
244C8×C61—42no—
245#24119—611nomaximal, Sylow
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61C2×#12011yescharacteristic
254C22×C61#54611yescharacteristic
255C22×C61#54711yescharacteristic
256C22×C61C2×#13511yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C61#57711yescharacteristic
259C22×C61#57811yescharacteristic
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C4×C61—22no—
268C4×C61—42no—
269C4×C61#44914yes—
270C4×C61#52511yescharacteristic
271C4×C61#53711yescharacteristic
272C4×C61#54811yescharacteristic
273C4×C61#55011yescharacteristic
274Dic61—42no—
275Dic61—44no—
276Dic61—42no—
277Dic61—81no—
278Dic61—82no—
279C2×#556—611no—
280C2×#474—611no—
281C2×C4×SD16—611no—
282#2537—611no—
283#2036—612no—
284#2000—612no—
285#1990—611no—
286#1954D6111yescharacteristic, complemented
287#1875—612no—
288#1854—612no—
289#1803—611no—
290C2×C61—82no—
291C2×C61#240711yescharacteristic
292C2×C61C2×#44911yescharacteristic
293C2×C61#301311yescharacteristic
294C2×C61#330211yescharacteristic
295C2×C61#332411yescharacteristic
296C2×C61#334011yescharacteristic
297C2×C61#335311yescharacteristic
298D61—82no—
299C22×SD16—611no—
300#556—1224no—
301#556—1224no—
302C2×C4⋊Q8—611no—
303C2×#126—1222no—
304C2×#124—1221no—
305C2×C22⋊Q8—612no—
306C2×#121—612no—
307C2×#121—1221no—
308C2×C4×Q8—611no—
309C2×C4×Q8—1221no—
310C2×C4×D4—611no—
311C2×C4×D4C2×D6111yescharacteristic
312C2×#117—612no—
313#474—1224no—
314C4×SD16—1224no—
315C2×#107—612no—
316C2×#106—611no—
317C2×C4⋊C8C2×D6111yescharacteristic
318C2×#103—611no—
319C2×#103—612no—
320C2×#102—611no—
321C2×#102—612no—
322C2×C22⋊C8C2×D6112yes—
323C2×C22⋊C8C2×D6112yes—
324C2×C4×C8C2×D6111yescharacteristic
325#394—1222no—
326#384—611no—
327#380—612no—
328#380—1221no—
329#340—612no—
330#340—612no—
331#340—612no—
332C61#2411911yescharacteristic, complemented, Sylow
333C22×Q8—611no—
334C22×Q8—1221no—
335C22×D4—611no—
336C22×D4—1222no—
337C22×D4D24411yescharacteristic
338C23×C4—1221no—
339C23×C4C22×D6112yes—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C2×SD16—1222no—
343C22×C8C22×D6111yescharacteristic
344C22×C8C22×D6112yes—
345C22×C8C22×D6111yescharacteristic
346C4⋊Q8—614no—
347C4⋊Q8—1222no—
348#126—12216no—
349#124—1228no—
350C22⋊Q8—1224no—
351C22⋊Q8—1224no—
352#121—1224no—
353#121—2442no—
354#121—2444no—
355C4×Q8—614no—
356C4×Q8—1228no—
357C4×Q8—1222no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—1222no—
361C4×D4—1222no—
362#117—1224no—
363#117—1224no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4D24411yescharacteristic
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4D24412yes—
380C2×C42—1222no—
381C2×C42—1221no—
382C2×C42C22×D6111yescharacteristic
383#107—1224no—
384#106—1222no—
385C4⋊C8—22no—
386#103—1222no—
387#103—2442no—
388#102—1222no—
389#102—1224no—
390C22⋊C8—24no—
391C22⋊C8—42no—
392C4×C8—22no—
393#95—1222no—
394C24—1221no—
395C24C2×D24412yes—
396C2×Q8—612no—
397C2×Q8—1221no—
398C2×Q8—1224no—
399C2×Q8—1221no—
400C2×Q8—1221no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×Q8—2442no—
404C2×Q8—2442no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—612no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—4882no—
415C2×D4—2444no—
416C2×D4—2444no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—1221no—
424C22×C4—1222no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1222no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4C2×D24412yes—
432C22×C4C2×D24411yescharacteristic
433C22×C4C2×D24411yescharacteristic
434C22×C4C23×D6111yescharacteristic
435SD16—1228no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—21no—
444C2×C8#1175822412yes—
445C4⋊C4—22no—
446C4⋊C4—612no—
447C4⋊C4—614no—
448C4⋊C4—1228no—
449C4⋊C4—1221no—
450C4⋊C4—1222no—
451C4⋊C4—1224no—
452C4⋊C4—1224no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—4881no—
458C4⋊C4—2442no—
459C22⋊C4—24no—
460C22⋊C4—1228no—
461C22⋊C4—1228no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C22⋊C4—2442no—
465C42—1228no—
466C42—1224no—
467C42D4×D6112yes—
468C42D4×D6112yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—1222no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23—2441no—
482C23—2441no—
483C23C22×D24411yescharacteristic
484Q8—1222no—
485Q8—1222no—
486Q8—2441no—
487Q8—2448no—
488Q8—2441no—
489D4—44no—
490D4—1222no—
491D4—1222no—
492D4—4888no—
493D4—2441no—
494D4—2441no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—1222no—
505C2×C4—1224no—
506C2×C4—1222no—
507C2×C4—4884no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4#1451974712yes—
521C2×C4#1451974712yes—
522C2×C4#1451979912yes—
523C2×C4#1451980512yes—
524C2×C4C2×D4×D6111yescharacteristic
525C2×C4C2×D4×D6111yescharacteristic
526C2×C4#41×D6111yescharacteristic
527C2×C4C2×#1175822411yescharacteristic
528C2×C4#1451982211yescharacteristic
529C2×C4#1451982411yescharacteristic
530C8—22no—
531C8—41no—
532C8—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—4881no—
541C22—4881no—
542C22—4881no—
543C22—4881no—
544C22—2442no—
545C22—2442no—
546C22—2442no—
547C22C2×#1451974711yescharacteristic
548C22#1684040711yescharacteristic
549C22#1684041611yescharacteristic
550C22C2×#1451979911yescharacteristic
551C22C2×#1451980511yescharacteristic
552C22#1684057411yescharacteristic
553C22#1684057911yescharacteristic
554C4—22no—
555C4—42no—
556C4—4882no—
557C4—4881no—
558C4—2442no—
559C4—2444no—
560C4—2442no—
561C4#1683990914yes—
562C4#1684027211yescharacteristic
563C4#1684034111yescharacteristic
564C4#1684042011yescharacteristic
565C4#1684043211yescharacteristic
566C2—82no—
567C2—4882no—
568C2#2144916211yescharacteristic
569C2C2×#1683990911yescharacteristic
570C2#2145513711yescharacteristic
571C2#2145773711yescharacteristic
572C2#2145805111yescharacteristic
573C2#2145823311yescharacteristic
574C2#2145840411yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1954C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenter#1954D61
hypocenterC61#24119
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
1. C2×#16840482C2semidirect
2. C2×#16840043C2semidirect
3. C2×C4×#11758190C2semidirect
5. #21447023C2semidirect
6. #21446824C2semidirect
7. #21446780C2semidirect
8. #21446139C2semidirect
9. #21446020C2semidirect
10. #21445825C2semidirect
11. C61×#1954C2semidirect
54. #14519777D4semidirect
63. C4×Dic122D4semidirect
286. #1954D61semidirect
332. C61#24119semidirect

Maximal subgroups

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

subgroupindex
C2×#168404822
C2×#168400432
C2×C4×#117581902
#214507322
#214470232
#214468242
#214467802
#214461392
#214460202
#214458252
C61×#19542
#2411961