Finite Group Databasebeta

← #217704

Subgroups — #217704

Order 512.

891 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22133C211yesmaximal, characteristic, complemented
2#21786C211yesmaximal, characteristic, complemented
3C4×#447C211yesmaximal, characteristic
4#9665C211yesmaximal, characteristic, complemented
5#9663C211yesmaximal, characteristic, complemented
6#9560C211yesmaximal, characteristic, complemented
7#9558C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9541C211yesmaximal, characteristic, complemented
10#8947C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8940C211yesmaximal, characteristic, complemented
13#8831C211yesmaximal, characteristic, complemented
14#8310C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#447C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic, complemented
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C4×#121C2211yescharacteristic
24#1954—21no—
25#1954—21no—
26#1954—21no—
27#1954C2211yescharacteristic
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic, complemented
32#1939C2211yescharacteristic, complemented
33#1923—21no—
34#1879C411yescharacteristic
35#1879C411yescharacteristic
36#1874C411yescharacteristic
37#1869—21no—
38#1862C411yescharacteristic
39#1862C411yescharacteristic
40#1852C411yescharacteristic
41#1850—21no—
42#1797C2211yescharacteristic
43#1786C411yescharacteristic
44#1779—21no—
45#1779—21no—
46C8×C4⋊C4C2211yescharacteristic, complemented
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic, complemented
52C2×#334C2211yescharacteristic, complemented
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic, complemented
58C2×D4⋊C8C2211yescharacteristic, complemented
59C2×#322C2211yescharacteristic, complemented
60C2×#322C2211yescharacteristic, complemented
61#1279C2211yescharacteristic, complemented
62#1279C2211yescharacteristic, complemented
63#1277C2211yescharacteristic, complemented
64#1277C2211yescharacteristic, complemented
65#1273C2211yescharacteristic, complemented
66#1273C2211yescharacteristic, complemented
67C23×D4C2×C411yescharacteristic
68C24×C4—21no—
69C22×D8C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C23×C8—21no—
72C2×#121C2×C411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—41no—
77C2×C4×D4—41no—
78C2×C4×D4—41no—
79C2×C4×D4—41no—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×C4×D4C2311yescharacteristic
84C2×C4×D4C2311yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C22⋊C4—21no—
87C22×C42C2311yescharacteristic
88#447—24no—
89C4×D8—41no—
90C4×D8—41no—
91C4×D8—41no—
92C4×D8—41no—
93C8×D4—41no—
94C8×D4—41no—
95#431—41no—
96#431—41no—
97#431—41no—
98#431—41no—
99C2×#107C2×C411yescharacteristic
100C2×#107C2×C411yescharacteristic
101C2×C4⋊C8—21no—
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8C2×C411yescharacteristic
132C2×C22⋊C8D411yescharacteristic
133C2×C22⋊C8D411yescharacteristic
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142#390C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#388—21no—
145#386—21no—
146#380C2×C411yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—21no—
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#335—41no—
160#334—41no—
161D4⋊C8—41no—
162D4⋊C8—41no—
163D4⋊C8—41no—
164D4⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167#322—41no—
168#322—41no—
169C25—21no—
170C22×D4—21no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4C22×C411yescharacteristic
179C22×D4C22×C411yescharacteristic
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—81no—
190C23×C4—81no—
191C23×C4—81no—
192C23×C4—81no—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4#4111yescharacteristic
196C2×D8—22no—
197C2×D8—22no—
198C2×D8—22no—
199C2×D8—22no—
200C2×D8—41no—
201C2×D8—41no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—42no—
209C22×C8—81no—
210C22×C8—81no—
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219#121—22no—
220#121M1614yes—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—81no—
242C4×D4—81no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42C2411yescharacteristic
272#107—41no—
273#107—41no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—41no—
283C4⋊C8—41no—
284C4⋊C8—81no—
285#102—22no—
286#102—22no—
287#102—22no—
288#102—22no—
289#102—41no—
290#102—41no—
291#102—41no—
292#102—41no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300C22⋊C8—81no—
301C22⋊C8—81no—
302C22⋊C8—81no—
303C22⋊C8—81no—
304C22⋊C8—81no—
305C22⋊C8—81no—
306C22⋊C8—81no—
307C22⋊C8—81no—
308C22⋊C8—81no—
309C22⋊C8—81no—
310C22⋊C8—81no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—42no—
324C4×C8—81no—
325#95—21no—
326#95—21no—
327#95—21no—
328C24—24no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—42no—
333C24—42no—
334C24—81no—
335C24—81no—
336C24—81no—
337C24—81no—
338C24C4×D411yescharacteristic
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—81no—
363C2×D4—81no—
364C2×D4C2×M1612yes—
365C2×D4#13112yes—
366C2×D4#13112yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
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—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C2×M1612yes—
421C22×C4#13112yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4111yescharacteristic
425D8—44no—
426D8—44no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—84no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4C2×M1612yes—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—41no—
501C42—41no—
502C42H(5,𝔽2)11yescharacteristic
503C42H(5,𝔽2)11yescharacteristic
504C42H(5,𝔽2)11yescharacteristic
505C42H(5,𝔽2)11yescharacteristic
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—48no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23#43512yes—
541C23C2×C4×D411yescharacteristic
542D4—22no—
543D4—22no—
544D4—22no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—48no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—84no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4#43512yes—
607C2×C4#51811yescharacteristic
608C2×C4#51811yescharacteristic
609C2×C4#53411yescharacteristic
610C2×C4#53511yescharacteristic
611C2×C4#53811yescharacteristic
612C2×C4#53811yescharacteristic
613C2×C4C22×M1611yescharacteristic
614C2×C4C2×#13111yescharacteristic
615C2×C4C2×#13111yescharacteristic
616C2×C4C2×#13111yescharacteristic
617C2×C4#57611yescharacteristic
618C2×C4#57611yescharacteristic
619C2×C4#57611yescharacteristic
620C2×C4#57711yescharacteristic
621C8—42no—
622C8—41no—
623C8—41no—
624C8—41no—
625C8—41no—
626C8—41no—
627C8—41no—
628C8—41no—
629C8—41no—
630C8—81no—
631C8—81no—
632C8—81no—
633C8—81no—
634C8—81no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—816no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22#232111yescharacteristic
665C22C2×#43511yescharacteristic
666C22#292911yescharacteristic
667C22#294611yescharacteristic
668C22#294811yescharacteristic
669C22#295611yescharacteristic
670C22#295711yescharacteristic
671C4—22no—
672C4—21no—
673C4—21no—
674C4—41no—
675C4—41no—
676C4—42no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4#297411yescharacteristic
682C4#297411yescharacteristic
683C4#297711yescharacteristic
684C4#297711yescharacteristic
685C4#319811yescharacteristic
686C4#319911yescharacteristic
687C4#322011yescharacteristic
688C4#322411yescharacteristic
689C2—42no—
690C2—82no—
691C2—82no—
692C2—82no—
693C2#1787511yescharacteristic
694C2#1787711yescharacteristic
695C2#1791811yescharacteristic
696C2#1792211yescharacteristic
697C2#2203011yescharacteristic
698C2#2205611yescharacteristic
699C2#2206211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#447C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22133C2semidirect
2. #21786C2semidirect
4. #9665C2semidirect
5. #9663C2semidirect
6. #9560C2semidirect
7. #9558C2semidirect
8. #9542C2semidirect
9. #9541C2semidirect
10. #8947C2semidirect
11. #8947C2semidirect
12. #8940C2semidirect
13. #8831C2semidirect
14. #8310C2semidirect
15. #8149C2semidirect
20. C2×C8×D4C22semidirect
31. #1954C22semidirect
32. #1939C22semidirect
46. C8×C4⋊C4C22semidirect
51. C2×#335C22semidirect
52. C2×#334C22semidirect
57. C2×D4⋊C8C22semidirect
58. C2×D4⋊C8C22semidirect
59. C2×#322C22semidirect
60. C2×#322C22semidirect
61. #1279C22semidirect
62. #1279C22semidirect
63. #1277C22semidirect
64. #1277C22semidirect
65. #1273C22semidirect
66. #1273C22semidirect

Maximal subgroups

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

subgroupindex
#221332
#217862
C4×#4472
#96652
#96632
#95602
#95582
#95422
#95412
#89472
#89402
#88312
#83102
#81492