Finite Group Databasebeta

← #409848

Subgroups — #409848

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2092C211yesmaximal, characteristic
2C2×#2059C211yesmaximal, characteristic, complemented
3C2×#1991C211yesmaximal, characteristic, complemented
4C2×#1934C211yesmaximal, characteristic
5#15239C211yesmaximal, characteristic, complemented
6#15229C211yesmaximal, characteristic
7#11378C211yesmaximal, characteristic, complemented
8#11377C211yesmaximal, characteristic, complemented
9#11123C211yesmaximal, characteristic
10#11121C211yesmaximal, characteristic
11#10925C211yesmaximal, characteristic
12#10925C211yesmaximal, characteristic
13#10862C211yesmaximal, characteristic, complemented
14#10861C211yesmaximal, characteristic, complemented
15C8×#95C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#382C2211yescharacteristic
33#2092—24no—
34#2059—24no—
35#1991—24no—
36#1934—24no—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48#1443—22no—
49#1442—22no—
50#1439—22no—
51#1439—22no—
52C4⋊#95C2211yescharacteristic
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C23×D4C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#125—22no—
65C2×#125—22no—
66C2×#125—22no—
67C2×#125—22no—
68C2×#125—41no—
69C2×#125—41no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—41no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106—41no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—41no—
109C2×#102—41no—
110C2×#102Q814yes—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—41no—
116C2×C4×C8—41no—
117C2×C4×C8—41no—
118C2×C4×C8—41no—
119C2×C4×C8D414yes—
120#390—22no—
121#390—22no—
122#390—22no—
123#390—22no—
124#390—41no—
125#390—41no—
126#389—22no—
127#389—22no—
128#382—22no—
129#382—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95D412yes—
140C2×#95D412yes—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#336—22no—
150#336—22no—
151#336D414yes—
152C25—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C2×Q812yes—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2411yescharacteristic
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8C2×D412yes—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×Q812yes—
202C22×C8#4112yes—
203C22×C8#4112yes—
204C22×C8#4112yes—
205C22×C8#4112yes—
206C22×C8#4112yes—
207#125—28no—
208#125—44no—
209#125—44no—
210#125—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
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—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—82no—
258C2×C4⋊C4C2×Q812yes—
259C2×C4⋊C4#4112yes—
260C2×C4⋊C4#4112yes—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—82no—
268C2×C22⋊C4—82no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289#107—44no—
290#107—44no—
291#106—44no—
292#106—44no—
293#102—28no—
294#102—44no—
295#102—44no—
296#102—44no—
297C4×C8—44no—
298C4×C8—44no—
299C4×C8—44no—
300#95—24no—
301#95—24no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—82no—
308#95—82no—
309#95—82no—
310#95—82no—
311C24—22no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24C22×D411yescharacteristic
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4#12712yes—
407C22×C4#12712yes—
408C22×C4#13514yes—
409C22×C4#13514yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×Q811yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8C22⋊Q814yes—
450C2×C8#12414yes—
451C2×C8#12414yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—84no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—82no—
505C23—82no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—81no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23—81no—
516C23—81no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—81no—
522C23C2×#12711yescharacteristic
523C23#53211yescharacteristic
524C23#53211yescharacteristic
525C23#54611yescharacteristic
526C23#54611yescharacteristic
527C23#54711yescharacteristic
528C23D4×Q811yescharacteristic
529C23C2×#13512yes—
530C23C2×#13512yes—
531C23#57612yes—
532C23#57712yes—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—81no—
538D4—81no—
539D4—81no—
540D4—81no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4C2×C22⋊Q811yescharacteristic
601C2×C4#52511yescharacteristic
602C2×C4C2×#12411yescharacteristic
603C2×C4C2×#12411yescharacteristic
604C2×C4#52911yescharacteristic
605C2×C4#52911yescharacteristic
606C2×C4#53211yescharacteristic
607C2×C4#53211yescharacteristic
608C2×C4#53311yescharacteristic
609C2×C4#53311yescharacteristic
610C2×C4#53411yescharacteristic
611C2×C4#53611yescharacteristic
612C2×C4#53911yescharacteristic
613C2×C4#53911yescharacteristic
614C2×C4#54211yescharacteristic
615C2×C4#54211yescharacteristic
616C2×C4#54611yescharacteristic
617C2×C4#54611yescharacteristic
618C2×C4#54711yescharacteristic
619C2×C4D4×Q811yescharacteristic
620C2×C4#55111yescharacteristic
621C2×C4#55111yescharacteristic
622C2×C4#55211yescharacteristic
623C2×C4#55211yescharacteristic
624C2×C4#55411yescharacteristic
625C2×C4#55411yescharacteristic
626C2×C4#55611yescharacteristic
627C2×C4#55611yescharacteristic
628C8—24no—
629C8—24no—
630C8—24no—
631C8—42no—
632C22—84no—
633C22—84no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—162no—
638C22—162no—
639C22—161no—
640C22—161no—
641C22—161no—
642C22—161no—
643C22#252111yescharacteristic
644C22#252111yescharacteristic
645C22#253911yescharacteristic
646C22#254211yescharacteristic
647C22#255611yescharacteristic
648C22#255611yescharacteristic
649C22#285811yescharacteristic
650C22#314711yescharacteristic
651C22#314711yescharacteristic
652C22#315411yescharacteristic
653C22#315511yescharacteristic
654C22#316812yes—
655C22#316812yes—
656C22#317912yes—
657C22#317912yes—
658C22#329612yes—
659C22#329712yes—
660C22#330412yes—
661C22#330512yes—
662C22#333712yes—
663C22#333912yes—
664C22#338012yes—
665C22#338112yes—
666C4—44no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—161no—
674C4—161no—
675C4—161no—
676C4—161no—
677C4#239611yescharacteristic
678C4#248811yescharacteristic
679C4#248811yescharacteristic
680C4#249211yescharacteristic
681C4#249211yescharacteristic
682C4#256411yescharacteristic
683C4#260411yescharacteristic
684C4#266311yescharacteristic
685C2—162no—
686C2#1506511yescharacteristic
687C2#2806712yes—
688C2#2806812yes—
689C2#2833711yescharacteristic
690C2#2834011yescharacteristic
691C2#2853311yescharacteristic
692C2#2853411yescharacteristic
693C2#2894412yes—
694C2#2895012yes—
695C2#3324011yescharacteristic
696C2#3324311yescharacteristic
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
2. C2×#2059C2semidirect
3. C2×#1991C2semidirect
5. #15239C2semidirect
7. #11378C2semidirect
8. #11377C2semidirect
13. #10862C2semidirect
14. #10861C2semidirect
15. C8×#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20922
C2×#20592
C2×#19912
C2×#19342
#152392
#152292
#113782
#113772
#111232
#111212
#109252
#108622
#108612
C8×#952