Finite Group Databasebeta

← #209209

Subgroups — #209209

Order 512.

1240 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×M16C211yesmaximal, characteristic
2#21596C211yesmaximal, characteristic, complemented
3C4×#450C211yesmaximal, characteristic
4C2×#1583C211yesmaximal, characteristic, complemented
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#8949C211yesmaximal, characteristic, complemented
7#8945C211yesmaximal, characteristic, complemented
8#8942C211yesmaximal, characteristic, complemented
9#8849C211yesmaximal, characteristic, complemented
10#8845C211yesmaximal, characteristic, complemented
11#8833C211yesmaximal, characteristic, complemented
12#8310C211yesmaximal, characteristic
13#8206C211yesmaximal, characteristic, complemented
14#8202C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C23×M16—21no—
17C22×C4×D4C2211yescharacteristic
18C2×#450C411yescharacteristic
19C2×C8×Q8C2211yescharacteristic, complemented
20C2×C4×SD16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22D4×M16—24no—
23D4×M16—24no—
24C2×#436C2211yescharacteristic
25C2×#436C2211yescharacteristic
26C2×#435—21no—
27C2×#435—21no—
28C2×C8×D4C2211yescharacteristic
29C2×C8×D4C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C2×#407—21no—
32C2×C4×M16C2211yescharacteristic
33C4×C22⋊Q8C2211yescharacteristic
34#1984C2211yescharacteristic, complemented
35#1939C2211yescharacteristic, complemented
36#1917—21no—
37#1877C411yescharacteristic
38#1877C411yescharacteristic
39#1872C411yescharacteristic
40#1864C411yescharacteristic
41#1864C411yescharacteristic
42#1853C411yescharacteristic
43#1851—21no—
44#1797C2211yescharacteristic
45#1786C411yescharacteristic
46#1771—21no—
47#1770C2211yescharacteristic, complemented
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1753—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1583—24no—
54C8×SD16—24no—
55C2×#334C2211yescharacteristic, complemented
56C2×#334C2211yescharacteristic, complemented
57C2×Q8⋊C8C2211yescharacteristic, complemented
58C2×Q8⋊C8C2211yescharacteristic, complemented
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic, complemented
64C2×C82C2211yescharacteristic, complemented
65#1279C2211yescharacteristic, complemented
66#1279C2211yescharacteristic, complemented
67#1277C2211yescharacteristic, complemented
68#1277C2211yescharacteristic, complemented
69#1273C2211yescharacteristic, complemented
70#1273C2211yescharacteristic, complemented
71C23×D4C2×C411yescharacteristic
72C24×C4—21no—
73C22×SD16C2×C411yescharacteristic
74C22×SD16C2×C411yescharacteristic
75C22×M16—28no—
76C22×M16—21no—
77C22×M16—21no—
78C22×M16—21no—
79C22×M16—44no—
80C22×M16—44no—
81C22×M16D411yescharacteristic
82C22×M16D411yescharacteristic
83C23×C8—21no—
84C23×C8—21no—
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×C4×D4—22no—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4C2×C411yescharacteristic
94C2×C4×D4C2×C411yescharacteristic
95C2×C4×D4C2311yescharacteristic
96C2×C4×D4C2311yescharacteristic
97C22×C4⋊C4C2×C411yescharacteristic
98C22×C22⋊C4—21no—
99C22×C42C2311yescharacteristic
100#450—24no—
101C8×Q8—22no—
102C8×Q8—22no—
103C4×SD16—22no—
104C4×SD16—22no—
105C4×SD16—22no—
106C4×SD16—22no—
107#436—22no—
108#436—22no—
109#436—22no—
110#436D414yes—
111#435—28no—
112#435—28no—
113C8×D4—22no—
114C8×D4—22no—
115C8×D4—22no—
116C8×D4D414yes—
117C2×#106C2×C411yescharacteristic
118C2×#106C2×C411yescharacteristic
119#423—22no—
120#423—22no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136#407—28no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—21no—
143C2×C22⋊C8C2×C411yescharacteristic
144C2×C22⋊C8C2×C411yescharacteristic
145C4×M16—22no—
146C4×M16—22no—
147C2×#97—21no—
148C2×#97D411yescharacteristic
149C2×#97D411yescharacteristic
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157#391C2×C411yescharacteristic
158#389C2×C411yescharacteristic
159#387—21no—
160#385—21no—
161#380C2×C411yescharacteristic
162C4×C4⋊C4—21no—
163C4×C4⋊C4C2311yescharacteristic
164C4×C22⋊C4—21no—
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336C2×C411yescharacteristic
176#336C2×C411yescharacteristic
177#334—22no—
178#334—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183D4⋊C8—22no—
184D4⋊C8—22no—
185#322—22no—
186C82—22no—
187C25—21no—
188C22×Q8C22×C411yescharacteristic
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4C22×C411yescharacteristic
197C22×D4C22×C411yescharacteristic
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C22×C411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4#4111yescharacteristic
212C2×SD16—22no—
213C2×SD16—22no—
214C2×SD16—22no—
215C2×SD16—22no—
216C2×SD16—41no—
217C2×SD16M1614yescomplemented
218C2×M16—24no—
219C2×M16—24no—
220C2×M16—22no—
221C2×M16—22no—
222C2×M16—24no—
223C2×M16—22no—
224C2×M16—22no—
225C2×M16—41no—
226C2×M16—42no—
227C2×M16—432no—
228C2×M16—42no—
229C2×M16—42no—
230C2×M16—42no—
231C2×M16—42no—
232C2×M16—42no—
233C2×M16SD1614yescomplemented
234C22×C8—24no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—24no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—42no—
243C22×C8—42no—
244C22×C8—42no—
245C22×C8—42no—
246C22×C8C22×C411yescharacteristic
247C22×C8C22×C411yescharacteristic
248C22×C8C22×C411yescharacteristic
249C22×C8C22×C411yescharacteristic
250C22×C8C2×D411yescharacteristic
251C22×C8C2×D411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C22×C8C2×D411yescharacteristic
254C22⋊Q8—22no—
255C22⋊Q8—41no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C2×D412yes—
273C4×D4C2×D412yes—
274C4×D4#4112yes—
275C4×D4#4112yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42C2411yescharacteristic
305#106—22no—
306#106—22no—
307C4⋊C8—24no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316C4⋊C8C2×D412yes—
317C4⋊C8C2×D412yes—
318#103—22no—
319#103—22no—
320#102—22no—
321#102—22no—
322#102—22no—
323#102—22no—
324#102—22no—
325#102—41no—
326C22⋊C8—24no—
327C22⋊C8—24no—
328C22⋊C8—24no—
329C22⋊C8—24no—
330C22⋊C8—22no—
331C22⋊C8—41no—
332C22⋊C8—42no—
333C22⋊C8—42no—
334#97—22no—
335#97—22no—
336#97—42no—
337C4×C8—21no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—42no—
348C4×C8C2×D412yes—
349C4×C8C2×D412yes—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355C24—21no—
356C24—24no—
357C24—21no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—81no—
363C24—81no—
364C24C4×D411yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8C2×M1612yescomplemented
371C2×D4—24no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4C4×D412yes—
387C2×D4C4×D412yes—
388C2×D4C2×M1612yes—
389C2×D4#13112yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C2×SD1612yes—
439C22×C4#13512yes—
440C22×C4C23×C411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4111yescharacteristic
443SD16—28no—
444SD16—44no—
445M16—28no—
446M16—44no—
447M16—48no—
448M16—48no—
449M16—48no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—22no—
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—48no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—48no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8#12012yes—
489C2×C8#12012yes—
490C2×C8C2×M1612yescomplemented
491C2×C8C2×SD1612yes—
492C2×C8C2×SD1612yes—
493C2×C8AGL(1,ℤ/8)12yes—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—81no—
507C4⋊C4C4×D412yes—
508C4⋊C4C4×D412yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—42no—
516C22⋊C4—81no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42C22×D411yescharacteristic
528C42C22×D411yescharacteristic
529C42C2×#4111yescharacteristic
530C42C2×#4111yescharacteristic
531C23—21no—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—21no—
540C23—22no—
541C23—48no—
542C23—48no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23C4×SD1612yes—
559C23C2×C4×D411yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—48no—
570D4—41no—
571D4—41no—
572D4#43514yes—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4C4×SD1612yes—
623C2×C4C2×C4×D411yescharacteristic
624C2×C4C2×C4×D411yescharacteristic
625C2×C4C2×#12011yescharacteristic
626C2×C4#52511yescharacteristic
627C2×C4#52511yescharacteristic
628C2×C4#52511yescharacteristic
629C2×C4C22×M1611yescharacteristic
630C2×C4C2×#13111yescharacteristic
631C2×C4#56811yescharacteristic
632C2×C4#56811yescharacteristic
633C2×C4C22×SD1611yescharacteristic
634C2×C4C2×#13511yescharacteristic
635C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
636C2×C4#57511yescharacteristic
637C8—22no—
638C8—22no—
639C8—22no—
640C8—42no—
641C8—41no—
642C8—41no—
643C8—41no—
644C8—42no—
645C8—42no—
646C8—41no—
647C8—81no—
648C8—81no—
649C8—81no—
650C8#45014yescomplemented
651C22—22no—
652C22—22no—
653C22—24no—
654C22—22no—
655C22—42no—
656C22—41no—
657C22—42no—
658C22—42no—
659C22—41no—
660C22—42no—
661C22—41no—
662C22—42no—
663C22—41no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—816no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22C4×#12011yescharacteristic
674C22#293311yescharacteristic
675C22D4×M1611yescharacteristic
676C22C2×C4×SD1611yescharacteristic
677C22C4×AGL(1,ℤ/8)11yescharacteristic
678C22#295611yescharacteristic
679C22#295711yescharacteristic
680C4—22no—
681C4—21no—
682C4—21no—
683C4—41no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—41no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4C2×#43511yescharacteristic
692C4#293011yescharacteristic
693C4#293311yescharacteristic
694C4D4×M1611yescharacteristic
695C4C2×#45011yescharacteristic
696C4#300211yescharacteristic
697C4#310111yescharacteristic
698C4#310511yescharacteristic
699C2—42no—
700C2—44no—
701C2—82no—
702C2#1799511yescharacteristic
703C2#1800811yescharacteristic
704C2M16×SD1611yescharacteristic
705C2#1804811yescharacteristic
706C2#2143911yescharacteristic
707C2C4×#45011yescharacteristic
708C2#2146811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D4C2×C4
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
2. #21596C2semidirect
4. C2×#1583C2semidirect
5. C2×C8×SD16C2semidirect
6. #8949C2semidirect
7. #8945C2semidirect
8. #8942C2semidirect
9. #8849C2semidirect
10. #8845C2semidirect
11. #8833C2semidirect
13. #8206C2semidirect
14. #8202C2semidirect
15. #8146C2semidirect
19. C2×C8×Q8C22semidirect
34. #1984C22semidirect
35. #1939C22semidirect
47. #1770C22semidirect
55. C2×#334C22semidirect
56. C2×#334C22semidirect
57. C2×Q8⋊C8C22semidirect
58. C2×Q8⋊C8C22semidirect
63. C2×#322C22semidirect
64. C2×C82C22semidirect
65. #1279C22semidirect
66. #1279C22semidirect
67. #1277C22semidirect
68. #1277C22semidirect
69. #1273C22semidirect
70. #1273C22semidirect
217. C2×SD16M16semidirect
233. C2×M16SD16semidirect
370. C2×Q8C2×M16semidirect
490. C2×C8C2×M16semidirect
650. C8#450semidirect

Maximal subgroups

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

subgroupindex
C2×D4×M162
#215962
C4×#4502
C2×#15832
C2×C8×SD162
#89492
#89452
#89422
#88492
#88452
#88332
#83102
#82062
#82022
#81462