Finite Group Databasebeta

← #29729743

Subgroups — #29729743

Order 15616.

856 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839974C211yesmaximal, characteristic
2C2×#16839893C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic, complemented
4#21447200C211yesmaximal, characteristic, complemented
5#21447193C211yesmaximal, characteristic, complemented
6#21447134C211yesmaximal, characteristic, complemented
7#21447129C211yesmaximal, characteristic
8#21446694C211yesmaximal, characteristic
9#21446676C211yesmaximal, characteristic, complemented
10#21446636C211yesmaximal, characteristic, complemented
11#21446582C211yesmaximal, characteristic, complemented
12#21446512C211yesmaximal, characteristic, complemented
13#21446505C211yesmaximal, characteristic, complemented
14#21445647C211yesmaximal, characteristic, complemented
15C61×#2592C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C2×#14519753—21no—
19C2×C4×D244C2211yescharacteristic
20#16839974—24no—
21#16839893—24no—
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×#14519688C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839684—21no—
39#16839684—21no—
40#16839664C2211yescharacteristic
41#16839664C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839632C2211yescharacteristic
45#16839619—21no—
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C2×#124×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#397—21no—
53C61×#393—21no—
54C61×#391C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#388—21no—
59C61×#388—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758201—22no—
64C2×#11758201—22no—
65C2×#11758201—41no—
66C2×#11758199—22no—
67C2×#11758199—22no—
68C2×#11758199—41no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519753—44no—
74C4×D244—22no—
75C4×D244—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88#14519708—41no—
89#14519708—41no—
90C4⋊#92418—22no—
91C4⋊#92418—22no—
92C4⋊#92418D414yes—
93#14519701—22no—
94#14519701—22no—
95#14519701—41no—
96#14519698—22no—
97#14519698—22no—
98#14519698—41no—
99#14519697—22no—
100#14519697—22no—
101#14519688—22no—
102#14519688—22no—
103C4×#92418—22no—
104#14519684—21no—
105#14519684—21no—
106C22×Q8×C61D411yescharacteristic
107C22×Q8×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109#124×C61—22no—
110#124×C61—22no—
111C4×Q8×C61—22no—
112C4×Q8×C61—22no—
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
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×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132#95×C61—21no—
133C24×C61—21no—
134C23×D61—21no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—41no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244#4112yes—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C22⋊C4×C61—42no—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—81no—
184C22⋊C4×C61—81no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—81no—
190C42×C61C2×D412yes—
191C42×C61#4112yes—
192#11758201—44no—
193#11758199—44no—
194#11758198—42no—
195#11758198—42no—
196#11758198—42no—
197#11758198—81no—
198#11758198—81no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206C4⋊Dic61—81no—
207#11758196—42no—
208#11758196—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D412yes—
219C2×#92418C2×D412yes—
220C2×#92418#4112yes—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C22×D411yescharacteristic
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236Q8×C61—44no—
237Q8×C61—41no—
238Q8×C61—41no—
239Q8×C61—41no—
240Q8×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245C2×C4×C61—21no—
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—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61#12112yes—
270C2×C4×C61#12112yes—
271C2×C4×C61AGL(1,ℤ/8)12yes—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61H(5,𝔽2)11yescharacteristic
277C2×C4×C612−1+411yescharacteristic
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—41no—
287D244—22no—
288D244—22no—
289D244—41no—
290D244—41no—
291C4×D61—84no—
292#92418—22no—
293#92418—22no—
294#92418—22no—
295#92418—41no—
296#25309—611nomaximal, Sylow
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61C2×#12111yescharacteristic
305C22×C61#54811yescharacteristic
306C22×C61#55011yescharacteristic
307C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C61#57511yescharacteristic
309C22×C61#57711yescharacteristic
310C22×C61#57711yescharacteristic
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C4×C61—22no—
319C4×C61—22no—
320C4×C61—22no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—81no—
325C4×C61—81no—
326C4×C61—81no—
327C4×C61#52911yescharacteristic
328C4×C61#54211yescharacteristic
329C4×C61#54611yescharacteristic
330C4×C61#54811yescharacteristic
331Dic61—81no—
332Dic61—81no—
333Dic61—81no—
334C2×#463—611no—
335C2×#446—611no—
336#2592D6111yescharacteristic, complemented
337#2538—611no—
338#2066—611no—
339#2065—611no—
340#2055—611no—
341#2054—611no—
342#1972—611no—
343#1968—611no—
344#1959—611no—
345#1948—611no—
346#1935—611no—
347#1934—611no—
348#1762—611no—
349C2×C61—161no—
350C2×C61#242611yescharacteristic
351C2×C61#305511yescharacteristic
352C2×C61#306011yescharacteristic
353C2×C61#337311yescharacteristic
354C2×C61#337611yescharacteristic
355C2×C61#339711yescharacteristic
356C2×C61#340111yescharacteristic
357D61—82no—
358C22×SD16—611no—
359C22×D8—611no—
360C2×#124C2×D6111yescharacteristic
361C2×C22⋊Q8—1221no—
362C2×C4×Q8C2×D6111yescharacteristic
363C2×C4×D4—611no—
364#463—1224no—
365#446—1224no—
366C2×#106—611no—
367C2×#106—611no—
368C2×C4⋊C8—611no—
369C2×C4⋊C8—611no—
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#97—611no—
380C2×#97—611no—
381C2×C4×C8—611no—
382#397—21no—
383#396—1221no—
384#396—1221no—
385#393—21no—
386#391C2×D6111yescharacteristic
387#391C2×D6111yescharacteristic
388#390—611no—
389#390—611no—
390#390C2×D6111yescharacteristic
391#390C2×D6111yescharacteristic
392#389—611no—
393#389—611no—
394#388—21no—
395#388—21no—
396#384—611no—
397#380—1221no—
398C4×C4⋊C4C2×D6111yescharacteristic
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403C61#2530911yescharacteristic, complemented, Sylow
404C22×Q8C22⋊D6111yescharacteristic
405C22×Q8C22×D6111yescharacteristic
406C22×D4—611no—
407C22×D4C22×D6111yescharacteristic
408C23×C4—1221no—
409C2×SD16—1222no—
410C2×SD16—1222no—
411C2×SD16—2441no—
412C2×D8—1222no—
413C2×D8—1222no—
414C2×D8—2441no—
415C22×C8—611no—
416C22×C8—611no—
417C22×C8—611no—
418C22×C8—611no—
419#124—22no—
420#124—22no—
421C22⋊Q8—2444no—
422C4×Q8—22no—
423C4×Q8—22no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4C22⋊D6111yescharacteristic
434C2×C4⋊C4C22⋊D6111yescharacteristic
435C2×C4⋊C4C22⋊D6111yescharacteristic
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C42—21no—
450C2×C42C22×D6111yescharacteristic
451C2×C42C22×D6111yescharacteristic
452C2×C42C22×D6111yescharacteristic
453#106—2441no—
454#106—2441no—
455C4⋊C8—614no—
456C4⋊C8—1222no—
457C4⋊C8—1222no—
458#103—1222no—
459#103—1222no—
460#103—2441no—
461#102—1222no—
462#102—1222no—
463#102—1222no—
464#102—1222no—
465#102—2441no—
466#97—1222no—
467#97—1222no—
468C4×C8—1222no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—1221no—
475#95—1221no—
476C24—21no—
477C24—1221no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—41no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—612no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C2×C22⋊D6111yescharacteristic
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C23×D6111yescharacteristic
515SD16—2444no—
516D8—2444no—
517C2×C8—612no—
518C2×C8—612no—
519C2×C8—612no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—4881no—
543C4⋊C4—2442no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—81no—
552C22⋊C4—4881no—
553C22⋊C4—4881no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—81no—
562C42D4×D6112yes—
563C42#1175822412yes—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—1222no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23C22×C22⋊D6111yescharacteristic
579Q8—44no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583Q8—41no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—1222no—
589D4—1222no—
590D4—2441no—
591D4—2441no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—4884no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4#1451974912yes—
627C2×C4#1451974912yes—
628C2×C4#1451980212yes—
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4#41×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4C2×#1175822411yescharacteristic
633C2×C4#1451982311yescharacteristic
634C2×C4#1451982411yescharacteristic
635C8—1222no—
636C8—1222no—
637C8—1222no—
638C8—2441no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—4881no—
647C22—4881no—
648C22—4881no—
649C22—4881no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22C2×#1451974911yescharacteristic
654C22#1684041911yescharacteristic
655C22#1684043111yescharacteristic
656C22C2×#1451980211yescharacteristic
657C22#1684056011yescharacteristic
658C22#1684057011yescharacteristic
659C22#1684057311yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—81no—
667C4—81no—
668C4—81no—
669C4—4881no—
670C4—4881no—
671C4—4881no—
672C4#1684029111yescharacteristic
673C4#1684038111yescharacteristic
674C4#1684040511yescharacteristic
675C4#1684042011yescharacteristic
676C2—161no—
677C2—4882no—
678C2#2144938811yescharacteristic
679C2#2145544011yescharacteristic
680C2#2145547811yescharacteristic
681C2#2145860711yescharacteristic
682C2#2145864011yescharacteristic
683C2#2145883911yescharacteristic
684C2#2145887911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2592C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2592D61
hypocenterC61#25309
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
2. C2×#16839893C2semidirect
3. #21450737C2semidirect
4. #21447200C2semidirect
5. #21447193C2semidirect
6. #21447134C2semidirect
9. #21446676C2semidirect
10. #21446636C2semidirect
11. #21446582C2semidirect
12. #21446512C2semidirect
13. #21446505C2semidirect
14. #21445647C2semidirect
15. C61×#2592C2semidirect
336. #2592D61semidirect
403. C61#25309semidirect

Maximal subgroups

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

subgroupindex
C2×#168399742
C2×#168398932
#214507372
#214472002
#214471932
#214471342
#214471292
#214466942
#214466762
#214466362
#214465822
#214465122
#214465052
#214456472
C61×#25922
#2530961