Finite Group Databasebeta

← #6447029

Subgroups — #6447029

Order 512.

1988 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic
3#24980C214yesmaximal, complemented
4#24979C214yesmaximal, complemented
5#24972C212yesmaximal, complemented
6#24971C212yesmaximal, complemented
7#24970C212yesmaximal, complemented
8#24969C212yesmaximal, complemented
9#23641C212yesmaximal, complemented
10#23487C212yesmaximal, complemented
11#21189C211yesmaximal, characteristic
12#21169C211yesmaximal, characteristic, complemented
13#20913C212yesmaximal, complemented
14#20764C211yesmaximal, characteristic, complemented
15#20703C212yesmaximal, complemented
16C2×#1803C211yesmaximal, characteristic, complemented
17C4×#419C211yesmaximal, characteristic, complemented
18C2×#536C2212yes—
19C2×#532C2211yescharacteristic
20C2×#520C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C2×#427C2212yes—
23C22×C4⋊C8C2211yescharacteristic
24C2×#419C2211yescharacteristic
25C2×#419C2211yescharacteristic
26C22×#103C2212yes—
27C2×#408C2212yes—
28C2×#408C2212yes—
29C2×C4×M16C2211yescharacteristic
30#2854—22no—
31#2618—22no—
32#2548—21no—
33#2546—21no—
34#2545—22no—
35#2538C2212yes—
36#2538C2212yes—
37#2536C2214yes—
38#2361—22no—
39#2361—22no—
40#2354C2212yes—
41#2354C2212yes—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44#2326—22no—
45#2311C2212yes—
46#2311C2212yes—
47C4×C4⋊Q8—21no—
48C4×#127—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2077C2214yes—
53#2077C2214yes—
54#2076C2214yes—
55#2075C2214yes—
56#2036C2214yes—
57#2036C2214yes—
58#2035C2214yes—
59#2035C2214yes—
60#1955C2214yes—
61#1954C2212yes—
62#1954C2212yes—
63#1949C2212yes—
64#1949C2212yes—
65#1948C2211yescharacteristic
66#1948C2211yescharacteristic
67#1947C2211yescharacteristic
68#1947C2211yescharacteristic
69#1873C2214yes—
70#1873C2214yes—
71#1872C2214yes—
72#1872C2214yes—
73#1836C2214yes—
74#1834C2212yes—
75#1834C2212yes—
76#1825C2214yes—
77#1825C2214yes—
78#1824C2214yes—
79#1823C2214yes—
80#1803C2212yes—
81#1803C2212yes—
82#1803C2214yes—
83#1763C2212yes—
84#1763C2212yes—
85C4×#103C2212yes—
86C4×#103C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2212yes—
92C2×#340C2212yes—
93C22×#41C2312yes—
94C23×Q8C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#536—28no—
99#536—44no—
100#532—24no—
101#532—24no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8D411yescharacteristic
104C2×#127D411yescharacteristic
105C2×#124—22no—
106C2×#124D412yes—
107C2×#123—24no—
108C2×C22⋊Q8C2314yes—
109C2×C22⋊Q8C2314yes—
110C2×#121D412yes—
111C2×#121D412yes—
112#520—28no—
113#520—44no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—24no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2312yes—
129C2×#117C2314yes—
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2314yes—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#427—44no—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×C4⋊C8—24no—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2312yes—
148#419—24no—
149#419—24no—
150C2×#103—44no—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157#408—44no—
158#408—44no—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163C4×M16—24no—
164C2×#97C2312yes—
165C2×C4×C8C2312yes—
166#398—24no—
167#396—24no—
168#396—24no—
169#394—24no—
170#391—22no—
171#391—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#388—24no—
183#388—24no—
184#388—24no—
185#388—24no—
186#386—24no—
187#386—24no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382C2312yes—
197#380—21no—
198#380—21no—
199#380C2314yes—
200#380C2314yes—
201#380C2314yes—
202#380C2314yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213#376—21no—
214C43—21no—
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229C2×#41—24no—
230C2×#41—24no—
231C2×#41—24no—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—24no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8C2×D412yes—
242C22×Q8#4114yes—
243C22×Q8C2412yes—
244C22×D4C2×D412yes—
245C22×D4C2×D414yes—
246C22×D4C2×D412yes—
247C22×D4C2×D412yes—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—24no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2414yes—
255C23×C4C2411yescharacteristic
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—48no—
271#127—24no—
272#124—28no—
273#124—48no—
274#123—416no—
275C22⋊Q8—28no—
276C22⋊Q8—28no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279#121—28no—
280#121—28no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—48no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—48no—
293C4×D4—416no—
294C4×D4—84no—
295C4×D4—84no—
296#117—22no—
297#117—24no—
298#117—22no—
299#117—24no—
300#117—22no—
301#117—28no—
302#117—44no—
303#117#4114yes—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4C2×D414yes—
349C2×C22⋊C4C2×D414yes—
350C2×C42—21no—
351C2×C42—24no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42#4114yes—
364C2×C42#4114yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—42no—
368#107—42no—
369#106—42no—
370#106—42no—
371C4⋊C8—22no—
372C4⋊C8—22no—
373C4⋊C8—24no—
374#103—24no—
375#103—24no—
376#103—24no—
377#103—24no—
378#103—44no—
379#103—44no—
380C22⋊C8—44no—
381C22⋊C8—44no—
382C22⋊C8—44no—
383C22⋊C8—44no—
384#97—24no—
385C4×C8—24no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396C24C22×D414yes—
397C24C22×D411yescharacteristic
398#41—48no—
399#41—48no—
400#41—48no—
401#41—48no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—28no—
408C2×Q8—28no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8C2×#4114yes—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—42no—
429C2×D4—42no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4#12314yes—
462C22×C4#12314yes—
463C22×C4C22×D414yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4C2511yescharacteristic
476M16—44no—
477M16—44no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
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—48no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4C2×#4112yes—
523C4⋊C4C2×#4112yes—
524C22⋊C4—24no—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—24no—
537C42—48no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—44no—
542C42—42no—
543C42—44no—
544C42—42no—
545C42—42no—
546C42#12314yes—
547C42#12314yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C42C2×#4112yes—
551C42C2×#4112yes—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23C2×#12312yes—
564C23#54612yes—
565C23#54612yes—
566C23C23×D411yescharacteristic
567Q8—24no—
568Q8—24no—
569Q8—48no—
570Q8—48no—
571Q8—42no—
572Q8—42no—
573D4—48no—
574D4—416no—
575D4—48no—
576D4—48no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4—168no—
622C2×C4—164no—
623C2×C4C2×#12312yes—
624C2×C4C2×#12312yes—
625C2×C4C2×#12312yes—
626C2×C4C2×#12312yes—
627C2×C4C2×#12312yes—
628C2×C4#52914yes—
629C2×C4#54014yes—
630C2×C4#54612yes—
631C2×C4#54612yes—
632C2×C4#55314yes—
633C2×C4#55514yes—
634C2×C4#57712yes—
635C2×C4#57812yes—
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C8—42no—
643C8—42no—
644C8—42no—
645C8—42no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22#242014yes—
660C22C22×#12311yescharacteristic
661C22C2×#54611yescharacteristic
662C22C2×#54611yescharacteristic
663C22C2×#57711yescharacteristic
664C22C2×#57811yescharacteristic
665C22#358511yescharacteristic
666C22#358611yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—81no—
675C4—82no—
676C4—81no—
677C4—82no—
678C4—84no—
679C4—82no—
680C4#242014yes—
681C4#309714yes—
682C4C2×#52911yescharacteristic
683C4C2×#54011yescharacteristic
684C4C2×#55311yescharacteristic
685C4C2×#55511yescharacteristic
686C2—24no—
687C2—84no—
688C2C2×#242011yescharacteristic
689C2C2×#309711yescharacteristic
690C2#6051711yescharacteristic
691C2#6120211yescharacteristic
692C2#6120311yescharacteristic
693C2#6120611yescharacteristic
694C2#6121011yescharacteristic
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. #24980C2semidirect
4. #24979C2semidirect
5. #24972C2semidirect
6. #24971C2semidirect
7. #24970C2semidirect
8. #24969C2semidirect
9. #23641C2semidirect
10. #23487C2semidirect
12. #21169C2semidirect
13. #20913C2semidirect
14. #20764C2semidirect
15. #20703C2semidirect
16. C2×#1803C2semidirect
17. C4×#419C2semidirect

Maximal subgroups

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

subgroupindex
#351782
#342072
#249802
#249792
#249722
#249712
#249702
#249692
#236412
#234872
#211892
#211692
#209132
#207642
#207032
C2×#18032
C4×#4192