Finite Group Databasebeta

← #6458582

Subgroups — #6458582

Order 512.

1820 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#35286C211yesmaximal, characteristic, complemented
2#35178C211yesmaximal, characteristic
3#27765C214yesmaximal, complemented
4#27761C214yesmaximal, complemented
5#27758C212yesmaximal, complemented
6#27757C212yesmaximal, complemented
7#27750C212yesmaximal, complemented
8#27749C212yesmaximal, complemented
9#23634C212yesmaximal, complemented
10#23512C212yesmaximal, complemented
11#21257C211yesmaximal, characteristic
12#21205C212yesmaximal, complemented
13#21200C211yesmaximal, characteristic
14#21092C212yesmaximal, complemented
15C2×#1803C211yesmaximal, characteristic, complemented
16#20441C211yesmaximal, characteristic, complemented
17C4×#419C211yesmaximal, characteristic, complemented
18C2×#536C2212yes—
19C2×#532C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#417C2212yes—
23C2×#417C2212yes—
24C2×#419C2211yescharacteristic
25C2×#419C2211yescharacteristic
26C2×#416C2212yes—
27C2×#416C2212yes—
28C22×#103C2212yes—
29C2×#405C2212yes—
30C2×C4×M16C2211yescharacteristic
31#2859—21no—
32#2859—21no—
33#2854—22no—
34#2647—22no—
35#2618—22no—
36#2571C2212yes—
37#2571C2212yes—
38#2552C2212yes—
39#2552C2212yes—
40#2548C2212yes—
41#2546C2212yes—
42#2545C2214yes—
43#2538—21no—
44#2538—21no—
45#2536—22no—
46#2442—22no—
47#2442—22no—
48#2317C2212yes—
49#2316C2212yes—
50#2311C2212yes—
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2074C2214yes—
55#2073C2214yes—
56#2072C2214yes—
57#2071C2214yes—
58#2043C2214yes—
59#2043C2214yes—
60#2042C2214yes—
61#2042C2214yes—
62#1983C2212yes—
63#1983C2212yes—
64#1982C2211yescharacteristic
65#1982C2211yescharacteristic
66#1981C2211yescharacteristic
67#1981C2211yescharacteristic
68#1960C2214yes—
69#1959C2214yes—
70#1959C2214yes—
71#1958C2214yes—
72#1957C2212yes—
73#1956C2212yes—
74#1956C2214yes—
75#1936C2214yes—
76#1935C2214yes—
77#1934C2214yes—
78#1933C2214yes—
79#1803C2212yes—
80#1803C2212yes—
81#1803C2214yes—
82#1771C2214yes—
83#1770C2212yes—
84C8×C4⋊C4C2212yes—
85#1763C2212yes—
86#1763C2212yes—
87C4×#103C2212yes—
88C4×#103C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91C2×#340C2212yes—
92#1729C2212yes—
93C22×#41C2312yes—
94C23×Q8C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#536—28no—
99#536—44no—
100#532—24no—
101#532—24no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2314yes—
104C2×#127C2311yescharacteristic
105C2×#125—22no—
106C2×#125—22no—
107C2×#124C2312yes—
108C2×#124C2314yes—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×#121—21no—
113C2×#121—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—24no—
117C2×C4×Q8—24no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C2×C4⋊C8—24no—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136#419—24no—
137#419—24no—
138#417—28no—
139#417—44no—
140#416—44no—
141#416—44no—
142C2×#103—44no—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161#405—44no—
162C4×M16—24no—
163C2×#97C2312yes—
164C2×#97C2314yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2314yes—
167#400—24no—
168#400—24no—
169#398—24no—
170#398—24no—
171#398—24no—
172#396—24no—
173#396—24no—
174#394—24no—
175#391C2314yes—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#388—24no—
192#388—24no—
193#386—24no—
194#385—24no—
195#384—21no—
196#384—21no—
197#384D411yescharacteristic
198#384D411yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#384C2314yes—
202#382—22no—
203#382—22no—
204#382D412yes—
205#382C2314yes—
206#380—22no—
207#380—22no—
208#380—24no—
209#380D412yes—
210#380D412yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219#376—21no—
220C43—21no—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2314yes—
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×Q8#4114yes—
242C22×Q8C2412yes—
243C22×Q8C2412yes—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16#4114yes—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C4⋊Q8—28no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—44no—
272#127—22no—
273#127—22no—
274#125—84no—
275#125—84no—
276#124—28no—
277#124—24no—
278#124—24no—
279#124—44no—
280#123—416no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283#121—44no—
284#121—44no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—44no—
292C4×D4—44no—
293#117—24no—
294#117—24no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D414yes—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—44no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42#4114yes—
355C2×C42#4114yes—
356C2×C42C2414yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C4⋊C8—24no—
360C4⋊C8—22no—
361C4⋊C8—22no—
362#103—24no—
363#103—24no—
364#103—28no—
365#103—24no—
366#103—24no—
367#103—44no—
368#103—44no—
369#103—42no—
370#103—42no—
371#103—42no—
372#103—42no—
373#103—44no—
374#102—28no—
375#102—42no—
376#102—42no—
377#102—42no—
378#102—42no—
379#102—44no—
380#97—24no—
381#97—44no—
382C4×C8—24no—
383C4×C8—44no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95C2×D414yes—
395#95C2×D414yes—
396C24—22no—
397C24C22×D411yescharacteristic
398#41—48no—
399#41—48no—
400#41—48no—
401#41—48no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—44no—
420C2×Q8C2×#4114yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—81no—
435C2×D4—81no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4#12414yes—
461C22×C4#12414yes—
462C22×C4C22×D414yes—
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—28no—
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—48no—
490C2×C8C2×#4114yes—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—24no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C22⋊C4—28no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—24no—
535C42—24no—
536C42—42no—
537C42—42no—
538C42—44no—
539C42—44no—
540C42—42no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42#12414yes—
545C42#12414yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C42C2×#4112yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23C2×#12412yes—
562C23#54612yes—
563C23#55012yes—
564C23C23×D411yescharacteristic
565Q8—24no—
566Q8—24no—
567Q8—42no—
568Q8—42no—
569Q8—48no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—84no—
579D4—88no—
580D4—84no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4—164no—
623C2×C4C2×#12412yes—
624C2×C4C2×#12412yes—
625C2×C4C2×#12412yes—
626C2×C4C2×#12412yes—
627C2×C4C2×#12412yes—
628C2×C4#52914yes—
629C2×C4#54014yes—
630C2×C4#54612yes—
631C2×C4#55012yes—
632C2×C4#55314yes—
633C2×C4#55314yes—
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—24no—
643C8—42no—
644C8—42no—
645C8—42no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22#249014yes—
660C22C22×#12411yescharacteristic
661C22C2×#54611yescharacteristic
662C22C2×#55011yescharacteristic
663C22C2×#57711yescharacteristic
664C22C2×#57811yescharacteristic
665C22#358511yescharacteristic
666C22#358611yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—81no—
674C4—81no—
675C4—82no—
676C4—82no—
677C4—81no—
678C4—84no—
679C4—81no—
680C4#249014yes—
681C4#313714yes—
682C4C2×#52911yescharacteristic
683C4C2×#54011yescharacteristic
684C4C2×#55311yescharacteristic
685C4C2×#55311yescharacteristic
686C2—24no—
687C2—162no—
688C2C2×#249011yescharacteristic
689C2C2×#313711yescharacteristic
690C2#6062111yescharacteristic
691C2#6120211yescharacteristic
692C2#6120311yescharacteristic
693C2#6142111yescharacteristic
694C2#6142211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #27765C2semidirect
4. #27761C2semidirect
5. #27758C2semidirect
6. #27757C2semidirect
7. #27750C2semidirect
8. #27749C2semidirect
9. #23634C2semidirect
10. #23512C2semidirect
12. #21205C2semidirect
14. #21092C2semidirect
15. C2×#1803C2semidirect
16. #20441C2semidirect
17. C4×#419C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#351782
#277652
#277612
#277582
#277572
#277502
#277492
#236342
#235122
#212572
#212052
#212002
#210922
C2×#18032
#204412
C4×#4192