Finite Group Databasebeta

← #6506758

Subgroups — #6506758

Order 512.

1710 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2442C211yesmaximal, characteristic
3#29298C212yesmaximal, complemented
4#29298C212yesmaximal, complemented
5#28772C214yesmaximal, complemented
6#28770C214yesmaximal, complemented
7#27952C212yesmaximal, complemented
8#26067C214yesmaximal, complemented
9#26062C214yesmaximal, complemented
10#25815C212yesmaximal, complemented
11#21940C211yesmaximal, characteristic
12#21938C211yesmaximal, characteristic
13C2×#1853C211yesmaximal, characteristic
14C2×#1852C211yesmaximal, characteristic
15#20523C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#499C2214yescomplemented
19C22×#107C2212yes—
20C22×#106C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2851C2212yes—
24#2851C2212yes—
25#2646—21no—
26#2646—21no—
27#2569—22no—
28#2569—22no—
29#2442—22no—
30#2442—22no—
31#2442C2212yes—
32#2442C2214yes—
33#2442C2212yes—
34#2442C2214yes—
35#2442C2212yes—
36#2442C2212yes—
37C2×#400—21no—
38C2×#398—21no—
39#2315—21no—
40#2315—21no—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#385—21no—
44C2×#382C2211yescharacteristic
45C2×#380—21no—
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2085C2214yescomplemented
51#2085C2214yescomplemented
52#2085C2214yescomplemented
53#2085C2214yescomplemented
54#2059C2214yescomplemented
55#2059C2214yescomplemented
56#2059C2214yescomplemented
57#2059C2214yescomplemented
58#1991C2214yescomplemented
59#1991C2214yescomplemented
60#1942C2214yescomplemented
61#1942C2214yescomplemented
62#1941C2214yescomplemented
63#1941C2214yescomplemented
64#1853C2212yes—
65#1853C2212yes—
66#1853C2212yes—
67#1853C2214yes—
68#1853C2212yes—
69#1853C2212yes—
70#1853C2212yes—
71#1852C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1852C2214yes—
75#1852C2212yes—
76#1852C2212yes—
77#1852C2212yes—
78#1846C2214yescomplemented
79#1808C2214yes—
80#1807C2214yes—
81C8×C22⋊C4C2214yes—
82C8×C22⋊C4C2214yes—
83C4×C22⋊C8C2212yes—
84C4×C22⋊C8C2212yes—
85C2×#340C2212yes—
86C2×#340C2212yes—
87C24×C4—21no—
88C24×C4C2311yescharacteristic
89C23×C8C2312yes—
90C23×C8C2312yes—
91C2×#125—22no—
92C2×#125—44no—
93C2×#125C2314yes—
94C2×#125C2312yes—
95C2×#125C2312yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#499—48no—
122C2×#107—44no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106—44no—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C22⋊C8Q818yes—
133C2×C22⋊C8Q818yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C4×C8C2314yes—
139C2×C4×C8C2314yes—
140#400—22no—
141#400—22no—
142#400—22no—
143#400—22no—
144#400—24no—
145#400—24no—
146#398—22no—
147#398—22no—
148#398—22no—
149#398—22no—
150#398—24no—
151#398—24no—
152#389—22no—
153#389—22no—
154#389C2312yes—
155#389C2312yes—
156#389C2314yes—
157#389C2314yes—
158#389C2312yes—
159#389C2312yes—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#385—22no—
165#385—22no—
166#385—22no—
167#385—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—24no—
173#385—24no—
174#382—22no—
175#382C2312yes—
176#382C2312yes—
177#382C2314yes—
178#380—21no—
179#380—21no—
180#380—21no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—22no—
185#380—22no—
186#380—22no—
187#380—22no—
188#380—24no—
189#380—24no—
190#380C2312yes—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4D412yes—
198C4×C22⋊C4C2312yes—
199C4×C22⋊C4C2312yes—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C25C2×D411yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—82no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×Q814yes—
243C23×C4C2×Q814yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C22×C8—42no—
248C22×C8—42no—
249C22×C8—42no—
250C22×C8—42no—
251C22×C8C2×Q818yes—
252C22×C8C2×Q818yes—
253C22×C8C2414yes—
254C22×C8C2414yes—
255#125—28no—
256#125—44no—
257#125—44no—
258#125—44no—
259#125—48no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330#107—28no—
331#107—44no—
332#107—44no—
333#107—44no—
334#106—28no—
335#106—44no—
336#106—44no—
337#106—44no—
338C22⋊C8C2×Q818yes—
339C22⋊C8C2×Q818yes—
340C22⋊C8C2×Q818yes—
341C22⋊C8C2×Q818yes—
342C4×C8—44no—
343C4×C8—44no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—41no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24C4⋊Q814yes—
371C24#13514yes—
372C24#13514yes—
373C24C22×D411yescharacteristic
374C24C22×D411yescharacteristic
375C24C22×D411yescharacteristic
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
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—44no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—816no—
441C22×C4C4⋊Q814yes—
442C22×C4C4⋊Q814yes—
443C22×C4C4⋊Q814yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×Q812yes—
452C22×C4C22×Q812yes—
453C22×C4C22×Q812yes—
454C22×C4C22×Q812yes—
455C22×C4C22×Q812yes—
456C22×C4C22×Q812yes—
457C22×C4C2511yescharacteristic
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—48no—
463C2×C8—48no—
464C2×C8C22×Q818yes—
465C2×C8C22×Q818yes—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4H(5,𝔽2)14yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C42—24no—
509C42—24no—
510C42—42no—
511C42—42no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23#499116yescomplemented
531C23C2×C4⋊Q812yes—
532C23C2×C4⋊Q812yes—
533C23C2×C4⋊Q812yes—
534C23C2×#13512yes—
535C23C2×#13512yes—
536C23C2×#13512yes—
537C23C2×#13512yes—
538C23C2×#13512yes—
539C23C2×#13512yes—
540C23C23×D411yescharacteristic
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—88no—
598C2×C4—88no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—162no—
603C2×C4—162no—
604C2×C4—162no—
605C2×C4—162no—
606C2×C4C2×C4⋊Q812yes—
607C2×C4C2×C4⋊Q812yes—
608C2×C4C2×C4⋊Q812yes—
609C2×C4C23×Q811yescharacteristic
610C2×C4C23×Q811yescharacteristic
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4#58512yes—
613C8—28no—
614C8—28no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—416no—
634C22C2×#49914yes—
635C22C2×#49914yes—
636C22C2×#49914yes—
637C22C22×C4⋊Q811yescharacteristic
638C22#348411yescharacteristic
639C22#348511yescharacteristic
640C22C22×#13511yescharacteristic
641C22C22×#13511yescharacteristic
642C22#358511yescharacteristic
643C22#358611yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4#350112yes—
660C4#350112yes—
661C2—24no—
662C2—24no—
663C2—24no—
664C2#3539111yescharacteristic
665C2C22×#49911yescharacteristic
666C2#6066411yescharacteristic
667C2#6156111yescharacteristic
668C2#6156411yescharacteristic
669C2#6169611yescharacteristic
670C2#6169911yescharacteristic
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. #29298C2semidirect
4. #29298C2semidirect
5. #28772C2semidirect
6. #28770C2semidirect
7. #27952C2semidirect
8. #26067C2semidirect
9. #26062C2semidirect
10. #25815C2semidirect
18. C2×#499C22semidirect
50. #2085C22semidirect
51. #2085C22semidirect
52. #2085C22semidirect
53. #2085C22semidirect
54. #2059C22semidirect
55. #2059C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
58. #1991C22semidirect
59. #1991C22semidirect
60. #1942C22semidirect
61. #1942C22semidirect
62. #1941C22semidirect
63. #1941C22semidirect
78. #1846C22semidirect
530. C23#499semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#24422
#292982
#287722
#287702
#279522
#260672
#260622
#258152
#219402
#219382
C2×#18532
C2×#18522
#205232