Finite Group Databasebeta

← #29729731

Subgroups — #29729731

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×#16839978C211yesmaximal, characteristic
2#21451307C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic, complemented
4#21447199C211yesmaximal, characteristic, complemented
5#21447192C211yesmaximal, characteristic, complemented
6#21447137C211yesmaximal, characteristic, complemented
7#21447130C211yesmaximal, characteristic
8#21446695C211yesmaximal, characteristic
9#21446673C211yesmaximal, characteristic, complemented
10#21446635C211yesmaximal, characteristic, complemented
11#21446583C211yesmaximal, characteristic, complemented
12#21446509C211yesmaximal, characteristic, complemented
13#21446502C211yesmaximal, characteristic, complemented
14#21445645C211yesmaximal, characteristic, complemented
15C2×C61×#446C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×D488C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519753—21no—
20C2×C4×D244C2211yescharacteristic
21#16839978—24no—
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32#16839688—21no—
33#16839684—21no—
34#16839684—21no—
35#16839673—21no—
36#16839668C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839664C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839652—21no—
45#16839652—21no—
46#16839632C2211yescharacteristic
47#16839619—21no—
48C4×C4⋊Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C61×#446—24no—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×#97×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190—41no—
68C2×D488—22no—
69C2×D488—22no—
70C2×D488—41no—
71#14519765—22no—
72#14519765—22no—
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×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758198—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×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97#14519709—41no—
98#14519709—41no—
99#14519702—22no—
100#14519702—22no—
101#14519702—41no—
102#14519699—22no—
103#14519699—22no—
104#14519699—22no—
105#14519699—22no—
106#14519699—41no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114C22×Q8×C61D411yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×Q8×C61—22no—
120C4×Q8×C61—22no—
121C2×C4⋊C4×C61D411yescharacteristic
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C4⋊C8×C61—22no—
128C4⋊C8×C61—22no—
129C4⋊C8×C61D414yes—
130#97×C61—22no—
131#97×C61—22no—
132C4×C8×C61—22no—
133C23×D61—21no—
134C23×D61—21no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—41no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C2×D244—21no—
152C2×D244—21no—
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#4112yes—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D412yes—
182C2×C8×C61C2×D412yes—
183C2×C8×C61#4112yes—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61C2×D412yes—
192C42×C61#4112yes—
193#11758198—42no—
194#11758198—42no—
195#11758198—42no—
196#11758198—42no—
197#11758198—42no—
198#11758198—42no—
199#11758198—42no—
200#11758198—81no—
201#11758198—81no—
202#11758198—81no—
203#11758198—81no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—42no—
215C4⋊Dic61—81no—
216#11758196—42no—
217#11758196—42no—
218C4×Dic61—81no—
219D488—44no—
220#11758190—44no—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
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—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236Q8×C61—44no—
237C2×C4×C61—21no—
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#12112yes—
253C2×C4×C61#12112yes—
254C2×C4×C61AGL(1,ℤ/8)12yes—
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—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
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—
279D244—22no—
280D244—22no—
281D244—41no—
282D244—41no—
283D244—41no—
284D244—41no—
285D244—41no—
286D244—41no—
287C4×D61—84no—
288Dic122—41no—
289Dic122—41no—
290Dic122—41no—
291Dic122—41no—
292C8×C61—22no—
293C8×C61—22no—
294C8×C61—22no—
295C8×C61—41no—
296#25309—611nomaximal, Sylow
297C22×C61C2×#12111yescharacteristic
298C22×C61#54811yescharacteristic
299C22×C61#55011yescharacteristic
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57711yescharacteristic
303C22×C61#57711yescharacteristic
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
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#52911yescharacteristic
325C4×C61#54211yescharacteristic
326C4×C61#54611yescharacteristic
327C4×C61#54811yescharacteristic
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333Dic61—81no—
334C2×#463—611no—
335C2×#446D6111yescharacteristic, complemented
336#2592—611no—
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#242611yescharacteristic
350C2×C61#305511yescharacteristic
351C2×C61#306011yescharacteristic
352C2×C61#337311yescharacteristic
353C2×C61#337611yescharacteristic
354C2×C61#339711yescharacteristic
355C2×C61#340111yescharacteristic
356D61—82no—
357D61—161no—
358C22×SD16—611no—
359C22×D8—611no—
360C2×#124—611no—
361C2×C22⋊Q8—1221no—
362C2×C4×Q8C2×D6111yescharacteristic
363C2×C4×D4—611no—
364#463—1224no—
365#446—24no—
366C2×#106—611no—
367C2×#106—611no—
368C2×C4⋊C8C2×D6111yescharacteristic
369C2×C4⋊C8C2×D6111yescharacteristic
370C2×C4⋊C8C2×D6111yescharacteristic
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×#97C2×D6111yescharacteristic
380C2×#97C2×D6111yescharacteristic
381C2×C4×C8C2×D6111yescharacteristic
382#397—1221no—
383#396—1221no—
384#396—1221no—
385#393—1221no—
386#391—611no—
387#391—611no—
388#390—611no—
389#390—611no—
390#390—611no—
391#390—611no—
392#389—611no—
393#389—611no—
394#388—1221no—
395#388—1221no—
396#384—611no—
397#380—1221no—
398C4×C4⋊C4—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403C61#2530911yescharacteristic, complemented, Sylow
404C22×Q8—611no—
405C22×Q8D24411yescharacteristic
406C22×D4—611no—
407C22×D4—611no—
408C23×C4—1221no—
409C2×SD16—1222no—
410C2×SD16—1222no—
411C2×SD16—2441no—
412C2×D8—1222no—
413C2×D8—1222no—
414C2×D8—2441no—
415C22×C8C22×D6111yescharacteristic
416C22×C8C22×D6111yescharacteristic
417C22×C8C22×D6111yescharacteristic
418C22×C8C22×D6111yescharacteristic
419#124—1222no—
420#124—1222no—
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—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4D24411yescharacteristic
436C2×C4⋊C4D24411yescharacteristic
437C2×C4⋊C4D24411yescharacteristic
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
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×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451C2×C42C22×D6111yescharacteristic
452C2×C42C22×D6111yescharacteristic
453#106—2441no—
454#106—2441no—
455C4⋊C8—22no—
456C4⋊C8—22no—
457C4⋊C8C22⋊D6114yes—
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—22no—
467#97—22no—
468C4×C8—22no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—1221no—
477C24—1221no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—41no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×D4—612no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4C2×D24411yescharacteristic
509C22×C4C2×D24411yescharacteristic
510C22×C4C2×D24411yescharacteristic
511C22×C4C2×D24411yescharacteristic
512C22×C4C2×D24411yescharacteristic
513C22×C4C2×D24411yescharacteristic
514C22×C4C23×D6111yescharacteristic
515SD16—2444no—
516D8—2444no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8#1175822612yes—
527C2×C8C2×C22⋊D6112yes—
528C2×C8C2×C22⋊D6112yes—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
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—4881no—
547C22⋊C4—4881no—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—4881no—
562C42#1175822212yes—
563C42C2×C22⋊D6112yes—
564C23—1222no—
565C23—2441no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23C22×D24411yescharacteristic
579Q8—44no—
580Q8—2441no—
581Q8—2441no—
582Q8—2441no—
583Q8—2441no—
584D4—1222no—
585D4—1222no—
586D4—2441no—
587D4—2441no—
588D4—2441no—
589D4—2441no—
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—4884no—
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—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4#1451975112yes—
627C2×C4#1451975112yes—
628C2×C4#1451980512yes—
629C2×C4C22×C22⋊D6111yescharacteristic
630C2×C4#41×D6111yescharacteristic
631C2×C4C2×#1175822211yescharacteristic
632C2×C4C2×#1175822611yescharacteristic
633C2×C4#1451982311yescharacteristic
634C2×C4#1451982511yescharacteristic
635C8—22no—
636C8—22no—
637C8—22no—
638C8—41no—
639C22—4881no—
640C22—4881no—
641C22—4881no—
642C22—4881no—
643C22—4881no—
644C22—4881no—
645C22—4881no—
646C22—4881no—
647C22—4881no—
648C22—4881no—
649C22—4881no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22C2×#1451975111yescharacteristic
654C22#1684042411yescharacteristic
655C22#1684043411yescharacteristic
656C22C2×#1451980511yescharacteristic
657C22#1684056211yescharacteristic
658C22#1684057411yescharacteristic
659C22#1684057411yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—4881no—
667C4—4881no—
668C4—4881no—
669C4—4881no—
670C4—4881no—
671C4—4881no—
672C4#1684029911yescharacteristic
673C4#1684038411yescharacteristic
674C4#1684040011yescharacteristic
675C4#1684042311yescharacteristic
676C2—9761no—
677C2—4882no—
678C2#2144937911yescharacteristic
679C2#2145543811yescharacteristic
680C2#2145548111yescharacteristic
681C2#2145861311yescharacteristic
682C2#2145864811yescharacteristic
683C2#2145884111yescharacteristic
684C2#2145888711yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#446C2
socleC23×C61C22×D4
prime radicalC22×D488C22
hypercenterC2×#446D61
hypocenterC61#25309
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #21451307C2semidirect
3. #21450737C2semidirect
4. #21447199C2semidirect
5. #21447192C2semidirect
6. #21447137C2semidirect
9. #21446673C2semidirect
10. #21446635C2semidirect
11. #21446583C2semidirect
12. #21446509C2semidirect
13. #21446502C2semidirect
14. #21445645C2semidirect
15. C2×C61×#446C2semidirect
335. C2×#446D61semidirect
403. C61#25309semidirect

Maximal subgroups

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

subgroupindex
C2×#168399782
#214513072
#214507372
#214471992
#214471922
#214471372
#214471302
#214466952
#214466732
#214466352
#214465832
#214465092
#214465022
#214456452
C2×C61×#4462
#2530961