Finite Group Databasebeta

← #29713060

Subgroups — #29713060

Order 15616.

1338 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840425C211yesmaximal, characteristic
2C22×#14519700C211yesmaximal, characteristic, complemented
3#21450121C211yesmaximal, characteristic
4#21447063C212yesmaximal, complemented
5#21447017C212yesmaximal, complemented
6#21446243C211yesmaximal, characteristic, complemented
7#21446109C211yesmaximal, characteristic
8#21446084C212yesmaximal, complemented
9#21445755C211yesmaximal, characteristic, complemented
10#21445752C212yesmaximal, complemented
11C2×C61×#483C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C22×Q8×D61C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15#16840425—24no—
16#16840425—24no—
17C2×#14519776—21no—
18C2×#14519775C2211yescharacteristic
19C2×#14519760—21no—
20C2×#14519757C2211yescharacteristic
21C2×#14519755—21no—
22C2×#14519754—21no—
23C2×#14519742C2211yescharacteristic
24C2×C4×C22⋊D61C2211yescharacteristic
25C2×#14519737—21no—
26C2×#14519702C2212yes—
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700—24no—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×#14519689C2212yes—
33#16839693—21no—
34#16839688—21no—
35#16839658C2212yes—
36#16839625—21no—
37#16839622C2211yescharacteristic
38#16839619C2212yes—
39#16839524C2212yes—
40#16839516C2212yes—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#483—24no—
44C2×#106×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#14519760—44no—
70#14519757—22no—
71#14519757—22no—
72#14519755—28no—
73#14519754—28no—
74#14519742—22no—
75#14519742—22no—
76C4×C22⋊D61—22no—
77C4×C22⋊D61—22no—
78#14519737—28no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2312yes—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—22no—
85C2×#11758196D411yescharacteristic
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61D411yescharacteristic
88C2×#11758198—21no—
89C2×#11758198—22no—
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519702—24no—
94#14519702—42no—
95#14519701—42no—
96#14519700—24no—
97#14519700—24no—
98#14519689—42no—
99#14519684—21no—
100#14519684—22no—
101#14519684—22no—
102C22×Q8×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61C2312yes—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61D414yes—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2312yes—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115#106×C61—24no—
116#103×C61—24no—
117C22⋊C8×C61—22no—
118C24×C61C2×D411yescharacteristic
119C23×D61C2×D411yescharacteristic
120C2×Q8×C61—21no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61C2×D412yes—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2411yescharacteristic
136C2×C22⋊D61—22no—
137C2×C22⋊D61—22no—
138C2×C22⋊D61SD1614yes—
139C22×Dic61—21no—
140C22×Dic61—24no—
141C22×Dic61—21no—
142C22×Dic61—22no—
143C22×Dic61—44no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147Q8×D61—48no—
148Q8×D61—44no—
149Q8×D61—44no—
150C2×C4×D61—24no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—44no—
156C2×Dic122—21no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—28no—
161C2×Dic122—24no—
162C2×Dic122—24no—
163C2×Dic122—21no—
164C2×Dic122—44no—
165C2×Dic122—48no—
166C2×Dic122—44no—
167C2×Dic122—44no—
168C2×Dic122C2×D412yes—
169C2×Dic122C2×D412yes—
170C2×Dic122#4114yes—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61C2×D412yes—
180C22⋊C4×C61—24no—
181C22⋊Dic61—22no—
182C22⋊Dic61—42no—
183#11758198—24no—
184#11758198—22no—
185#11758198—44no—
186C4⋊Dic61—24no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189#11758196—24no—
190#11758196—24no—
191#11758196—24no—
192#11758196—22no—
193#11758196—44no—
194C4×Dic61—24no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—44no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61#12012yes—
204C23×C61C2×SD1612yes—
205C23×C61#13712yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—21no—
208C22×D61—22no—
209C22×D61—21no—
210C22×D61—22no—
211C22×D61C2×SD1612yes—
212Q8×C61—24no—
213Q8×C61—41no—
214Q8×C61—41no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—84no—
227C2×C4×C61#12012yes—
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4112yes—
231C2×C4×C61H(5,𝔽2)12yes—
232C22⋊D61—28no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—24no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—48no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61C2×SD1612yes—
248C4×D61—48no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—42no—
252Dic122—22no—
253Dic122—24no—
254Dic122—22no—
255Dic122—22no—
256Dic122—48no—
257Dic122—48no—
258Dic122—42no—
259Dic122#12314yescomplemented
260C8×C61—42no—
261#92418—42no—
262#24041—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—24no—
266C22×C61—24no—
267C22×C61#44814yes—
268C22×C61C2×#12011yescharacteristic
269C22×C61#54612yes—
270C22×C61C22×SD1611yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C22×C61#57812yes—
273C2×D61—22no—
274C2×D61—24no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—42no—
278C4×C61—22no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61—82no—
282C4×C61C2×#12311yescharacteristic
283C4×C61#53411yescharacteristic
284C4×C61#54612yes—
285Dic61—24no—
286Dic61—22no—
287Dic61—42no—
288Dic61—41no—
289Dic61—41no—
290Dic61—82no—
291C2×D4×Q8—611no—
292C2×#483D6111yescharacteristic, complemented
293C22×#103—611no—
294#2486—611no—
295#2043—612no—
296#2035—612no—
297#1892—611no—
298#1869—611no—
299#1865—612no—
300#1787—612no—
301#1787—611no—
302C2×C61—24no—
303C2×C61#240511yescharacteristic
304C2×C61C2×#44811yescharacteristic
305C2×C61#301111yescharacteristic
306C2×C61#329812yes—
307C2×C61#331212yes—
308D61—42no—
309C23×Q8—611no—
310C23×Q8—611no—
311C23×C8—611no—
312D4×Q8—1224no—
313D4×Q8—1224no—
314C2×C4⋊Q8—611no—
315C2×C4⋊Q8—1221no—
316C2×#123—1221no—
317C2×C22⋊Q8—611no—
318C2×C22⋊Q8—1221no—
319C2×C22⋊Q8—1221no—
320C2×C22⋊Q8C2×D6111yescharacteristic
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C2×C4×D4—1221no—
324C22×C4⋊C4C2×D6111yescharacteristic
325#483—24no—
326C2×#106C2×D6112yes—
327C2×#103—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#103—1224no—
333C2×#103C2×D6112yes—
334C2×C22⋊C8—612no—
335C2×C22⋊C8C2×D6111yescharacteristic
336#398—1221no—
337#397—1221no—
338#389—612no—
339#381—1221no—
340#380—611no—
341#380—612no—
342#340—612no—
343#336—612no—
344C61#2404111yescharacteristic, complemented, Sylow
345C22×Q8—614no—
346C22×Q8—611no—
347C22×Q8—611no—
348C22×Q8—1221no—
349C22×Q8—1224no—
350C22×Q8—1222no—
351C22×Q8—1222no—
352C22×Q8—1221no—
353C22×Q8—1222no—
354C22×Q8—1224no—
355C22×Q8—2442no—
356C22×Q8—2442no—
357C22×Q8C22×D6111yescharacteristic
358C22×D4—611no—
359C23×C4—21no—
360C23×C4—611no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4C22×D6111yescharacteristic
364C22×C8—612no—
365C22×C8—1222no—
366C22×C8—2442no—
367C22×C8C22×D6112yes—
368C4⋊Q8—1228no—
369C4⋊Q8—1222no—
370C4⋊Q8—1222no—
371#123—2444no—
372C22⋊Q8—22no—
373C22⋊Q8—1228no—
374C22⋊Q8—1222no—
375C22⋊Q8—1228no—
376C22⋊Q8—1222no—
377C22⋊Q8C22⋊D6114yes—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×D4—1228no—
381C4×D4—1222no—
382C4×D4—1222no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4C22×D6112yes—
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—611no—
398C2×C22⋊C4—611no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4—1221no—
402C2×C42—611no—
403C2×C42—1221no—
404#106—24no—
405#103—24no—
406#103—1224no—
407#103—1224no—
408#103—1224no—
409#103—2442no—
410#103—2442no—
411C22⋊C8—22no—
412C22⋊C8—2442no—
413#95—1222no—
414#95—1222no—
415#95—1221no—
416C24—611no—
417C24D4×D6111yescharacteristic
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—612no—
422C2×Q8—612no—
423C2×Q8—614no—
424C2×Q8—1224no—
425C2×Q8—1228no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×Q8—1222no—
429C2×Q8—1224no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8—2444no—
433C2×Q8—2444no—
434C2×Q8—2448no—
435C2×Q8—2444no—
436C2×Q8—2448no—
437C2×Q8—2444no—
438C2×Q8—2444no—
439C2×Q8C2×C22⋊D6112yes—
440C2×D4—614no—
441C2×D4—1222no—
442C2×D4—1222no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—1224no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1224no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—2444no—
464C22×C4—2444no—
465C22×C4D4×D6112yes—
466C22×C4D4×D6111yescharacteristic
467C22×C4C2×C22⋊D6112yes—
468C22×C4C23×D6111yescharacteristic
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—2444no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—1224no—
482C4⋊C4—1224no—
483C4⋊C4—1224no—
484C4⋊C4—1224no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—2444no—
489C4⋊C4C2×C22⋊D6112yes—
490C22⋊C4—24no—
491C22⋊C4—1224no—
492C22⋊C4—1222no—
493C22⋊C4—1222no—
494C22⋊C4—2444no—
495C22⋊C4—2442no—
496C42—1224no—
497C42—1221no—
498C42—1221no—
499C23—22no—
500C23—22no—
501C23—612no—
502C23—1221no—
503C23—1221no—
504C23—1222no—
505C23—1222no—
506C23#1451974512yes—
507C23SD16×D6112yes—
508C23#1451980612yes—
509C23C2×D4×D6111yescharacteristic
510Q8—24no—
511Q8—41no—
512Q8—41no—
513Q8—614no—
514Q8—1222no—
515Q8—1224no—
516Q8—1222no—
517Q8—1222no—
518Q8—2448no—
519Q8—2448no—
520Q8—2442no—
521D4—1228no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—612no—
535C2×C4—1221no—
536C2×C4—1222no—
537C2×C4—1221no—
538C2×C4—1224no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—2448no—
545C2×C4—2448no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2442no—
553C2×C4#1451974512yes—
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4C22×C22⋊D6111yescharacteristic
556C2×C4#41×D6112yes—
557C2×C4#1451982312yes—
558C8—42no—
559C8—2442no—
560C22—22no—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—1224no—
565C22—1222no—
566C22—2441no—
567C22—2441no—
568C22—2442no—
569C22#1683989914yes—
570C22C2×#1451974511yescharacteristic
571C22#1684040412yes—
572C22C2×SD16×D6111yescharacteristic
573C22C2×#1451980611yescharacteristic
574C22#1684057812yes—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—82no—
579C4—1222no—
580C4—1224no—
581C4—4882no—
582C4—2441no—
583C4—2441no—
584C4—2442no—
585C4#123×D6111yescharacteristic
586C4#1684032311yescharacteristic
587C4#1684040012yes—
588C2—24no—
589C2—2442no—
590C2#2144914311yescharacteristic
591C2C2×#1683989911yescharacteristic
592C2#2145511311yescharacteristic
593C2#2145768212yes—
594C2#2145786812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#483C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenterC2×#483D61
hypocenterC61#24041
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
2. C22×#14519700C2semidirect
4. #21447063C2semidirect
5. #21447017C2semidirect
6. #21446243C2semidirect
8. #21446084C2semidirect
9. #21445755C2semidirect
10. #21445752C2semidirect
11. C2×C61×#483C2semidirect
259. Dic122#123semidirect
292. C2×#483D61semidirect
344. C61#24041semidirect

Maximal subgroups

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

subgroupindex
C2×#168404252
C22×#145197002
#214501212
#214470632
#214470172
#214462432
#214461092
#214460842
#214457552
#214457522
C2×C61×#4832
#2404161