Finite Group Databasebeta

← #6993548

Subgroups — #6993548

Order 512.

1731 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36287C211yesmaximal, characteristic, complemented
2#35317C211yesmaximal, characteristic
3#32819C212yesmaximal, complemented
4#32282C214yesmaximal, complemented
5#31932C214yesmaximal, complemented
6#30305C212yesmaximal, complemented
7#29983C212yesmaximal, complemented
8#29690C211yesmaximal, characteristic
9#29555C212yesmaximal, complemented
10#28530C212yesmaximal, complemented
11#27727C212yesmaximal, complemented
12#25811C211yesmaximal, characteristic, complemented
13#25451C212yesmaximal, complemented
14#24813C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic, complemented
16#23570C211yesmaximal, characteristic
17#22632C211yesmaximal, characteristic, complemented
18C2×#536—21no—
19C2×#536—21no—
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#498C2212yes—
23C2×#475C2212yes—
24C2×#476C2212yes—
25C2×C4×SD16C2212yes—
26#2868C2212yes—
27#2859C2212yes—
28#2858C2212yes—
29#2857C2211yescharacteristic
30#2658—22no—
31#2612—22no—
32#2567C2211yescharacteristic
33#2556—22no—
34#2552C2211yescharacteristic
35#2542—21no—
36#2542—21no—
37#2542C2212yes—
38#2542C2212yes—
39#2506—21no—
40#2501—22no—
41#2501C2214yes—
42#2497C2212yes—
43#2441—21no—
44#2423C2212yes—
45#2370—21no—
46#2369C2211yescharacteristic
47Q8×C4⋊C4C2211yescharacteristic
48#2346—21no—
49C4×C22⋊Q8C2211yescharacteristic
50#2295C2212yes—
51C4×#117C2211yescharacteristic
52#2099C2214yes—
53#2096C2214yes—
54#2094C2214yes—
55#2084C2214yes—
56#2084C2214yes—
57#2071C2214yes—
58#2065C2214yes—
59#2065C2214yes—
60#2058C2211yescharacteristic
61#2058C2212yes—
62#2058C2212yes—
63#2058C2212yes—
64#2058C2212yes—
65#2041C2211yescharacteristic
66#2041C2212yes—
67#2034C2212yes—
68#2034C2214yes—
69#2024C2211yescharacteristic
70#2024C2212yes—
71#2014C2211yescharacteristic
72#2000C2212yes—
73#1990C2212yes—
74#1988C2212yes—
75#1968C2214yes—
76#1948C2214yes—
77#1939C2212yes—
78#1934C2214yes—
79#1926C2214yes—
80#1923C2214yes—
81#1880C2214yes—
82#1872C2212yes—
83#1853C2212yes—
84#1853C2214yes—
85#1844C2214yes—
86#1834C2214yes—
87#1831C2212yes—
88#1803C2214yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C22×#41—21no—
96C23×Q8—21no—
97C22×SD16C2312yes—
98#536—48no—
99#536—48no—
100#532—44no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#125—21no—
112C2×#125—22no—
113C2×#124—22no—
114C2×#124—22no—
115C2×#124—21no—
116C2×#123—24no—
117C2×#123—24no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8C2311yescharacteristic
124C2×#121—21no—
125C2×#121C2312yes—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2312yes—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×#117D411yescharacteristic
131C2×#117D411yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135#498—44no—
136#476—44no—
137#475—44no—
138C4×SD16—44no—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C4×C8C2314yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#399—22no—
159#399—24no—
160#399—24no—
161#399—24no—
162#399—24no—
163#398—22no—
164#398—22no—
165#396—22no—
166#395—22no—
167#391—21no—
168#391—22no—
169#391C2312yes—
170#391C2311yescharacteristic
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2314yes—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2312yes—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#387—21no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#385—21no—
196#384—21no—
197#384—22no—
198#384—22no—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2314yes—
202#384C2312yes—
203#384C2314yes—
204#382—22no—
205#380—21no—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376—22no—
222#376D412yes—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C2×#41—44no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8C2411yescharacteristic
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—42no—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—44no—
266C4⋊Q8—42no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270C4⋊Q8—48no—
271#127—42no—
272#125—44no—
273#125—84no—
274#124—44no—
275#124—48no—
276#124—48no—
277#123—416no—
278#123—416no—
279C22⋊Q8—22no—
280C22⋊Q8—41no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C22⋊Q8—48no—
285C22⋊Q8—82no—
286#121—42no—
287#121—42no—
288#121—44no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—42no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—44no—
295#117—24no—
296#117SD1618yescomplemented
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359#106—42no—
360#106—44no—
361#106—44no—
362#106—44no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—42no—
367#103—44no—
368#102—24no—
369#102—44no—
370#102—42no—
371#102—42no—
372#102—44no—
373C22⋊C8—24no—
374C22⋊C8—42no—
375C4×C8—28no—
376C4×C8—24no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24C22×D411yescharacteristic
396#41—816no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×Q8—816no—
413C2×Q82−1+412yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—42no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4C2×SD1614yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463SD16—48no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C82−1+414yes—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—82no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4C2×SD1614yes—
509C4⋊C4C2×SD1614yes—
510C4⋊C4AGL(1,ℤ/8)14yes—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—81no—
520C22⋊C4—82no—
521C22⋊C4C2×SD1618yes—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—42no—
531C42—44no—
532C42—42no—
533C42—42no—
534C42—82no—
535C42C2×SD1618yes—
536C42H(5,𝔽2)14yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C22×SD1612yes—
551C23C23×D411yescharacteristic
552Q8—24no—
553Q8—42no—
554Q8—88no—
555Q8—84no—
556Q8—88no—
557Q8—162no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—84no—
563D4—84no—
564D4—88no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—88no—
612C2×C4—88no—
613C2×C4#53512yes—
614C2×C4#53512yes—
615C2×C4C22×SD1612yes—
616C2×C4C22×SD1614yes—
617C2×C4C22×SD1614yes—
618C2×C4C22×SD1612yes—
619C2×C4C2×AGL(1,ℤ/8)14yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4#57712yes—
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×2−1+411yescharacteristic
624C2×C4C2×2−1+412yes—
625C2×C4#58511yescharacteristic
626C8—24no—
627C8—42no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C2×#53511yescharacteristic
644C22#348911yescharacteristic
645C22#349911yescharacteristic
646C22C23×SD1611yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22C2×#57711yescharacteristic
649C22#358611yescharacteristic
650C4—24no—
651C4—24no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—24no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—81no—
660C4—82no—
661C4—84no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4—81no—
666C4#320814yes—
667C4#353812yes—
668C4#354712yes—
669C2—42no—
670C2—162no—
671C2#4299411yescharacteristic
672C2C2×#320811yescharacteristic
673C2#6081311yescharacteristic
674C2#6209911yescharacteristic
675C2#6215011yescharacteristic
676C2#6246911yescharacteristic
677C2#6250311yescharacteristic
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. #36287C2semidirect
3. #32819C2semidirect
4. #32282C2semidirect
5. #31932C2semidirect
6. #30305C2semidirect
7. #29983C2semidirect
9. #29555C2semidirect
10. #28530C2semidirect
11. #27727C2semidirect
12. #25811C2semidirect
13. #25451C2semidirect
14. #24813C2semidirect
15. #24301C2semidirect
17. #22632C2semidirect
296. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#362872
#353172
#328192
#322822
#319322
#303052
#299832
#296902
#295552
#285302
#277272
#258112
#254512
#248132
#243012
#235702
#226322