Finite Group Databasebeta

← #217764

Subgroups — #217764

Order 512.

946 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#21486C211yesmaximal, characteristic, complemented
2C4×#447C211yesmaximal, characteristic
3#21190C211yesmaximal, characteristic, complemented
4#9665C211yesmaximal, characteristic, complemented
5#9659C211yesmaximal, characteristic, complemented
6#9560C211yesmaximal, characteristic, complemented
7#9556C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9546C211yesmaximal, characteristic, complemented
10#8947C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8944C211yesmaximal, characteristic, complemented
13#8833C211yesmaximal, characteristic, complemented
14#8310C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#447C411yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#435—21no—
21C2×C8×D4C2211yescharacteristic, complemented
22C2×#423C2211yescharacteristic
23C2×#407—21no—
24C2×#407—21no—
25C2×C4×M16C2211yescharacteristic
26C4×#121C2211yescharacteristic
27#1955—21no—
28#1955—21no—
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic, complemented
34#1939C2211yescharacteristic, complemented
35#1924—21no—
36#1879C411yescharacteristic
37#1879C411yescharacteristic
38#1874C411yescharacteristic
39#1870—21no—
40#1862C411yescharacteristic
41#1862C411yescharacteristic
42#1852C411yescharacteristic
43#1797C2211yescharacteristic
44#1786C411yescharacteristic
45#1770C2211yescharacteristic, complemented
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1752—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic, 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
71C22×M16—21no—
72C22×M16D411yescharacteristic
73C22×M16D411yescharacteristic
74C23×C8—21no—
75C2×#121C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—41no—
80C2×C4×D4—41no—
81C2×C4×D4—41no—
82C2×C4×D4—41no—
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2311yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C42C2311yescharacteristic
91#447—24no—
92C4×D8—41no—
93C4×D8—41no—
94C4×D8—41no—
95C4×D8—41no—
96#435—44no—
97C8×D4—41no—
98C8×D4—41no—
99C2×#107C2×C411yescharacteristic
100C2×#107C2×C411yescharacteristic
101#423—41no—
102#423—41no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117#407—44no—
118#407—44no—
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⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8C2×C411yescharacteristic
133C2×C22⋊C8C2×C411yescharacteristic
134C4×M16—41no—
135C4×M16—41no—
136C2×#97—21no—
137C2×#97D411yescharacteristic
138C2×#97D411yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#390C2×C411yescharacteristic
146#389C2×C411yescharacteristic
147#388—21no—
148#386—21no—
149#380C2×C411yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C22⋊C4—21no—
152#340C2×C411yescharacteristic
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336C2×C411yescharacteristic
161#336C2×C411yescharacteristic
162#334—41no—
163#334—41no—
164D4⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168D4⋊C8—41no—
169D4⋊C8—41no—
170#322—41no—
171#322—41no—
172C25—21no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4C22×C411yescharacteristic
182C22×D4C22×C411yescharacteristic
183C22×D4C22×C411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C22×C411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4#4111yescharacteristic
199C2×D8—22no—
200C2×D8—22no—
201C2×D8—41no—
202C2×D8—41no—
203C2×D8—41no—
204C2×D8M1614yescomplemented
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—41no—
209C2×M16—41no—
210C2×M16—41no—
211C2×M16—42no—
212C2×M16—42no—
213C2×M16—42no—
214C22×C8—21no—
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×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C22×C411yescharacteristic
226C22×C8C22×C411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22×C8C2×D411yescharacteristic
231#121—22no—
232#121—41no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
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—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—41no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—42no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C22⋊C4—21no—
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—42no—
274C2×C22⋊C4—42no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42C2411yescharacteristic
284#107—41no—
285#107—41no—
286C4⋊C8—21no—
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—42no—
296#102—22no—
297#102—22no—
298#102—22no—
299#102—41no—
300#102—41no—
301#102—41no—
302#102—41no—
303#102—41no—
304C22⋊C8—22no—
305C22⋊C8—41no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308C22⋊C8—42no—
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—42no—
319#97—41no—
320#97—41no—
321#97—42no—
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—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—81no—
346C24—81no—
347C24C4×D411yescharacteristic
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—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—81no—
373C2×D4C2×M1612yes—
374C2×D4C2×M1612yes—
375C2×D4#13112yes—
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—21no—
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—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
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—42no—
421C22×C4—42no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4C4×D411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4AGL(1,ℤ/8)12yes—
430C22×C4AGL(1,ℤ/8)12yes—
431C22×C4C23×C411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4111yescharacteristic
434D8—44no—
435D8—44no—
436M16—44no—
437M16—44no—
438M16—84no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—22no—
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—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—41no—
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×C8C2×M1612yes—
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—21no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—41no—
513C42—41no—
514C42H(5,𝔽2)11yescharacteristic
515C42H(5,𝔽2)11yescharacteristic
516C42H(5,𝔽2)11yescharacteristic
517C42H(5,𝔽2)11yescharacteristic
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—48no—
528C23—41no—
529C23—41no—
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—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23#44212yes—
553C23C2×C4×D411yescharacteristic
554D4—22no—
555D4—22no—
556D4—22no—
557D4—22no—
558D4—22no—
559D4—22no—
560D4—48no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—84no—
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—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
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—81no—
602C2×C4—81no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
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#44212yes—
619C2×C4#51811yescharacteristic
620C2×C4#51811yescharacteristic
621C2×C4#53411yescharacteristic
622C2×C4#53511yescharacteristic
623C2×C4#53811yescharacteristic
624C2×C4#53811yescharacteristic
625C2×C4C22×M1611yescharacteristic
626C2×C4C2×#13111yescharacteristic
627C2×C4#56811yescharacteristic
628C2×C4#56811yescharacteristic
629C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
630C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
631C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
632C2×C4#57511yescharacteristic
633C8—22no—
634C8—22no—
635C8—42no—
636C8—41no—
637C8—41no—
638C8—41no—
639C8—41no—
640C8—41no—
641C8—41no—
642C8—81no—
643C8—81no—
644C8—81no—
645C8—81no—
646C8—81no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—816no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22#232111yescharacteristic
677C22#293311yescharacteristic
678C22D4×M1611yescharacteristic
679C22C2×#44211yescharacteristic
680C22C4×AGL(1,ℤ/8)11yescharacteristic
681C22#295611yescharacteristic
682C22#295711yescharacteristic
683C4—22no—
684C4—21no—
685C4—21no—
686C4—41no—
687C4—41no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4#297311yescharacteristic
694C4#297411yescharacteristic
695C4#297611yescharacteristic
696C4#297811yescharacteristic
697C4#319211yescharacteristic
698C4#319411yescharacteristic
699C4#321311yescharacteristic
700C4#321711yescharacteristic
701C2—42no—
702C2—82no—
703C2—82no—
704C2—82no—
705C2#1800111yescharacteristic
706C2#1800611yescharacteristic
707C2#1804411yescharacteristic
708C2#1804611yescharacteristic
709C2#2203611yescharacteristic
710C2#2204611yescharacteristic
711C2#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×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21486C2semidirect
3. #21190C2semidirect
4. #9665C2semidirect
5. #9659C2semidirect
6. #9560C2semidirect
7. #9556C2semidirect
8. #9546C2semidirect
9. #9546C2semidirect
10. #8947C2semidirect
11. #8947C2semidirect
12. #8944C2semidirect
13. #8833C2semidirect
14. #8310C2semidirect
15. #8147C2semidirect
21. C2×C8×D4C22semidirect
33. #1954C22semidirect
34. #1939C22semidirect
45. #1770C22semidirect
51. C2×#334C22semidirect
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
204. C2×D8M16semidirect

Maximal subgroups

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

subgroupindex
#214862
C4×#4472
#211902
#96652
#96592
#95602
#95562
#95462
#89472
#89442
#88332
#83102
#81472