Finite Group Databasebeta

← #29713940

Subgroups — #29713940

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×#16840485C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic, complemented
4#21450735C211yesmaximal, characteristic
5#21447021C212yesmaximal, complemented
6#21446821C212yesmaximal, complemented
7#21446779C211yesmaximal, characteristic, complemented
8#21446612C211yesmaximal, characteristic, complemented
9#21446140C212yesmaximal, complemented
10#21446021C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758201C2211yescharacteristic
13#16840485—24no—
14#16840485—24no—
15C2×#14519777C2211yescharacteristic
16C2×#14519772—22no—
17C2×#14519765—21no—
18C2×#14519753C2212yes—
19C2×#14519752—21no—
20C2×#14519752C2212yes—
21C2×C4×Dic122—21no—
22C2×C4×D244C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24C2×#14519735C2212yes—
25#16840043—24no—
26C4×#11758201—24no—
27C2×#14519711C2212yes—
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—21no—
50C22×C4×D61C2312yes—
51C2×#11758201—22no—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C22×#92418C2311yescharacteristic
55C22×#92418C2311yescharacteristic
56#14519777—22no—
57#14519777D414yescomplemented
58#14519772—216no—
59#14519765—28no—
60#14519753—24no—
61#14519753—24no—
62#14519752—24no—
63#14519752—42no—
64#14519752—44no—
65C4×Dic122—28no—
66C4×D244—22no—
67C4×D244—22no—
68C4×D244—22no—
69C4×D244—22no—
70#14519735—24no—
71#14519735—24no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2312yes—
76C2×#11758196—22no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198—21no—
81C2×#11758198—22no—
82C2×#11758198D412yes—
83C2×C4×Dic61—22no—
84#14519711—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—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—21no—
131C2×D244—44no—
132C2×D244—44no—
133C2×D244—82no—
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—42no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×Dic122—24no—
145C2×Dic122—42no—
146C2×Dic122—42no—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61—81no—
156C4⋊C4×C61C2×D412yes—
157C42×C61—24no—
158C42×C61C2×D412yes—
159C42×C61C2×D412yes—
160#11758201—28no—
161#11758198—28no—
162#11758198—28no—
163#11758198—24no—
164#11758198—44no—
165#11758198—44no—
166#11758198—42no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61C2×D414yes—
173#11758196—28no—
174C4×Dic61—28no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418#4112yes—
181C23×C61C22×D411yescharacteristic
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196Q8×C61—22no—
197Q8×C61—22no—
198Q8×C61—41no—
199Q8×C61—41no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61#12012yes—
212C2×C4×C61#12012yes—
213C2×C4×C61#13512yes—
214C2×C4×C61AGL(1,ℤ/8)12yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61H(5,𝔽2)11yescharacteristic
220C2×C4×C612−1+411yescharacteristic
221C2×Dic61—22no—
222C2×Dic61—24no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230D244—22no—
231D244—22no—
232D244—41no—
233D244—41no—
234D244—44no—
235D244—88no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—84no—
241Dic122—48no—
242C8×C61—42no—
243#92418—22no—
244#92418—41no—
245#24119—611nomaximal, Sylow
246C22×C61C2×#12011yescharacteristic
247C22×C61#54611yescharacteristic
248C22×C61#54711yescharacteristic
249C22×C61C2×#13511yescharacteristic
250C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C61#57711yescharacteristic
252C22×C61#57811yescharacteristic
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
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—42no—
270C4×C61—82no—
271C4×C61#44914yes—
272C4×C61#52511yescharacteristic
273C4×C61#53711yescharacteristic
274C4×C61#54811yescharacteristic
275C4×C61#55011yescharacteristic
276Dic61—42no—
277Dic61—44no—
278Dic61—81no—
279C2×#556—611no—
280C2×#474—611no—
281C2×C4×SD16—611no—
282#2537—611no—
283#2036—612no—
284#2000—612no—
285#1990—611no—
286#1954—611no—
287#1875—612no—
288#1854—612no—
289#1803D6111yescharacteristic, complemented
290C2×C61#240711yescharacteristic
291C2×C61C2×#44911yescharacteristic
292C2×C61#301311yescharacteristic
293C2×C61#330211yescharacteristic
294C2×C61#332411yescharacteristic
295C2×C61#334011yescharacteristic
296C2×C61#335311yescharacteristic
297D61—82no—
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—1221no—
309C2×C4×Q8C2×D6111yescharacteristic
310C2×C4×D4—611no—
311C2×C4×D4—611no—
312C2×#117—612no—
313#474—1224no—
314C4×SD16—1224no—
315C2×#107—612no—
316C2×#106—611no—
317C2×C4⋊C8C2×D6111yescharacteristic
318C2×#103—611no—
319C2×#103C2×D6112yes—
320C2×#102—612no—
321C2×#102—611no—
322C2×C22⋊C8—612no—
323C2×C22⋊C8—612no—
324C2×C4×C8—611no—
325#394—1222no—
326#384C2×D6111yescharacteristic
327#380—612no—
328#380—1221no—
329#340—612no—
330#340—612no—
331#340C2×D6112yes—
332C61#2411911yescharacteristic, complemented, Sylow
333C22×Q8—1221no—
334C22×Q8C22×D6111yescharacteristic
335C22×D4—611no—
336C22×D4—611no—
337C22×D4—1222no—
338C23×C4—612no—
339C23×C4—1221no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C2×SD16—1222no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8C22×D6112yes—
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—22no—
356C4×Q8—1228no—
357C4×Q8D24414yescomplemented
358C4×D4—1222no—
359C4×D4—1222no—
360C4×D4—1222no—
361C4×D4—1222no—
362#117—1224no—
363#117—1224no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22×D6112yes—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C22⋊C4—612no—
375C2×C22⋊C4—1222no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1221no—
380C2×C42—21no—
381C2×C42—1222no—
382C2×C42C22×D6111yescharacteristic
383#107—1224no—
384#106—1222no—
385C4⋊C8—22no—
386#103—42no—
387#103—1222no—
388#102—1224no—
389#102—1222no—
390C22⋊C8—1224no—
391C22⋊C8—2442no—
392C4×C8—1222no—
393#95—1222no—
394C24—612no—
395C24—1221no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—1224no—
402C2×Q8—2442no—
403C2×Q8—2442no—
404C2×Q8C2×D24412yes—
405C2×D4—612no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—4882no—
415C2×D4—2444no—
416C2×D4—2444no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—612no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1221no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4—2442no—
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4C23×D6111yescharacteristic
435SD16—1228no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—612no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C4⋊C4—21no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—81no—
451C4⋊C4—614no—
452C4⋊C4—1228no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—1224no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4C2×D24412yes—
459C22⋊C4—1228no—
460C22⋊C4—1228no—
461C22⋊C4—1224no—
462C22⋊C4—2444no—
463C22⋊C4—2444no—
464C22⋊C4—2442no—
465C42—24no—
466C42—1228no—
467C42C2×D24412yes—
468C42D4×D6112yes—
469C23—1222no—
470C23—1222no—
471C23—1222no—
472C23—1222no—
473C23—1222no—
474C23—1222no—
475C23—1222no—
476C23—1222no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23—2441no—
481C23—2441no—
482C23—2441no—
483C23C2×D4×D6111yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—41no—
487Q8—41no—
488Q8—2448no—
489D4—1222no—
490D4—1222no—
491D4—4888no—
492D4—2441no—
493D4—2444no—
494D4—2441no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—1222no—
507C2×C4—1224no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—4884no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4#1451974712yes—
521C2×C4#1451974712yes—
522C2×C4#1451980012yes—
523C2×C4#1451980412yes—
524C2×C4C2×D4×D6111yescharacteristic
525C2×C4C22×D24411yescharacteristic
526C2×C4#41×D6111yescharacteristic
527C2×C4#41×D6111yescharacteristic
528C2×C4#1451982211yescharacteristic
529C2×C4#1451982511yescharacteristic
530C8—42no—
531C8—1222no—
532C8—2441no—
533C22—4881no—
534C22—4881no—
535C22—4881no—
536C22—4881no—
537C22—2442no—
538C22—2442no—
539C22—2442no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22—2442no—
546C22—2442no—
547C22C2×#1451974711yescharacteristic
548C22#1684040911yescharacteristic
549C22#1684041411yescharacteristic
550C22C2×#1451980011yescharacteristic
551C22C2×#1451980411yescharacteristic
552C22#1684057211yescharacteristic
553C22#1684057811yescharacteristic
554C4—22no—
555C4—42no—
556C4—42no—
557C4—82no—
558C4—4881no—
559C4—2444no—
560C4—2442no—
561C4#1683991014yes—
562C4#1684027411yescharacteristic
563C4#1684034311yescharacteristic
564C4#1684042211yescharacteristic
565C4#1684043011yescharacteristic
566C2—4882no—
567C2—4882no—
568C2#2144916511yescharacteristic
569C2C2×#1683991011yescharacteristic
570C2#2145513611yescharacteristic
571C2#2145773911yescharacteristic
572C2#2145805311yescharacteristic
573C2#2145823511yescharacteristic
574C2#2145840611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radical#21450735C2
hypercenter#1803D61
hypocenterC61#24119
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×#16840485C2semidirect
2. C2×#16840043C2semidirect
3. C2×C4×#11758201C2semidirect
5. #21447021C2semidirect
6. #21446821C2semidirect
7. #21446779C2semidirect
8. #21446612C2semidirect
9. #21446140C2semidirect
10. #21446021C2semidirect
11. C61×#1803C2semidirect
57. #14519777D4semidirect
96. C4×Q8×C61D4semidirect
289. #1803D61semidirect
332. C61#24119semidirect
357. C4×Q8D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
C2×#168400432
C2×C4×#117582012
#214507352
#214470212
#214468212
#214467792
#214466122
#214461402
#214460212
C61×#18032
#2411961