Finite Group Databasebeta

← #227708

Subgroups — #227708

Order 512.

990 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23087C211yesmaximal, characteristic, complemented
2D4×C22⋊C8C211yesmaximal, characteristic, complemented
3C4×#447C211yesmaximal, characteristic
4C2×#1586C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic, complemented
7#9560C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9074C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic, complemented
13#8831C211yesmaximal, characteristic, complemented
14#8202C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#447C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#436C2211yescharacteristic, complemented
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C22×C22⋊C8—21no—
26C4×#121C2211yescharacteristic
27#1955—21no—
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic, complemented
31#1939C2211yescharacteristic, complemented
32#1923—21no—
33#1879C411yescharacteristic
34#1879C411yescharacteristic
35#1874C411yescharacteristic
36#1870—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#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1586—24no—
52C2×#335C2211yescharacteristic, complemented
53C2×#334C2211yescharacteristic, complemented
54#1478—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—
76C2×#121C2×C411yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—41no—
83C2×C4×D4—41no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C2×C4×D4C2311yescharacteristic
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C22⋊C4—21no—
91C22×C42C2311yescharacteristic
92#447—24no—
93C4×D8—22no—
94C4×D8—22no—
95C4×D8—41no—
96C4×D8—41no—
97#436—22no—
98#436—41no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—41no—
102C8×D4—41no—
103#431—22no—
104#431—22no—
105#431—41no—
106#431—41no—
107C2×#107C2×C411yescharacteristic
108C2×#107C2×C411yescharacteristic
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
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—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—44no—
136C2×C22⋊C8C2×C411yescharacteristic
137C2×C22⋊C8C2×C411yescharacteristic
138C2×C22⋊C8D411yescharacteristic
139C2×C22⋊C8D411yescharacteristic
140C2×#97—21no—
141C2×C4×C8—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—21no—
180C22×D4—21no—
181C22×D4—22no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4M1614yescomplemented
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×D8—41no—
209C2×M16—42no—
210C2×M16—42no—
211C2×M16—81no—
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—81no—
223C22×C8—81no—
224C22×C8C22×C411yescharacteristic
225C22×C8C22×C411yescharacteristic
226C22×C8C22×C411yescharacteristic
227C22×C8C22×C411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22×C8C2×D411yescharacteristic
231C22×C8C2×D411yescharacteristic
232#121—22no—
233#121—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4—41no—
249C4×D4—41no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—81no—
253C4×D4—81no—
254C4×D4C2×D412yes—
255C4×D4#4112yes—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
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—42no—
275C2×C22⋊C4—42no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42C22×C411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42C2411yescharacteristic
285#107—22no—
286#107—41no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—42no—
295C4⋊C8—81no—
296C4⋊C8C2×D412yes—
297#102—22no—
298#102—22no—
299#102—22no—
300#102—22no—
301#102—22no—
302#102—41no—
303#102—41no—
304#102—41no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—41no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—42no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—81no—
319C22⋊C8—81no—
320C22⋊C8—81no—
321#97—81no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
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—42no—
334C4×C8C2×D412yes—
335#95—21no—
336#95—21no—
337#95—21no—
338C24—21no—
339C24—21no—
340C24—24no—
341C24—21no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—81no—
347C24—81no—
348C24C4×D411yescharacteristic
349C2×D4—28no—
350C2×D4—21no—
351C2×D4—21no—
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—24no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4C4×D412yes—
374C2×D4C2×M1612yes—
375C2×D4C2×M1612yes—
376C2×D4#13112yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
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—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
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—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4C2×M1612yes—
431C22×C4#13112yes—
432C22×C4C23×C411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4111yescharacteristic
435D8—44no—
436D8—44no—
437M16—84no—
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—22no—
447C2×C8—21no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
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—41no—
464C2×C8—41no—
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—84no—
480C2×C8—84no—
481C2×C8AGL(1,ℤ/8)12yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—81no—
493C4⋊C4C4×D412yes—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—42no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—41no—
513C42—41no—
514C42C22×D411yescharacteristic
515C42C2×#4111yescharacteristic
516C42H(5,𝔽2)11yescharacteristic
517C42H(5,𝔽2)11yescharacteristic
518C23—22no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—22no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—48no—
529C23—48no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
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—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23#43512yes—
553C23C2×C4×D411yescharacteristic
554D4—22no—
555D4—22no—
556D4—22no—
557D4—22no—
558D4—22no—
559D4—48no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—84no—
567D4#43514yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4#43512yes—
619C2×C4C2×C4×D411yescharacteristic
620C2×C4#51811yescharacteristic
621C2×C4D4211yescharacteristic
622C2×C4#54611yescharacteristic
623C2×C4#54611yescharacteristic
624C2×C4#55111yescharacteristic
625C2×C4C22×M1611yescharacteristic
626C2×C4C2×#13111yescharacteristic
627C2×C4C2×#13111yescharacteristic
628C2×C4#56811yescharacteristic
629C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
630C2×C4#57511yescharacteristic
631C2×C4#57611yescharacteristic
632C2×C4#57611yescharacteristic
633C8—22no—
634C8—42no—
635C8—41no—
636C8—41no—
637C8—41no—
638C8—41no—
639C8—41no—
640C8—42no—
641C8—41no—
642C8—41no—
643C8—81no—
644C8—81no—
645C8—81no—
646C8—81no—
647C22—22no—
648C22—24no—
649C22—22no—
650C22—22no—
651C22—41no—
652C22—42no—
653C22—41no—
654C22—42no—
655C22—41no—
656C22—42no—
657C22—41no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—816no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22#235811yescharacteristic
677C22C2×#43511yescharacteristic
678C22#293311yescharacteristic
679C22C4×AGL(1,ℤ/8)11yescharacteristic
680C22#294811yescharacteristic
681C22#295511yescharacteristic
682C22#295711yescharacteristic
683C4—22no—
684C4—21no—
685C4—21no—
686C4—41no—
687C4—42no—
688C4—41no—
689C4—81no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4C2×#43511yescharacteristic
694C4#293311yescharacteristic
695C4#297311yescharacteristic
696C4#297711yescharacteristic
697C4#327311yescharacteristic
698C4#327411yescharacteristic
699C4#331811yescharacteristic
700C4#331911yescharacteristic
701C2—42no—
702C2—44no—
703C2—82no—
704C2—161no—
705C2#1786911yescharacteristic
706C2#1787711yescharacteristic
707C2#1800611yescharacteristic
708C2#1802611yescharacteristic
709C2#2284411yescharacteristic
710C2#2286811yescharacteristic
711C2#2288911yescharacteristic
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. #23087C2semidirect
2. D4×C22⋊C8C2semidirect
4. C2×#1586C2semidirect
5. C2×#1478C2semidirect
6. #9663C2semidirect
7. #9560C2semidirect
8. #9546C2semidirect
9. #9074C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8843C2semidirect
13. #8831C2semidirect
14. #8202C2semidirect
15. #8149C2semidirect
20. C2×#436C22semidirect
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
184. C22×D4M16semidirect

Maximal subgroups

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

subgroupindex
#230872
D4×C22⋊C82
C4×#4472
C2×#15862
C2×#14782
#96632
#95602
#95462
#90742
#89512
#89472
#88432
#88312
#82022
#81492