Finite Group Databasebeta

← #6447239

Subgroups — #6447239

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic
3#26315C214yesmaximal, complemented
4#26313C214yesmaximal, complemented
5#26303C212yesmaximal, complemented
6#26301C212yesmaximal, complemented
7#26298C212yesmaximal, complemented
8#26296C212yesmaximal, complemented
9#23643C212yesmaximal, complemented
10#23641C212yesmaximal, complemented
11#21324C211yesmaximal, characteristic, complemented
12C2×#1988C211yesmaximal, characteristic, complemented
13#21189C211yesmaximal, characteristic
14#21169C211yesmaximal, characteristic, complemented
15#20849C212yesmaximal, complemented
16#20703C212yesmaximal, complemented
17C4×#428C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2212yes—
21C2×#428C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#427C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C2×#408C2212yes—
26C2×#408C2212yes—
27C2×C4×M16C2211yescharacteristic
28#2862—21no—
29#2858—21no—
30#2640—22no—
31#2563C2212yes—
32#2563C2212yes—
33#2547C2212yes—
34#2547C2212yes—
35#2504—22no—
36#2436—22no—
37#2367—22no—
38#2366—21no—
39#2365—21no—
40#2354C2212yes—
41#2352C2214yes—
42D4×C4⋊C4C2212yes—
43#2325—22no—
44#2311C2212yes—
45#2311C2212yes—
46#2311C2212yes—
47C4×#121—22no—
48#2285C2211yescharacteristic
49C4×#117C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2077C2214yes—
52#2077C2214yes—
53#2077C2214yes—
54#2077C2214yes—
55#2076C2214yes—
56#2076C2214yes—
57#2075C2214yes—
58#2075C2214yes—
59#1993C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1988C2212yes—
63#1988C2212yes—
64#1988C2214yes—
65#1955C2214yes—
66#1954C2212yes—
67#1954C2212yes—
68#1949C2212yes—
69#1949C2212yes—
70#1948C2211yescharacteristic
71#1948C2211yescharacteristic
72#1947C2211yescharacteristic
73#1947C2211yescharacteristic
74#1854C2214yes—
75#1854C2214yes—
76#1853C2214yes—
77#1852C2214yes—
78#1825C2214yes—
79#1825C2214yes—
80#1824C2214yes—
81#1823C2214yes—
82#1776C2212yes—
83#1776C2212yes—
84C4×#107C2212yes—
85C4×#106C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1735C2212yes—
91C2×#340C2212yes—
92C2×#340C2212yes—
93C22×#41C2312yes—
94C23×Q8—21no—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#532—24no—
99#532—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#127D411yescharacteristic
108C2×#124—22no—
109C2×#124—22no—
110C2×#124D412yes—
111C2×#121D412yes—
112C2×#121D412yes—
113#520—28no—
114#520—44no—
115C2×C4×Q8—22no—
116C2×C4×D4—24no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2314yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#428—24no—
133#428—24no—
134#427—44no—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×C4⋊C8—24no—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2312yes—
147#408—28no—
148#408—44no—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C4×M16—24no—
154C2×#97C2312yes—
155C2×C4×C8C2312yes—
156#397—24no—
157#397—24no—
158#393—24no—
159#393—24no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390—22no—
171#389—22no—
172#388—24no—
173#388—24no—
174#387—24no—
175#386—24no—
176#386—24no—
177#386—24no—
178#386—24no—
179#385—24no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#382—22no—
183#382—22no—
184#382C2312yes—
185#380—21no—
186#380—21no—
187#380C2314yes—
188#380C2314yes—
189#380C2314yes—
190#380C2314yes—
191#380C2314yes—
192#380C2314yes—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C22⋊C4—21no—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—24no—
207#376—21no—
208C2×#95—21no—
209C43—21no—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—24no—
231C2×#41—24no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—44no—
235C22×Q8—82no—
236C22×Q8—82no—
237C22×Q8C2×D412yes—
238C22×D4C2×D412yes—
239C22×D4C2×D414yes—
240C22×D4C2×D412yes—
241C22×D4C2×D412yes—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—21no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2414yes—
249C23×C4C2411yescharacteristic
250C2×M16—22no—
251C2×M16—22no—
252C2×M16—22no—
253C2×M16—22no—
254C2×M16—22no—
255C2×M16#4114yes—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#127—24no—
271#124—28no—
272#124—84no—
273#124—84no—
274#121—28no—
275#121—28no—
276C4×Q8—48no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—48no—
282C4×D4—416no—
283C4×D4—84no—
284#117—24no—
285#117—22no—
286#117—22no—
287#117—24no—
288#117—22no—
289#117—22no—
290#117—28no—
291#117—44no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4#4114yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D414yes—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—82no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42#4114yes—
348C2×C42#4114yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—24no—
352#107—24no—
353#107—42no—
354#107—42no—
355#106—24no—
356#106—24no—
357#106—42no—
358#106—42no—
359C4⋊C8—22no—
360C4⋊C8—22no—
361C4⋊C8—24no—
362C22⋊C8—28no—
363C22⋊C8—28no—
364C22⋊C8—44no—
365C22⋊C8—44no—
366#97—24no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382C24C22×D414yes—
383C24C22×D411yescharacteristic
384#41—48no—
385#41—48no—
386#41—48no—
387#41—48no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—816no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—42no—
409C2×D4—42no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4#12314yes—
442C22×C4#12314yes—
443C22×C4C22×D414yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456M16—28no—
457M16—44no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—48no—
470C2×C8C2×#4114yes—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4C2×#4114yes—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—48no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—48no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—44no—
518C42—42no—
519C42—44no—
520C42—42no—
521C42—42no—
522C42—82no—
523C42#12314yes—
524C42#12314yes—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23C2×#12312yes—
541C23#54612yes—
542C23#54712yes—
543C23C23×D411yescharacteristic
544Q8—48no—
545Q8—88no—
546Q8—88no—
547D4—416no—
548D4—48no—
549D4—48no—
550D4—84no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—88no—
596C2×C4—168no—
597C2×C4C2×#12312yes—
598C2×C4C2×#12312yes—
599C2×C4C2×#12312yes—
600C2×C4C2×#12312yes—
601C2×C4C2×#12312yes—
602C2×C4#52914yes—
603C2×C4#54214yes—
604C2×C4#54612yes—
605C2×C4#54712yes—
606C2×C4#55614yes—
607C2×C4#55914yes—
608C2×C4#57612yes—
609C2×C4#57712yes—
610C2×C4C22×#4111yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C8—24no—
617C8—42no—
618C8—42no—
619C8—42no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22#244714yes—
634C22C22×#12311yescharacteristic
635C22C2×#54611yescharacteristic
636C22C2×#54711yescharacteristic
637C22C2×#57611yescharacteristic
638C22C2×#57711yescharacteristic
639C22#358611yescharacteristic
640C22#358611yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—44no—
645C4—44no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4—84no—
653C4—81no—
654C4#244714yes—
655C4#309614yes—
656C4C2×#52911yescharacteristic
657C4C2×#54211yescharacteristic
658C4C2×#55611yescharacteristic
659C4C2×#55911yescharacteristic
660C2—24no—
661C2—84no—
662C2C2×#244711yescharacteristic
663C2C2×#309611yescharacteristic
664C2#6051911yescharacteristic
665C2#6119911yescharacteristic
666C2#6120211yescharacteristic
667C2#6127811yescharacteristic
668C2#6128111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26315C2semidirect
4. #26313C2semidirect
5. #26303C2semidirect
6. #26301C2semidirect
7. #26298C2semidirect
8. #26296C2semidirect
9. #23643C2semidirect
10. #23641C2semidirect
11. #21324C2semidirect
12. C2×#1988C2semidirect
14. #21169C2semidirect
15. #20849C2semidirect
16. #20703C2semidirect
17. C4×#428C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#342062
#263152
#263132
#263032
#263012
#262982
#262962
#236432
#236412
#213242
C2×#19882
#211892
#211692
#208492
#207032
C4×#4282