Finite Group Databasebeta

← #228018

Subgroups — #228018

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22986C211yesmaximal, characteristic, complemented
2#22401C211yesmaximal, characteristic, complemented
3C4×#447C211yesmaximal, characteristic
4C2×#1586C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic, complemented
6#9659C211yesmaximal, characteristic, complemented
7#9560C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9076C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8849C211yesmaximal, characteristic, complemented
13#8833C211yesmaximal, characteristic, complemented
14#8208C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#447C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#435—21no—
23C2×C8×D4C2211yescharacteristic, complemented
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#407—21no—
27C4×#121C2211yescharacteristic
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic, complemented
31#1939C2211yescharacteristic, complemented
32#1924—21no—
33#1879C411yescharacteristic
34#1879C411yescharacteristic
35#1874C411yescharacteristic
36#1870—21no—
37#1870—21no—
38#1870—21no—
39#1862C411yescharacteristic
40#1862C411yescharacteristic
41#1852C411yescharacteristic
42#1850—21no—
43#1797C2211yescharacteristic
44#1786C411yescharacteristic
45#1779—21no—
46#1770C2211yescharacteristic, complemented
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1586—24no—
52C2×#334C2211yescharacteristic, complemented
53C2×#334C2211yescharacteristic, complemented
54#1486—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—
74C22×M16—21no—
75C22×M16—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—
98#436—22no—
99#436—22no—
100#436—41no—
101#436—41no—
102#435—44no—
103C8×D4—22no—
104C8×D4—41no—
105#431—22no—
106#431—41no—
107C2×#107C2×C411yescharacteristic
108C2×#107C2×C411yescharacteristic
109#423—22no—
110#423—41no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×#102C2×C411yescharacteristic
125C2×#102C2×C411yescharacteristic
126#407—44no—
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—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8C2×C411yescharacteristic
139C2×C22⋊C8C2×C411yescharacteristic
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×#97—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#390C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#388—21no—
152#386—21no—
153#380C2×C411yescharacteristic
154C4×C4⋊C4C2311yescharacteristic
155C4×C22⋊C4—21no—
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#336—21no—
161#336—21no—
162#336—21no—
163#336C2×C411yescharacteristic
164#336C2×C411yescharacteristic
165#336D411yescharacteristic
166#336D411yescharacteristic
167#334—22no—
168#334—41no—
169D4⋊C8—22no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—22no—
173D4⋊C8—41no—
174D4⋊C8D414yescomplemented
175#322—22no—
176#322—41no—
177C25—21no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4C22×C411yescharacteristic
187C22×D4C22×C411yescharacteristic
188C22×D4C22×C411yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—81no—
200C23×C4—81no—
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4#4111yescharacteristic
204C2×D8—22no—
205C2×D8—22no—
206C2×D8—41no—
207C2×D8—41no—
208C2×D8—41no—
209C2×D8M1614yescomplemented
210C2×M16—42no—
211C2×M16—42no—
212C2×M16—42no—
213C2×M16—42no—
214C2×M16—42no—
215C2×M16—42no—
216C2×M16—81no—
217C2×M16—81no—
218C2×M16—81no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8—81no—
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C22×C411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235C22×C8C2×D411yescharacteristic
236#121—22no—
237#121—41no—
238C4×D4—22no—
239C4×D4—22no—
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—21no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4C2×D412yes—
259C4×D4#4112yes—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
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—21no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42C2411yescharacteristic
289#107—22no—
290#107—41no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301C4⋊C8C2×D412yes—
302#102—22no—
303#102—22no—
304#102—22no—
305#102—22no—
306#102—41no—
307#102—41no—
308#102—41no—
309#102—41no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—41no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—81no—
323C22⋊C8—81no—
324C22⋊C8—81no—
325#97—42no—
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×C8—21no—
337C4×C8C2×D412yes—
338#95—21no—
339#95—21no—
340#95—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—81no—
350C24—81no—
351C24C4×D411yescharacteristic
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—21no—
366C2×D4—21no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—81no—
377C2×D4C4×D412yes—
378C2×D4C2×M1612yes—
379C2×D4C2×M1612yes—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
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—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
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—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—81no—
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4AGL(1,ℤ/8)12yes—
434C22×C4AGL(1,ℤ/8)12yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438D8—44no—
439D8—44no—
440M16—84no—
441M16—84no—
442M16—84no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—22no—
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—21no—
466C2×C8—21no—
467C2×C8—21no—
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—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—84no—
485C2×C8C2×M1612yes—
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—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4C4×D412yes—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—42no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—41no—
517C42—41no—
518C42C22×D411yescharacteristic
519C42C2×#4111yescharacteristic
520C42H(5,𝔽2)11yescharacteristic
521C42H(5,𝔽2)11yescharacteristic
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—48no—
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—41no—
548C23—41no—
549C23—41no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23#44212yes—
557C23C2×C4×D411yescharacteristic
558D4—22no—
559D4—22no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4—22no—
564D4—48no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—84no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
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—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—81no—
610C2×C4—81no—
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×C4#44212yes—
623C2×C4C2×C4×D411yescharacteristic
624C2×C4#51811yescharacteristic
625C2×C4D4211yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#54611yescharacteristic
628C2×C4#55111yescharacteristic
629C2×C4C22×M1611yescharacteristic
630C2×C4#56811yescharacteristic
631C2×C4#56811yescharacteristic
632C2×C4#56811yescharacteristic
633C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
634C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
635C2×C4#57611yescharacteristic
636C2×C4#57711yescharacteristic
637C8—22no—
638C8—42no—
639C8—41no—
640C8—41no—
641C8—41no—
642C8—41no—
643C8—42no—
644C8—41no—
645C8—41no—
646C8—41no—
647C8—81no—
648C8—81no—
649C8—81no—
650C8—81no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—816no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22#235811yescharacteristic
681C22#293011yescharacteristic
682C22D4×M1611yescharacteristic
683C22C2×#44211yescharacteristic
684C22#294611yescharacteristic
685C22#295511yescharacteristic
686C22#295711yescharacteristic
687C4—22no—
688C4—21no—
689C4—21no—
690C4—41no—
691C4—42no—
692C4—41no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4#293011yescharacteristic
698C4D4×M1611yescharacteristic
699C4#297311yescharacteristic
700C4#297811yescharacteristic
701C4#326411yescharacteristic
702C4#326511yescharacteristic
703C4#330611yescharacteristic
704C4#331811yescharacteristic
705C2—42no—
706C2—82no—
707C2—82no—
708C2—82no—
709C2#1793511yescharacteristic
710C2#1794611yescharacteristic
711C2#1804411yescharacteristic
712C2#1805011yescharacteristic
713C2#2285011yescharacteristic
714C2#2285811yescharacteristic
715C2#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. #22986C2semidirect
2. #22401C2semidirect
4. C2×#1586C2semidirect
5. C2×#1486C2semidirect
6. #9659C2semidirect
7. #9560C2semidirect
8. #9546C2semidirect
9. #9076C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8849C2semidirect
13. #8833C2semidirect
14. #8208C2semidirect
15. #8149C2semidirect
23. C2×C8×D4C22semidirect
30. #1954C22semidirect
31. #1939C22semidirect
46. #1770C22semidirect
52. C2×#334C22semidirect
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
174. D4⋊C8D4semidirect
209. C2×D8M16semidirect

Maximal subgroups

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

subgroupindex
#229862
#224012
C4×#4472
C2×#15862
C2×#14862
#96592
#95602
#95462
#90762
#89512
#89472
#88492
#88332
#82082
#81492