Finite Group Databasebeta

← #29729503

Subgroups — #29729503

Order 15616.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840128C211yesmaximal, characteristic
2#21450818C211yesmaximal, characteristic
3#21448919C211yesmaximal, characteristic, complemented
4#21447202C211yesmaximal, characteristic, complemented
5#21447188C211yesmaximal, characteristic, complemented
6#21447057C211yesmaximal, characteristic, complemented
7#21447057C211yesmaximal, characteristic, complemented
8#21446748C211yesmaximal, characteristic, complemented
9#21446748C211yesmaximal, characteristic, complemented
10#21446745C211yesmaximal, characteristic
11#21446540C211yesmaximal, characteristic, complemented
12#21446445C211yesmaximal, characteristic, complemented
13#21446438C211yesmaximal, characteristic, complemented
14#21445649C211yesmaximal, characteristic, complemented
15C2×C61×#479C211yesmaximal, characteristic, complemented
16C2×#14519774C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×#14519766C2211yescharacteristic
19#16840128—24no—
20C2×#14519706C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33#16839679—21no—
34#16839679—21no—
35#16839664—21no—
36#16839664—21no—
37#16839663C2211yescharacteristic
38#16839663C2211yescharacteristic
39#16839661—21no—
40#16839661C2211yescharacteristic
41#16839661C2211yescharacteristic
42#16839648—21no—
43#16839648—21no—
44C4⋊C4×Dic61—21no—
45C4⋊C4×Dic61C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48C4×C4⋊Dic61C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#479—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C22×D244—21no—
62C22×D244—21no—
63C22×D244C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519774—22no—
68#14519774—22no—
69#14519766—22no—
70#14519766—22no—
71#14519766—41no—
72#14519766—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519706—41no—
93#14519706—41no—
94#14519701—22no—
95#14519701—22no—
96#14519699—22no—
97#14519699—22no—
98#14519699—41no—
99#14519699—41no—
100#14519697—22no—
101#14519697—22no—
102#14519697—22no—
103#14519697—22no—
104#14519688—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115#125×C61—22no—
116#125×C61D414yes—
117C4×Q8×C61—22no—
118C4×Q8×C61—22no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#107×C61—22no—
128#106×C61—22no—
129C4⋊C8×C61—22no—
130#103×C61—22no—
131#103×C61—22no—
132C23×D61—21no—
133C23×D61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61#4112yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—42no—
166C2×D244—42no—
167C2×D244—42no—
168C2×D244—42no—
169C2×D244—81no—
170C2×D244—81no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61C2×D412yes—
185C4⋊C4×C61C2×D412yes—
186C42×C61—42no—
187C42×C61C2×D412yes—
188C42×C61#4112yes—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192#11758198—42no—
193#11758198—81no—
194#11758198—81no—
195#11758198—81no—
196#11758198—81no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61—42no—
203C4⋊Dic61—81no—
204C4⋊Dic61#4112yes—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—42no—
210C4×Dic61—81no—
211C4×Dic61—81no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C23×C61C22×D411yescharacteristic
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233Q8×C61—22no—
234Q8×C61—22no—
235Q8×C61—41no—
236Q8×C61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61#12112yes—
254C2×C4×C61#12112yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279D244—41no—
280D244—41no—
281D244—41no—
282D244—41no—
283D244—41no—
284D244—41no—
285D244—41no—
286D244—41no—
287D244—84no—
288D244—84no—
289C8×C61—41no—
290C8×C61—41no—
291#92418—41no—
292#92418—41no—
293#25291—611nomaximal, Sylow
294C22×C61C2×#12111yescharacteristic
295C22×C61#54611yescharacteristic
296C22×C61#54811yescharacteristic
297C22×C61#57511yescharacteristic
298C22×C61#57511yescharacteristic
299C22×C61#57611yescharacteristic
300C22×C61#57711yescharacteristic
301C2×D61—81no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61—81no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—81no—
321C4×C61#52911yescharacteristic
322C4×C61#54211yescharacteristic
323C4×C61#54811yescharacteristic
324C4×C61#55011yescharacteristic
325Dic61—42no—
326Dic61—41no—
327Dic61—41no—
328Dic61—41no—
329Dic61—41no—
330Dic61—81no—
331C2×#490—611no—
332C2×#479D6111yescharacteristic, complemented
333#2548—611no—
334#2370—611no—
335#2067—611no—
336#2065—611no—
337#2042—611no—
338#2042—611no—
339#1983—611no—
340#1983—611no—
341#1982—611no—
342#1940—611no—
343#1926—611no—
344#1925—611no—
345#1763—611no—
346C2×C61#242611yescharacteristic
347C2×C61#304011yescharacteristic
348C2×C61#304511yescharacteristic
349C2×C61#330911yescharacteristic
350C2×C61#331711yescharacteristic
351C2×C61#337011yescharacteristic
352C2×C61#337111yescharacteristic
353D61—161no—
354D61—161no—
355C2×#127—611no—
356C2×#125C2×D6111yescharacteristic
357C2×#124—611no—
358C2×#124—611no—
359C2×C4×Q8C2×D6111yescharacteristic
360#490—1224no—
361#479—24no—
362C2×#107C2×D6111yescharacteristic
363C2×#106C2×D6111yescharacteristic
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C4⋊C8C2×D6111yescharacteristic
367C2×#103—611no—
368C2×#103—611no—
369C2×#103C2×D6111yescharacteristic
370C2×#103C2×D6111yescharacteristic
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#97—611no—
380#394—1221no—
381#394—1221no—
382#390—611no—
383#390—611no—
384#390—1221no—
385#390—1221no—
386#389—611no—
387#389—611no—
388#389—1221no—
389#387—1221no—
390#387—1221no—
391C4×C4⋊C4—611no—
392C4×C4⋊C4—611no—
393C4×C4⋊C4—611no—
394C4×C4⋊C4—611no—
395C4×C4⋊C4—1221no—
396#340—611no—
397#340—611no—
398#336—611no—
399#336—611no—
400C61#2529111yescharacteristic, complemented, Sylow
401C22×Q8C22×D6111yescharacteristic
402C22×D4—611no—
403C22×D4—611no—
404C22×D4—1221no—
405C22×D4—1221no—
406C22×C8—611no—
407C22×C8—611no—
408C22×C8C22×D6111yescharacteristic
409C22×C8C22×D6111yescharacteristic
410#127—1222no—
411#127—1222no—
412#125—22no—
413#125C22⋊D6114yes—
414#124—1222no—
415#124—1222no—
416#124—2441no—
417#124—2441no—
418C4×Q8—22no—
419C4×Q8—22no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C42—21no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42C22×D6111yescharacteristic
447#107—22no—
448#106—22no—
449C4⋊C8—22no—
450C4⋊C8—2441no—
451C4⋊C8—2441no—
452#103—22no—
453#103—22no—
454#103—1222no—
455#103—1222no—
456#102—1222no—
457#102—1222no—
458#102—1222no—
459#102—1222no—
460#102—1222no—
461#102—1222no—
462#102—2441no—
463#102—2441no—
464#97—1222no—
465#95—611no—
466#95—611no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—1221no—
473C24—1221no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8#1175822612yes—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—4881no—
493C2×D4—4881no—
494C2×D4—2442no—
495C2×D4—2442no—
496C2×D4—2442no—
497C2×D4—2442no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—612no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—4881no—
535C4⋊C4—2441no—
536C4⋊C4—2441no—
537C4⋊C4—2442no—
538C4⋊C4C2×C22⋊D6112yes—
539C4⋊C4C2×C22⋊D6112yes—
540C22⋊C4—4881no—
541C22⋊C4—4881no—
542C22⋊C4—4881no—
543C22⋊C4—4881no—
544C22⋊C4—2442no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4—2442no—
548C42—42no—
549C42—1221no—
550C42—1221no—
551C42—1221no—
552C42—1221no—
553C42—4881no—
554C42—4881no—
555C42—2442no—
556C42#1175822412yes—
557C42C2×C22⋊D6112yes—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23—2441no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23C2×D4×D6111yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577D4—4884no—
578D4—4884no—
579D4—2441no—
580D4—2441no—
581D4—2441no—
582D4—2441no—
583D4—2441no—
584D4—2441no—
585D4—2441no—
586D4—2441no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4#1451974912yes—
622C2×C4#1451974912yes—
623C2×C4C22×C22⋊D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822411yescharacteristic
626C2×C4C2×#1175822611yescharacteristic
627C2×C4#1451982311yescharacteristic
628C2×C4#1451982511yescharacteristic
629C8—41no—
630C8—41no—
631C8—2441no—
632C8—2441no—
633C22—4881no—
634C22—4881no—
635C22—4881no—
636C22—4881no—
637C22—4881no—
638C22—4881no—
639C22—4881no—
640C22—4881no—
641C22—4881no—
642C22—4881no—
643C22—4881no—
644C22—4881no—
645C22—4881no—
646C22—4881no—
647C22C2×#1451974911yescharacteristic
648C22#1684040311yescharacteristic
649C22#1684042211yescharacteristic
650C22#1684056111yescharacteristic
651C22#1684056111yescharacteristic
652C22#1684056411yescharacteristic
653C22#1684056911yescharacteristic
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—81no—
660C4—4881no—
661C4—2441no—
662C4—2441no—
663C4—2442no—
664C4—2441no—
665C4—2441no—
666C4#1684029811yescharacteristic
667C4#1684038511yescharacteristic
668C4#1684042311yescharacteristic
669C4#1684043111yescharacteristic
670C2—9761no—
671C2—9761no—
672C2#2144938711yescharacteristic
673C2#2145531511yescharacteristic
674C2#2145535611yescharacteristic
675C2#2145782211yescharacteristic
676C2#2145794211yescharacteristic
677C2#2145857511yescharacteristic
678C2#2145859011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#479C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenterC2×#479D61
hypocenterC61#25291
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
3. #21448919C2semidirect
4. #21447202C2semidirect
5. #21447188C2semidirect
6. #21447057C2semidirect
7. #21447057C2semidirect
8. #21446748C2semidirect
9. #21446748C2semidirect
11. #21446540C2semidirect
12. #21446445C2semidirect
13. #21446438C2semidirect
14. #21445649C2semidirect
15. C2×C61×#479C2semidirect
332. C2×#479D61semidirect
400. C61#25291semidirect

Maximal subgroups

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

subgroupindex
C2×#168401282
#214508182
#214489192
#214472022
#214471882
#214470572
#214467482
#214467452
#214465402
#214464452
#214464382
#214456492
C2×C61×#4792
#2529161