Finite Group Databasebeta

← #6434361

Subgroups — #6434361

Order 512.

1750 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic, complemented
2#34275C211yesmaximal, characteristic
3#30088C214yesmaximal, complemented
4#30085C214yesmaximal, complemented
5#30081C214yesmaximal, complemented
6#30077C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23628C211yesmaximal, characteristic, complemented
12#23627C211yesmaximal, characteristic, complemented
13#23511C211yesmaximal, characteristic
14#23511C211yesmaximal, characteristic
15#20882C211yesmaximal, characteristic, complemented
16#20881C211yesmaximal, characteristic, complemented
17#20876C211yesmaximal, characteristic, complemented
18#20876C211yesmaximal, characteristic, complemented
19#20774C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35#2864C2214yescomplemented
36#2647C2214yescomplemented
37#2570—22no—
38#2570—22no—
39#2569—24no—
40#2363—22no—
41#2361—22no—
42#2358—24no—
43#2328C2212yes—
44#2326C2212yes—
45#2317C2211yescharacteristic
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50C4×#127C2212yes—
51C4×#124C2212yes—
52#2297C2211yescharacteristic
53C2×#390C2211yescharacteristic
54#2285C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2092C2214yes—
60#2092C2214yes—
61#2092C2212yescomplemented
62#2092C2212yescomplemented
63#2085C2214yescomplemented
64#2085C2214yescomplemented
65#2084C2214yescomplemented
66#2083C2214yescomplemented
67#2074C2212yescomplemented
68#2073C2214yes—
69#2073C2212yescomplemented
70#2072C2212yescomplemented
71#2071C2214yes—
72#2071C2212yescomplemented
73#2042C2214yes—
74#2042C2214yes—
75#2041C2214yes—
76#2041C2214yes—
77#1865C2214yescomplemented
78#1864C2214yes—
79#1864C2214yes—
80#1864C2214yescomplemented
81#1863C2214yescomplemented
82#1862C2214yes—
83#1862C2214yes—
84#1862C2214yescomplemented
85#1840C2214yescomplemented
86#1839C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1729C2211yescharacteristic
92#1729C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#535—24no—
104#535—24no—
105C2×#127C2312yes—
106C2×#124C2312yes—
107C2×#121—22no—
108C2×#120—22no—
109#520—42no—
110#520—42no—
111#518—42no—
112#518—42no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—24no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C42C2311yescharacteristic
134#428—42no—
135#428—42no—
136C2×#107—42no—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#106—42no—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144#418—24no—
145#418—24no—
146#417—42no—
147#417—42no—
148#416—24no—
149#416—24no—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#102—42no—
155C2×#102—42no—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164#400—24no—
165#400—24no—
166#398—24no—
167#398—24no—
168#391C2311yescharacteristic
169#391C2311yescharacteristic
170#390—24no—
171#390—22no—
172#390C2311yescharacteristic
173#390C2312yes—
174#390C2312yes—
175#390C2311yescharacteristic
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2314yes—
180#389C2314yes—
181#389C2314yes—
182#389C2314yes—
183#388—24no—
184#388—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#387—24no—
189#386—22no—
190#386—24no—
191#386—24no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—24no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2312yes—
200#382—21no—
201#382—21no—
202#382—22no—
203#382C2312yes—
204#382C2312yes—
205#382C2312yes—
206#382C2312yes—
207#380—22no—
208#380—22no—
209#380—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D412yes—
213C4×C22⋊C4—24no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D414yes—
216#376C2312yes—
217C2×#95—22no—
218C43C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#336C2312yes—
226#336C2312yes—
227#336C2312yes—
228#336C2312yes—
229#336C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233C25—21no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41#4114yes—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—24no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4#4114yes—
250C22×D4C2412yes—
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277#127—24no—
278#127—24no—
279#124—24no—
280#124—24no—
281#121—216no—
282#120—216no—
283C4×Q8—42no—
284C4×Q8—42no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—48no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—88no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—44no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—84no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42C2×D412yes—
356C2×C42C2×D414yes—
357C2×C42C2×D414yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—44no—
363#107—42no—
364#107—42no—
365#106—44no—
366#106—42no—
367#106—42no—
368#103—24no—
369#103—24no—
370#103—42no—
371#103—42no—
372#102—24no—
373#102—24no—
374#102—28no—
375#102—28no—
376#102—44no—
377#102—44no—
378#102—42no—
379#102—42no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95C2×D414yes—
392#95C2×D414yes—
393C24—22no—
394C24—21no—
395C24—24no—
396C24—42no—
397C24—42no—
398C24—82no—
399C24—82no—
400C24C22×D411yescharacteristic
401#41—28no—
402#41—28no—
403#41—44no—
404#41—44no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8C2×#4112yes—
411C2×D4—24no—
412C2×D4—216no—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4C2×#4114yes—
435C2×D4C2×#4112yes—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C22×C4C2511yescharacteristic
475M16—44no—
476M16—44no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—44no—
490C2×C8—44no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—82no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C22⋊C4—216no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—82no—
531C22⋊C4—84no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—44no—
542C42—44no—
543C42—44no—
544C42—42no—
545C42—42no—
546C42—42no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—22no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23—41no—
556C23—42no—
557C23—41no—
558C23—42no—
559C23—48no—
560C23—88no—
561C23—162no—
562C23—162no—
563C23—164no—
564C23#52512yes—
565C23#53512yes—
566C23#53612yes—
567C23C23×D411yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572D4—24no—
573D4—24no—
574D4—22no—
575D4—22no—
576D4—48no—
577D4—48no—
578D4—42no—
579D4—42no—
580D4—41no—
581D4—41no—
582D4—816no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—164no—
618C2×C4—164no—
619C2×C4—164no—
620C2×C4—162no—
621C2×C4—162no—
622C2×C4—162no—
623C2×C4#52512yes—
624C2×C4#53512yes—
625C2×C4#53612yes—
626C2×C4#53914yes—
627C2×C4#53914yes—
628C2×C4#53914yes—
629C2×C4#53914yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C8—42no—
637C8—42no—
638C8—42no—
639C8—42no—
640C22—22no—
641C22—22no—
642C22—24no—
643C22—24no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—82no—
649C22—81no—
650C22—81no—
651C22—82no—
652C22—82no—
653C22—84no—
654C22—168no—
655C22#263314yes—
656C22C2×#52511yescharacteristic
657C22C2×#53511yescharacteristic
658C22C2×#53611yescharacteristic
659C22#358412yes—
660C22#358512yes—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—82no—
674C4#263314yes—
675C4C2×#53911yescharacteristic
676C4C2×#53911yescharacteristic
677C4C2×#53911yescharacteristic
678C4C2×#53911yescharacteristic
679C2—24no—
680C2—82no—
681C2—84no—
682C2C2×#263311yescharacteristic
683C2#6060112yes—
684C2#6081911yescharacteristic
685C2#6082711yescharacteristic
686C2#6087512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35339C2semidirect
3. #30088C2semidirect
4. #30085C2semidirect
5. #30081C2semidirect
6. #30077C2semidirect
7. #23733C2semidirect
8. C2×#2092C2semidirect
9. #23661C2semidirect
10. #23660C2semidirect
11. #23628C2semidirect
12. #23627C2semidirect
15. #20882C2semidirect
16. #20881C2semidirect
17. #20876C2semidirect
18. #20876C2semidirect
19. #20774C2semidirect
35. #2864C22semidirect
36. #2647C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
61. #2092C22semidirect
62. #2092C22semidirect
63. #2085C22semidirect
64. #2085C22semidirect
65. #2084C22semidirect
66. #2083C22semidirect
67. #2074C22semidirect
69. #2073C22semidirect
70. #2072C22semidirect
72. #2071C22semidirect
77. #1865C22semidirect
80. #1864C22semidirect
81. #1863C22semidirect
84. #1862C22semidirect
85. #1840C22semidirect
86. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#353392
#342752
#300882
#300852
#300812
#300772
#237332
C2×#20922
#236612
#236602
#236282
#236272
#235112
#208822
#208812
#208762
#207742