Finite Group Databasebeta

← #29737624

Subgroups — #29737624

Order 15616.

702 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
1#21449506C211yesmaximal, characteristic
2#21448711C211yesmaximal, characteristic
3#21447390C211yesmaximal, characteristic, complemented
4#21447320C211yesmaximal, characteristic, complemented
5#21447314C211yesmaximal, characteristic, complemented
6#21446772C211yesmaximal, characteristic, complemented
7#21446584C211yesmaximal, characteristic, complemented
8#21446412C211yesmaximal, characteristic, complemented
9#21446392C211yesmaximal, characteristic, complemented
10#21446015C211yesmaximal, characteristic
11#21445998C211yesmaximal, characteristic
12#21445912C211yesmaximal, characteristic
13#21445845C211yesmaximal, characteristic
14#21445534C211yesmaximal, characteristic
15C61×#2057C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C2×#14519735C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519689C2211yescharacteristic
24#16839684—21no—
25#16839673—21no—
26#16839669—21no—
27#16839661C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839660C2211yescharacteristic
30#16839654—21no—
31#16839652—21no—
32#16839643—21no—
33#16839629—21no—
34#16839627C2211yescharacteristic
35#16839619C2211yescharacteristic
36#16839619C2211yescharacteristic
37C4⋊C4×Dic61—21no—
38C4⋊C4×Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C4×#11758198C2211yescharacteristic
41#16839525C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×C4×D61C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C2×C8×D61—22no—
63C2×C8×D61—22no—
64C2×C8×D61—41no—
65#14519735—22no—
66#14519735—22no—
67#14519735—41no—
68#14519735Q814yescomplemented
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198D411yescharacteristic
81C2×#11758198D411yescharacteristic
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87#14519711—22no—
88#14519708—41no—
89C4⋊#92418—41no—
90#14519706—22no—
91#14519689—22no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103C22×Q8×C61—21no—
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C4⋊Q8×C61—22no—
107C4⋊Q8×C61—41no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#107×C61—22no—
116#107×C61—41no—
117#95×C61—21no—
118C23×D61C2×D411yescharacteristic
119C2×Q8×C61—42no—
120C2×Q8×C61—42no—
121C2×Q8×C61—81no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61C2×Q812yes—
141C2×C4×D61#4112yes—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61#4112yes—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61—42no—
156C4⋊C4×C61C2×Q812yes—
157C4⋊C4×C61#4112yes—
158C42×C61—21no—
159C42×C61—21no—
160#11758198—24no—
161#11758198—22no—
162#11758198—41no—
163#11758198—42no—
164#11758198—81no—
165#11758198—81no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—42no—
174C4⋊Dic61—81no—
175C4⋊Dic61C2×Q812yes—
176#11758196—42no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—24no—
180C4×Dic61—42no—
181C4×Dic61—81no—
182C4×Dic61—81no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C8×D61—44no—
190C23×C61C22×D411yescharacteristic
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61C22⋊Q812yes—
198Q8×C61—84no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61#13512yes—
217C2×C4×C61C22×Q811yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—22no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61C22⋊Q812yes—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C8×C61—22no—
249C8×C61—41no—
250#92418—41no—
251#92418—41no—
252#25989—611nomaximal, Sylow
253C22×C61C2×C22⋊Q811yescharacteristic
254C22×C61#54611yescharacteristic
255C22×C61#54711yescharacteristic
256C22×C61C2×#13511yescharacteristic
257C22×C61#57511yescharacteristic
258C22×C61#57611yescharacteristic
259C22×C61#57811yescharacteristic
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—41no—
264C2×D61—41no—
265C2×D61—41no—
266C2×D61—41no—
267C4×C61—22no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—41no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61#53311yescharacteristic
274C4×C61#54411yescharacteristic
275C4×C61#55111yescharacteristic
276C4×C61#55211yescharacteristic
277Dic61—22no—
278Dic61—42no—
279Dic61—41no—
280Dic61—41no—
281Dic61—81no—
282Dic61—81no—
283Dic61—81no—
284#2439—611no—
285#2347—611no—
286#2101—611no—
287#2088—611no—
288#2087—611no—
289#2057D6111yescharacteristic, complemented
290#1988—611no—
291#1949—611no—
292#1921—611no—
293#1918—611no—
294#1854—611no—
295#1850—611no—
296#1825—611no—
297#1808—611no—
298#1735—611no—
299C2×C61#243811yescharacteristic
300C2×C61#307611yescharacteristic
301C2×C61#308311yescharacteristic
302C2×C61#329711yescharacteristic
303C2×C61#331311yescharacteristic
304C2×C61#334211yescharacteristic
305C2×C61#334711yescharacteristic
306D61—42no—
307C23×C8—611no—
308C2×C4⋊Q8C2×D6111yescharacteristic
309C2×#117—611no—
310C2×#117—611no—
311C2×#107—611no—
312C2×#107C2×D6111yescharacteristic
313C2×#107C2×D6111yescharacteristic
314C2×#106—611no—
315C2×C4⋊C8—611no—
316C2×C4⋊C8—611no—
317C2×C22⋊C8—611no—
318#396—1221no—
319#393—1221no—
320#391—1221no—
321#389—611no—
322#389—611no—
323#389—611no—
324#389C2×D6111yescharacteristic
325#388—1221no—
326#388—1221no—
327#387—1221no—
328#383—1221no—
329#382—611no—
330#380—611no—
331#380—611no—
332C4×C4⋊C4—611no—
333C4×C4⋊C4—611no—
334C4×C4⋊C4—1221no—
335C4×C22⋊C4—611no—
336#340—611no—
337#340—611no—
338#340—611no—
339#340—611no—
340#340—611no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340—611no—
345#340—611no—
346#340C2×D6111yescharacteristic
347#340C2×D6111yescharacteristic
348#336—611no—
349#336C2×D6111yescharacteristic
350C61#2598911yescharacteristic, complemented, Sylow
351C22×Q8—21no—
352C23×C4—611no—
353C22×C8—611no—
354C22×C8—611no—
355C22×C8—1222no—
356C22×C8—1222no—
357C22×C8—2441no—
358C22×C8C22×D6111yescharacteristic
359C22×C8C22×D6111yescharacteristic
360C4⋊Q8—22no—
361C4⋊Q8—41no—
362#117—614no—
363#117—1222no—
364#117—1222no—
365#117—2441no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—611no—
380C2×C22⋊C4—611no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—611no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42—1221no—
389C2×C42—1221no—
390C2×C42C22×D6111yescharacteristic
391#107—22no—
392#107—41no—
393#107—1222no—
394#106—2441no—
395C4⋊C8—1222no—
396C4⋊C8—2441no—
397C22⋊C8—1222no—
398#95—21no—
399#95—611no—
400#95—611no—
401#95—1221no—
402#95—1221no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410C24—611no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—81no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—611no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4D4×D6111yescharacteristic
432C22×C4D4×D6111yescharacteristic
433C22×C4C23×D6111yescharacteristic
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—2444no—
446C2×C8#1175822612yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—612no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—4881no—
464C4⋊C4—2442no—
465C4⋊C4—2442no—
466C4⋊C4#1175822412yes—
467C4⋊C4Q8×D6112yes—
468C22⋊C4—1224no—
469C22⋊C4—1222no—
470C22⋊C4—4881no—
471C22⋊C4—4881no—
472C22⋊C4—2441no—
473C22⋊C4—2442no—
474C42—21no—
475C42—21no—
476C42—1224no—
477C42—1221no—
478C42—1221no—
479C42—4881no—
480C42—4881no—
481C42—2442no—
482C23—612no—
483C23—1221no—
484C23—1221no—
485C23—1221no—
486C23—1221no—
487C23—1221no—
488C23—1221no—
489C23C2×D4×D6111yescharacteristic
490Q8—84no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—612no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4#1451979612yes—
534C2×C4C2×Q8×D6111yescharacteristic
535C2×C4#41×D6111yescharacteristic
536C2×C4C2×#1175822411yescharacteristic
537C2×C4C2×#1175822611yescharacteristic
538C2×C4#1451982211yescharacteristic
539C2×C4#1451982511yescharacteristic
540C8—22no—
541C8—41no—
542C8—2441no—
543C8—2441no—
544C22—1222no—
545C22—1222no—
546C22—1222no—
547C22—2441no—
548C22—2441no—
549C22—2441no—
550C22—2441no—
551C22C22⋊Q8×D6111yescharacteristic
552C22#1684039911yescharacteristic
553C22#1684041511yescharacteristic
554C22C2×#1451979611yescharacteristic
555C22#1684056111yescharacteristic
556C22#1684056611yescharacteristic
557C22#1684057511yescharacteristic
558C4—22no—
559C4—42no—
560C4—41no—
561C4—42no—
562C4—81no—
563C4—81no—
564C4—1222no—
565C4—4881no—
566C4—4881no—
567C4—4881no—
568C4—2442no—
569C4—2441no—
570C4—2441no—
571C4#1684031711yescharacteristic
572C4#1684038911yescharacteristic
573C4#1684043911yescharacteristic
574C4#1684044611yescharacteristic
575C2—2442no—
576C2#2144948811yescharacteristic
577C2#2145560011yescharacteristic
578C2#2145566711yescharacteristic
579C2#2145767311yescharacteristic
580C2#2145787811yescharacteristic
581C2#2145824911yescharacteristic
582C2#2145831811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2057C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2057D61
hypocenterC61#25989
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
3. #21447390C2semidirect
4. #21447320C2semidirect
5. #21447314C2semidirect
6. #21446772C2semidirect
7. #21446584C2semidirect
8. #21446412C2semidirect
9. #21446392C2semidirect
15. C61×#2057C2semidirect
68. #14519735Q8semidirect
289. #2057D61semidirect
350. C61#25989semidirect

Maximal subgroups

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

subgroupindex
#214495062
#214487112
#214473902
#214473202
#214473142
#214467722
#214465842
#214464122
#214463922
#214460152
#214459982
#214459122
#214458452
#214455342
C61×#20572
#2598961