Finite Group Databasebeta

← #29748195

Subgroups — #29748195

Order 15616.

824 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450169C211yesmaximal, characteristic, complemented
2#21448862C211yesmaximal, characteristic, complemented
3#21446999C211yesmaximal, characteristic, complemented
4#21446965C211yesmaximal, characteristic, complemented
5#21446814C211yesmaximal, characteristic
6#21446774C211yesmaximal, characteristic, complemented
7#21446499C211yesmaximal, characteristic, complemented
8#21446436C211yesmaximal, characteristic, complemented
9#21446424C211yesmaximal, characteristic, complemented
10#21446409C211yesmaximal, characteristic, complemented
11#21446169C211yesmaximal, characteristic, complemented
12#21446138C211yesmaximal, characteristic, complemented
13#21445913C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic, complemented
15C61×#2054C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×#14519748C2211yescharacteristic
18C2×#14519748C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839690—21no—
30#16839677—21no—
31#16839665C2211yescharacteristic
32#16839662—21no—
33#16839662—21no—
34#16839658C2211yescharacteristic
35#16839656—21no—
36#16839650—21no—
37#16839639—21no—
38#16839619C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C22⋊C4×Dic61—21no—
42C4×#11758198C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×D8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×C22⋊D61—21no—
60C22×C22⋊D61—21no—
61C23×Dic61—21no—
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519769—22no—
66#14519769—41no—
67#14519748—22no—
68#14519748—22no—
69#14519748—22no—
70#14519748—41no—
71D4×Dic61—22no—
72D4×Dic61—41no—
73#14519735—22no—
74#14519735—41no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198D411yescharacteristic
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95#14519710—22no—
96#14519698—22no—
97#14519698—22no—
98#14519698—22no—
99#14519698—41no—
100#14519689—41no—
101#14519689—41no—
102C8×Dic61—41no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C2×D8×C61—22no—
112C2×D8×C61—41no—
113C2×D8×C61—41no—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C42×C61C2311yescharacteristic
123C4⋊C8×C61—41no—
124#102×C61—22no—
125#102×C61—41no—
126C24×C61—21no—
127C24×C61—21no—
128C23×D61C2×D411yescharacteristic
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61#4112yes—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—24no—
147C2×C22⋊D61—42no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—81no—
150C2×C22⋊D61—81no—
151C2×C22⋊D61—81no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—81no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168D8×C61—44no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C22⋊C4×C61—42no—
180C22⋊C4×C61—81no—
181C22⋊C4×C61—81no—
182C22⋊C4×C61—81no—
183C42×C61—21no—
184C42×C61—21no—
185C22⋊Dic61—24no—
186C22⋊Dic61—42no—
187C22⋊Dic61—42no—
188C22⋊Dic61—42no—
189C22⋊Dic61—81no—
190#11758198—22no—
191#11758198—41no—
192#11758198—42no—
193#11758198—81no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61#4112yes—
200C4⋊Dic61#4112yes—
201C4⋊Dic61#4112yes—
202#11758196—42no—
203#11758196—42no—
204#11758196—81no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C4×Dic61—81no—
209C4×Dic61—81no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C23×C61—22no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C22×D411yescharacteristic
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—41no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C61H(5,𝔽2)11yescharacteristic
264C22⋊D61—84no—
265C22⋊D61—84no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—84no—
287C2×Dic61#13512yes—
288C4×D61—41no—
289C4×D61—41no—
290C4×D61—41no—
291C4×D61—41no—
292C8×C61—41no—
293C8×C61—41no—
294#92418—41no—
295#92418—41no—
296#26730—611nomaximal, Sylow
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61#52511yescharacteristic
312C22×C61#54611yescharacteristic
313C22×C61#54611yescharacteristic
314C22×C61C2×#13511yescharacteristic
315C22×C61#57511yescharacteristic
316C22×C61#57611yescharacteristic
317C22×C61#57611yescharacteristic
318C2×D61—41no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—41no—
322C2×D61—41no—
323C2×D61—41no—
324C2×D61—41no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—81no—
328C4×C61—81no—
329C4×C61#52911yescharacteristic
330C4×C61#53811yescharacteristic
331C4×C61#55511yescharacteristic
332C4×C61#55911yescharacteristic
333Dic61—22no—
334Dic61—42no—
335Dic61—41no—
336Dic61—42no—
337Dic61—41no—
338Dic61—42no—
339Dic61—81no—
340#2490—611no—
341#2363—611no—
342#2054D6111yescharacteristic, complemented
343#2033—611no—
344#2025—611no—
345#1999—611no—
346#1989—611no—
347#1934—611no—
348#1925—611no—
349#1923—611no—
350#1921—611no—
351#1879—611no—
352#1875—611no—
353#1825—611no—
354C4×#102—611no—
355C2×C61—82no—
356C2×C61—161no—
357C2×C61#245911yescharacteristic
358C2×C61#310211yescharacteristic
359C2×C61#312211yescharacteristic
360C2×C61#329711yescharacteristic
361C2×C61#330511yescharacteristic
362C2×C61#330911yescharacteristic
363C2×C61#332211yescharacteristic
364D61—81no—
365C22×D8C2×D6111yescharacteristic
366C2×#125—611no—
367C2×#121—611no—
368C2×#121—611no—
369C2×C4×D4—611no—
370C2×#117—611no—
371C2×#107—611no—
372C2×C4⋊C8C2×D6111yescharacteristic
373C2×#102—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102C2×D6111yescharacteristic
378C2×#102C2×D6111yescharacteristic
379C2×C22⋊C8—611no—
380C2×C22⋊C8—611no—
381C2×C4×C8—611no—
382#397—1221no—
383#394—1221no—
384#390—611no—
385#390—1221no—
386#390—1221no—
387#390C2×D6111yescharacteristic
388#390C2×D6111yescharacteristic
389#389—611no—
390#388—1221no—
391#388—1221no—
392#386—1221no—
393#380—611no—
394C4×C4⋊C4—611no—
395C4×C4⋊C4—611no—
396C4×C22⋊C4—611no—
397C4×C22⋊C4—1221no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#340—611no—
405#340C2×D6111yescharacteristic
406#336—611no—
407#336—611no—
408C61#2673011yescharacteristic, complemented, Sylow
409C22×D4—1221no—
410C22×D4—1221no—
411C22×D4C22×D6111yescharacteristic
412C22×D4C22×D6111yescharacteristic
413C23×C4—611no—
414C23×C4—1221no—
415C2×D8—22no—
416C2×D8—41no—
417C2×D8—41no—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8C22×D6111yescharacteristic
421C22×C8C22×D6111yescharacteristic
422#125—1222no—
423#125—2441no—
424#121—1222no—
425#121—1222no—
426#121—1222no—
427#121—2441no—
428C4×D4—1222no—
429C4×D4—2441no—
430#117—1222no—
431#117—2441no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—611no—
446C2×C22⋊C4—611no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C42—611no—
455C2×C42—1221no—
456C2×C42—1221no—
457C2×C42—1221no—
458C2×C42C22×D6111yescharacteristic
459#107—1222no—
460C4⋊C8—41no—
461#102—22no—
462#102—41no—
463#102—1222no—
464#102—1222no—
465#102—1222no—
466#102—2441no—
467C22⋊C8—2441no—
468C22⋊C8—2441no—
469C4×C8—2441no—
470#95—611no—
471#95—611no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—21no—
477C24—21no—
478C24—611no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—1224no—
491C2×D4—4881no—
492C2×D4—4881no—
493C2×D4—4881no—
494C2×D4—2442no—
495C2×D4—2442no—
496C2×D4#1175822412yes—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—4881no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4D4×D6111yescharacteristic
516C22×C4D4×D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518D8—44no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—612no—
536C4⋊C4—612no—
537C4⋊C4—612no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—4881no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C22⋊C4—1222no—
551C22⋊C4—1224no—
552C22⋊C4—4881no—
553C22⋊C4—4881no—
554C22⋊C4—2441no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C42—21no—
560C42—21no—
561C42—1221no—
562C42—1221no—
563C42—4881no—
564C42—4881no—
565C42—2442no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—1221no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23—1221no—
586C23—1221no—
587C23C2×D4×D6111yescharacteristic
588D4—22no—
589D4—22no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—4884no—
597D4—4884no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—612no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1222no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1222no—
622C2×C4—4884no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4#41×D6111yescharacteristic
637C2×C4#41×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4C2×#1175822411yescharacteristic
640C2×C4#1451982211yescharacteristic
641C2×C4#1451982211yescharacteristic
642C8—41no—
643C8—41no—
644C8—2441no—
645C8—2441no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—2441no—
661C22—2441no—
662C22—2441no—
663C22—2441no—
664C22—2441no—
665C22—2441no—
666C22—2441no—
667C22#1684027211yescharacteristic
668C22#1684040111yescharacteristic
669C22#1684040911yescharacteristic
670C22#135×D6111yescharacteristic
671C22#1684055811yescharacteristic
672C22#1684056311yescharacteristic
673C22#1684056611yescharacteristic
674C4—41no—
675C4—41no—
676C4—81no—
677C4—81no—
678C4—1222no—
679C4—4881no—
680C4—2442no—
681C4—2441no—
682C4—2442no—
683C4—2441no—
684C4—2442no—
685C4#1684028911yescharacteristic
686C4#1684034711yescharacteristic
687C4#1684047411yescharacteristic
688C4#1684049611yescharacteristic
689C2—82no—
690C2—161no—
691C2—4881no—
692C2#2144975711yescharacteristic
693C2#2145583411yescharacteristic
694C2#2145606011yescharacteristic
695C2#2145767611yescharacteristic
696C2#2145776411yescharacteristic
697C2#2145783211yescharacteristic
698C2#2145801511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2054C2
socleC23×C61C22×D4
prime radical#21446814C2
hypercenter#2054D61
hypocenterC61#26730
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2054

Complements

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

NH ≅ G/Ntype
1. #21450169C2semidirect
2. #21448862C2semidirect
3. #21446999C2semidirect
4. #21446965C2semidirect
6. #21446774C2semidirect
7. #21446499C2semidirect
8. #21446436C2semidirect
9. #21446424C2semidirect
10. #21446409C2semidirect
11. #21446169C2semidirect
12. #21446138C2semidirect
13. #21445913C2semidirect
14. #102×Dic61C2semidirect
15. C61×#2054C2semidirect
342. #2054D61semidirect
408. C61#26730semidirect

Maximal subgroups

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

subgroupindex
#214501692
#214488622
#214469992
#214469652
#214468142
#214467742
#214464992
#214464362
#214464242
#214464092
#214461692
#214461382
#214459132
#102×Dic612
C61×#20542
#2673061