Finite Group Databasebeta

← #209265

Subgroups — #209265

Order 512.

1228 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#21598C211yesmaximal, characteristic, complemented
3C4×#450C211yesmaximal, characteristic
4C2×#1592C211yesmaximal, characteristic, complemented
5C2×#1585C211yesmaximal, characteristic, complemented
6#8949C211yesmaximal, characteristic, complemented
7#8945C211yesmaximal, characteristic, complemented
8#8942C211yesmaximal, characteristic, complemented
9#8847C211yesmaximal, characteristic, complemented
10#8843C211yesmaximal, characteristic, complemented
11#8832C211yesmaximal, characteristic, complemented
12#8310C211yesmaximal, characteristic
13#8212C211yesmaximal, characteristic, complemented
14#8200C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C23×M16—21no—
17C22×C4×D4C2211yescharacteristic
18C2×#450C411yescharacteristic
19C2×#446C2211yescharacteristic, 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#1770—21no—
47C8×C4⋊C4C2211yescharacteristic, complemented
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1753—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1592—24no—
54#1585—24no—
55C2×#335C2211yescharacteristic, complemented
56C2×#335C2211yescharacteristic, 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×#322C2211yescharacteristic, 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—
101#446—22no—
102#446—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#335—22no—
178#335—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183D4⋊C8—22no—
184D4⋊C8—22no—
185#322—22no—
186#322—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—22no—
217C2×SD16—41no—
218C2×M16—22no—
219C2×M16—24no—
220C2×M16—24no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—24no—
224C2×M16—22no—
225C2×M16—22no—
226C2×M16—41no—
227C2×M16—42no—
228C2×M16—42no—
229C2×M16—42no—
230C2×M16—432no—
231C2×M16—42no—
232C2×M16—42no—
233C2×M16—42no—
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—41no—
325#102M1614yescomplemented
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×Q8—21no—
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×C4AGL(1,ℤ/8)12yes—
439C22×C4#13712yes—
440C22×C4C23×C411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4111yescharacteristic
443SD16—44no—
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—21no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—48no—
474C2×C8—48no—
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—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8#12012yes—
491C2×C8#12012yes—
492C2×C8C2×M1612yescomplemented
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—42no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—81no—
506C4⋊C4C4×D412yes—
507C4⋊C4C4×D412yes—
508C4⋊C4C2×M1612yescomplemented
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—
558C23#44012yes—
559C23C2×C4×D411yescharacteristic
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—48no—
570D4—41no—
571D4—41no—
572D4#43614yes—
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×C4#44012yes—
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×C4C2×AGL(1,ℤ/8)11yescharacteristic
634C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
635C2×C4C2×#13711yescharacteristic
636C2×C4#57511yescharacteristic
637C8—22no—
638C8—22no—
639C8—22no—
640C8—22no—
641C8—42no—
642C8—41no—
643C8—41no—
644C8—41no—
645C8—42no—
646C8—42no—
647C8—41no—
648C8—81no—
649C8—81no—
650C8—81no—
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×#44011yescharacteristic
677C22C4×AGL(1,ℤ/8)11yescharacteristic
678C22#295611yescharacteristic
679C22#295711yescharacteristic
680C4—22no—
681C4—21no—
682C4—21no—
683C4—42no—
684C4—41no—
685C4—42no—
686C4—42no—
687C4—41no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4C2×#43611yescharacteristic
692C4#293011yescharacteristic
693C4#293311yescharacteristic
694C4D4×M1611yescharacteristic
695C4#300311yescharacteristic
696C4#300711yescharacteristic
697C4#311011yescharacteristic
698C4#311411yescharacteristic
699C2—42no—
700C2—44no—
701C2—82no—
702C2#1800011yescharacteristic
703C2#1800811yescharacteristic
704C2#1804011yescharacteristic
705C2#1804611yescharacteristic
706C2#2144111yescharacteristic
707C2#2145311yescharacteristic
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. #21598C2semidirect
4. C2×#1592C2semidirect
5. C2×#1585C2semidirect
6. #8949C2semidirect
7. #8945C2semidirect
8. #8942C2semidirect
9. #8847C2semidirect
10. #8843C2semidirect
11. #8832C2semidirect
13. #8212C2semidirect
14. #8200C2semidirect
15. #8147C2semidirect
19. C2×#446C22semidirect
34. #1984C22semidirect
35. #1939C22semidirect
47. C8×C4⋊C4C22semidirect
55. C2×#335C22semidirect
56. C2×#335C22semidirect
57. C2×Q8⋊C8C22semidirect
58. C2×Q8⋊C8C22semidirect
63. C2×#322C22semidirect
64. C2×#322C22semidirect
65. #1279C22semidirect
66. #1279C22semidirect
67. #1277C22semidirect
68. #1277C22semidirect
69. #1273C22semidirect
70. #1273C22semidirect
325. #102M16semidirect
492. C2×C8C2×M16semidirect
508. C4⋊C4C2×M16semidirect

Maximal subgroups

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

subgroupindex
C2×D4×M162
#215982
C4×#4502
C2×#15922
C2×#15852
#89492
#89452
#89422
#88472
#88432
#88322
#83102
#82122
#82002
#81472