Finite Group Databasebeta

← #227698

Subgroups — #227698

Order 512.

1049 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22989C211yesmaximal, characteristic, complemented
2D4×C22⋊C8C211yesmaximal, characteristic, complemented
3C4×#447C211yesmaximal, characteristic
4C2×#1586C211yesmaximal, characteristic, complemented
5C22×D4⋊C8C211yesmaximal, characteristic, complemented
6#9665C211yesmaximal, characteristic, complemented
7#9560C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9072C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8830C211yesmaximal, characteristic, complemented
14#8204C211yesmaximal, characteristic, complemented
15#8144C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#447C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#436C2211yescharacteristic, complemented
21C2×#435—21no—
22C2×C8×D4C2211yescharacteristic
23C2×C8×D4C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C22⋊C8—21no—
26C22×C4×C8C2211yescharacteristic
27C4×#121C2211yescharacteristic
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#1851—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
54C2×D4⋊C8—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—
77C23×C8—21no—
78C2×#121C2×C411yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—41no—
85C2×C4×D4—41no—
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4C2311yescharacteristic
89C2×C4×D4C2311yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C22×C4⋊C4C2×C411yescharacteristic
92C22×C22⋊C4—21no—
93C22×C42C2311yescharacteristic
94#447—24no—
95C4×D8—22no—
96C4×D8—22no—
97C4×D8—41no—
98C4×D8—41no—
99#436—22no—
100#436—41no—
101#435—44no—
102C8×D4—22no—
103C8×D4—22no—
104C8×D4—41no—
105C8×D4—41no—
106C2×#107C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×C4⋊C8—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—41no—
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
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—44no—
135C2×C22⋊C8C2×C411yescharacteristic
136C2×C22⋊C8C2×C411yescharacteristic
137C2×C22⋊C8D411yescharacteristic
138C2×C22⋊C8D411yescharacteristic
139C2×#97—21no—
140C2×C4×C8—22no—
141C2×C4×C8—21no—
142C2×C4×C8—41no—
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—22no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4M1614yescomplemented
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—22no—
207C2×D8—22no—
208C2×D8—41no—
209C2×D8—41no—
210C2×M16—42no—
211C2×M16—42no—
212C2×M16—81no—
213C22×C8—21no—
214C22×C8—24no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—42no—
221C22×C8—42no—
222C22×C8—42no—
223C22×C8—42no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—81no—
229C22×C8—81no—
230C22×C8—81no—
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C411yescharacteristic
234C22×C8C22×C411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239#121—22no—
240#121—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—42no—
254C4×D4—42no—
255C4×D4—41no—
256C4×D4—41no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4C2×D412yes—
262C4×D4#4112yes—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
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—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42C22×C411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42C2411yescharacteristic
292#107—22no—
293#107—41no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—42no—
302C4⋊C8—42no—
303C4⋊C8C2×D412yes—
304#102—22no—
305#102—22no—
306#102—22no—
307#102—22no—
308#102—22no—
309#102—41no—
310#102—41no—
311#102—41no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—42no—
323C22⋊C8—81no—
324C22⋊C8—81no—
325C22⋊C8—81no—
326#97—42no—
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×C8—21no—
338C4×C8—42no—
339C4×C8C2×D412yes—
340#95—21no—
341#95—21no—
342#95—21no—
343C24—21no—
344C24—21no—
345C24—24no—
346C24—21no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—81no—
352C24—81no—
353C24C4×D411yescharacteristic
354C2×D4—28no—
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—21no—
368C2×D4—21no—
369C2×D4—24no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4C4×D412yes—
379C2×D4C2×M1612yes—
380C2×D4C2×M1612yes—
381C2×D4#13112yes—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
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×C4C2×M1612yes—
434C22×C4#13112yes—
435C22×C4AGL(1,ℤ/8)12yes—
436C22×C4AGL(1,ℤ/8)12yes—
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4111yescharacteristic
440D8—44no—
441D8—44no—
442M16—84no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
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—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—48no—
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—41no—
485C2×C8—41no—
486C2×C8—84no—
487C2×C8—84no—
488C2×C8—84no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4C4×D412yes—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—41no—
520C42—41no—
521C42C22×D411yescharacteristic
522C42C2×#4111yescharacteristic
523C42H(5,𝔽2)11yescharacteristic
524C42H(5,𝔽2)11yescharacteristic
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—48no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—48no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23#43512yes—
558C23#44212yes—
559C23#44312yes—
560C23C2×C4×D411yescharacteristic
561D4—22no—
562D4—22no—
563D4—22no—
564D4—22no—
565D4—22no—
566D4—48no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—84no—
574D4#43514yescomplemented
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
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—41no—
612C2×C4—41no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4#43512yes—
625C2×C4#44212yes—
626C2×C4C2×C4×D411yescharacteristic
627C2×C4#51811yescharacteristic
628C2×C4D4211yescharacteristic
629C2×C4#54611yescharacteristic
630C2×C4#54611yescharacteristic
631C2×C4#55111yescharacteristic
632C2×C4C22×M1611yescharacteristic
633C2×C4C2×#13111yescharacteristic
634C2×C4C2×#13111yescharacteristic
635C2×C4#56811yescharacteristic
636C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
637C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
638C2×C4#57611yescharacteristic
639C2×C4#57711yescharacteristic
640C8—24no—
641C8—42no—
642C8—41no—
643C8—41no—
644C8—41no—
645C8—41no—
646C8—41no—
647C8—42no—
648C8—41no—
649C8—41no—
650C8—41no—
651C8—81no—
652C8—81no—
653C8—81no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—24no—
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—81no—
680C22—81no—
681C22—81no—
682C22#158614yescomplemented
683C22#235811yescharacteristic
684C22C2×#43511yescharacteristic
685C22#293311yescharacteristic
686C22C2×#44211yescharacteristic
687C22#294611yescharacteristic
688C22C2×#44311yescharacteristic
689C22#295611yescharacteristic
690C4—22no—
691C4—22no—
692C4—21no—
693C4—21no—
694C4—42no—
695C4—41no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4C2×#43511yescharacteristic
701C4#293311yescharacteristic
702C4#297311yescharacteristic
703C4#297711yescharacteristic
704C4#326411yescharacteristic
705C4#326511yescharacteristic
706C4#330611yescharacteristic
707C4#331811yescharacteristic
708C2—24no—
709C2—44no—
710C2—82no—
711C2—161no—
712C2C2×#158611yescharacteristic
713C2#1787311yescharacteristic
714C2#1800111yescharacteristic
715C2#1801411yescharacteristic
716C2#2284411yescharacteristic
717C2#2285811yescharacteristic
718C2#2288811yescharacteristic
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. #22989C2semidirect
2. D4×C22⋊C8C2semidirect
4. C2×#1586C2semidirect
5. C22×D4⋊C8C2semidirect
6. #9665C2semidirect
7. #9560C2semidirect
8. #9546C2semidirect
9. #9072C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8845C2semidirect
13. #8830C2semidirect
14. #8204C2semidirect
15. #8144C2semidirect
20. C2×#436C22semidirect
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
185. C22×D4M16semidirect
574. D4#435semidirect
682. C22#1586semidirect

Maximal subgroups

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

subgroupindex
#229892
D4×C22⋊C82
C4×#4472
C2×#15862
C22×D4⋊C82
#96652
#95602
#95462
#90722
#89512
#89472
#88452
#88302
#82042
#81442