Finite Group Databasebeta

← #6500558

Subgroups — #6500558

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3#29244C212yesmaximal, complemented
4#29243C212yesmaximal, complemented
5#27686C212yesmaximal, complemented
6#26418C214yesmaximal, complemented
7#26346C214yesmaximal, complemented
8#26002C214yesmaximal, complemented
9#25965C214yesmaximal, complemented
10#24307C212yesmaximal, complemented
11#21940C211yesmaximal, characteristic
12#21930C211yesmaximal, characteristic
13#21759C211yesmaximal, characteristic
14C2×#1872C211yesmaximal, characteristic
15C2×#1853C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#479C2214yescomplemented
20C22×#106C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2442—22no—
25#2442—22no—
26#2442C2214yes—
27#2442C2214yes—
28C2×#400—21no—
29C2×#398—21no—
30#2368—22no—
31#2360C2212yes—
32#2360C2212yes—
33#2345—22no—
34#2345C2212yes—
35Q8×C22⋊C4C2212yes—
36#2329—21no—
37#2329—21no—
38#2315—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#385—21no—
44C2×#382C2211yescharacteristic
45C2×#380—21no—
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2214yescomplemented
51#2097C2214yescomplemented
52#2088C2214yescomplemented
53#2086C2214yescomplemented
54#2068C2214yescomplemented
55#2066C2214yescomplemented
56#2059C2214yescomplemented
57#2026C2214yescomplemented
58#1989C2214yescomplemented
59#1949C2214yescomplemented
60#1920C2214yescomplemented
61#1918C2214yescomplemented
62#1892C2214yes—
63#1872C2212yes—
64#1872C2212yes—
65#1872C2214yes—
66#1872C2212yes—
67#1872C2212yes—
68#1865C2214yescomplemented
69#1863C2214yescomplemented
70#1853C2212yes—
71#1853C2212yes—
72#1853C2214yes—
73#1852C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1850C2214yes—
78#1850C2214yes—
79#1824C2212yes—
80#1808C2214yes—
81#1805C2214yescomplemented
82#1797C2212yes—
83#1797C2212yes—
84#1787C2212yes—
85C2×#340C2212yes—
86C2×#340C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C23×C8C2312yes—
91C23×C8C2312yes—
92C2×#125—22no—
93C2×#125—44no—
94C2×#125C2314yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8Q818yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—44no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#479—48no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106—44no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103—44no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—24no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147#400—24no—
148#400—24no—
149#398—24no—
150#398—24no—
151#391—22no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2312yes—
160#389C2314yes—
161#389C2314yes—
162#389C2312yes—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—24no—
174#385—24no—
175#382—22no—
176#382C2314yes—
177#380—21no—
178#380—21no—
179#380—21no—
180#380—21no—
181#380—22no—
182#380—22no—
183#380—24no—
184#380—24no—
185#380C2312yes—
186#380C2312yes—
187#380C2312yes—
188#380C2312yes—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4C2312yes—
202C4×C22⋊C4C2312yes—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×Q814yes—
227C22×Q8C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×Q814yes—
251C23×C4#4114yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—42no—
258C22×C8—42no—
259C22×C8—42no—
260C22×C8—42no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263#125—28no—
264#125—44no—
265#125—48no—
266C22⋊Q8—28no—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8C2×Q818yes—
271C22⋊Q8C2×Q818yes—
272C4×Q8—44no—
273C4×Q8—44no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×Q814yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—44no—
346#107—44no—
347#106—28no—
348#106—44no—
349C4⋊C8—44no—
350C4⋊C8—44no—
351#103—28no—
352#103—44no—
353#103—44no—
354#103—44no—
355C22⋊C8—24no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—41no—
383C24C22⋊Q814yes—
384C24#13514yes—
385C24#13714yes—
386C24C22×D411yescharacteristic
387C24C22×D411yescharacteristic
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—48no—
394C2×Q8—81no—
395C2×Q8—81no—
396C2×Q8—84no—
397C2×Q8C22×Q814yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
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—41no—
434C22×C4—41no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—816no—
463C22×C4C22⋊Q814yes—
464C22×C4C22⋊Q814yes—
465C22×C4C22⋊Q814yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q812yes—
474C22×C4C22×Q812yes—
475C22×C4C22×Q812yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2511yescharacteristic
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—48no—
487C2×C8—84no—
488C4⋊C4—24no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4C22×Q814yes—
516C4⋊C4H(5,𝔽2)14yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C42—24no—
528C42—24no—
529C42—42no—
530C42—42no—
531C42—82no—
532C42—82no—
533C42—82no—
534C42—82no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23#479116yescomplemented
554C23C2×C22⋊Q812yes—
555C23C2×C22⋊Q812yes—
556C23C2×C22⋊Q812yes—
557C23C2×#13512yes—
558C23C2×#13512yes—
559C23C2×#13512yes—
560C23C2×#13712yes—
561C23C2×#13712yes—
562C23C2×#13712yes—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—44no—
566Q8—84no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—88no—
626C2×C4—88no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4C2×C22⋊Q812yes—
633C2×C4C2×C22⋊Q812yes—
634C2×C4C2×C22⋊Q812yes—
635C2×C4C23×Q811yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4#58512yes—
639C8—44no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—416no—
660C22C2×#47914yes—
661C22C2×#47914yes—
662C22C2×#47914yes—
663C22C22×C22⋊Q811yescharacteristic
664C22#348411yescharacteristic
665C22#348511yescharacteristic
666C22C22×#13511yescharacteristic
667C22C22×#13711yescharacteristic
668C22#358511yescharacteristic
669C22#358611yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#350112yes—
686C4#350512yes—
687C2—24no—
688C2—24no—
689C2—24no—
690C2#3534911yescharacteristic
691C2C22×#47911yescharacteristic
692C2#6049311yescharacteristic
693C2#6156411yescharacteristic
694C2#6157011yescharacteristic
695C2#6169911yescharacteristic
696C2#6170511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29244C2semidirect
4. #29243C2semidirect
5. #27686C2semidirect
6. #26418C2semidirect
7. #26346C2semidirect
8. #26002C2semidirect
9. #25965C2semidirect
10. #24307C2semidirect
19. C2×#479C22semidirect
50. #2101C22semidirect
51. #2097C22semidirect
52. #2088C22semidirect
53. #2086C22semidirect
54. #2068C22semidirect
55. #2066C22semidirect
56. #2059C22semidirect
57. #2026C22semidirect
58. #1989C22semidirect
59. #1949C22semidirect
60. #1920C22semidirect
61. #1918C22semidirect
68. #1865C22semidirect
69. #1863C22semidirect
81. #1805C22semidirect
553. C23#479semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
#342122
#292442
#292432
#276862
#264182
#263462
#260022
#259652
#243072
#219402
#219302
#217592
C2×#18722
C2×#18532