Finite Group Databasebeta

← #222130

Subgroups — #222130

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic
2#22751C211yesmaximal, characteristic, complemented
3#22648C211yesmaximal, characteristic, complemented
4C2×#1560C211yesmaximal, characteristic, complemented
5C2×#1551C211yesmaximal, characteristic, complemented
6#9702C211yesmaximal, characteristic, complemented
7#9691C211yesmaximal, characteristic, complemented
8#9136C211yesmaximal, characteristic, complemented
9#9135C211yesmaximal, characteristic, complemented
10#9135C211yesmaximal, characteristic, complemented
11#9127C211yesmaximal, characteristic, complemented
12#8588C211yesmaximal, characteristic, complemented
13#8344C211yesmaximal, characteristic, complemented
14#8344C211yesmaximal, characteristic, complemented
15#8234C211yesmaximal, characteristic, complemented
16C2×#547C412yes—
17C2×#532C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#435—21no—
20C2×#432C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24#2868C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2547C412yes—
28#2537—21no—
29#2537—21no—
30#2432C412yes—
31#2432C412yes—
32#2422—21no—
33#2422—21no—
34#2422—21no—
35#2422—21no—
36D4×C4⋊C4—21no—
37D4×C4⋊C4—21no—
38C4×#127C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1940—21no—
41#1923—21no—
42#1917—21no—
43#1870—21no—
44#1851—21no—
45#1850—21no—
46#1842C2211yescharacteristic
47#1831C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1753—21no—
55#1560—24no—
56#1551—24no—
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#333C2211yescharacteristic
64C2×#333C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C2×#331C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#1296C2211yescharacteristic
72#1296C2211yescharacteristic
73C22×#41C2×C412yes—
74C22×#41C2×C412yes—
75C22×M16—21no—
76C22×M16—21no—
77C23×C8—21no—
78#547—48no—
79#532—22no—
80#532—22no—
81#532—22no—
82#532—22no—
83#532—41no—
84#532—41no—
85#532—41no—
86#532—41no—
87C2×C4⋊Q8C2×C412yes—
88C2×C4⋊Q8C2×C412yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#124C2×C412yes—
93C2×#124C2×C412yes—
94C2×#123—22no—
95C2×#123—22no—
96C2×C22⋊Q8C2×C412yes—
97C2×C22⋊Q8C2×C412yes—
98C2×#121—22no—
99C2×#121—21no—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121—21no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4C2×C412yes—
110C2×C4×D4C2×C411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D411yescharacteristic
113C22×C4⋊C4—22no—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#435—44no—
117#432—22no—
118#432—41no—
119#423—22no—
120#423—22no—
121#423—41no—
122#423—41no—
123C2×C4⋊C8—22no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—41no—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—21no—
143C2×#97—21no—
144C2×#97—21no—
145C2×C4×C8—21no—
146#397—22no—
147#397—22no—
148#394—22no—
149#394—22no—
150#394—22no—
151#394—22no—
152#391C2×C412yes—
153#391C2×C412yes—
154#390—21no—
155#390—21no—
156#390C2×C412yes—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#385—22no—
162#385—22no—
163#385—22no—
164#385—22no—
165#384—21no—
166#384—21no—
167#384—21no—
168#384—21no—
169#384C2311yescharacteristic
170#380—21no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—22no—
175#380—22no—
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C22⋊C4C2×C411yescharacteristic
180C4×C22⋊C4C2×C411yescharacteristic
181C4×C22⋊C4D411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C43C2311yescharacteristic
186#336—21no—
187#336—21no—
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#333—22no—
195#333—22no—
196#333—22no—
197#333—22no—
198#333—41no—
199#333—41no—
200#333—41no—
201#333—41no—
202#331—22no—
203#331—22no—
204#331—22no—
205#331D414yescomplemented
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—42no—
209C2×#41—42no—
210C2×#41—44no—
211C2×#41—44no—
212C2×#41—82no—
213C2×#41—82no—
214C22×Q8C22×C412yes—
215C22×Q8C22×C412yes—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4C22⋊C411yescharacteristic
221C22×D4C22×C412yes—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4C22×C411yescharacteristic
226C23×C4C22×C411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C2×M16—42no—
232C2×M16—42no—
233C2×M16—42no—
234C2×M16—42no—
235C2×M16—81no—
236C2×M16—81no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—42no—
246C22×C8—42no—
247C22×C8—81no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—44no—
257C4⋊Q8—82no—
258#127—21no—
259#127—21no—
260#127C2×D412yes—
261#127#4112yes—
262#124—24no—
263#124—24no—
264#124—42no—
265#124—42no—
266#123—48no—
267#123—48no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270#121—42no—
271#121—42no—
272#121—42no—
273#121—42no—
274#121—42no—
275#121—42no—
276#121—42no—
277#121—42no—
278#121—48no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—41no—
283C4×D4—41no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—44no—
295C4×D4—42no—
296C4×D4—42no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4C22⋊C411yescharacteristic
311C2×C4⋊C4C22⋊C411yescharacteristic
312C2×C4⋊C4C22⋊C411yescharacteristic
313C2×C4⋊C4C22×C412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4C22⋊C411yescharacteristic
324C2×C22⋊C4C22⋊C411yescharacteristic
325C2×C22⋊C4C22⋊C411yescharacteristic
326C2×C22⋊C4C22⋊C411yescharacteristic
327C2×C22⋊C4C22⋊C411yescharacteristic
328C2×C22⋊C4C22⋊C411yescharacteristic
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42C22×C411yescharacteristic
338C2×C42C22×C411yescharacteristic
339C2×C42C22×C411yescharacteristic
340C2×C42C22×C411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2411yescharacteristic
350C4⋊C8—21no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8—21no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8—21no—
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364C4⋊C8—42no—
365C4⋊C8—42no—
366C4⋊C8—42no—
367C4⋊C8—42no—
368C4⋊C8—81no—
369C4⋊C8C2×D412yes—
370C4⋊C8C2×D412yes—
371C22⋊C8—42no—
372C22⋊C8—42no—
373C22⋊C8—42no—
374C22⋊C8—42no—
375C22⋊C8—81no—
376#97—42no—
377#97—42no—
378C4×C8—81no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387#95C22⋊C411yescharacteristic
388#95C22⋊C411yescharacteristic
389C24—22no—
390C24C2×C22⋊C411yescharacteristic
391C24C2×C22⋊C411yescharacteristic
392#41—88no—
393#41—88no—
394#41—88no—
395#41—88no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—41no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4C4×D414yes—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4C2×C22⋊C411yescharacteristic
474C22×C4C2×C22⋊C411yescharacteristic
475C22×C4C2×C22⋊C411yescharacteristic
476C22×C4C2×C22⋊C411yescharacteristic
477C22×C4C2×C22⋊C411yescharacteristic
478C22×C4C2×C22⋊C411yescharacteristic
479C22×C4C2×C22⋊C411yescharacteristic
480C22×C4C2×C22⋊C411yescharacteristic
481C22×C4C2×C22⋊C411yescharacteristic
482C22×C4C2×C22⋊C411yescharacteristic
483C22×C4C23×C411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486M16—84no—
487M16—84no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—84no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—41no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4#10014yes—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—22no—
569C42—41no—
570C42AGL(1,ℤ/8)12yes—
571C42#13712yes—
572C42C22×D411yescharacteristic
573C42C2×#4111yescharacteristic
574C42H(5,𝔽2)11yescharacteristic
575C42H(5,𝔽2)11yescharacteristic
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—24no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23C2×#10012yes—
597C23C22×C22⋊C411yescharacteristic
598Q8—82no—
599Q8—82no—
600Q8—82no—
601Q8—82no—
602D4—44no—
603D4—44no—
604D4—44no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—88no—
614D4—88no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—88no—
673C2×C4—88no—
674C2×C4—88no—
675C2×C4C2×#10012yes—
676C2×C4C2×#10012yes—
677C2×C4#41712yes—
678C2×C4#41712yes—
679C2×C4C2×C4×D411yescharacteristic
680C2×C4#51811yescharacteristic
681C2×C4D4211yescharacteristic
682C2×C4D4211yescharacteristic
683C2×C4#54611yescharacteristic
684C2×C4#54611yescharacteristic
685C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
686C2×C4C2×#13711yescharacteristic
687C2×C4#57511yescharacteristic
688C2×C4#57511yescharacteristic
689C2×C4#57711yescharacteristic
690C2×C4#57711yescharacteristic
691C2×C4#57811yescharacteristic
692C2×C4#57811yescharacteristic
693C8—42no—
694C8—42no—
695C8—81no—
696C8—81no—
697C8—81no—
698C8—81no—
699C8—81no—
700C8—81no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—44no—
714C22—41no—
715C22—44no—
716C22—41no—
717C22—44no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22D4×C22⋊C411yescharacteristic
723C22C22×#10011yescharacteristic
724C22#288811yescharacteristic
725C22C2×#41711yescharacteristic
726C22#289611yescharacteristic
727C22#289811yescharacteristic
728C22#289811yescharacteristic
729C4—22no—
730C4—22no—
731C4—22no—
732C4—21no—
733C4—21no—
734C4—22no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4#326711yescharacteristic
745C4#326911yescharacteristic
746C4#327211yescharacteristic
747C4#327611yescharacteristic
748C4#330711yescharacteristic
749C4#331511yescharacteristic
750C4#333011yescharacteristic
751C4#333111yescharacteristic
752C2—42no—
753C2—81no—
754C2—84no—
755C2#1775011yescharacteristic
756C2#1776411yescharacteristic
757C2#1776811yescharacteristic
758C2#1777911yescharacteristic
759C2D4×#10011yescharacteristic
760C2#2242011yescharacteristic
761C2#2243111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22751C2semidirect
3. #22648C2semidirect
4. C2×#1560C2semidirect
5. C2×#1551C2semidirect
6. #9702C2semidirect
7. #9691C2semidirect
8. #9136C2semidirect
9. #9135C2semidirect
10. #9135C2semidirect
11. #9127C2semidirect
12. #8588C2semidirect
13. #8344C2semidirect
14. #8344C2semidirect
15. #8234C2semidirect
205. #331D4semidirect

Maximal subgroups

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

subgroupindex
#361272
#227512
#226482
C2×#15602
C2×#15512
#97022
#96912
#91362
#91352
#91272
#85882
#83442
#82342