Finite Group Databasebeta

← #6484434

Subgroups — #6484434

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic, complemented
2#34208C211yesmaximal, characteristic
3#32965C212yesmaximal, complemented
4#32963C212yesmaximal, complemented
5#29965C212yesmaximal, complemented
6#29964C212yesmaximal, complemented
7#29266C212yesmaximal, complemented
8#29265C212yesmaximal, complemented
9#29198C212yesmaximal, complemented
10#29197C212yesmaximal, complemented
11#29146C211yesmaximal, characteristic, complemented
12#29146C211yesmaximal, characteristic, complemented
13#28520C211yesmaximal, characteristic, complemented
14#28519C211yesmaximal, characteristic, complemented
15#28291C212yesmaximal, complemented
16#28290C212yesmaximal, complemented
17#21986C212yesmaximal, complemented
18#21809C212yesmaximal
19#20811C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#477C2212yes—
24C2×#475C2212yes—
25C2×#476C2212yes—
26C2×#474C2212yes—
27C2×#432C2212yes—
28#2868C2211yescharacteristic
29#2859C2212yes—
30#2859C2211yescharacteristic
31#2629C2212yes—
32#2628C2212yes—
33#2570—22no—
34#2550—22no—
35#2441—22no—
36#2439—22no—
37#2364—22no—
38#2364—22no—
39#2352—22no—
40#2352—22no—
41#2331C2212yes—
42#2330C2212yes—
43#2328C2211yescharacteristic
44#2328C2211yescharacteristic
45#2327C2211yescharacteristic
46#2327C2211yescharacteristic
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49C4×#124C2211yescharacteristic
50C4×#124C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2211yescharacteristic
53#2285C2212yes—
54C2×#382C2211yescharacteristic
55C4×#117C2211yescharacteristic
56#2096C2212yes—
57#2095C2212yes—
58#2094C2212yes—
59#2093C2212yes—
60#2068C2214yes—
61#2067C2214yes—
62#2066C2214yes—
63#2065C2214yes—
64#2026C2212yes—
65#2026C2212yes—
66#2025C2212yes—
67#2025C2212yes—
68#1982C2214yes—
69#1981C2214yes—
70#1959C2212yes—
71#1959C2212yes—
72#1959C2212yes—
73#1959C2212yes—
74#1949C2214yes—
75#1949C2214yes—
76#1865C2214yes—
77#1864C2214yes—
78#1863C2214yes—
79#1862C2214yes—
80#1853C2214yes—
81#1852C2214yes—
82#1847C2212yes—
83#1847C2212yes—
84#1843C2212yes—
85#1836C2212yes—
86#1836C2212yes—
87#1832C2212yes—
88#1805C2212yes—
89#1805C2212yes—
90#1804C2212yes—
91#1804C2212yes—
92#1798C2212yes—
93#1798C2212yes—
94#1788C2214yes—
95#1788C2214yes—
96#1767C2212yes—
97#1767C2211yescharacteristic
98#1767C2211yescharacteristic
99C22×#41C2311yescharacteristic
100C22×#41C2311yescharacteristic
101#532—24no—
102#532Q818yescomplemented
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#125—22no—
108C2×#125C2312yes—
109C2×#124C2311yescharacteristic
110C2×#124C2311yescharacteristic
111C2×#121—22no—
112#520—44no—
113#520—44no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117—21no—
129C2×#117—21no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#477—44no—
138#476—44no—
139#475—44no—
140#474—44no—
141#432—28no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2314yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161#400—22no—
162#399—22no—
163#398—24no—
164#396—24no—
165#395—22no—
166#393—22no—
167#391—24no—
168#391C2311yescharacteristic
169#391C2311yescharacteristic
170#391C2311yescharacteristic
171#391C2311yescharacteristic
172#390C2311yescharacteristic
173#390C2311yescharacteristic
174#390C2311yescharacteristic
175#390C2311yescharacteristic
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—24no—
181#389C2314yes—
182#389C2314yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—24no—
188#387—24no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#384—22no—
200#384C2312yes—
201#384C2312yes—
202#384C2312yes—
203#382—22no—
204#382—22no—
205#382—22no—
206#382C2312yes—
207#382C2311yescharacteristic
208#382C2311yescharacteristic
209#382C2311yescharacteristic
210#382C2311yescharacteristic
211#380—22no—
212#380—22no—
213#380—22no—
214#380C2314yes—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225#376C2311yescharacteristic
226#376C2311yescharacteristic
227C2×#95—22no—
228C43C2311yescharacteristic
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—42no—
240C2×#41—42no—
241C2×#41—42no—
242C2×#41—42no—
243C2×#41C2×Q814yes—
244C2×#41C2×Q814yes—
245C22×Q8—22no—
246C22×Q8C2411yescharacteristic
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—22no—
251C22×D4C2411yescharacteristic
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C22×C8C2414yes—
264C22×C8C2414yes—
265C4⋊Q8—22no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—84no—
269C4⋊Q8C2×Q814yes—
270#127—22no—
271#127C2×Q814yes—
272#125—44no—
273#125—84no—
274#124—22no—
275#124—22no—
276#124C2×Q814yes—
277#124C2×Q814yes—
278#121—216no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×D4—216no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—42no—
287C4×D4—42no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—44no—
292#117—24no—
293#117—42no—
294#117—44no—
295#117—44no—
296#117—84no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2414yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D411yescharacteristic
338C2×C22⋊C4C2×D411yescharacteristic
339C2×C22⋊C4C2×D412yes—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×Q814yes—
351C2×C42#4114yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—44no—
359#107—44no—
360#106—44no—
361#106—44no—
362C4⋊C8—28no—
363C4⋊C8—24no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#102—44no—
371#102—44no—
372#102—44no—
373#102—44no—
374C22⋊C8—28no—
375#97—24no—
376#97—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24—21no—
397C24C22×D411yescharacteristic
398#41—48no—
399#41—48no—
400#41—48no—
401#41—48no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—84no—
410C2×Q8C22×Q812yes—
411C2×Q8C22×Q812yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4C22×Q812yes—
428C2×D4C22×Q812yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4C22⋊Q818yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×Q812yes—
470C22×C4C2×#4112yes—
471C22×C4C2511yescharacteristic
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C4⋊C4—28no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—84no—
531C42C22⋊Q818yes—
532C42C22×Q812yes—
533C42C22×Q812yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C42H(5,𝔽2)14yes—
537C23—22no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C2×C22⋊Q812yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—24no—
555Q8—42no—
556Q8—42no—
557Q8—164no—
558D4—24no—
559D4—24no—
560D4—48no—
561D4—48no—
562D4—42no—
563D4—42no—
564D4—88no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—162no—
609C2×C4—162no—
610C2×C4—162no—
611C2×C4—162no—
612C2×C4—162no—
613C2×C4—162no—
614C2×C4C2×C22⋊Q812yes—
615C2×C4C2×C22⋊Q814yes—
616C2×C4C2×C22⋊Q814yes—
617C2×C4#53512yes—
618C2×C4#53512yes—
619C2×C4C23×Q811yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)12yes—
622C2×C4#58512yes—
623C8—44no—
624C8—44no—
625C22—24no—
626C22—22no—
627C22—42no—
628C22—41no—
629C22—41no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—82no—
635C22—82no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22C22×C22⋊Q811yescharacteristic
641C22C2×#53511yescharacteristic
642C22#344911yescharacteristic
643C22#358512yes—
644C22#358612yes—
645C4—24no—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—41no—
655C4—41no—
656C4—84no—
657C4—81no—
658C4—81no—
659C4—84no—
660C4—81no—
661C4—81no—
662C4#350112yes—
663C4#350612yes—
664C2—42no—
665C2—82no—
666C2—82no—
667C2#3517211yescharacteristic
668C2#6049312yes—
669C2#6082511yescharacteristic
670C2#6082611yescharacteristic
671C2#6102112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #35294C2semidirect
3. #32965C2semidirect
4. #32963C2semidirect
5. #29965C2semidirect
6. #29964C2semidirect
7. #29266C2semidirect
8. #29265C2semidirect
9. #29198C2semidirect
10. #29197C2semidirect
11. #29146C2semidirect
12. #29146C2semidirect
13. #28520C2semidirect
14. #28519C2semidirect
15. #28291C2semidirect
16. #28290C2semidirect
17. #21986C2semidirect
19. #20811C2semidirect
102. #532Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#342082
#329652
#329632
#299652
#299642
#292662
#292652
#291982
#291972
#291462
#285202
#285192
#282912
#282902
#219862
#218092
#208112