Finite Group Databasebeta

← #6470457

Subgroups — #6470457

Order 512.

2434 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35079C211yesmaximal, characteristic, complemented
2C2×#2341C211yesmaximal, characteristic
3#32960C212yesmaximal, complemented
4#32957C212yesmaximal, complemented
5#29956C212yesmaximal, complemented
6#29950C212yesmaximal, complemented
7#25975C214yesmaximal, complemented
8#25969C214yesmaximal, complemented
9#25961C214yesmaximal, complemented
10#25959C214yesmaximal, complemented
11#21944C211yesmaximal, characteristic, complemented
12#21940C211yesmaximal, characteristic, complemented
13#21932C211yesmaximal, characteristic, complemented
14#20601C211yesmaximal, characteristic
15#20597C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C4×Q8C2212yes—
18C2×#479C2214yescomplemented
19C2×#477C2214yescomplemented
20C2×#417C2214yes—
21C2×#416C2214yes—
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2857C2212yes—
25#2851C2212yes—
26#2667—21no—
27#2665—21no—
28#2569—21no—
29#2549—21no—
30#2442C2212yescomplemented
31#2441C2212yescomplemented
32#2440C2212yescomplemented
33#2439C2212yescomplemented
34#2437—22no—
35#2341—22no—
36#2341—21no—
37#2341—21no—
38#2341C2214yes—
39#2341C2212yes—
40#2341C2212yes—
41#2329—22no—
42C2×#390—22no—
43C2×#390C2212yes—
44#2280C2211yescharacteristic
45C2×#380—21no—
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2093C2214yescomplemented
51#2092C2214yescomplemented
52#2067C2214yescomplemented
53#2067C2214yescomplemented
54#2067C2214yescomplemented
55#2065C2214yescomplemented
56#2025C2214yescomplemented
57#2023C2214yescomplemented
58#1865C2214yescomplemented
59#1863C2214yescomplemented
60#1863C2214yescomplemented
61#1863C2214yescomplemented
62#1854C2214yescomplemented
63#1853C2212yes—
64#1852C2212yes—
65#1852C2212yes—
66#1852C2212yes—
67#1852C2214yescomplemented
68#1851C2212yescomplemented
69#1851C2212yescomplemented
70#1850C2214yescomplemented
71#1809C2214yescomplemented
72#1808C2212yescomplemented
73#1808C2212yescomplemented
74#1805C2214yescomplemented
75#1803C2214yescomplemented
76#1797C2212yes—
77#1797C2212yes—
78#1788C2212yes—
79#1788C2212yes—
80#1788C2214yes—
81#1787C2211yescharacteristic
82#1787C2211yescharacteristic
83#1787C2212yes—
84#1786C2211yescharacteristic
85#1786C2212yes—
86#1786C2211yescharacteristic
87C22×#41—21no—
88C22×#41—21no—
89C22×#41—24no—
90C22×#41—48no—
91C22×#41Q818yescomplemented
92C22×#41C2312yes—
93C22×#41C2312yes—
94C23×Q8C2311yescharacteristic
95C23×D4—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C24×C4C2311yescharacteristic
99C22×M16C2312yes—
100C23×C8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#125C2312yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—48no—
108C2×C4×Q8C2314yes—
109C2×C4×Q8C2314yes—
110C2×#117C2312yes—
111C22×C4⋊C4—22no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—22no—
118C22×C22⋊C4D412yes—
119C22×C42—22no—
120C22×C42C2312yes—
121#479—48no—
122#477—48no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129#417—48no—
130#416—48no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139#407—24no—
140C2×C22⋊C8—24no—
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2311yescharacteristic
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146#400—22no—
147#399—22no—
148#398—22no—
149#398—22no—
150#396—22no—
151#396—22no—
152#395—22no—
153#393—22no—
154#391—22no—
155#391—22no—
156#390—24no—
157#390—24no—
158#390—24no—
159#390—24no—
160#390—22no—
161#390—22no—
162#390C2314yes—
163#390C2314yes—
164#389—22no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384C2312yes—
179#382C2312yes—
180#381—21no—
181#380—22no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—22no—
187#380—24no—
188#380—22no—
189#380—22no—
190#380C2312yes—
191#380C2311yescharacteristic
192#380C2311yescharacteristic
193#380C2312yes—
194#380C2312yes—
195#380C2311yescharacteristic
196#380C2312yes—
197#380C2311yescharacteristic
198#379—21no—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—24no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4D412yes—
205C4×C22⋊C4D411yescharacteristic
206C4×C22⋊C4D411yescharacteristic
207C4×C22⋊C4C2312yes—
208C4×C22⋊C4C2311yescharacteristic
209C4×C22⋊C4C2311yescharacteristic
210C2×#95—21no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220C25—21no—
221C25C2×D411yescharacteristic
222C2×#41—28no—
223C2×#41—28no—
224C2×#41—24no—
225C2×#41—24no—
226C2×#41—28no—
227C2×#41—28no—
228C2×#41—44no—
229C2×#41—44no—
230C2×#41—432no—
231C2×#41—432no—
232C2×#41—48no—
233C2×#41—48no—
234C2×#41C2×Q818yes—
235C2×#41C2×Q818yes—
236C22×Q8—28no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8C2×Q814yes—
241C22×Q8C2412yes—
242C22×D4—28no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—44no—
248C22×D4—48no—
249C22×D4C2×Q814yes—
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—28no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—44no—
258C23×C4—48no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×Q814yes—
265C23×C4#4114yes—
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×M16—24no—
270C2×M16—44no—
271C22×C8—24no—
272C22×C8—44no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C4⋊Q8—44no—
277#125—44no—
278C22⋊Q8—48no—
279C22⋊Q8—84no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—48no—
285C4×Q8—48no—
286#117—44no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4—82no—
324C2×C22⋊C4—84no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—44no—
334C2×C42—48no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339#107—44no—
340#107—44no—
341#107—44no—
342#106—44no—
343C4⋊C8—44no—
344C4⋊C8—44no—
345#103—28no—
346#103—28no—
347#103—28no—
348#103—28no—
349#102—44no—
350#102—44no—
351#102—44no—
352#102—44no—
353C22⋊C8—22no—
354C22⋊C8—22no—
355C22⋊C8—24no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D411yescharacteristic
374#95C2×D411yescharacteristic
375#95C2×D412yes—
376#95C2×D411yescharacteristic
377#95C2×D411yescharacteristic
378C24—22no—
379C24—22no—
380C24—24no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—44no—
385C24—44no—
386C24—42no—
387C24—42no—
388C24C22⋊Q814yes—
389C24C22×D411yescharacteristic
390C24C22×D411yescharacteristic
391C24C22×D411yescharacteristic
392#41—216no—
393#41—216no—
394#41—48no—
395#41—48no—
396#41—416no—
397#41—416no—
398#41—432no—
399#41—432no—
400C2×Q8—28no—
401C2×Q8—216no—
402C2×Q8—216no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8C22×Q814yes—
410C2×Q8H(5,𝔽2)18yes—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—432no—
421C2×D4—44no—
422C2×D4—48no—
423C2×D4C22×Q814yes—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—432no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4C22⋊Q814yes—
458C22×C4C22⋊Q814yes—
459C22×C4C22⋊Q814yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C22×Q812yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2511yescharacteristic
474M16—84no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—84no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—48no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—24no—
520C42—24no—
521C42—42no—
522C42—42no—
523C42—48no—
524C42—48no—
525C42—82no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—28no—
537C23—24no—
538C23—44no—
539C23—44no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—84no—
548C23—84no—
549C23—88no—
550C23—88no—
551C23C2×C22⋊Q812yes—
552C23C2×C22⋊Q812yes—
553C23C2×C22⋊Q812yes—
554C23#57612yes—
555C23#57812yes—
556C23C23×D411yescharacteristic
557Q8—216no—
558Q8—24no—
559Q8—28no—
560Q8—164no—
561Q8#54318yes—
562D4—24no—
563D4—24no—
564D4—416no—
565D4—416no—
566D4—48no—
567D4—44no—
568C2×C4—216no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—84no—
613C2×C4—88no—
614C2×C4—164no—
615C2×C4C2×C22⋊Q812yes—
616C2×C4C2×C22⋊Q812yes—
617C2×C4C2×C22⋊Q812yes—
618C2×C4#57514yes—
619C2×C4C23×Q811yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)14yes—
622C8—42no—
623C8—42no—
624C8—44no—
625C22—22no—
626C22—24no—
627C22—24no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—88no—
643C22—816no—
644C22—84no—
645C22—84no—
646C22#308314yes—
647C22C22×C22⋊Q811yescharacteristic
648C22#348412yes—
649C22C2×#57512yes—
650C22C2×#57611yescharacteristic
651C22C2×#57811yescharacteristic
652C4—24no—
653C4—28no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4C2×#54312yes—
666C4#349012yes—
667C2—24no—
668C2—42no—
669C2—42no—
670C2—84no—
671C2#3498211yescharacteristic
672C2#6048411yescharacteristic
673C2C2×#308311yescharacteristic
674C2#6160312yes—
675C2#6160512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
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
1. #35079C2semidirect
3. #32960C2semidirect
4. #32957C2semidirect
5. #29956C2semidirect
6. #29950C2semidirect
7. #25975C2semidirect
8. #25969C2semidirect
9. #25961C2semidirect
10. #25959C2semidirect
11. #21944C2semidirect
12. #21940C2semidirect
13. #21932C2semidirect
18. C2×#479C22semidirect
19. C2×#477C22semidirect
30. #2442C22semidirect
31. #2441C22semidirect
32. #2440C22semidirect
33. #2439C22semidirect
50. #2093C22semidirect
51. #2092C22semidirect
52. #2067C22semidirect
53. #2067C22semidirect
54. #2067C22semidirect
55. #2065C22semidirect
56. #2025C22semidirect
57. #2023C22semidirect
58. #1865C22semidirect
59. #1863C22semidirect
60. #1863C22semidirect
61. #1863C22semidirect
62. #1854C22semidirect
67. #1852C22semidirect
68. #1851C22semidirect
69. #1851C22semidirect
70. #1850C22semidirect
71. #1809C22semidirect
72. #1808C22semidirect
73. #1808C22semidirect
74. #1805C22semidirect
75. #1803C22semidirect
91. C22×#41Q8semidirect

Maximal subgroups

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

subgroupindex
#350792
C2×#23412
#329602
#329572
#299562
#299502
#259752
#259692
#259612
#259592
#219442
#219402
#219322
#206012
#205972