Finite Group Databasebeta

← #29713962

Subgroups — #29713962

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×#11758200C211yesmaximal, characteristic, complemented
4#21450742C211yesmaximal, characteristic, complemented
5#21447000C212yesmaximal, complemented
6#21446823C212yesmaximal, complemented
7#21446779C211yesmaximal, characteristic, complemented
8#21446612C211yesmaximal, characteristic, complemented
9#21446130C212yesmaximal, complemented
10#21446021C212yesmaximal, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758200C2211yescharacteristic
13#16840477—24no—
14#16840477—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519763—22no—
17C2×#14519757C2212yes—
18C2×#14519751—21no—
19C2×#14519748C2212yes—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519736—21no—
23C2×#14519735C2212yes—
24#16840058—24no—
25C4×#11758200—24no—
26C2×#14519711C2212yes—
27C2×#14519708C2211yescharacteristic
28C2×#14519702C2212yes—
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519689C2212yes—
32C2×#14519689C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839684—22no—
35#16839663—21no—
36#16839661—21no—
37#16839619C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—21no—
46C22×Dic122C2311yescharacteristic
47C22×D244D411yescharacteristic
48C22×C22⋊D61—22no—
49C22×C4×D61C2312yes—
50C2×#11758200—22no—
51C2×#11758200—22no—
52C2×#11758200—22no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519777—22no—
56#14519777D414yes—
57#14519763—216no—
58#14519757—24no—
59#14519757—42no—
60#14519751—28no—
61#14519748—24no—
62#14519748—24no—
63C4×Dic122—22no—
64C4×Dic122—22no—
65C4×D244—22no—
66C4×D244—22no—
67#14519736—28no—
68#14519735—24no—
69#14519735—24no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2312yes—
75C2×#11758196—21no—
76C2×C22⋊Dic61—22no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198D412yes—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83#14519711—24no—
84#14519708—22no—
85#14519702—24no—
86#14519700—22no—
87#14519698—22no—
88#14519689—24no—
89#14519689—42no—
90C4×#92418—22no—
91#14519684—22no—
92C22×D4×C61C2311yescharacteristic
93C23×C4×C61—21no—
94C22×C8×C61C2312yes—
95C4×D4×C61—22no—
96C4×D4×C61D414yes—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2312yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—22no—
104#102×C61—42no—
105C24×C61—21no—
106C23×D61C2×D412yes—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C2×D412yes—
113C22×C4×C61—21no—
114C22×C4×C61—24no—
115C22×C4×C61—22no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—28no—
122C2×C22⋊D61—82no—
123C2×C22⋊D61—82no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61C2×D412yes—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
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—22no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61—81no—
154C4⋊C4×C61C2×D412yes—
155C22⋊C4×C61—24no—
156C42×C61C2×D412yes—
157C42×C61C2×D412yes—
158C22⋊Dic61—28no—
159#11758200—28no—
160#11758198—28no—
161#11758198—24no—
162#11758198—44no—
163#11758198—82no—
164C4⋊Dic61—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—44no—
170C4⋊Dic61C2×D414yes—
171#11758196—42no—
172C4×Dic61—28no—
173C4×Dic61—42no—
174C2×#92418—21no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418#4112yes—
180C23×C61—22no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195D4×C61—22no—
196D4×C61—22no—
197D4×C61—41no—
198D4×C61—41no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—48no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61#13512yes—
211C2×C4×C61#13712yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61H(5,𝔽2)11yescharacteristic
217C2×C4×C612−1+411yescharacteristic
218C22⋊D61—88no—
219C2×Dic61—22no—
220C2×Dic61—24no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231D244—44no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236Dic122—22no—
237Dic122—22no—
238Dic122—41no—
239Dic122—41no—
240Dic122—84no—
241C8×C61—42no—
242#92418—22no—
243#92418—41no—
244#24121—611nomaximal, Sylow
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
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—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—82no—
269C4×C61#45214yes—
270C4×C61#52511yescharacteristic
271C4×C61#53711yescharacteristic
272C4×C61#54811yescharacteristic
273C4×C61#55011yescharacteristic
274Dic61—42no—
275Dic61—42no—
276Dic61—44no—
277Dic61—81no—
278C2×#555—611no—
279C2×#477—611no—
280C2×C4×SD16—611no—
281#2538—611no—
282#2033—612no—
283#2000—612no—
284#1990—611no—
285#1954—611no—
286#1873—612no—
287#1854—612no—
288#1802D6111yescharacteristic, complemented
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×#102C2×D6112yes—
320C2×C22⋊C8—612no—
321C2×C22⋊C8—612no—
322C2×C4×C8—611no—
323#396—1222no—
324#390—1221no—
325#389—1221no—
326#384C2×D6111yescharacteristic
327#380—612no—
328#340—612no—
329#340—612no—
330#340C2×D6112yes—
331C61#2412111yescharacteristic, complemented, Sylow
332C22×Q8—611no—
333C22×Q8—1221no—
334C22×D4—611no—
335C22×D4—1222no—
336C22×D4C22×D6111yescharacteristic
337C23×C4—21no—
338C23×C4—612no—
339C2×SD16—1222no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C22×C8—611no—
343C22×C8—611no—
344C22×C8C22×D6112yes—
345C4⋊Q8—614no—
346C4⋊Q8—1222no—
347#124—12216no—
348C22⋊Q8—1224no—
349C22⋊Q8—2442no—
350#121—1224no—
351#121—1228no—
352#121—1224no—
353C4×Q8—1222no—
354C4×Q8—1222no—
355C4×D4—22no—
356C4×D4—1222no—
357C4×D4—1222no—
358C4×D4D24414yes—
359#117—1224no—
360#117—1228no—
361#117—1224no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4C22×D6112yes—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—612no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C42—1222no—
379C2×C42—1221no—
380C2×C42C22×D6111yescharacteristic
381#107—1224no—
382#106—1222no—
383C4⋊C8—22no—
384#103—1224no—
385#103—1222no—
386#102—42no—
387#102—1222no—
388C22⋊C8—1224no—
389C22⋊C8—2442no—
390C4×C8—1222no—
391#95—1222no—
392C24—21no—
393C24—612no—
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—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—1228no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—4882no—
413C2×D4—4882no—
414C2×D4C2×D24412yes—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—612no—
421C22×C4—1222no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1221no—
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433SD16—1228no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—612no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—42no—
447C4⋊C4—81no—
448C4⋊C4—614no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1224no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—2444no—
455C4⋊C4—2442no—
456C4⋊C4C2×D24412yes—
457C22⋊C4—24no—
458C22⋊C4—1228no—
459C22⋊C4—1228no—
460C22⋊C4—1224no—
461C22⋊C4—4882no—
462C22⋊C4—2444no—
463C42—1228no—
464C42—2442no—
465C42C2×D24412yes—
466C42D4×D6112yes—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—1222no—
475C23—1222no—
476C23—1222no—
477C23—1222no—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23C2×D4×D6111yescharacteristic
482Q8—1222no—
483Q8—1222no—
484Q8—4884no—
485Q8—2441no—
486Q8—2441no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—4888no—
492D4—2444no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—1222no—
503C2×C4—1224no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4#1451974712yes—
519C2×C4#1451974712yes—
520C2×C4#1451979612yes—
521C2×C4#1451980612yes—
522C2×C4C2×D4×D6111yescharacteristic
523C2×C4C22×D24411yescharacteristic
524C2×C4#41×D6111yescharacteristic
525C2×C4#41×D6111yescharacteristic
526C2×C4#1451982211yescharacteristic
527C2×C4#1451982511yescharacteristic
528C8—42no—
529C8—1222no—
530C8—2441no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22C2×#1451974711yescharacteristic
546C22#1684040911yescharacteristic
547C22#1684041411yescharacteristic
548C22C2×#1451979611yescharacteristic
549C22C2×#1451980611yescharacteristic
550C22#1684056611yescharacteristic
551C22#1684057111yescharacteristic
552C4—22no—
553C4—42no—
554C4—82no—
555C4—4881no—
556C4—2444no—
557C4—2442no—
558C4—2442no—
559C4#1683992514yes—
560C4#1684027411yescharacteristic
561C4#1684034311yescharacteristic
562C4#1684042211yescharacteristic
563C4#1684043011yescharacteristic
564C2—82no—
565C2—4882no—
566C2#2144916511yescharacteristic
567C2C2×#1683992511yescharacteristic
568C2#2145512511yescharacteristic
569C2#2145777211yescharacteristic
570C2#2145811611yescharacteristic
571C2#2145822911yescharacteristic
572C2#2145844811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC2×#16840477C2
hypercenter#1802D61
hypocenterC61#24121
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#16840058C2semidirect
3. C2×C4×#11758200C2semidirect
4. #21450742C2semidirect
5. #21447000C2semidirect
6. #21446823C2semidirect
7. #21446779C2semidirect
8. #21446612C2semidirect
9. #21446130C2semidirect
10. #21446021C2semidirect
11. C61×#1802C2semidirect
288. #1802D61semidirect
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×#117582002
#214507422
#214470002
#214468232
#214467792
#214466122
#214461302
#214460212
C61×#18022
#2412161