Finite Group Databasebeta

← #29729562

Subgroups — #29729562

Order 15616.

848 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×#16839977C211yesmaximal, characteristic
2C2×Q8×#92418C211yesmaximal, characteristic, complemented
3#21451307C211yesmaximal, characteristic, complemented
4#21447211C211yesmaximal, characteristic, complemented
5#21447190C211yesmaximal, characteristic, complemented
6#21447136C211yesmaximal, characteristic, complemented
7#21447129C211yesmaximal, characteristic
8#21446684C211yesmaximal, characteristic
9#21446672C211yesmaximal, characteristic, complemented
10#21446628C211yesmaximal, characteristic, complemented
11#21446585C211yesmaximal, characteristic, complemented
12#21446501C211yesmaximal, characteristic, complemented
13#21446494C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic, complemented
15C61×#2580C211yesmaximal, characteristic, complemented
16C22×#11758200C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19#16839977—24no—
20Q8×#92418—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35C2×C4×#92418C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37#16839688—21no—
38#16839673—21no—
39#16839668C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839664C2211yescharacteristic
42#16839664C2211yescharacteristic
43#16839652—21no—
44#16839652—21no—
45C4×C4⋊Dic61C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C61×#399—21no—
53C61×#393—21no—
54C61×#390C2211yescharacteristic
55C61×#390C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#385—21no—
59C61×#385—21no—
60C61×#382C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758200—41no—
66C2×#11758199—22no—
67C2×#11758199—22no—
68C2×#11758199—41no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519765—22no—
74#14519765—22no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—21no—
84#14519710—41no—
85#14519708—41no—
86C4⋊#92418—22no—
87C4⋊#92418—22no—
88C4⋊#92418D414yes—
89#14519700—22no—
90#14519700—22no—
91#14519700—41no—
92#14519698—22no—
93#14519698—22no—
94#14519697—22no—
95#14519697—22no—
96#14519697—41no—
97C4×#92418—22no—
98C4×#92418—22no—
99C4×#92418—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×Q8×C61D411yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C23×C4×C61—21no—
108C4×Q8×C61—22no—
109C4×Q8×C61—22no—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×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—
132C24×C61—21no—
133C23×D61—21no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61#4112yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—81no—
148C22×C4×C61—81no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—81no—
181C4⋊C4×C61—81no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—81no—
186C22⋊C4×C61—81no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61C2×D412yes—
192C42×C61#4112yes—
193#11758200—44no—
194#11758199—44no—
195#11758198—42no—
196#11758198—42no—
197#11758198—42no—
198#11758198—42no—
199#11758198—81no—
200#11758198—81no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4×Dic61—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C2×#92418C2×D412yes—
217C2×#92418#4112yes—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61C22×D411yescharacteristic
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233Q8×C61—28no—
234D4×C61—22no—
235D4×C61—22no—
236D4×C61—41no—
237D4×C61—41no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
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—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—84no—
262C2×C4×C61#12112yes—
263C2×C4×C61#12112yes—
264C2×C4×C61#13512yes—
265C2×C4×C61#13512yes—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C612−1+411yescharacteristic
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281D244—41no—
282D244—41no—
283D244—41no—
284D244—41no—
285Dic122—41no—
286Dic122—41no—
287Dic122—41no—
288Dic122—41no—
289#92418—22no—
290#92418—22no—
291#92418—22no—
292#92418—41no—
293#25296—611nomaximal, Sylow
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61C2×#12111yescharacteristic
302C22×C61#54811yescharacteristic
303C22×C61#55011yescharacteristic
304C22×C61C2×#13511yescharacteristic
305C22×C61C2×#13511yescharacteristic
306C22×C61#57611yescharacteristic
307C22×C61#57711yescharacteristic
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—22no—
318C4×C61—22no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61#52911yescharacteristic
325C4×C61#54211yescharacteristic
326C4×C61#54611yescharacteristic
327C4×C61#54811yescharacteristic
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#463—611no—
332C2×C8×Q8—611no—
333#2592—611no—
334#2580D6111yescharacteristic, complemented
335#2068—611no—
336#2065—611no—
337#2055—611no—
338#2054—611no—
339#1970—611no—
340#1968—611no—
341#1958—611no—
342#1949—611no—
343#1934—611no—
344#1933—611no—
345C4×#102—611no—
346C2×C61—82no—
347C2×C61#242611yescharacteristic
348C2×C61#305011yescharacteristic
349C2×C61#306211yescharacteristic
350C2×C61#336011yescharacteristic
351C2×C61#336111yescharacteristic
352C2×C61#339411yescharacteristic
353C2×C61#339611yescharacteristic
354D61—161no—
355C22×SD16—611no—
356C22×D8—611no—
357C2×#124—611no—
358C2×C4×Q8C2×D6111yescharacteristic
359C2×C4×D4C2×D6111yescharacteristic
360#463—1224no—
361C8×Q8—1224no—
362C2×#107—611no—
363C2×#106—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C4⋊C8—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×C4×C8—611no—
376C2×C4×C8—611no—
377C2×C4×C8—611no—
378#399—21no—
379#397—1221no—
380#393—21no—
381#393—1221no—
382#391—611no—
383#391—611no—
384#390—611no—
385#390—611no—
386#390C2×D6111yescharacteristic
387#390C2×D6111yescharacteristic
388#389C2×D6111yescharacteristic
389#389C2×D6111yescharacteristic
390#388—1221no—
391#388—1221no—
392#385—21no—
393#385—21no—
394#382C2×D6111yescharacteristic
395C4×C4⋊C4—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400C61#2529611yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×Q8C22⋊D6111yescharacteristic
403C22×D4—611no—
404C22×D4C22×D6111yescharacteristic
405C23×C4—21no—
406C2×SD16—1222no—
407C2×SD16—1222no—
408C2×SD16—2441no—
409C2×D8—1222no—
410C2×D8—1222no—
411C2×D8—2441no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8—611no—
415C22×C8—611no—
416#124—1222no—
417#124—1222no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×D4—22no—
421C4×D4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4C22⋊D6111yescharacteristic
427C2×C4⋊C4C22⋊D6111yescharacteristic
428C2×C4⋊C4C22⋊D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445C2×C42C22×D6111yescharacteristic
446C2×C42C22×D6111yescharacteristic
447#107—2441no—
448#106—2441no—
449C4⋊C8—614no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452#103—1222no—
453#103—1222no—
454#103—2441no—
455#102—1222no—
456#102—1222no—
457#102—1222no—
458#102—1222no—
459#102—2441no—
460C4×C8—1222no—
461C4×C8—1222no—
462C4×C8—1222no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—21no—
473C24—1221no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4#1175822612yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C2×C22⋊D6111yescharacteristic
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—2444no—
512D8—2444no—
513C2×C8—612no—
514C2×C8—612no—
515C2×C8—612no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—81no—
535C4⋊C4—81no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C22⋊C4—4881no—
546C22⋊C4—4881no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—4881no—
556C42D4×D6112yes—
557C42#1175822412yes—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—2441no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23C22×C22⋊D6111yescharacteristic
573Q8—28no—
574Q8—2441no—
575Q8—2441no—
576Q8—2441no—
577Q8—2441no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582D4—2441no—
583D4—2441no—
584D4—2441no—
585D4—2441no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
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—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4#1451974912yes—
620C2×C4#1451974912yes—
621C2×C4#1451979712yes—
622C2×C4#1451979712yes—
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822411yescharacteristic
626C2×C4C2×#1175822611yescharacteristic
627C2×C4#1451982311yescharacteristic
628C2×C4#1451982511yescharacteristic
629C8—1222no—
630C8—1222no—
631C8—1222no—
632C8—2441no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—4881no—
641C22—4881no—
642C22—4881no—
643C22—4881no—
644C22—4881no—
645C22—4881no—
646C22—4881no—
647C22C2×#1451974911yescharacteristic
648C22#1684042311yescharacteristic
649C22#1684043111yescharacteristic
650C22C2×#1451979711yescharacteristic
651C22C2×#1451979711yescharacteristic
652C22#1684056511yescharacteristic
653C22#1684057311yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4#1684029811yescharacteristic
667C4#1684038511yescharacteristic
668C4#1684040311yescharacteristic
669C4#1684042211yescharacteristic
670C2—82no—
671C2—9761no—
672C2#2144938711yescharacteristic
673C2#2145539711yescharacteristic
674C2#2145549511yescharacteristic
675C2#2145846811yescharacteristic
676C2#2145847911yescharacteristic
677C2#2145881111yescharacteristic
678C2#2145882711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2580C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenter#2580D61
hypocenterC61#25296
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#92418C2semidirect
3. #21451307C2semidirect
4. #21447211C2semidirect
5. #21447190C2semidirect
6. #21447136C2semidirect
9. #21446672C2semidirect
10. #21446628C2semidirect
11. #21446585C2semidirect
12. #21446501C2semidirect
13. #21446494C2semidirect
14. C4×#14519698C2semidirect
15. C61×#2580C2semidirect
334. #2580D61semidirect
400. C61#25296semidirect

Maximal subgroups

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

subgroupindex
C2×#168399772
C2×Q8×#924182
#214513072
#214472112
#214471902
#214471362
#214471292
#214466842
#214466722
#214466282
#214465852
#214465012
#214464942
C4×#145196982
C61×#25802
#2529661