Finite Group Databasebeta

← #6475519

Subgroups — #6475519

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic, complemented
2#35273C211yesmaximal, characteristic
3#33059C211yesmaximal, characteristic, complemented
4#33058C211yesmaximal, characteristic, complemented
5#33053C211yesmaximal, characteristic, complemented
6#33052C211yesmaximal, characteristic, complemented
7#30195C214yesmaximal, complemented
8#29342C214yesmaximal, complemented
9#29120C214yesmaximal, complemented
10#27983C211yesmaximal, characteristic, complemented
11#27982C211yesmaximal, characteristic, complemented
12#27966C211yesmaximal, characteristic, complemented
13#27965C211yesmaximal, characteristic, complemented
14#25911C211yesmaximal, characteristic, complemented
15#25910C211yesmaximal, characteristic, complemented
16#25862C211yesmaximal, characteristic, complemented
17#25862C211yesmaximal, characteristic, complemented
18#21772C214yesmaximal, complemented
19#20375C211yesmaximal, characteristic, complemented
20C2×#537C2212yes—
21C2×#533C2211yescharacteristic
22C2×#532C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#491C2211yescharacteristic
25C2×#491C2211yescharacteristic
26C2×#490C2211yescharacteristic
27C2×#490C2211yescharacteristic
28C2×#408C2212yes—
29C2×C4×M16C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2856C2211yescharacteristic
34#2856C2211yescharacteristic
35#2851C2211yescharacteristic
36#2851C2211yescharacteristic
37#2851C2211yescharacteristic
38#2851C2211yescharacteristic
39#2628C2212yes—
40#2628C2212yes—
41#2620C2212yes—
42#2620C2212yes—
43#2581—22no—
44#2580—22no—
45#2564—22no—
46#2564—22no—
47#2538—22no—
48#2537—22no—
49#2416—22no—
50#2416—22no—
51#2316C2212yes—
52#2311C2212yes—
53#2311C2212yes—
54C2×#384C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2100C2214yes—
57#2099C2214yes—
58#2074C2214yes—
59#2073C2214yes—
60#2042C2214yes—
61#2042C2214yes—
62#2024C2214yes—
63#2023C2214yes—
64#1992C2211yescharacteristic
65#1992C2211yescharacteristic
66#1992C2211yescharacteristic
67#1992C2211yescharacteristic
68#1961C2211yescharacteristic
69#1961C2211yescharacteristic
70#1960C2211yescharacteristic
71#1960C2211yescharacteristic
72#1936C2214yes—
73#1935C2214yes—
74#1934C2214yes—
75#1933C2214yes—
76#1895C2214yes—
77#1894C2214yes—
78#1893C2214yes—
79#1892C2214yes—
80#1875C2214yes—
81#1874C2214yes—
82#1873C2214yes—
83#1872C2214yes—
84#1836C2211yescharacteristic
85#1836C2211yescharacteristic
86#1836C2211yescharacteristic
87#1836C2211yescharacteristic
88#1824C2214yes—
89#1823C2214yes—
90#1788C2214yes—
91#1788C2214yes—
92#1750C2212yes—
93#1750C2212yes—
94#1749C2211yescharacteristic
95#1749C2211yescharacteristic
96#1749C2211yescharacteristic
97#1749C2211yescharacteristic
98#1729C2212yes—
99C22×#41C2312yes—
100C22×#41C2311yescharacteristic
101C22×M16C2312yes—
102C23×C8C2312yes—
103#537—216no—
104#533—42no—
105#533—42no—
106#532—42no—
107#532—42no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×#127C2311yescharacteristic
111C2×#125C2311yescharacteristic
112C2×#125C2311yescharacteristic
113C2×#124C2312yes—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×#121—22no—
119C2×#121—22no—
120C2×#121—22no—
121C2×#121—22no—
122C2×#121—22no—
123C2×#121—22no—
124C2×#121C2312yes—
125C2×#121C2312yes—
126C4×#41—42no—
127C4×#41—42no—
128C2×C4×Q8—22no—
129C2×C4×Q8D412yes—
130C2×C4×D4—22no—
131C2×C4×D4D412yes—
132C2×C4×D4D412yes—
133C2×C4×D4D412yes—
134C2×#117C2312yes—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—22no—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#491—42no—
145#491—42no—
146#490—42no—
147#490—42no—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156#408—28no—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C4×M16—42no—
162C2×#97—42no—
163C2×#97C2311yescharacteristic
164C2×#97C2311yescharacteristic
165C2×#97C2311yescharacteristic
166C2×#97C2311yescharacteristic
167C2×#97C2312yes—
168C2×C4×C8C2312yes—
169#400—22no—
170#400—22no—
171#399—22no—
172#399—22no—
173#396—24no—
174#396—24no—
175#395—22no—
176#395—22no—
177#394—24no—
178#394—24no—
179#393—22no—
180#393—22no—
181#391—22no—
182#391C2312yes—
183#391C2312yes—
184#390—22no—
185#390—22no—
186#390—22no—
187#390C2312yes—
188#390C2312yes—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#386—22no—
202#386—22no—
203#386—24no—
204#385—22no—
205#385—22no—
206#385—22no—
207#385—22no—
208#384—22no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2312yes—
212#382—22no—
213#382C2312yes—
214#382C2311yescharacteristic
215#382C2311yescharacteristic
216#380—22no—
217#380—22no—
218#380—24no—
219#380—22no—
220#380—22no—
221#380C2312yes—
222#380C2312yes—
223#380C2312yes—
224#380C2312yes—
225C2×#95—22no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238C2×#41—28no—
239C2×#41—41no—
240C2×#41—41no—
241C2×#41—42no—
242C2×#41—42no—
243C2×#41—42no—
244C2×#41—44no—
245C2×#41#4118yes—
246C22×Q8—22no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2412yes—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2×D411yescharacteristic
255C22×D4C2×D411yescharacteristic
256C22×D4C2412yes—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×M16—24no—
267C2×M16—42no—
268C2×M16—42no—
269C22×C8—24no—
270C22×C8—42no—
271C22×C8—42no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8—42no—
279C4⋊Q8—42no—
280#127—22no—
281#127—22no—
282#125—22no—
283#125—22no—
284#125—22no—
285#125—22no—
286#124—42no—
287#124—42no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290C22⋊Q8—48no—
291C22⋊Q8—48no—
292#121—28no—
293#121—28no—
294#121—48no—
295#121—48no—
296#121—48no—
297#121—48no—
298#121—84no—
299#121—84no—
300C4×Q8—44no—
301C4×Q8—48no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—48no—
306#117—41no—
307#117—41no—
308#117—41no—
309#117—41no—
310#117—41no—
311#117—41no—
312#117—42no—
313#117—42no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374#102—44no—
375#102—44no—
376#102—44no—
377#102—44no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382#97—22no—
383#97—22no—
384#97—22no—
385#97—22no—
386#97—42no—
387C4×C8—42no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402C24—22no—
403C24C22×D411yescharacteristic
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406#41—48no—
407#41—48no—
408#41—88no—
409#41—88no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8—44no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8#12414yes—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4#12414yes—
441C2×D4#12414yes—
442C2×D4#12414yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—84no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C22×C4C2511yescharacteristic
487M16—48no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—48no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C42—24no—
543C42—24no—
544C42—24no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—44no—
552C42H(5,𝔽2)12yes—
553C42H(5,𝔽2)12yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—44no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23C2×#12412yes—
572C23C2×#12412yes—
573C23C2×#12412yes—
574C23C23×D411yescharacteristic
575Q8—44no—
576Q8—44no—
577Q8—44no—
578Q8—88no—
579D4—44no—
580D4—44no—
581D4—44no—
582D4—44no—
583D4—44no—
584D4—88no—
585D4—88no—
586D4—164no—
587D4—164no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4—164no—
635C2×C4—164no—
636C2×C4C2×#12412yes—
637C2×C4C2×#12412yes—
638C2×C4C2×#12412yes—
639C2×C4#53414yes—
640C2×C4C22×#4112yes—
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4#58512yes—
644C8—42no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—84no—
664C22—84no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22C22×#12411yescharacteristic
669C22C2×#53411yescharacteristic
670C22#344911yescharacteristic
671C22#358512yes—
672C22#358612yes—
673C4—24no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4#350512yes—
688C4#350512yes—
689C2—42no—
690C2—42no—
691C2—42no—
692C2—162no—
693C2#3501911yescharacteristic
694C2#6062611yescharacteristic
695C2#6062711yescharacteristic
696C2#6075512yes—
697C2#6101612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35334C2semidirect
3. #33059C2semidirect
4. #33058C2semidirect
5. #33053C2semidirect
6. #33052C2semidirect
7. #30195C2semidirect
8. #29342C2semidirect
9. #29120C2semidirect
10. #27983C2semidirect
11. #27982C2semidirect
12. #27966C2semidirect
13. #27965C2semidirect
14. #25911C2semidirect
15. #25910C2semidirect
16. #25862C2semidirect
17. #25862C2semidirect
18. #21772C2semidirect
19. #20375C2semidirect

Maximal subgroups

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

subgroupindex
#353342
#352732
#330592
#330582
#330532
#330522
#301952
#293422
#291202
#279832
#279822
#279662
#279652
#259112
#259102
#258622
#217722
#203752