Finite Group Databasebeta

← #29718378

Subgroups — #29718378

Order 15616.

1232 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840081C211yesmaximal, characteristic, complemented
2C22×#14519698C211yesmaximal, characteristic, complemented
3#21449452C211yesmaximal, characteristic
4#21448698C211yesmaximal, characteristic, complemented
5#21447191C212yesmaximal, complemented
6#21447051C212yesmaximal, complemented
7#21446744C212yesmaximal, complemented
8#21446448C212yesmaximal, complemented
9#21446245C211yesmaximal, characteristic, complemented
10#21445996C211yesmaximal, characteristic, complemented
11C61×#1786C211yesmaximal, characteristic, complemented
12C23×#92418C2211yescharacteristic
13C2×#14519773C2212yes—
14C2×#14519751C2211yescharacteristic
15C2×#14519746—21no—
16C2×#14519746—21no—
17C22×C4⋊Dic61C2211yescharacteristic
18#16840081—24no—
19C2×#14519709C2211yescharacteristic, complemented
20C2×#14519709C2211yescharacteristic, complemented
21C2×#14519706C2212yescomplemented
22C2×#14519699C2211yescharacteristic, complemented
23C2×#14519699C2211yescharacteristic, complemented
24C2×#14519698—24no—
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×#14519697C2212yescomplemented
28#16839670—22no—
29#16839665C2212yes—
30#16839658C2212yescomplemented
31#16839647—22no—
32#16839620—21no—
33C4⋊C4×Dic61C2212yescomplemented
34C22⋊C4×Dic61—21no—
35C4×C22⋊Dic61C2211yescharacteristic
36#16839524C2212yescomplemented
37#16839516C2212yescomplemented
38C23×D4×C61C2211yescharacteristic
39C2×#102×C61C2212yes—
40C2×#102×C61C2212yes—
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C22⋊D61—21no—
45C22×C22⋊D61—22no—
46C22×C22⋊D61—22no—
47C23×Dic61—21no—
48C22×#92418—22no—
49C22×#92418—42no—
50C22×#92418C2312yes—
51#14519773—24no—
52#14519773—42no—
53#14519751—22no—
54#14519751—22no—
55#14519746—44no—
56#14519746—44no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61D414yescomplemented
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2311yescharacteristic
63C2×#11758196—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61D411yescharacteristic
67C2×C22⋊Dic61D411yescharacteristic
68C2×#11758198—21no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61C2312yes—
71#14519709—22no—
72#14519709—22no—
73#14519706—42no—
74#14519699—22no—
75#14519699—22no—
76#14519698—24no—
77#14519698—24no—
78#14519697—42no—
79#14519684—22no—
80#14519684—22no—
81#14519684D412yes—
82C25×C61—21no—
83C22×D4×C61—24no—
84C22×D4×C61—22no—
85C22×D4×C61—21no—
86C22×D4×C61—22no—
87C22×D4×C61—42no—
88C22×D4×C61C2312yes—
89C23×C4×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91C22×C8×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2312yes—
93C2×C22⋊C4×C61—21no—
94#102×C61—24no—
95#102×C61—42no—
96C22⋊C8×C61—22no—
97#95×C61—21no—
98C24×C61—22no—
99C24×C61—44no—
100C24×C61—42no—
101C24×C61—42no—
102C24×C61—42no—
103C24×C61—82no—
104C24×C61C2×D411yescharacteristic
105C23×D61—21no—
106C2×D4×C61—28no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—48no—
113C2×D4×C61—44no—
114C2×D4×C61—44no—
115C2×D4×C61—42no—
116C2×D4×C61#4114yes—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61#4112yes—
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—44no—
125C2×C22⋊D61—44no—
126C2×C22⋊D61—44no—
127C2×C22⋊D61—42no—
128C2×C22⋊D61—42no—
129C2×C22⋊D61—42no—
130C2×C22⋊D61—44no—
131C2×C22⋊D61—82no—
132C2×C22⋊D61—82no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—24no—
136C22×Dic61—21no—
137C22×Dic61—82no—
138C22×Dic61C2×D412yes—
139C22×Dic61C2×D412yes—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—22no—
143C2×D244—22no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C22⋊C4×C61—42no—
151C22⋊Dic61—22no—
152C22⋊Dic61—44no—
153C22⋊Dic61—44no—
154C22⋊Dic61SD1614yes—
155#11758198—42no—
156C4⋊Dic61—24no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61C2×D412yes—
162C4⋊Dic61C2×D412yes—
163#11758196—82no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C4×Dic61—82no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—44no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—24no—
174C23×C61—48no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—88no—
182C23×C61—88no—
183C23×C61—84no—
184C23×C61—84no—
185C23×C61#12112yes—
186C23×C61C2×SD1612yes—
187C23×C61AGL(1,ℤ/8)12yes—
188C23×C61C22×D411yescharacteristic
189C22×D61—41no—
190C22×D61—42no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—42no—
194D4×C61—24no—
195D4×C61—24no—
196D4×C61—48no—
197D4×C61—42no—
198D4×C61—42no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—24no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61#12112yes—
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4112yes—
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61H(5,𝔽2)12yes—
211C22⋊D61—88no—
212C22⋊D61—88no—
213C22⋊D61—84no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—84no—
227C2×Dic61C2×SD1614yes—
228D244—41no—
229D244—41no—
230D244—42no—
231C8×C61—41no—
232C8×C61—41no—
233#92418—42no—
234#24478—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—24no—
237C22×C61—22no—
238C22×C61—24no—
239C22×C61—44no—
240C22×C61—44no—
241C22×C61—44no—
242C22×C61—816no—
243C22×C61—82no—
244C22×C61—82no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61#46014yes—
248C22×C61C2×#12111yescharacteristic
249C22×C61#54612yes—
250C22×C61C22×SD1611yescharacteristic
251C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
252C22×C61#57612yes—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—82no—
256C2×D61—81no—
257C2×D61—82no—
258C4×C61—22no—
259C4×C61—82no—
260C4×C61#52511yescharacteristic
261C4×C61#53511yescharacteristic
262C4×C61#55112yes—
263Dic61—24no—
264Dic61—42no—
265Dic61—42no—
266Dic61—42no—
267Dic61—81no—
268C2×#481—611no—
269C22×#102—611no—
270#2433—611no—
271#2345—611no—
272#2065—612no—
273#2041—612no—
274#1982—612no—
275#1926—612no—
276#1893—611no—
277#1850—611no—
278#1786D6111yescharacteristic, complemented
279C2×C61—24no—
280C2×C61—84no—
281C2×C61#241111yescharacteristic
282C2×C61C2×#46011yescharacteristic
283C2×C61#304111yescharacteristic
284C2×C61#329512yes—
285C2×C61#331812yes—
286D61—161no—
287C23×D4C2×D6111yescharacteristic
288C23×C8—611no—
289C2×#127—612no—
290C2×#121—611no—
291C2×#120—1221no—
292C2×#120—1221no—
293C22×C4⋊C4—611no—
294#481—1224no—
295C2×#106—611no—
296C2×#106—611no—
297C2×C4⋊C8—612no—
298C2×#102—612no—
299C2×#102—612no—
300C2×#102—611no—
301C2×#102—611no—
302C2×#102—612no—
303C2×#102—1224no—
304C2×#102C2×D6112yes—
305C2×#102C2×D6112yes—
306C2×C22⋊C8C2×D6111yescharacteristic
307#392—1222no—
308#390—612no—
309#389—612no—
310#387—1222no—
311#380—1221no—
312#380C2×D6111yescharacteristic
313C4×C4⋊C4—612no—
314C4×C22⋊C4—611no—
315C4×C22⋊C4—1221no—
316#340—612no—
317#336—612no—
318C61#2447811yescharacteristic, complemented, Sylow
319C25—21no—
320C22×D4—24no—
321C22×D4—22no—
322C22×D4—22no—
323C22×D4—21no—
324C22×D4—42no—
325C22×D4—611no—
326C22×D4—1222no—
327C22×D4—1222no—
328C22×D4—1221no—
329C22×D4C22×D6112yes—
330C23×C4—1221no—
331C23×C4C22×D6111yescharacteristic
332C22×C8—612no—
333C22×C8—1222no—
334C22×C8—2442no—
335C22×C8C22×D6111yescharacteristic
336C22×C8C22×D6111yescharacteristic
337#127—1224no—
338#127—2442no—
339#121—1222no—
340#121—1222no—
341#120—2444no—
342#120—2444no—
343C2×C4⋊C4—614no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4C22×D6112yes—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—611no—
353C2×C22⋊C4—611no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1222no—
356C2×C22⋊C4—1221no—
357C2×C42—612no—
358C2×C42—1222no—
359#106—1222no—
360#106—1222no—
361C4⋊C8—2442no—
362#102—24no—
363#102—42no—
364#102—1222no—
365#102—1222no—
366#102—1224no—
367#102—1224no—
368#102—2442no—
369C22⋊C8—22no—
370#95—21no—
371#95—612no—
372#95—1222no—
373#95—1222no—
374C24—22no—
375C24—44no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—82no—
380C24—1221no—
381C24D4×D6111yescharacteristic
382C2×D4—28no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—48no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—1221no—
393C2×D4—1222no—
394C2×D4—1221no—
395C2×D4—1222no—
396C2×D4—4882no—
397C2×D4—4882no—
398C2×D4—2444no—
399C2×D4—2444no—
400C2×D4—2444no—
401C2×D4—2444no—
402C2×D4—2442no—
403C2×D4—2442no—
404C2×D4—2442no—
405C2×D4#1175822414yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—1224no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—4882no—
416C22×C4#1175822212yes—
417C22×C4D4×D6112yes—
418C22×C4D4×D6111yescharacteristic
419C22×C4C23×D6111yescharacteristic
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—1222no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8—2444no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—612no—
431C4⋊C4—612no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1224no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—4882no—
438C22⋊C4—42no—
439C22⋊C4—614no—
440C22⋊C4—1222no—
441C22⋊C4—2444no—
442C22⋊C4—2444no—
443C22⋊C4—2442no—
444C42—1222no—
445C42—1222no—
446C42—4882no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—48no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—88no—
458C23—88no—
459C23—84no—
460C23—84no—
461C23—2441no—
462C23—2441no—
463C23—2441no—
464C23—2442no—
465C23—2442no—
466C23#1451975012yes—
467C23SD16×D6112yes—
468C23#1451980112yes—
469C23C2×D4×D6111yescharacteristic
470D4—24no—
471D4—24no—
472D4—48no—
473D4—42no—
474D4—42no—
475D4—4888no—
476D4—4884no—
477D4—4888no—
478D4—2441no—
479D4—2441no—
480D4—2442no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—614no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—4884no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451975012yes—
503C2×C4C2×D4×D6111yescharacteristic
504C2×C4C2×#1175822412yes—
505C2×C4C2×#1175822211yescharacteristic
506C2×C4#1451982212yes—
507C8—41no—
508C8—41no—
509C8—2442no—
510C22—24no—
511C22—22no—
512C22—24no—
513C22—22no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22—816no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—4882no—
523C22—4881no—
524C22—4881no—
525C22—4882no—
526C22—4881no—
527C22#1683996114yes—
528C22C2×#1451975011yescharacteristic
529C22#1684039912yes—
530C22C2×SD16×D6111yescharacteristic
531C22C2×#1451980111yescharacteristic
532C22#1684056612yes—
533C4—22no—
534C4—82no—
535C4—1224no—
536C4—4881no—
537C4—2442no—
538C4—2442no—
539C4—2442no—
540C4#1684026811yescharacteristic
541C4#1684032711yescharacteristic
542C4#1684043812yes—
543C2—24no—
544C2—84no—
545C2—9761no—
546C2#2144920611yescharacteristic
547C2C2×#1683996111yescharacteristic
548C2#2145532111yescharacteristic
549C2#2145763612yes—
550C2#2145796412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1786C2
socleC23×C61C22×D4
prime radical#21449452C2
hypercenter#1786D61
hypocenterC61#24478
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
1. C2×#16840081C2semidirect
2. C22×#14519698C2semidirect
4. #21448698C2semidirect
5. #21447191C2semidirect
6. #21447051C2semidirect
7. #21446744C2semidirect
8. #21446448C2semidirect
9. #21446245C2semidirect
10. #21445996C2semidirect
11. C61×#1786C2semidirect
19. C2×#14519709C22semidirect
20. C2×#14519709C22semidirect
21. C2×#14519706C22semidirect
22. C2×#14519699C22semidirect
23. C2×#14519699C22semidirect
27. C2×#14519697C22semidirect
30. #16839658C22semidirect
33. C4⋊C4×Dic61C22semidirect
36. #16839524C22semidirect
37. #16839516C22semidirect
59. C2×C4⋊Dic61D4semidirect
278. #1786D61semidirect
318. C61#24478semidirect

Maximal subgroups

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

subgroupindex
C2×#168400812
C22×#145196982
#214494522
#214486982
#214471912
#214470512
#214467442
#214464482
#214462452
#214459962
C61×#17862
#2447861