Finite Group Databasebeta

← #29738178

Subgroups — #29738178

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450840C211yesmaximal, characteristic, complemented
2#21447207C211yesmaximal, characteristic
3#21447193C211yesmaximal, characteristic
4#21446965C211yesmaximal, characteristic
5#21446955C211yesmaximal, characteristic
6#21446776C211yesmaximal, characteristic, complemented
7#21446771C211yesmaximal, characteristic, complemented
8#21446585C211yesmaximal, characteristic, complemented
9#21446538C211yesmaximal, characteristic, complemented
10#21446411C211yesmaximal, characteristic, complemented
11#21446404C211yesmaximal, characteristic, complemented
12#21445949C211yesmaximal, characteristic, complemented
13#21445829C211yesmaximal, characteristic
14C4×#14519698C211yesmaximal, characteristic
15C61×#2357C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×C4×#92418C2211yescharacteristic
28#16839695—21no—
29#16839682—21no—
30#16839666—21no—
31#16839660C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839659—21no—
34#16839628C2211yescharacteristic
35C4×C4⋊Dic61C2211yescharacteristic
36#16839524C2211yescharacteristic
37#16839524C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C2×C4×D4×C61C2211yescharacteristic
49C61×#390C2211yescharacteristic
50C61×#390C2211yescharacteristic
51C61×#388—21no—
52C61×#388—21no—
53C61×#382C2211yescharacteristic
54C61×#381—21no—
55C4×C4⋊C4×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61—21no—
59C22×Dic122—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519777—22no—
65#14519777—41no—
66#14519769—22no—
67#14519769—41no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C4×Dic61—21no—
78#14519710—22no—
79#14519708—22no—
80C4⋊#92418—22no—
81C4⋊#92418—41no—
82C4⋊#92418—41no—
83#14519698—22no—
84#14519698—22no—
85#14519698—41no—
86#14519698—41no—
87C4×#92418—22no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92#14519684—21no—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95C4×D4×C61—22no—
96C4×D4×C61Q814yescomplemented
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112#95×C61—21no—
113#95×C61—21no—
114#95×C61—21no—
115#95×C61—21no—
116#95×C61D411yescharacteristic
117#95×C61D411yescharacteristic
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—24no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—81no—
130C22×C4×C61—81no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C2×Dic122—42no—
143C2×Dic122—42no—
144C2×Dic122—81no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—41no—
152C4⋊C4×C61C2×Q812yes—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—42no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—81no—
157C22⋊C4×C61—81no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—81no—
163C42×C61—81no—
164C42×C61C2×Q812yes—
165C42×C61#4112yes—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—42no—
173C4⋊Dic61#4112yes—
174C4⋊Dic61#4112yes—
175#11758196—42no—
176#11758196—42no—
177#11758196—42no—
178#11758196—81no—
179C4×Dic61—81no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
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—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—84no—
229C2×C4×C61C22⋊Q812yes—
230C2×C4×C61C22⋊Q812yes—
231C2×C4×C61#13512yes—
232C2×C4×C61C22×Q811yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C2×C4×C612−1+411yescharacteristic
238C2×Dic61—22no—
239C2×Dic61—22no—
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—84no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#26028—611nomaximal, Sylow
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61C2×C22⋊Q811yescharacteristic
264C22×C61#54611yescharacteristic
265C22×C61#54811yescharacteristic
266C22×C61C2×#13511yescharacteristic
267C22×C61#57511yescharacteristic
268C22×C61#57611yescharacteristic
269C22×C61#57711yescharacteristic
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61#53311yescharacteristic
280C4×C61#54411yescharacteristic
281C4×C61#55511yescharacteristic
282C4×C61#55611yescharacteristic
283Dic61—42no—
284Dic61—42no—
285Dic61—81no—
286Dic61—81no—
287#2550—611no—
288#2357D6111yescharacteristic
289#2067—611no—
290#2065—611no—
291#2025—611no—
292#2023—611no—
293#1989—611no—
294#1988—611no—
295#1949—611no—
296#1939—611no—
297#1921—611no—
298#1920—611no—
299#1835—611no—
300#1804—611no—
301C4×#102—611no—
302C2×C61—82no—
303C2×C61#243911yescharacteristic
304C2×C61#307611yescharacteristic
305C2×C61#308111yescharacteristic
306C2×C61#329711yescharacteristic
307C2×C61#332611yescharacteristic
308C2×C61#336511yescharacteristic
309C2×C61#337011yescharacteristic
310C2×C4⋊Q8—611no—
311C2×#125—611no—
312C2×C4×D4C2×D6111yescharacteristic
313C2×#107—611no—
314C2×#106—611no—
315C2×C4⋊C8—611no—
316C2×C4⋊C8—611no—
317C2×C4⋊C8—611no—
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—611no—
321C2×#102—611no—
322C2×C4×C8—611no—
323#398—1221no—
324#395—1221no—
325#391—1221no—
326#390C2×D6111yescharacteristic
327#390C2×D6111yescharacteristic
328#389—611no—
329#389—611no—
330#389—1221no—
331#388—21no—
332#388—21no—
333#382—611no—
334#382C2×D6111yescharacteristic
335#381—21no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4C2×D6111yescharacteristic
338C4×C4⋊C4C2×D6111yescharacteristic
339C4×C4⋊C4C2×D6111yescharacteristic
340C4×C22⋊C4—21no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#336—611no—
352#336—611no—
353C61#2602811yescharacteristic, complemented, Sylow
354C22×Q8—1221no—
355C22×D4C22×D6111yescharacteristic
356C23×C4—21no—
357C22×C8—611no—
358C22×C8—611no—
359C22×C8—611no—
360C22×C8—611no—
361C4⋊Q8—1222no—
362C4⋊Q8—2441no—
363#125—1222no—
364#125—2441no—
365C4×D4—22no—
366C4×D4Dic12214yescomplemented
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4C22⋊D6111yescharacteristic
377C2×C4⋊C4C22⋊D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—1221no—
389C2×C42C22×D6111yescharacteristic
390C2×C42C22×D6111yescharacteristic
391C2×C42C22×D6111yescharacteristic
392#107—1222no—
393#106—1222no—
394C4⋊C8—1222no—
395C4⋊C8—2441no—
396C4⋊C8—2441no—
397#102—1222no—
398#102—1222no—
399#102—2441no—
400#102—2441no—
401C4×C8—1222no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95C22⋊D6111yescharacteristic
412#95C22⋊D6111yescharacteristic
413C24—21no—
414C2×Q8—4881no—
415C2×Q8—2442no—
416C2×Q8—2442no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Dic12212yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—612no—
460C4⋊C4—612no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—4881no—
468C4⋊C4—2442no—
469C4⋊C4—2442no—
470C4⋊C4—2442no—
471C4⋊C4—2442no—
472C4⋊C4C2×Dic12212yes—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—81no—
483C42—81no—
484C42—4881no—
485C42C2×Dic12212yes—
486C42#1175822212yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C22×C22⋊D6111yescharacteristic
495Q8—4884no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—84no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—2441no—
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#1451975612yes—
538C2×C4#1451975612yes—
539C2×C4#1451979712yes—
540C2×C4C22×Dic12211yescharacteristic
541C2×C4#41×D6111yescharacteristic
542C2×C4#41×D6111yescharacteristic
543C2×C4C2×#1175822211yescharacteristic
544C2×C4#1451982311yescharacteristic
545C2×C4#1451982411yescharacteristic
546C8—2441no—
547C8—2441no—
548C8—2441no—
549C8—2441no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1451975611yescharacteristic
558C22#1684040011yescharacteristic
559C22#1684041911yescharacteristic
560C22C2×#1451979711yescharacteristic
561C22#1684056011yescharacteristic
562C22#1684056511yescharacteristic
563C22#1684057311yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—4881no—
574C4—4881no—
575C4—2442no—
576C4—2442no—
577C4#1684031511yescharacteristic
578C4#1684039011yescharacteristic
579C4#1684047511yescharacteristic
580C4#1684048011yescharacteristic
581C2—82no—
582C2#2144950411yescharacteristic
583C2#2145560711yescharacteristic
584C2#2145565011yescharacteristic
585C2#2145766311yescharacteristic
586C2#2145806611yescharacteristic
587C2#2145851111yescharacteristic
588C2#2145856611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2357C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2357D61
hypocenterC61#26028
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #21450840C2semidirect
6. #21446776C2semidirect
7. #21446771C2semidirect
8. #21446585C2semidirect
9. #21446538C2semidirect
10. #21446411C2semidirect
11. #21446404C2semidirect
12. #21445949C2semidirect
96. C4×D4×C61Q8semidirect
353. C61#26028semidirect
366. C4×D4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214508402
#214472072
#214471932
#214469652
#214469552
#214467762
#214467712
#214465852
#214465382
#214464112
#214464042
#214459492
#214458292
C4×#145196982
C61×#23572
#2602861