Finite Group Databasebeta

← #29730298

Subgroups — #29730298

Order 15616.

860 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839832C211yesmaximal, characteristic
2#21450815C211yesmaximal, characteristic
3#21450724C211yesmaximal, characteristic
4#21447248C211yesmaximal, characteristic, complemented
5#21447243C211yesmaximal, characteristic, complemented
6#21447021C211yesmaximal, characteristic, complemented
7#21447014C211yesmaximal, characteristic, complemented
8#21446793C211yesmaximal, characteristic, complemented
9#21446589C211yesmaximal, characteristic, complemented
10#21446249C211yesmaximal, characteristic, complemented
11#21446239C211yesmaximal, characteristic, complemented
12#21445853C211yesmaximal, characteristic, complemented
13#21445845C211yesmaximal, characteristic, complemented
14#21445649C211yesmaximal, characteristic, complemented
15C2×C61×#477C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519766—21no—
19C2×#14519753C2211yescharacteristic
20C2×#14519753C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22#16839832—24no—
23C2×#14519711C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30#16839692—21no—
31#16839688—21no—
32#16839688—21no—
33#16839679—21no—
34#16839669—21no—
35#16839653—21no—
36#16839653—21no—
37#16839628C2211yescharacteristic
38#16839619C2211yescharacteristic
39#16839619C2211yescharacteristic
40#16839619C2211yescharacteristic
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#477—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C22×D244D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C2×#11758189—22no—
65C2×#11758189—22no—
66C2×#11758189—41no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C2×C8×D61—22no—
70C2×C8×D61—22no—
71C2×C8×D61—41no—
72#14519766—44no—
73#14519753—22no—
74#14519753—22no—
75#14519753—41no—
76#14519753—41no—
77C4×D244—22no—
78C4×D244—22no—
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198D411yescharacteristic
95C2×#11758198D411yescharacteristic
96C2×C4×Dic61—21no—
97#14519711—41no—
98#14519709—41no—
99#14519701—22no—
100#14519701—22no—
101#14519689—22no—
102#14519689—22no—
103#14519688—22no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C22×Q8×C61—21no—
114C22×Q8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C4⋊Q8×C61—22no—
118C4⋊Q8×C61—22no—
119C4×Q8×C61—22no—
120C4×Q8×C61—22no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#107×C61—22no—
129#107×C61—22no—
130C4⋊C8×C61D414yes—
131#103×C61—22no—
132#103×C61—22no—
133C23×D61C2×D411yescharacteristic
134C23×D61C2×D411yescharacteristic
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—42no—
141C2×Q8×C61—42no—
142C2×Q8×C61—42no—
143C2×Q8×C61#4112yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C2×D244—22no—
158C2×D244—22no—
159C2×D244—41no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D412yes—
177C2×C8×C61C2×D412yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61—42no—
186C42×C61—42no—
187C42×C61C2×D412yes—
188C42×C61#4112yes—
189#11758198—22no—
190#11758198—22no—
191#11758198—42no—
192#11758198—42no—
193#11758198—42no—
194#11758198—42no—
195#11758198—42no—
196#11758198—42no—
197#11758198—42no—
198#11758198—42no—
199C4⋊Dic61—22no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208#11758196—42no—
209#11758196—42no—
210C4×Dic61—42no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418#4112yes—
217#11758189—44no—
218C8×D61—44no—
219C23×C61C22×D411yescharacteristic
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61AGL(1,ℤ/8)12yes—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238Q8×C61—84no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
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—41no—
254C2×C4×C61—41no—
255C2×C4×C61#12112yes—
256C2×C4×C61#12112yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×C4×C612−1+411yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
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—44no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—41no—
288C4×D61—41no—
289C4×D61—41no—
290C8×C61—22no—
291C8×C61—22no—
292#92418—22no—
293#92418—41no—
294#25352—611nomaximal, Sylow
295C22×C61C2×#12111yescharacteristic
296C22×C61#54611yescharacteristic
297C22×C61#54711yescharacteristic
298C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C61#57511yescharacteristic
300C22×C61#57811yescharacteristic
301C22×C61#57811yescharacteristic
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—41no—
316C4×C61—22no—
317C4×C61—22no—
318C4×C61—42no—
319C4×C61—81no—
320C4×C61—81no—
321C4×C61—81no—
322C4×C61#52911yescharacteristic
323C4×C61#54211yescharacteristic
324C4×C61#54611yescharacteristic
325C4×C61#54811yescharacteristic
326Dic61—41no—
327Dic61—41no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331Dic61—81no—
332C2×#477D6111yescharacteristic, complemented
333C2×#435—611no—
334#2547—611no—
335#2536—611no—
336#2076—611no—
337#2075—611no—
338#2036—611no—
339#2035—611no—
340#1993—611no—
341#1949—611no—
342#1894—611no—
343#1892—611no—
344#1809—611no—
345#1808—611no—
346#1763—611no—
347C2×C61#242711yescharacteristic
348C2×C61#305511yescharacteristic
349C2×C61#305911yescharacteristic
350C2×C61#330811yescharacteristic
351C2×C61#331311yescharacteristic
352C2×C61#334311yescharacteristic
353C2×C61#334911yescharacteristic
354D61—42no—
355D61—81no—
356C22×M16—611no—
357C23×C8—611no—
358C2×C4⋊Q8C2×D6111yescharacteristic
359C2×#124—1221no—
360C2×C22⋊Q8—611no—
361C2×C22⋊Q8—611no—
362C2×C4×Q8C2×D6111yescharacteristic
363C2×C4×D4—611no—
364#477—24no—
365#435—1224no—
366C2×#107—611no—
367C2×#107C2×D6111yescharacteristic
368C2×#107C2×D6111yescharacteristic
369C2×#106—611no—
370C2×C4⋊C8C2×D6111yescharacteristic
371C2×#103—611no—
372C2×#103—611no—
373C2×#103C2×D6111yescharacteristic
374C2×#103C2×D6111yescharacteristic
375C2×C22⋊C8—611no—
376C2×C22⋊C8—611no—
377C2×#97—611no—
378#398—1221no—
379#397—1221no—
380#397—1221no—
381#394—1221no—
382#391—1221no—
383#388—1221no—
384#388—1221no—
385#382—611no—
386#380—611no—
387#380—611no—
388#380—611no—
389#380—611no—
390#380—611no—
391#380—611no—
392C4×C4⋊C4—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401C61#2535211yescharacteristic, complemented, Sylow
402C22×Q8—21no—
403C22×Q8C22×D6111yescharacteristic
404C22×D4—611no—
405C23×C4—611no—
406C23×C4—611no—
407C2×M16—1222no—
408C2×M16—1222no—
409C2×M16—2441no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8—1222no—
413C22×C8—1222no—
414C22×C8—2441no—
415C22×C8C22×D6111yescharacteristic
416C22×C8C22×D6111yescharacteristic
417C4⋊Q8—22no—
418C4⋊Q8—22no—
419#124—2444no—
420C22⋊Q8—1222no—
421C22⋊Q8—1222no—
422C22⋊Q8—2441no—
423C22⋊Q8—2441no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×D4—1222no—
427C4×D4—1222no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—611no—
441C2×C22⋊C4—611no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C42—21no—
451C2×C42—1221no—
452C2×C42C22×D6111yescharacteristic
453#107—22no—
454#107—22no—
455#107—2441no—
456#106—2441no—
457C4⋊C8C22⋊D6114yes—
458#103—22no—
459#103—22no—
460#103—1222no—
461#103—1222no—
462C22⋊C8—1222no—
463C22⋊C8—1222no—
464#97—1222no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474C24—611no—
475C24—611no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—42no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8#1175822612yes—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4—2441no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
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×C4D4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512C22×C4C23×D6111yescharacteristic
513M16—2444no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—612no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—2444no—
525C2×C8C2×C22⋊D6112yes—
526C2×C8C2×C22⋊D6112yes—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—1222no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C42—42no—
557C42—2442no—
558C42#1175822412yes—
559C42C2×C22⋊D6112yes—
560C23—612no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23—1221no—
573C23—1221no—
574C23C2×D4×D6111yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579Q8—84no—
580D4—2444no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
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—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4#1451974812yes—
624C2×C4#1451974812yes—
625C2×C4C22×C22⋊D6111yescharacteristic
626C2×C4#41×D6111yescharacteristic
627C2×C4C2×#1175822411yescharacteristic
628C2×C4C2×#1175822611yescharacteristic
629C2×C4#1451982311yescharacteristic
630C2×C4#1451982511yescharacteristic
631C8—22no—
632C8—22no—
633C8—1222no—
634C8—2441no—
635C22—1222no—
636C22—1222no—
637C22—1222no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22—2441no—
642C22—2441no—
643C22—2441no—
644C22—2441no—
645C22—2441no—
646C22—2441no—
647C22—2441no—
648C22—2441no—
649C22C2×#1451974811yescharacteristic
650C22#1684040311yescharacteristic
651C22#1684041411yescharacteristic
652C22AGL(1,ℤ/8)×D6111yescharacteristic
653C22#1684055811yescharacteristic
654C22#1684057511yescharacteristic
655C22#1684057511yescharacteristic
656C4—22no—
657C4—22no—
658C4—42no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—4881no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4—2441no—
667C4—2441no—
668C4#1684029811yescharacteristic
669C4#1684038511yescharacteristic
670C4#1684040011yescharacteristic
671C4#1684042311yescharacteristic
672C2—4881no—
673C2—2442no—
674C2#2144939211yescharacteristic
675C2#2145543211yescharacteristic
676C2#2145547011yescharacteristic
677C2#2145780711yescharacteristic
678C2#2145788211yescharacteristic
679C2#2145825611yescharacteristic
680C2#2145834611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#477C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenterC2×#477D61
hypocenterC61#25352
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
4. #21447248C2semidirect
5. #21447243C2semidirect
6. #21447021C2semidirect
7. #21447014C2semidirect
8. #21446793C2semidirect
9. #21446589C2semidirect
10. #21446249C2semidirect
11. #21446239C2semidirect
12. #21445853C2semidirect
13. #21445845C2semidirect
14. #21445649C2semidirect
15. C2×C61×#477C2semidirect
332. C2×#477D61semidirect
401. C61#25352semidirect

Maximal subgroups

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

subgroupindex
C2×#168398322
#214508152
#214507242
#214472482
#214472432
#214470212
#214470142
#214467932
#214465892
#214462492
#214462392
#214458532
#214458452
#214456492
C2×C61×#4772
#2535261