Finite Group Databasebeta

← #227900

Subgroups — #227900

Order 512.

1009 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23084C211yesmaximal, characteristic, complemented
2D4×C22⋊C8C211yesmaximal, characteristic, complemented
3C4×#447C211yesmaximal, characteristic
4C2×#1580C211yesmaximal, characteristic, complemented
5C2×#1490C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic, complemented
7#9556C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9078C211yesmaximal, characteristic, complemented
10#8947C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic, complemented
13#8834C211yesmaximal, characteristic, complemented
14#8202C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#447C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic, complemented
23C2×#432C2211yescharacteristic
24C22×C22⋊C8—21no—
25C2×C4×M16C2211yescharacteristic
26C4×#121C2211yescharacteristic
27#1955—21no—
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic, complemented
31#1939C2211yescharacteristic, complemented
32#1924—21no—
33#1879C411yescharacteristic
34#1879C411yescharacteristic
35#1874C411yescharacteristic
36#1869—21no—
37#1869—21no—
38#1869—21no—
39#1862C411yescharacteristic
40#1862C411yescharacteristic
41#1852C411yescharacteristic
42#1797C2211yescharacteristic
43#1786C411yescharacteristic
44#1779—21no—
45#1770C2211yescharacteristic, complemented
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1752—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1580—24no—
52C2×#335C2211yescharacteristic, complemented
53C2×#334C2211yescharacteristic, complemented
54#1490—24no—
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic, complemented
60C2×D4⋊C8C2211yescharacteristic, complemented
61C2×#322C2211yescharacteristic, complemented
62C2×#322C2211yescharacteristic, complemented
63#1279C2211yescharacteristic, complemented
64#1279C2211yescharacteristic, complemented
65#1277C2211yescharacteristic, complemented
66#1277C2211yescharacteristic, complemented
67#1273C2211yescharacteristic, complemented
68#1273C2211yescharacteristic, complemented
69C23×D4C2×C411yescharacteristic
70C24×C4—21no—
71C22×D8C2×C411yescharacteristic
72C22×D8C2×C411yescharacteristic
73C22×M16—21no—
74C23×C8—21no—
75C23×C8—21no—
76C23×C8—21no—
77C2×#121C2×C411yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—41no—
84C2×C4×D4—41no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×C4×D4C2311yescharacteristic
89C2×C4×D4C2311yescharacteristic
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C22⋊C4—21no—
92C22×C42C2311yescharacteristic
93#447—24no—
94C4×D8—22no—
95C4×D8—22no—
96C4×D8—41no—
97C4×D8—41no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—41no—
102C8×D4—41no—
103C8×D4—41no—
104#432—22no—
105#432—41no—
106C2×#107C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
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⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—44no—
133C2×C22⋊C8C2×C411yescharacteristic
134C2×C22⋊C8C2×C411yescharacteristic
135C2×C22⋊C8D411yescharacteristic
136C2×C22⋊C8D411yescharacteristic
137C4×M16—22no—
138C4×M16—41no—
139C2×#97—21no—
140C2×#97—21no—
141C2×#97—21no—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#390C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#388—21no—
151#386—21no—
152#380C2×C411yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154C4×C22⋊C4—21no—
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#336—21no—
160#336—21no—
161#336—21no—
162#336C2×C411yescharacteristic
163#336C2×C411yescharacteristic
164#336D411yescharacteristic
165#336D411yescharacteristic
166#335—41no—
167#334—22no—
168D4⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—41no—
173D4⋊C8D414yescomplemented
174#322—22no—
175#322—41no—
176C25—21no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4C22×C411yescharacteristic
186C22×D4C22×C411yescharacteristic
187C22×D4C22×C411yescharacteristic
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—81no—
199C23×C4—81no—
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4#4111yescharacteristic
203C2×D8—22no—
204C2×D8—22no—
205C2×D8—22no—
206C2×D8—22no—
207C2×D8—41no—
208C2×D8M1614yescomplemented
209C2×M16—42no—
210C2×M16—42no—
211C2×M16—42no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8—42no—
221C22×C8—42no—
222C22×C8—42no—
223C22×C8—42no—
224C22×C8—81no—
225C22×C8—81no—
226C22×C8—81no—
227C22×C8C22×C411yescharacteristic
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235#121—22no—
236#121—41no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—41no—
252C4×D4—41no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—81no—
256C4×D4—81no—
257C4×D4C2×D412yes—
258C4×D4#4112yes—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C22×C411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42C2411yescharacteristic
288#107—22no—
289#107—41no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—81no—
298C4⋊C8C2×D412yes—
299#102—22no—
300#102—22no—
301#102—22no—
302#102—22no—
303#102—22no—
304#102—22no—
305#102—41no—
306#102—41no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—41no—
310C22⋊C8—41no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—42no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—81no—
320C22⋊C8—81no—
321C22⋊C8—81no—
322#97—42no—
323#97—42no—
324#97—81no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8C2×D412yes—
337#95—21no—
338#95—21no—
339#95—21no—
340C24—21no—
341C24—24no—
342C24—21no—
343C24—21no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—81no—
349C24—81no—
350C24C4×D411yescharacteristic
351C2×D4—24no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—81no—
374C2×D4C4×D412yes—
375C2×D4C2×M1612yes—
376C2×D4C2×M1612yes—
377C2×D4#13112yes—
378C2×D4#13112yes—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4111yescharacteristic
437D8—28no—
438D8—44no—
439M16—48no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—22no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—84no—
481C2×C8—84no—
482C2×C8—84no—
483C2×C8C2×M1612yes—
484C2×C8AGL(1,ℤ/8)12yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4C4×D412yes—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—41no—
517C42C22×D411yescharacteristic
518C42C2×#4111yescharacteristic
519C42H(5,𝔽2)11yescharacteristic
520C42H(5,𝔽2)11yescharacteristic
521C23—22no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—21no—
526C23—22no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—48no—
533C23—48no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23#44412yes—
556C23C2×C4×D411yescharacteristic
557D4—22no—
558D4—22no—
559D4—22no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4—22no—
564D4—48no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—84no—
570D4#43514yes—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
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×C4C2×C4×D411yescharacteristic
623C2×C4#51811yescharacteristic
624C2×C4D4211yescharacteristic
625C2×C4#54611yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#55111yescharacteristic
628C2×C4C22×M1611yescharacteristic
629C2×C4C2×#13111yescharacteristic
630C2×C4C2×#13111yescharacteristic
631C2×C4#56811yescharacteristic
632C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
633C2×C4#57511yescharacteristic
634C2×C4#57611yescharacteristic
635C2×C4#57611yescharacteristic
636C8—22no—
637C8—22no—
638C8—42no—
639C8—42no—
640C8—41no—
641C8—41no—
642C8—41no—
643C8—41no—
644C8—42no—
645C8—41no—
646C8—41no—
647C8—81no—
648C8—81no—
649C8—81no—
650C22—24no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—42no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—42no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—816no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22#235811yescharacteristic
680C22#292911yescharacteristic
681C22D4×M1611yescharacteristic
682C22C4×AGL(1,ℤ/8)11yescharacteristic
683C22#294811yescharacteristic
684C22C2×#44411yescharacteristic
685C22#295611yescharacteristic
686C4—21no—
687C4—21no—
688C4—41no—
689C4—41no—
690C4—42no—
691C4—41no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4C2×#43511yescharacteristic
697C4#293311yescharacteristic
698C4#297411yescharacteristic
699C4#297611yescharacteristic
700C4#327311yescharacteristic
701C4#327411yescharacteristic
702C4#331811yescharacteristic
703C4#331911yescharacteristic
704C2—42no—
705C2—44no—
706C2—82no—
707C2—82no—
708C2#1791411yescharacteristic
709C2#1792211yescharacteristic
710C2#1804611yescharacteristic
711C2#1806411yescharacteristic
712C2#2284611yescharacteristic
713C2#2286811yescharacteristic
714C2#2288611yescharacteristic
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 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23084C2semidirect
2. D4×C22⋊C8C2semidirect
4. C2×#1580C2semidirect
5. C2×#1490C2semidirect
6. #9663C2semidirect
7. #9556C2semidirect
8. #9546C2semidirect
9. #9078C2semidirect
10. #8947C2semidirect
11. #8947C2semidirect
12. #8843C2semidirect
13. #8834C2semidirect
14. #8202C2semidirect
15. #8147C2semidirect
22. C2×C8×D4C22semidirect
30. #1954C22semidirect
31. #1939C22semidirect
45. #1770C22semidirect
52. C2×#335C22semidirect
53. C2×#334C22semidirect
59. C2×D4⋊C8C22semidirect
60. C2×D4⋊C8C22semidirect
61. C2×#322C22semidirect
62. C2×#322C22semidirect
63. #1279C22semidirect
64. #1279C22semidirect
65. #1277C22semidirect
66. #1277C22semidirect
67. #1273C22semidirect
68. #1273C22semidirect
173. D4⋊C8D4semidirect
208. C2×D8M16semidirect

Maximal subgroups

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

subgroupindex
#230842
D4×C22⋊C82
C4×#4472
C2×#15802
C2×#14902
#96632
#95562
#95462
#90782
#89472
#88432
#88342
#82022
#81472