Finite Group Databasebeta

← #29752629

Subgroups — #29752629

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449685C211yesmaximal, characteristic
2#21448914C211yesmaximal, characteristic, complemented
3#21447205C211yesmaximal, characteristic
4#21447160C211yesmaximal, characteristic, complemented
5#21447159C211yesmaximal, characteristic, complemented
6#21446963C211yesmaximal, characteristic
7#21446773C211yesmaximal, characteristic, complemented
8#21446559C211yesmaximal, characteristic, complemented
9#21446449C211yesmaximal, characteristic
10#21446442C211yesmaximal, characteristic
11#21446399C211yesmaximal, characteristic, complemented
12#21446391C211yesmaximal, characteristic, complemented
13#21446381C211yesmaximal, characteristic, complemented
14#21445644C211yesmaximal, characteristic
15C61×#1864C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519688C2211yescharacteristic
25#16839698—21no—
26#16839685—21no—
27#16839665C2211yescharacteristic
28#16839665C2211yescharacteristic
29#16839661C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839657—21no—
33#16839647—21no—
34#16839644—21no—
35#16839635—21no—
36#16839622—21no—
37C4⋊C4×Dic61—21no—
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4×#11758196C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61#14519768—22no—
62#14519768—22no—
63#14519768—22no—
64#14519768—41no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196D411yescharacteristic
77C2×#11758196D411yescharacteristic
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519711—22no—
88#14519710—41no—
89#14519708—22no—
90#14519706—41no—
91#14519698—22no—
92#14519698Q814yes—
93#14519688—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684D411yescharacteristic
101#14519684D411yescharacteristic
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C4×D4×C61—22no—
107C4×D4×C61—41no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C42×C61C2311yescharacteristic
114#106×C61—22no—
115#102×C61—22no—
116#102×C61—41no—
117#95×C61—21no—
118C24×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61C2×Q812yes—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61—81no—
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
141C22×Dic61C2×D411yescharacteristic
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61#4112yes—
154C4⋊C4×C61#4112yes—
155C22⋊C4×C61—42no—
156C42×C61—21no—
157C42×C61—21no—
158C22⋊Dic61—42no—
159C22⋊Dic61—81no—
160C22⋊Dic61—81no—
161C22⋊Dic61—81no—
162C4⋊Dic61—24no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—42no—
170C4⋊Dic61—81no—
171C4⋊Dic61C2×Q812yes—
172C4⋊Dic61#4112yes—
173#11758196—24no—
174#11758196—41no—
175#11758196—41no—
176#11758196—42no—
177#11758196—81no—
178#11758196—81no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—42no—
184C4×Dic61—81no—
185C4×Dic61—81no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418C2×Q812yes—
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—22no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
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×C61C22×Q811yescharacteristic
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—22no—
229C2×Dic61—21no—
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—
250C8×C61—41no—
251C8×C61—41no—
252#92418—22no—
253#92418—41no—
254#27027—611nomaximal, Sylow
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61#52511yescharacteristic
263C22×C61#54611yescharacteristic
264C22×C61D4×Q811yescharacteristic
265C22×C61#57511yescharacteristic
266C22×C61#57511yescharacteristic
267C22×C61#57611yescharacteristic
268C22×C61#57711yescharacteristic
269C4×C61—42no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—42no—
273C4×C61—81no—
274C4×C61#53311yescharacteristic
275C4×C61#54211yescharacteristic
276C4×C61#55411yescharacteristic
277C4×C61#55611yescharacteristic
278Dic61—41no—
279Dic61—41no—
280Dic61—42no—
281Dic61—42no—
282Dic61—41no—
283Dic61—41no—
284Dic61—81no—
285Dic61—81no—
286#2454—611no—
287#2370—611no—
288#2067—611no—
289#2059—611no—
290#2059—611no—
291#2025—611no—
292#1989—611no—
293#1944—611no—
294#1926—611no—
295#1925—611no—
296#1919—611no—
297#1918—611no—
298#1917—611no—
299#1864D6111yescharacteristic
300#1762—611no—
301C2×C61—82no—
302C2×C61#246511yescharacteristic
303C2×C61#311111yescharacteristic
304C2×C61#311911yescharacteristic
305C2×C61#330911yescharacteristic
306C2×C61#332311yescharacteristic
307C2×C61#338811yescharacteristic
308C2×C61#338911yescharacteristic
309C2×#125—611no—
310C2×#125—611no—
311C2×C4×D4C2×D6111yescharacteristic
312C2×#107—611no—
313C2×#107—611no—
314C2×#106—611no—
315C2×#106C2×D6111yescharacteristic
316C2×C4⋊C8—611no—
317C2×#102—611no—
318C2×#102—611no—
319C2×#102C2×D6111yescharacteristic
320C2×#102C2×D6111yescharacteristic
321C2×#97—611no—
322#399—1221no—
323#396—1221no—
324#390—611no—
325#390—611no—
326#389—611no—
327#389—611no—
328#389—611no—
329#389—1221no—
330#389C2×D6111yescharacteristic
331#387—1221no—
332#387—1221no—
333#385—1221no—
334#380—1221no—
335C4×C4⋊C4—611no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4—611no—
338C4×C4⋊C4—611no—
339C4×C4⋊C4—1221no—
340#340—611no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340C2×D6111yescharacteristic
348#336—611no—
349#336—611no—
350#336—611no—
351#336C2×D6111yescharacteristic
352C61#2702711yescharacteristic, complemented, Sylow
353C22×D4C22×D6111yescharacteristic
354C23×C4—21no—
355C22×C8—611no—
356C22×C8—611no—
357C22×C8C22×D6111yescharacteristic
358C22×C8C22×D6111yescharacteristic
359#125—1222no—
360#125—1222no—
361#125—1222no—
362#125—2441no—
363C4×D4—22no—
364C4×D4—41no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
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⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—1222no—
394#107—2441no—
395#106—22no—
396#106—1222no—
397C4⋊C8—2441no—
398#102—22no—
399#102—41no—
400#102—614no—
401#102—1222no—
402#97—2441no—
403#95—21no—
404#95—611no—
405#95—611no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4Q8×D6112yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—81no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4D4×D6111yescharacteristic
434C22×C4D4×D6111yescharacteristic
435C22×C4C23×D6111yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—612no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—612no—
454C4⋊C4—612no—
455C4⋊C4—1224no—
456C4⋊C4—1224no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—4881no—
464C4⋊C4—4881no—
465C4⋊C4—4881no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2442no—
469C4⋊C4—2442no—
470C4⋊C4#1175822212yes—
471C4⋊C4#1175822412yes—
472C22⋊C4—42no—
473C22⋊C4—4881no—
474C22⋊C4—4881no—
475C22⋊C4—4881no—
476C22⋊C4—2442no—
477C42—21no—
478C42—21no—
479C42—1221no—
480C42—1221no—
481C42—1221no—
482C42—1221no—
483C42—4881no—
484C42—4881no—
485C42—2442no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C2×D4×D6111yescharacteristic
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1222no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1222no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4C2×Q8×D6111yescharacteristic
539C2×C4#41×D6111yescharacteristic
540C2×C4C2×#1175822411yescharacteristic
541C2×C4C2×#1175822211yescharacteristic
542C2×C4#1451982211yescharacteristic
543C2×C4#1451982411yescharacteristic
544C8—41no—
545C8—41no—
546C8—1222no—
547C8—2441no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22#1684027111yescharacteristic
556C22#1684039911yescharacteristic
557C22#1684042511yescharacteristic
558C22#1684055811yescharacteristic
559C22#1684055811yescharacteristic
560C22#1684056411yescharacteristic
561C22#1684056811yescharacteristic
562C4—42no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—81no—
567C4—4881no—
568C4—4881no—
569C4—2441no—
570C4—2442no—
571C4—2441no—
572C4—2441no—
573C4—2442no—
574C4—2441no—
575C4#1684031411yescharacteristic
576C4#1684038011yescharacteristic
577C4#1684046711yescharacteristic
578C4#1684048211yescharacteristic
579C2—82no—
580C2#2144984411yescharacteristic
581C2#2145592411yescharacteristic
582C2#2145602411yescharacteristic
583C2#2145781811yescharacteristic
584C2#2145803911yescharacteristic
585C2#2145875711yescharacteristic
586C2#2145877611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1864C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1864D61
hypocenterC61#27027
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #21448914C2semidirect
4. #21447160C2semidirect
5. #21447159C2semidirect
7. #21446773C2semidirect
8. #21446559C2semidirect
11. #21446399C2semidirect
12. #21446391C2semidirect
13. #21446381C2semidirect
352. C61#27027semidirect

Maximal subgroups

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

subgroupindex
#214496852
#214489142
#214472052
#214471602
#214471592
#214469632
#214467732
#214465592
#214464492
#214464422
#214463992
#214463912
#214463812
#214456442
C61×#18642
#2702761