Finite Group Databasebeta

← #431066

Subgroups — #431066

Order 512.

1432 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#2094C211yesmaximal, characteristic
3C2×#2088C211yesmaximal, characteristic, complemented
4C2×#2059C211yesmaximal, characteristic, complemented
5C2×#1862C211yesmaximal, characteristic
6C2×#1846C211yesmaximal, characteristic, complemented
7#15228C211yesmaximal, characteristic, complemented
8#13979C211yesmaximal, characteristic
9#11378C211yesmaximal, characteristic, complemented
10#11325C211yesmaximal, characteristic, complemented
11#11324C211yesmaximal, characteristic, complemented
12#11123C211yesmaximal, characteristic
13#11051C211yesmaximal, characteristic
14#10925C211yesmaximal, characteristic
15#10876C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#478—24no—
19C2×#478—24no—
20C2×#478—24no—
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#382C2211yescharacteristic
40#2094—24no—
41#2088—24no—
42#2059—24no—
43#1862—24no—
44#1846—24no—
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#340C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55#1444—22no—
56#1441—22no—
57#1439—22no—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61C22⋊#95—22no—
62C4×#95C2211yescharacteristic
63C23×D4C2311yescharacteristic
64C24×C4—21no—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×#125—22no—
70C2×#125—22no—
71C2×#125—22no—
72C2×#125—22no—
73C2×#125—41no—
74C2×#125—41no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—41no—
80C2×C4×D4—41no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—22no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—22no—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#478—232no—
100C2×#107—22no—
101C2×#107—22no—
102C2×#107—22no—
103C2×#107—22no—
104C2×#107—22no—
105C2×#107—22no—
106C2×#107—22no—
107C2×#107—41no—
108C2×#107—41no—
109C2×#106—22no—
110C2×#106—22no—
111C2×#106—22no—
112C2×#106—22no—
113C2×#106—22no—
114C2×#106—22no—
115C2×#106—22no—
116C2×#106—41no—
117C2×#106D414yes—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8D414yes—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×#102—22no—
127C2×#102—22no—
128C2×#102—22no—
129C2×#102—22no—
130C2×#102—41no—
131C2×#102D414yes—
132C2×#102Q814yes—
133C2×C4×C8—22no—
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136#390—22no—
137#390—22no—
138#390—41no—
139#390Q814yes—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#384—22no—
145#384—41no—
146#382—22no—
147#382—22no—
148#382—41no—
149#382—41no—
150C2×#95—21no—
151C2×#95—22no—
152C2×#95—22no—
153C2×#95—21no—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95D412yes—
158C2×#95D412yes—
159#340—22no—
160#340—22no—
161#340—22no—
162#340—22no—
163#340—22no—
164#340—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336D414yes—
169C25—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4C2×D412yes—
182C22×D4C2×Q812yes—
183C22×D4C2×Q812yes—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—82no—
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D412yes—
200C23×C4C2411yescharacteristic
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8C2×D412yes—
227C22×C8C2×Q812yes—
228C22×C8#4112yes—
229#125—28no—
230#125—44no—
231#125—44no—
232#125—44no—
233C4×D4—28no—
234C4×D4—44no—
235C4×D4—44no—
236C4×D4—44no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×Q812yes—
291C2×C4⋊C4C2×Q812yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—82no—
299C2×C22⋊C4—82no—
300C2×C22⋊C4—82no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×Q812yes—
318C2×C42#4112yes—
319#107—28no—
320#107—44no—
321#107—44no—
322#106—28no—
323#106—44no—
324#106—44no—
325C4⋊C8—28no—
326#102—28no—
327#102—28no—
328#102—28no—
329#102—44no—
330C4×C8—28no—
331#95—24no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—42no—
337#95—44no—
338#95—82no—
339#95—82no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—41no—
344C24—41no—
345C24—41no—
346C24—41no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24C22×D411yescharacteristic
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4C4⋊Q814yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—48no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4C22⋊Q814yes—
441C22×C4#12712yes—
442C22×C4#12712yes—
443C22×C4#13514yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×Q811yescharacteristic
449C22×C4C22×Q811yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4H(5,𝔽2)11yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C42−1+411yescharacteristic
455C22×C42−1+411yescharacteristic
456C22×C42−1+411yescharacteristic
457C22×C42−1+411yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8#12014yes—
485C2×C8C22⋊Q814yes—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—88no—
519C22⋊C4—48no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C23—24no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—82no—
548C23—82no—
549C23—81no—
550C23—81no—
551C23—81no—
552C23—81no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23—82no—
557C23C2×C22⋊Q811yescharacteristic
558C23C2×#12711yescharacteristic
559C23#53611yescharacteristic
560C23D4211yescharacteristic
561C23D4211yescharacteristic
562C23#54711yescharacteristic
563C23D4×Q811yescharacteristic
564C23C2×#13512yes—
565C23#57512yes—
566C23#57612yes—
567C23#57712yes—
568D4—24no—
569D4—24no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—816no—
611C2×C4—82no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4C2×#12011yescharacteristic
637C2×C4C2×C22⋊Q811yescharacteristic
638C2×C4#52511yescharacteristic
639C2×C4C2×C4⋊Q811yescharacteristic
640C2×C4#53511yescharacteristic
641C2×C4#53611yescharacteristic
642C2×C4#53611yescharacteristic
643C2×C4#53611yescharacteristic
644C2×C4#53911yescharacteristic
645C2×C4#53911yescharacteristic
646C2×C4#54411yescharacteristic
647C2×C4#54411yescharacteristic
648C2×C4D4211yescharacteristic
649C2×C4D4211yescharacteristic
650C2×C4#54711yescharacteristic
651C2×C4#54711yescharacteristic
652C2×C4#54711yescharacteristic
653C2×C4#54711yescharacteristic
654C2×C4#54711yescharacteristic
655C2×C4D4×Q811yescharacteristic
656C2×C4D4×Q811yescharacteristic
657C2×C4D4×Q811yescharacteristic
658C2×C4D4×Q811yescharacteristic
659C2×C4D4×Q811yescharacteristic
660C2×C4#55411yescharacteristic
661C2×C4#55411yescharacteristic
662C2×C4#56311yescharacteristic
663C2×C4#56311yescharacteristic
664C8—24no—
665C8—24no—
666C8—42no—
667C8—42no—
668C22—44no—
669C22—48no—
670C22—82no—
671C22—82no—
672C22—84no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22—84no—
679C22#243311yescharacteristic
680C22#244011yescharacteristic
681C22#244911yescharacteristic
682C22#254211yescharacteristic
683C22#254211yescharacteristic
684C22#267711yescharacteristic
685C22#268011yescharacteristic
686C22#307612yes—
687C22#308112yes—
688C22#314711yescharacteristic
689C22#315311yescharacteristic
690C22#315411yescharacteristic
691C22#315511yescharacteristic
692C22#323212yes—
693C22#324112yes—
694C22#326812yes—
695C22#327312yes—
696C22#328212yes—
697C22#328412yes—
698C22#333912yes—
699C22#334812yes—
700C22#338012yes—
701C22#338812yes—
702C4—44no—
703C4—42no—
704C4—42no—
705C4—44no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—161no—
712C4—161no—
713C4#240611yescharacteristic
714C4#243411yescharacteristic
715C4#244011yescharacteristic
716C4#244911yescharacteristic
717C4#267711yescharacteristic
718C4#268011yescharacteristic
719C4#272011yescharacteristic
720C4#272011yescharacteristic
721C2—84no—
722C2#1543011yescharacteristic
723C2#2577511yescharacteristic
724C2#2579311yescharacteristic
725C2#2605112yes—
726C2#2634512yes—
727C2#2851311yescharacteristic
728C2#2852511yescharacteristic
729C2#2853411yescharacteristic
730C2#2854911yescharacteristic
731C2#3095112yes—
732C2#3108812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2088C2semidirect
4. C2×#2059C2semidirect
6. C2×#1846C2semidirect
7. #15228C2semidirect
9. #11378C2semidirect
10. #11325C2semidirect
11. #11324C2semidirect
15. #10876C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#20942
C2×#20882
C2×#20592
C2×#18622
C2×#18462
#152282
#139792
#113782
#113252
#113242
#111232
#110512
#109252
#108762