Finite Group Databasebeta

← #29752745

Subgroups — #29752745

Order 15616.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450224C211yesmaximal, characteristic
2#21448820C211yesmaximal, characteristic
3#21447240C211yesmaximal, characteristic, complemented
4#21447212C211yesmaximal, characteristic, complemented
5#21447021C211yesmaximal, characteristic
6#21447013C211yesmaximal, characteristic
7#21446590C211yesmaximal, characteristic, complemented
8#21446552C211yesmaximal, characteristic, complemented
9#21446425C211yesmaximal, characteristic
10#21446404C211yesmaximal, characteristic, complemented
11#21446388C211yesmaximal, characteristic, complemented
12#21446073C211yesmaximal, characteristic, complemented
13#21446008C211yesmaximal, characteristic
14#21445904C211yesmaximal, characteristic
15C4×#103×C61C211yesmaximal, characteristic, complemented
16C2×#14519753C2211yescharacteristic
17C2×C4⋊C4×D61C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27#16839703—21no—
28#16839691—21no—
29#16839669C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839657—21no—
34#16839653—21no—
35#16839646—21no—
36#16839633—21no—
37#16839630—21no—
38#16839627C2211yescharacteristic
39#16839619C2211yescharacteristic
40#16839619C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758198—21no—
43C4×#11758198C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×C4×D61—21no—
60C22×C4×D61C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519753—22no—
64#14519753—22no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C4⋊C4×D61—22no—
75C4⋊C4×D61—41no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198D411yescharacteristic
80C2×#11758198D411yescharacteristic
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83#14519711—22no—
84#14519711—41no—
85#14519710—22no—
86#14519708—22no—
87#14519706—22no—
88#14519701—22no—
89#14519701—41no—
90#14519689—22no—
91#14519689—41no—
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#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×Q8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C4×Q8×C61—22no—
108C4×Q8×C61—41no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C42×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#103×C61—22no—
117#103×C61—22no—
118C4×C8×C61—41no—
119#95×C61—21no—
120C23×D61C2×D411yescharacteristic
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61#4112yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—81no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61#4112yes—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—42no—
162C4⋊C4×C61#4112yes—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—42no—
166C42×C61—42no—
167#11758198—24no—
168#11758198—22no—
169#11758198—22no—
170#11758198—42no—
171#11758198—81no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61#4112yes—
180#11758196—24no—
181#11758196—42no—
182#11758196—81no—
183#11758196—81no—
184#11758196—81no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—81no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C23×C61C22×D411yescharacteristic
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61#13512yes—
202Q8×C61—22no—
203Q8×C61—22no—
204Q8×C61—41no—
205Q8×C61—41no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61#13712yes—
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C2×Dic61—21no—
231C2×Dic61—22no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—84no—
256C8×C61—22no—
257C8×C61—41no—
258#92418—41no—
259#92418—41no—
260#27035—611nomaximal, Sylow
261C22×C61#52511yescharacteristic
262C22×C61#54611yescharacteristic
263C22×C61#54711yescharacteristic
264C22×C61C2×#13511yescharacteristic
265C22×C61C2×#13711yescharacteristic
266C22×C61#57611yescharacteristic
267C22×C61#57811yescharacteristic
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—41no—
275C4×C61—22no—
276C4×C61—42no—
277C4×C61—42no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61#52911yescharacteristic
282C4×C61#53911yescharacteristic
283C4×C61#55311yescharacteristic
284C4×C61#55611yescharacteristic
285Dic61—41no—
286Dic61—41no—
287Dic61—42no—
288Dic61—41no—
289Dic61—81no—
290Dic61—81no—
291Dic61—81no—
292#2495—611no—
293#2360—611no—
294#2074—611no—
295#2068—611no—
296#2036—611no—
297#2035—611no—
298#1949—611no—
299#1942—611no—
300#1923—611no—
301#1920—611no—
302#1918—611no—
303#1863—611no—
304#1852—611no—
305#1823—611no—
306C4×#103D6111yescharacteristic, complemented
307C2×C61#246611yescharacteristic
308C2×C61#310811yescharacteristic
309C2×C61#312411yescharacteristic
310C2×C61#329711yescharacteristic
311C2×C61#331211yescharacteristic
312C2×C61#333611yescharacteristic
313C2×C61#335711yescharacteristic
314D61—42no—
315C2×C22⋊Q8—611no—
316C2×C4×Q8C2×D6111yescharacteristic
317C22×C4⋊C4—611no—
318C2×#107—611no—
319C2×#107—611no—
320C2×#107—611no—
321C2×#106—611no—
322C2×C4⋊C8—611no—
323C2×#103—611no—
324C2×#103—611no—
325C2×#103C2×D6111yescharacteristic
326C2×#103C2×D6111yescharacteristic
327C2×C22⋊C8—611no—
328C2×C22⋊C8—611no—
329C2×C4×C8C2×D6111yescharacteristic
330#400—1221no—
331#397—1221no—
332#391—611no—
333#389—611no—
334#389—611no—
335#389—611no—
336#389—1221no—
337#388—1221no—
338#387—1221no—
339#385—1221no—
340#383—1221no—
341#382—611no—
342#380—611no—
343#380—611no—
344C4×C4⋊C4—611no—
345C4×C4⋊C4C2×D6111yescharacteristic
346C4×C22⋊C4—611no—
347C4×C22⋊C4—1221no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340C2×D6111yescharacteristic
355#340C2×D6111yescharacteristic
356#336—611no—
357#336—611no—
358C61#2703511yescharacteristic, complemented, Sylow
359C22×Q8C22×D6111yescharacteristic
360C23×C4—611no—
361C23×C4—1221no—
362C22×C8—611no—
363C22×C8—611no—
364C22×C8C22×D6111yescharacteristic
365C22×C8C22×D6111yescharacteristic
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C4×Q8—22no—
369C4×Q8—41no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—2441no—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C22⋊C4—611no—
386C2×C22⋊C4—611no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—611no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395#107—1222no—
396#107—1222no—
397#107—2441no—
398#106—1222no—
399C4⋊C8—1222no—
400#103—22no—
401#103—22no—
402#103—1222no—
403#103—2441no—
404C22⋊C8—1222no—
405C22⋊C8—2441no—
406C4×C8—41no—
407#95—21no—
408#95—611no—
409#95—611no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420C24—611no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1175822212yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—4881no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4D4×D6111yescharacteristic
448C22×C4D4×D6111yescharacteristic
449C22×C4C23×D6111yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8#1175822212yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—612no—
469C4⋊C4—1224no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—4881no—
478C4⋊C4—4881no—
479C4⋊C4—4881no—
480C4⋊C4—2442no—
481C4⋊C4#1175822212yes—
482C22⋊C4—1222no—
483C22⋊C4—1224no—
484C22⋊C4—1222no—
485C22⋊C4—4881no—
486C22⋊C4—2442no—
487C42—21no—
488C42—21no—
489C42—42no—
490C42—42no—
491C42—1221no—
492C42—1221no—
493C42—4881no—
494C23—612no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23—1221no—
501C23C2×D4×D6111yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—1222no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—4884no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4#1451980612yes—
550C2×C4#41×D6111yescharacteristic
551C2×C4C2×#1175822211yescharacteristic
552C2×C4C2×#1175822211yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C2×C4#1451982211yescharacteristic
555C2×C4#1451982411yescharacteristic
556C8—22no—
557C8—41no—
558C8—2441no—
559C8—2441no—
560C22—1222no—
561C22—1222no—
562C22—1222no—
563C22—2441no—
564C22—2441no—
565C22—2441no—
566C22—2441no—
567C22#1684027111yescharacteristic
568C22#1684040211yescharacteristic
569C22#1684041711yescharacteristic
570C22#135×D6111yescharacteristic
571C22C2×#1451980611yescharacteristic
572C22#1684056411yescharacteristic
573C22#1684057511yescharacteristic
574C4—22no—
575C4—41no—
576C4—42no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—4881no—
581C4—4881no—
582C4—4881no—
583C4—2441no—
584C4—2441no—
585C4—2441no—
586C4—2442no—
587C4#1684029611yescharacteristic
588C4#1684035411yescharacteristic
589C4#1684046111yescharacteristic
590C4#1684048011yescharacteristic
591C2—2442no—
592C2#2144985511yescharacteristic
593C2#2145589711yescharacteristic
594C2#2145608711yescharacteristic
595C2#2145767011yescharacteristic
596C2#2145786611yescharacteristic
597C2#2145818911yescharacteristic
598C2#2145844711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC4×#103D61
hypocenterC61#27035
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. #21447240C2semidirect
4. #21447212C2semidirect
7. #21446590C2semidirect
8. #21446552C2semidirect
10. #21446404C2semidirect
11. #21446388C2semidirect
12. #21446073C2semidirect
15. C4×#103×C61C2semidirect
306. C4×#103D61semidirect
358. C61#27035semidirect

Maximal subgroups

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

subgroupindex
#214502242
#214488202
#214472402
#214472122
#214470212
#214470132
#214465902
#214465522
#214464252
#214464042
#214463882
#214460732
#214460082
#214459042
C4×#103×C612
#2703561