Finite Group Databasebeta

← #6463629

Subgroups — #6463629

Order 512.

1810 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic, complemented
3#29267C214yesmaximal, complemented
4#29264C214yesmaximal, complemented
5#29261C214yesmaximal, complemented
6#29257C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic
8C2×#2097C211yesmaximal, characteristic, complemented
9#23455C211yesmaximal, characteristic, complemented
10#23451C211yesmaximal, characteristic
11#21262C211yesmaximal, characteristic
12#21262C211yesmaximal, characteristic
13#21174C211yesmaximal, characteristic, complemented
14#21173C211yesmaximal, characteristic, complemented
15#20882C211yesmaximal, characteristic
16#20881C211yesmaximal, characteristic
17#20877C211yesmaximal, characteristic, complemented
18#20877C211yesmaximal, characteristic, complemented
19#20754C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#419C2211yescharacteristic
34C2×#416C2211yescharacteristic
35C2×#416C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38#2368—24no—
39#2362—22no—
40#2360—22no—
41#2359—24no—
42#2355—22no—
43#2355—22no—
44#2331C2214yes—
45#2327C2212yes—
46#2326C2212yes—
47#2317C2211yescharacteristic
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×#125C2214yes—
52C4×#124C2212yes—
53#2297C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2274C2211yescharacteristic
57#2274C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2097C2212yes—
60#2097C2214yes—
61#2097C2212yes—
62#2097C2214yes—
63#2092C2212yes—
64#2092C2212yes—
65#2026C2212yes—
66#2026C2214yes—
67#2025C2212yes—
68#2024C2212yes—
69#2024C2214yes—
70#2023C2212yes—
71#1983C2214yes—
72#1983C2214yes—
73#1981C2214yes—
74#1981C2214yes—
75#1949C2214yes—
76#1949C2214yes—
77#1948C2214yes—
78#1947C2214yes—
79#1865C2214yes—
80#1865C2214yes—
81#1865C2214yes—
82#1864C2214yes—
83#1863C2214yes—
84#1863C2214yes—
85#1863C2214yes—
86#1862C2214yes—
87#1832C2214yes—
88#1831C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#536—24no—
104#536Q818yescomplemented
105#533—24no—
106#533—24no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#125C2312yes—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114#520—42no—
115#520—42no—
116#519—42no—
117#519—42no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117—22no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#428—42no—
139#428—42no—
140C2×#107—42no—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#106—42no—
145C2×#106C2312yes—
146C2×#106C2312yes—
147C2×#106C2314yes—
148#424—24no—
149#424—24no—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152#419—24no—
153#419—24no—
154#417—42no—
155#417—42no—
156#416—24no—
157#416—24no—
158C2×#103—42no—
159C2×#103—42no—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172#391—22no—
173#391—24no—
174#391—24no—
175#391C2311yescharacteristic
176#391C2312yes—
177#391C2312yes—
178#391C2311yescharacteristic
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#389—22no—
182#389—24no—
183#389C2314yes—
184#389C2314yes—
185#388—24no—
186#388—24no—
187#387—24no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#386—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#385—24no—
198#384—21no—
199#384—21no—
200#384C2312yes—
201#382—21no—
202#382—21no—
203#382—22no—
204#382C2312yes—
205#382C2312yes—
206#382C2312yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—24no—
210C4×C4⋊C4—24no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2314yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—24no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D414yes—
221#376C2312yes—
222C2×#95—22no—
223C2×#95—22no—
224C43C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41C2×Q814yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8C2×Q814yes—
251C22×Q8C2412yes—
252C22×Q8C2411yescharacteristic
253C22×D4—21no—
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8C2×Q818yes—
283#125—24no—
284#125—24no—
285#124—24no—
286#124C2×Q818yes—
287#123—216no—
288C22⋊Q8—216no—
289C22⋊Q8—216no—
290C4×Q8—216no—
291C4×Q8—42no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×Q8—42no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—48no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—48no—
309#117—44no—
310#117C2×Q814yes—
311#117#4114yes—
312#117#4114yes—
313#117#4114yes—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
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—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42C2×D412yes—
364C2×C42C2×D414yes—
365C2×C42C2×D414yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370#107—44no—
371#107—42no—
372#107—42no—
373#106—44no—
374#106—42no—
375#106—42no—
376C4⋊C8—28no—
377C4⋊C8—28no—
378#103—24no—
379#103—24no—
380#103—28no—
381#103—28no—
382#103—44no—
383#103—44no—
384#103—42no—
385#103—42no—
386#102—24no—
387#102—24no—
388#102—42no—
389#102—42no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95C2×D414yes—
402#95C2×D414yes—
403C24—21no—
404C24C22×D411yescharacteristic
405#41—28no—
406#41—28no—
407#41—44no—
408#41—44no—
409C2×Q8—28no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—28no—
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8C22×Q814yes—
430C2×Q8C22×Q812yes—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C22×Q812yes—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—84no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×Q812yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4C2511yescharacteristic
479M16—44no—
480M16—44no—
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—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—44no—
494C2×C8—44no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4C22⋊Q818yes—
523C4⋊C4C2×#4114yes—
524C4⋊C4C2×#4114yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—216no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—82no—
533C22⋊C4C22⋊Q818yes—
534C42—24no—
535C42—24no—
536C42—24no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—22no—
542C42—22no—
543C42—44no—
544C42—44no—
545C42—44no—
546C42—44no—
547C42—44no—
548C42—42no—
549C42—84no—
550C42—84no—
551C42C22×Q814yes—
552C42C2×#4114yes—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23C2×C22⋊Q812yes—
561C23#52512yes—
562C23#53512yes—
563C23C23×D411yescharacteristic
564Q8—24no—
565Q8—24no—
566Q8—22no—
567Q8—22no—
568Q8—48no—
569Q8—42no—
570Q8—42no—
571Q8—41no—
572Q8—41no—
573Q8—88no—
574D4—22no—
575D4—22no—
576D4—48no—
577D4—41no—
578D4—41no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—84no—
616C2×C4—164no—
617C2×C4—162no—
618C2×C4—162no—
619C2×C4—162no—
620C2×C4C2×C22⋊Q818yes—
621C2×C4C2×C22⋊Q812yes—
622C2×C4#52512yes—
623C2×C4#53512yes—
624C2×C4#55414yes—
625C2×C4#55414yes—
626C2×C4#55914yes—
627C2×C4#55914yes—
628C2×C4C23×Q811yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C8—42no—
635C8—42no—
636C8—42no—
637C8—42no—
638C22—24no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—42no—
643C22—44no—
644C22—81no—
645C22—81no—
646C22—82no—
647C22#258014yes—
648C22C22×C22⋊Q811yescharacteristic
649C22C2×#52511yescharacteristic
650C22C2×#53511yescharacteristic
651C22#358412yes—
652C22#358612yes—
653C4—28no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—44no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4—81no—
666C4—82no—
667C4#258014yes—
668C4C2×#55411yescharacteristic
669C4C2×#55411yescharacteristic
670C4C2×#55911yescharacteristic
671C4C2×#55911yescharacteristic
672C2—24no—
673C2—82no—
674C2C2×#258011yescharacteristic
675C2#6049912yes—
676C2#6058612yes—
677C2#6081911yescharacteristic
678C2#6082811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
2. #34215C2semidirect
3. #29267C2semidirect
4. #29264C2semidirect
5. #29261C2semidirect
6. #29257C2semidirect
8. C2×#2097C2semidirect
9. #23455C2semidirect
13. #21174C2semidirect
14. #21173C2semidirect
17. #20877C2semidirect
18. #20877C2semidirect
19. #20754C2semidirect
104. #536Q8semidirect

Maximal subgroups

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

subgroupindex
#342762
#342152
#292672
#292642
#292612
#292572
#237332
C2×#20972
#234552
#234512
#212622
#211742
#211732
#208822
#208812
#208772
#207542