Finite Group Databasebeta

← #29713071

Subgroups — #29713071

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840425C211yesmaximal, characteristic
2C22×#14519700C211yesmaximal, characteristic, complemented
3#21450130C211yesmaximal, characteristic
4#21447063C212yesmaximal, complemented
5#21447024C212yesmaximal, complemented
6#21446243C211yesmaximal, characteristic, complemented
7#21446109C211yesmaximal, characteristic
8#21446070C212yesmaximal, complemented
9#21445755C211yesmaximal, characteristic, complemented
10#21445752C212yesmaximal, complemented
11C2×C61×#484C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C22×Q8×D61C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840425—24no—
16#16840425—24no—
17C2×#14519776—21no—
18C2×#14519775C2211yescharacteristic
19C2×#14519757C2211yescharacteristic
20C2×#14519755—21no—
21C2×#14519754—21no—
22C2×#14519742C2211yescharacteristic
23C2×C4×C22⋊D61C2211yescharacteristic
24C2×#14519737—21no—
25C2×#14519702C2212yes—
26C2×#14519702C2212yes—
27C2×#14519701C2212yes—
28C2×#14519700—24no—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519689C2212yes—
32#16839700—21no—
33#16839697—21no—
34#16839658C2212yes—
35#16839641—21no—
36#16839635—21no—
37#16839622C2211yescharacteristic
38#16839619C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#484—24no—
44C2×#107×C61C2212yes—
45C2×#103×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C22×Dic122—24no—
48C22×Dic122—21no—
49C22×Dic122—22no—
50C22×Dic122—21no—
51C22×Dic122—42no—
52C22×Dic122D414yes—
53C22×Dic122C2311yescharacteristic
54C22×Dic122C2311yescharacteristic
55C2×Q8×D61—22no—
56C2×Q8×D61—22no—
57C2×Q8×D61—24no—
58C2×Q8×D61—42no—
59C22×C22⋊D61D411yescharacteristic
60C23×Dic61—21no—
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C22×#92418—22no—
64C22×#92418—42no—
65C22×#92418C2312yes—
66#14519776—28no—
67#14519775—22no—
68#14519775—22no—
69#14519757—22no—
70#14519757—22no—
71#14519755—28no—
72#14519754—28no—
73#14519742—22no—
74#14519742—22no—
75C4×C22⋊D61—22no—
76C4×C22⋊D61—22no—
77#14519737—28no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2312yes—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—22no—
84C2×#11758196D411yescharacteristic
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61D411yescharacteristic
87C2×#11758198—21no—
88C2×#11758198—22no—
89C2×#11758198D411yescharacteristic
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92#14519702—24no—
93#14519702—42no—
94#14519701—42no—
95#14519700—24no—
96#14519700—24no—
97#14519689—42no—
98#14519684—21no—
99#14519684—22no—
100#14519684—22no—
101C22×Q8×C61C2311yescharacteristic
102C23×C4×C61—21no—
103C23×C4×C61C2311yescharacteristic
104C22×C8×C61C2312yes—
105C22⋊Q8×C61—22no—
106C22⋊Q8×C61D414yes—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2312yes—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114#107×C61—24no—
115#103×C61—24no—
116C22⋊C8×C61—22no—
117C24×C61C2×D411yescharacteristic
118C23×D61C2×D411yescharacteristic
119C2×Q8×C61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61C2×D412yes—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—42no—
129C22×C4×C61—82no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—22no—
135C2×C22⋊D61—22no—
136C2×C22⋊D61Q1614yes—
137C22×Dic61—21no—
138C22×Dic61—24no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—44no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145Q8×D61—48no—
146Q8×D61—44no—
147Q8×D61—44no—
148C2×C4×D61—24no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—44no—
154C2×Dic122—21no—
155C2×Dic122—22no—
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—28no—
159C2×Dic122—24no—
160C2×Dic122—24no—
161C2×Dic122—21no—
162C2×Dic122—44no—
163C2×Dic122—48no—
164C2×Dic122—44no—
165C2×Dic122—44no—
166C2×Dic122C2×D412yes—
167C2×Dic122C2×D412yes—
168C2×Dic122#4114yes—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61C2×D412yes—
178C22⋊C4×C61—24no—
179C22⋊Dic61—22no—
180C22⋊Dic61—42no—
181#11758198—24no—
182#11758198—22no—
183#11758198—82no—
184C4⋊Dic61—24no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187#11758196—24no—
188#11758196—24no—
189#11758196—24no—
190#11758196—22no—
191#11758196—82no—
192C4×Dic61—24no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—44no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61#12012yes—
202C23×C61C2×Q1612yes—
203C23×C61#13712yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—21no—
206C22×D61—22no—
207C22×D61—21no—
208C22×D61—22no—
209C22×D61C2×Q1612yes—
210Q8×C61—24no—
211Q8×C61—41no—
212Q8×C61—41no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—84no—
225C2×C4×C61#12012yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4112yes—
229C2×C4×C61H(5,𝔽2)12yes—
230C22⋊D61—28no—
231C2×Dic61—21no—
232C2×Dic61—22no—
233C2×Dic61—24no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—48no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61C2×Q1612yes—
246C4×D61—48no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—42no—
250Dic122—22no—
251Dic122—24no—
252Dic122—22no—
253Dic122—22no—
254Dic122—48no—
255Dic122—48no—
256Dic122—42no—
257Dic122#12314yes—
258C8×C61—42no—
259#92418—42no—
260#24042—611nomaximal, Sylow
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—24no—
264C22×C61—24no—
265C22×C61#45114yes—
266C22×C61C2×#12011yescharacteristic
267C22×C61#54612yes—
268C22×C61C22×Q1611yescharacteristic
269C22×C61C2×#13711yescharacteristic
270C22×C61#57712yes—
271C2×D61—22no—
272C2×D61—24no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—42no—
276C4×C61—22no—
277C4×C61—42no—
278C4×C61—42no—
279C4×C61—82no—
280C4×C61C2×#12311yescharacteristic
281C4×C61#53411yescharacteristic
282C4×C61#54612yes—
283Dic61—24no—
284Dic61—22no—
285Dic61—42no—
286Dic61—41no—
287Dic61—41no—
288Dic61—82no—
289C2×D4×Q8—611no—
290C2×#484D6111yescharacteristic, complemented
291C22×#103—611no—
292#2487—611no—
293#2043—612no—
294#2036—612no—
295#1892—611no—
296#1869—611no—
297#1863—612no—
298#1787—612no—
299#1787—611no—
300C2×C61—24no—
301C2×C61#240511yescharacteristic
302C2×C61C2×#45111yescharacteristic
303C2×C61#301011yescharacteristic
304C2×C61#329912yes—
305C2×C61#332012yes—
306D61—42no—
307C23×Q8—611no—
308C23×Q8—611no—
309C23×C8—611no—
310D4×Q8—1224no—
311D4×Q8—1224no—
312C2×C4⋊Q8—611no—
313C2×C4⋊Q8—1221no—
314C2×C22⋊Q8—611no—
315C2×C22⋊Q8—1221no—
316C2×C22⋊Q8—1221no—
317C2×C22⋊Q8C2×D6111yescharacteristic
318C2×C4×Q8—611no—
319C2×C4×D4—611no—
320C2×C4×D4—1221no—
321C22×C4⋊C4C2×D6111yescharacteristic
322#484—24no—
323C2×#107C2×D6112yes—
324C2×#103—612no—
325C2×#103—612no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103—612no—
329C2×#103—1224no—
330C2×#103C2×D6112yes—
331C2×C22⋊C8—612no—
332C2×C22⋊C8C2×D6111yescharacteristic
333#400—1221no—
334#399—1221no—
335#389—612no—
336#386—1221no—
337#385—1221no—
338#380—611no—
339#380—612no—
340#340—612no—
341#336—612no—
342C61#2404211yescharacteristic, complemented, Sylow
343C22×Q8—614no—
344C22×Q8—611no—
345C22×Q8—611no—
346C22×Q8—1221no—
347C22×Q8—1224no—
348C22×Q8—1222no—
349C22×Q8—1222no—
350C22×Q8—1221no—
351C22×Q8—1222no—
352C22×Q8—1224no—
353C22×Q8—2442no—
354C22×Q8—2442no—
355C22×Q8C22×D6111yescharacteristic
356C22×D4—611no—
357C23×C4—21no—
358C23×C4—611no—
359C23×C4—1221no—
360C23×C4—1221no—
361C23×C4C22×D6111yescharacteristic
362C22×C8—612no—
363C22×C8—1222no—
364C22×C8—2442no—
365C22×C8C22×D6112yes—
366C4⋊Q8—1228no—
367C4⋊Q8—1222no—
368C4⋊Q8—1222no—
369C22⋊Q8—22no—
370C22⋊Q8—1228no—
371C22⋊Q8—1222no—
372C22⋊Q8—1228no—
373C22⋊Q8—1222no—
374C22⋊Q8C22⋊D6114yes—
375C4×Q8—1222no—
376C4×Q8—1222no—
377C4×D4—1228no—
378C4×D4—1222no—
379C4×D4—1222no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1222no—
391C2×C4⋊C4C22×D6112yes—
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—611no—
395C2×C22⋊C4—611no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1221no—
399C2×C42—611no—
400C2×C42—1221no—
401#107—24no—
402#103—24no—
403#103—1224no—
404#103—1224no—
405#103—1224no—
406#103—2442no—
407#103—2442no—
408C22⋊C8—22no—
409C22⋊C8—2442no—
410#95—1222no—
411#95—1222no—
412#95—1221no—
413C24—611no—
414C24D4×D6111yescharacteristic
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—612no—
419C2×Q8—612no—
420C2×Q8—614no—
421C2×Q8—1224no—
422C2×Q8—1228no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1222no—
426C2×Q8—1224no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—2444no—
430C2×Q8—2444no—
431C2×Q8—2448no—
432C2×Q8—2444no—
433C2×Q8—2448no—
434C2×Q8—2444no—
435C2×Q8—2444no—
436C2×Q8C2×C22⋊D6112yes—
437C2×D4—614no—
438C2×D4—1222no—
439C2×D4—1222no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—42no—
446C22×C4—82no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1224no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1224no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—2444no—
460C22×C4—2444no—
461C22×C4D4×D6112yes—
462C22×C4D4×D6111yescharacteristic
463C22×C4C2×C22⋊D6112yes—
464C22×C4C23×D6111yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—2444no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—1224no—
478C4⋊C4—1224no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—4882no—
485C4⋊C4C2×C22⋊D6112yes—
486C22⋊C4—24no—
487C22⋊C4—1224no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C22⋊C4—4882no—
491C22⋊C4—2442no—
492C42—1224no—
493C42—1221no—
494C42—1221no—
495C23—22no—
496C23—22no—
497C23—612no—
498C23—1221no—
499C23—1221no—
500C23—1222no—
501C23—1222no—
502C23#1451974512yes—
503C23Q16×D6112yes—
504C23#1451980912yes—
505C23C2×D4×D6111yescharacteristic
506Q8—24no—
507Q8—41no—
508Q8—41no—
509Q8—614no—
510Q8—1222no—
511Q8—1224no—
512Q8—1222no—
513Q8—1222no—
514Q8—2448no—
515Q8—2448no—
516Q8—2442no—
517D4—1228no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4—612no—
531C2×C4—1221no—
532C2×C4—1222no—
533C2×C4—1221no—
534C2×C4—1224no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—2448no—
541C2×C4—2448no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4#1451974512yes—
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4C22×C22⋊D6111yescharacteristic
552C2×C4#41×D6112yes—
553C2×C4#1451982312yes—
554C8—42no—
555C8—2442no—
556C22—22no—
557C22—24no—
558C22—24no—
559C22—22no—
560C22—1224no—
561C22—1222no—
562C22—2441no—
563C22—2442no—
564C22—2441no—
565C22#1683991614yes—
566C22C2×#1451974511yescharacteristic
567C22#1684040412yes—
568C22C2×Q16×D6111yescharacteristic
569C22C2×#1451980911yescharacteristic
570C22#1684057212yes—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—82no—
575C4—1222no—
576C4—1224no—
577C4—4882no—
578C4—2441no—
579C4—2441no—
580C4—2442no—
581C4#123×D6111yescharacteristic
582C4#1684032311yescharacteristic
583C4#1684040012yes—
584C2—24no—
585C2—2442no—
586C2#2144914311yescharacteristic
587C2C2×#1683991611yescharacteristic
588C2#2145510411yescharacteristic
589C2#2145769312yes—
590C2#2145798812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#484C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenterC2×#484D61
hypocenterC61#24042
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C22×#14519700C2semidirect
4. #21447063C2semidirect
5. #21447024C2semidirect
6. #21446243C2semidirect
8. #21446070C2semidirect
9. #21445755C2semidirect
10. #21445752C2semidirect
11. C2×C61×#484C2semidirect
290. C2×#484D61semidirect
342. C61#24042semidirect

Maximal subgroups

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

subgroupindex
C2×#168404252
C22×#145197002
#214501302
#214470632
#214470242
#214462432
#214461092
#214460702
#214457552
#214457522
C2×C61×#4842
#2404261