Finite Group Databasebeta

← #29645927

Subgroups — #29645927

Order 15616.

1210 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840244C211yesmaximal, characteristic
2#21454847C212yesmaximal, complemented
3#21444163C211yesmaximal, characteristic, complemented
4#21444156C212yesmaximal, complemented
5#21444151C212yesmaximal, complemented
6#21444147C212yesmaximal, complemented
7#21444136C212yesmaximal, complemented
8#324×D61C212yesmaximal, complemented
9C61×#1467C211yesmaximal, characteristic, complemented
10C22×#11758226C411yescharacteristic
11#16840244—24no—
12#16840244—24no—
13C2×Q8×Dic61C411yescharacteristic
14C2×C4×D244C411yescharacteristic
15C2×C4×D244C411yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×#14519739C411yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519734C411yescharacteristic
21C2×#14519734C411yescharacteristic
22C2×#14519734C2211yescharacteristic
23C2×C4⋊C4×D61C411yescharacteristic
24C2×C4⋊C4×D61C2211yescharacteristic
25C2×C4⋊C4×D61C2211yescharacteristic
26#16839835—24no—
27#16839832—24no—
28#16839819C2212yes—
29#16839724C2214yes—
30C22⋊C8×D61C2214yes—
31#16839488C2212yes—
32#16839488C2212yes—
33#16839487C2214yes—
34#16839485C2214yes—
35C2×C4×Q8×C61C2211yescharacteristic
36C61×#432C2212yes—
37C61×#324C2212yes—
38C61×#324C2212yes—
39C2×#11758226—24no—
40C2×#11758226—24no—
41C2×#11758226—24no—
42C22×D244C2×C411yescharacteristic
43C22×D244C2×C411yescharacteristic
44C22×D244C2×C411yescharacteristic
45C22×C4×D61C2×C411yescharacteristic
46C22×C4×D61C2×C411yescharacteristic
47C22×C4×D61C2×C411yescharacteristic
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61C2×C411yescharacteristic
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758189—24no—
55C2×C8×D61—24no—
56Q8×Dic61—24no—
57C4×D244—22no—
58C4×D244—22no—
59C4×D244—22no—
60C4×D244—22no—
61C4×D244D414yes—
62C4×D244D414yes—
63#14519739—24no—
64#14519739—24no—
65#14519739—24no—
66#14519734—22no—
67#14519734—22no—
68#14519734—22no—
69#14519734—22no—
70#14519734—22no—
71#14519734—22no—
72C2×C4⋊Dic61C2×C411yescharacteristic
73C2×C4⋊Dic61C2×C411yescharacteristic
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×#11758196C2×C411yescharacteristic
76C2×#11758196C2×C411yescharacteristic
77C2×#11758196C2×C411yescharacteristic
78C2×#11758196C2311yescharacteristic
79C4⋊C4×D61—24no—
80C4⋊C4×D61—24no—
81C4⋊C4×D61—24no—
82C2×#11758198C2×C411yescharacteristic
83C2×#11758198C2×C411yescharacteristic
84C2×#11758198C2×C411yescharacteristic
85C2×#11758198C2×C411yescharacteristic
86C2×#11758198C2×C411yescharacteristic
87C2×#11758198C2311yescharacteristic
88C2×C4×Dic61C2×C411yescharacteristic
89C2×C4×Dic61C2×C411yescharacteristic
90C2×C4×Dic61C2×C411yescharacteristic
91C4⋊#92418—22no—
92C22⋊#92418C2314yes—
93#14519689—24no—
94#14519689—24no—
95#14519688—22no—
96C22×Q8×C61C2×C411yescharacteristic
97C4×Q8×C61—22no—
98C4×Q8×C61—22no—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C42×C61C2×C411yescharacteristic
103C2×C42×C61C2×C411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—22no—
106C22⋊C8×C61C2314yes—
107#97×C61—22no—
108C23×D61C22×C411yescharacteristic
109C23×D61C22×C411yescharacteristic
110C23×D61C22×C411yescharacteristic
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C22×C411yescharacteristic
123C22×Dic61C22×C411yescharacteristic
124C22×Dic61C22×C411yescharacteristic
125C22×Dic61C22×C411yescharacteristic
126#11758226—232no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244C22⋊C414yes—
135C2×D244M1614yescomplemented
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—81no—
149C2×C4×D61—81no—
150C2×C4×D61—81no—
151C2×C4×D61—81no—
152C2×C4×D61—81no—
153C2×C4×D61—81no—
154C2×C4×D61M1614yescomplemented
155C2×C4×D61C2×D414yes—
156C2×C4×D61C2×D414yes—
157C2×C8×C61—24no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61C2×D412yes—
166C42×C61C2×D412yes—
167#11758198—22no—
168#11758198—22no—
169#11758198—21no—
170#11758198—21no—
171#11758198—22no—
172#11758198C22⋊C412yes—
173#11758198C22⋊C412yes—
174#11758198C22⋊C412yes—
175#11758198C22⋊C412yes—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61C22⋊C414yes—
179#11758196—21no—
180#11758196—21no—
181#11758196—22no—
182#11758196—22no—
183#11758196—22no—
184C4×Dic61—22no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C2×#92418—24no—
188#11758189—84no—
189C8×D61—84no—
190C23×C61C23×C411yescharacteristic
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61C2×C22⋊C412yes—
202C22×D61C2×C22⋊C412yes—
203C22×D61C2×M1612yes—
204C22×D61C2×M1612yes—
205C22×D61C2×M1612yes—
206Q8×C61—28no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×M1612yes—
223C2×C4×C61C2×M1612yes—
224C2×C4×C61C22×D412yes—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×M1612yes—
236D244—28no—
237D244—28no—
238D244—28no—
239C4×D61—28no—
240C4×D61—28no—
241C4×D61—28no—
242C4×D61—44no—
243C4×D61—44no—
244C4×D61—82no—
245C4×D61—82no—
246C4×D61—82no—
247C4×D61—82no—
248C8×C61—44no—
249#92418—44no—
250#17368—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61C22×C22⋊C411yescharacteristic
254C22×C61C22×M1611yescharacteristic
255C22×C61C22×M1611yescharacteristic
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—42no—
270C2×D61#40714yes—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—22no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—41no—
278C4×C61—41no—
279Dic61—22no—
280Dic61—22no—
281Dic61—22no—
282Dic61—41no—
283Dic61—41no—
284C2×#520—611no—
285#2973—612no—
286#1467—611no—
287#1467D6111yescharacteristic, complemented
288#1465—612no—
289#1464—612no—
290#1463—612no—
291#1459—612no—
292C2×#324—612no—
293C2×C61—21no—
294C2×C61—21no—
295C2×C61C2×#40711yescharacteristic
296C2×C61#288311yescharacteristic
297C2×C61#288911yescharacteristic
298D61—24no—
299D61—42no—
300D61—42no—
301C22×#41—611no—
302#520—1224no—
303#520—1224no—
304C2×C4×Q8—611no—
305C2×C4×Q8C2×D6111yescharacteristic
306C2×C4×D4—611no—
307C2×C4×D4—611no—
308C2×C4×D4—611no—
309C2×C4×D4—611no—
310C2×C4×D4—611no—
311C2×C4×D4—611no—
312C2×#117—611no—
313C2×#117—611no—
314C2×#117—611no—
315C22×C4⋊C4—611no—
316C22×C4⋊C4—611no—
317C22×C4⋊C4—611no—
318#435—1224no—
319#435—1224no—
320#432—612no—
321#432C2×D6112yes—
322#407—614no—
323C2×C22⋊C8—614no—
324#324—612no—
325#324—612no—
326#324—614no—
327#324C2×D6112yes—
328#324C2×D6112yes—
329#323—614no—
330C61#1736811yescharacteristic, complemented, Sylow
331C2×#41—1224no—
332C2×#41—1224no—
333C2×#41—1224no—
334C22×Q8C4×D6111yescharacteristic
335C22×D4—611no—
336C22×D4—611no—
337C22×D4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C2×M16—1224no—
348C22×C8—1224no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—1224no—
352C4×D4—614no—
353C4×D4—614no—
354C4×D4—1222no—
355C4×D4—1222no—
356C4×D4—1222no—
357C4×D4—1222no—
358C4×D4—1224no—
359C4×D4—1224no—
360C4×D4—1224no—
361#117—1222no—
362#117—1222no—
363#117—1222no—
364#117—1222no—
365#117—1222no—
366#117—1222no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1224no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4C4×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C22⋊C4—611no—
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—611no—
384C2×C22⋊C4—611no—
385C2×C22⋊C4—611no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42C4×D6111yescharacteristic
390C2×C42C4×D6111yescharacteristic
391C2×C42C22×D6111yescharacteristic
392C4⋊C8—22no—
393C4⋊C8—1222no—
394C22⋊C8—614no—
395C22⋊C8—1224no—
396C22⋊C8—1224no—
397C22⋊C8C22×D6114yes—
398#97—22no—
399#97—1222no—
400C24—611no—
401C24—611no—
402C24—611no—
403#41—12232no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×D4—614no—
408C2×D4—614no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—614no—
420C22×C4—614no—
421C22×C4—614no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—4881no—
437C22×C4—4881no—
438C22×C4—4881no—
439C22×C4—4881no—
440C22×C4—4881no—
441C22×C4—4881no—
442C22×C4C2×C4×D6111yescharacteristic
443C22×C4C2×C4×D6111yescharacteristic
444C22×C4C2×C4×D6111yescharacteristic
445C22×C4C2×C4×D6111yescharacteristic
446C22×C4C2×C4×D6111yescharacteristic
447C22×C4C2×C4×D6111yescharacteristic
448C22×C4C23×D6111yescharacteristic
449M16—4884no—
450C2×C8—24no—
451C2×C8—1224no—
452C2×C8—4884no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—614no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C22⋊C4—612no—
465C22⋊C4—612no—
466C22⋊C4—612no—
467C22⋊C4—612no—
468C22⋊C4—1222no—
469C22⋊C4—1222no—
470C22⋊C4—1222no—
471C22⋊C4—1221no—
472C22⋊C4—1221no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—1222no—
478C42—1222no—
479C42—1222no—
480C42D4×D6112yes—
481C42D4×D6112yes—
482C23—612no—
483C23—612no—
484C23—612no—
485C23—612no—
486C23—612no—
487C23—1221no—
488C23—1221no—
489C23—1221no—
490C23—1221no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23C22×C4×D6111yescharacteristic
498Q8—28no—
499D4—1228no—
500D4—1228no—
501D4—1228no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—1228no—
518C2×C4—1228no—
519C2×C4—1228no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—4882no—
525C2×C4—4882no—
526C2×C4—4882no—
527C2×C4—4882no—
528C2×C4—2444no—
529C2×C4—2444no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4C22⋊C4×D6111yescharacteristic
535C2×C4C22⋊C4×D6111yescharacteristic
536C2×C4C22⋊C4×D6112yes—
537C2×C4M16×D6112yes—
538C2×C4M16×D6112yes—
539C2×C4C2×D4×D6112yes—
540C8—44no—
541C8—2444no—
542C22—21no—
543C22—21no—
544C22—614no—
545C22—1222no—
546C22—1222no—
547C22—1222no—
548C22—1222no—
549C22—1222no—
550C22—1222no—
551C22—2442no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22—2441no—
558C22—2441no—
559C22C2×C22⋊C4×D6111yescharacteristic
560C22C2×M16×D6111yescharacteristic
561C22C2×M16×D6111yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—21no—
567C4—21no—
568C4—41no—
569C4—41no—
570C4—1222no—
571C4—1222no—
572C4—1222no—
573C4—2441no—
574C4—2441no—
575C2—21no—
576C2—21no—
577C2—1224no—
578C2—2442no—
579C2—2442no—
580C2#407×D6111yescharacteristic
581C2#2145427811yescharacteristic
582C2#2145429911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1467C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#11758226C4
cosocleC22×C4×C61C24
hypercenter#1467D61
hypocenterC61#17368
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #21454847C2semidirect
3. #21444163C2semidirect
4. #21444156C2semidirect
5. #21444151C2semidirect
6. #21444147C2semidirect
7. #21444136C2semidirect
8. #324×D61C2semidirect
9. C61×#1467C2semidirect
135. C2×D244M16semidirect
154. C2×C4×D61M16semidirect
287. #1467D61semidirect
330. C61#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#168402442
#214548472
#214441632
#214441562
#214441512
#214441472
#214441362
#324×D612
C61×#14672
#1736861