Finite Group Databasebeta

← #29783231

Subgroups — #29783231

Order 15616.

674 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
1#21451787C211yesmaximal, characteristic, complemented
2#21448462C211yesmaximal, characteristic
3#21447323C211yesmaximal, characteristic, complemented
4#21447209C211yesmaximal, characteristic, complemented
5#21447205C211yesmaximal, characteristic
6#21447135C211yesmaximal, characteristic
7#21446674C211yesmaximal, characteristic
8#21446553C211yesmaximal, characteristic, complemented
9#21446518C211yesmaximal, characteristic
10#21446510C211yesmaximal, characteristic, complemented
11#21446461C211yesmaximal, characteristic, complemented
12#21446401C211yesmaximal, characteristic, complemented
13#21446153C211yesmaximal, characteristic
14#21445675C211yesmaximal, characteristic, complemented
15C61×#1862C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519711C2211yescharacteristic
18C2×#14519708C2211yescharacteristic
19C2×#14519706C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27C61⋊#401—21no—
28#16839704—21no—
29#16839701—21no—
30#16839700—21no—
31#16839668C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839665C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839655—21no—
41#16839655—21no—
42#16839625—21no—
43#16839622—21no—
44C4×C4⋊Dic61C2211yescharacteristic
45#16839608C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C2×#11758200—22no—
61C2×#11758200—22no—
62C2×#11758200—41no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61C2311yescharacteristic
78C2×C4×Dic61C2311yescharacteristic
79#14519711—41no—
80#14519708—22no—
81#14519706—22no—
82#14519706—22no—
83#14519702—22no—
84#14519702—41no—
85#14519700—22no—
86#14519698—41no—
87#14519687—41no—
88C8×Dic61—41no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
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#14519684D411yescharacteristic
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4×D4×C61—22no—
109C4×D4×C61—22no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#107×C61—22no—
117#102×C61—22no—
118#102×C61—22no—
119#95×C61—21no—
120C24×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61#4112yes—
127C22×C4×C61—21no—
128C22×C4×C61—24no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—81no—
132C22×C4×C61—81no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
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—81no—
162C4⋊C4×C61#4112yes—
163C22⋊C4×C61—24no—
164C42×C61—21no—
165C42×C61—21no—
166C22⋊Dic61—81no—
167C22⋊Dic61—81no—
168C22⋊Dic61—81no—
169C22⋊Dic61—81no—
170#11758200—44no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—81no—
178C4⋊Dic61—81no—
179C4⋊Dic61—81no—
180#11758196—81no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—81no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418#4112yes—
192C23×C61—22no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61C22×D411yescharacteristic
200D4×C61—22no—
201D4×C61—22no—
202D4×C61—41no—
203D4×C61—41no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
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—84no—
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—41no—
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—
250Dic122—41no—
251Dic122—41no—
252Dic122—41no—
253Dic122—41no—
254C8×C61—22no—
255C8×C61—41no—
256#92418—22no—
257#92418—41no—
258#29441—611nomaximal, Sylow
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61#52511yescharacteristic
267C22×C61#53211yescharacteristic
268C22×C61#53711yescharacteristic
269C22×C61#57611yescharacteristic
270C22×C61#57711yescharacteristic
271C22×C61#57711yescharacteristic
272C22×C61#57811yescharacteristic
273C4×C61—41no—
274C4×C61—42no—
275C4×C61—41no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61#55111yescharacteristic
279C4×C61#55211yescharacteristic
280C4×C61#55311yescharacteristic
281C4×C61#55611yescharacteristic
282Dic61—41no—
283Dic61—41no—
284Dic61—41no—
285Dic61—41no—
286Dic61—81no—
287Dic61—81no—
288Dic61—81no—
289Dic61—81no—
290#2645—611no—
291#2319—611no—
292#2088—611no—
293#2068—611no—
294#2067—611no—
295#2055—611no—
296#1968—611no—
297#1942—611no—
298#1936—611no—
299#1935—611no—
300#1928—611no—
301#1920—611no—
302#1877—611no—
303#1862D6111yescharacteristic
304#1768—611no—
305C2×C61—82no—
306C2×C61#260111yescharacteristic
307C2×C61#311811yescharacteristic
308C2×C61#312011yescharacteristic
309C2×C61#317411yescharacteristic
310C2×C61#317511yescharacteristic
311C2×C61#325711yescharacteristic
312C2×C61#325811yescharacteristic
313C22×SD16—611no—
314C2×C4×D4C2×D6111yescharacteristic
315C2×#107—611no—
316C2×#107C2×D6111yescharacteristic
317C2×#106—611no—
318C2×C4⋊C8—611no—
319C2×C4⋊C8—611no—
320C2×#103—611no—
321C2×#103—611no—
322C2×#103—611no—
323C2×#102—611no—
324C2×#102C2×D6111yescharacteristic
325C2×#102C2×D6111yescharacteristic
326C2×#97—611no—
327C2×C4×C8—611no—
328#401—1221no—
329#400—1221no—
330#400—1221no—
331#400—1221no—
332#391—611no—
333#391—611no—
334#391—611no—
335#390—611no—
336#390—611no—
337#389—611no—
338#389—611no—
339#389—611no—
340#389—611no—
341#389C2×D6111yescharacteristic
342#388—1221no—
343#388—1221no—
344#381—1221no—
345#380—1221no—
346C4×C4⋊C4—611no—
347#376—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340C2×D6111yescharacteristic
354#336—611no—
355#336C2×D6111yescharacteristic
356C61#2944111yescharacteristic, complemented, Sylow
357C22×Q8—611no—
358C22×D4C22×D6111yescharacteristic
359C23×C4—21no—
360C2×SD16—1222no—
361C2×SD16—1222no—
362C2×SD16—2441no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8C22×D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367C4×D4—22no—
368C4×D4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—1221no—
388C2×C42C22×D6111yescharacteristic
389#107—22no—
390#107—2441no—
391#106—1222no—
392C4⋊C8—1222no—
393C4⋊C8—1222no—
394#103—1222no—
395#103—1222no—
396#103—2441no—
397#102—22no—
398#102—22no—
399#102—2441no—
400#97—2441no—
401C4×C8—2441no—
402#95—21no—
403#95—611no—
404#95—611no—
405#95—611no—
406#95—611no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418C24—21no—
419C2×Q8—1221no—
420C2×Q8—1221no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4#1175822412yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4D4×D6111yescharacteristic
446C22×C4D4×D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448SD16—2444no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—612no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8#1175822212yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—81no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—4881no—
474C4⋊C4—4881no—
475C4⋊C4—4881no—
476C4⋊C4—4881no—
477C4⋊C4#1175822412yes—
478C22⋊C4—24no—
479C22⋊C4—4881no—
480C22⋊C4—4881no—
481C22⋊C4—4881no—
482C22⋊C4—4881no—
483C42—21no—
484C42—21no—
485C42—1221no—
486C42—1221no—
487C42—1221no—
488C42—1221no—
489C42—4881no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D6111yescharacteristic
498Q8—2441no—
499Q8—2441no—
500Q8—2441no—
501Q8—2441no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
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—84no—
522C2×C4—1221no—
523C2×C4—1221no—
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—1221no—
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#41×D6111yescharacteristic
547C2×C4C2×#1175822411yescharacteristic
548C2×C4C2×#1175822411yescharacteristic
549C2×C4C2×#1175822211yescharacteristic
550C2×C4#1451982311yescharacteristic
551C2×C4#1451982511yescharacteristic
552C8—22no—
553C8—41no—
554C8—1222no—
555C8—2441no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22#1684026811yescharacteristic
564C22#1684030911yescharacteristic
565C22#1684034211yescharacteristic
566C22#1684056311yescharacteristic
567C22#1684057111yescharacteristic
568C22#1684057211yescharacteristic
569C22#1684057611yescharacteristic
570C4—41no—
571C4—42no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—4881no—
576C4—4881no—
577C4—4881no—
578C4—4881no—
579C4—2441no—
580C4—2441no—
581C4—2441no—
582C4—2441no—
583C4#1684043511yescharacteristic
584C4#1684044711yescharacteristic
585C4#1684045811yescharacteristic
586C4#1684048311yescharacteristic
587C2—82no—
588C2#2145142011yescharacteristic
589C2#2145599611yescharacteristic
590C2#2145603411yescharacteristic
591C2#2145644511yescharacteristic
592C2#2145645411yescharacteristic
593C2#2145723711yescharacteristic
594C2#2145724911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1862C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1862D61
hypocenterC61#29441
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
1. #21451787C2semidirect
3. #21447323C2semidirect
4. #21447209C2semidirect
8. #21446553C2semidirect
10. #21446510C2semidirect
11. #21446461C2semidirect
12. #21446401C2semidirect
14. #21445675C2semidirect
356. C61#29441semidirect

Maximal subgroups

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

subgroupindex
#214517872
#214484622
#214473232
#214472092
#214472052
#214471352
#214466742
#214465532
#214465182
#214465102
#214464612
#214464012
#214461532
#214456752
C61×#18622
#2944161