Finite Group Databasebeta

← #7534795

Subgroups — #7534795

Order 512.

1961 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59815C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#31570C214yesmaximal, complemented
4#28716C211yesmaximal, characteristic, complemented
5#28508C211yesmaximal, characteristic, complemented
6#28397C212yesmaximal, complemented
7#28263C212yesmaximal, complemented
8#27190C214yesmaximal, complemented
9#25272C212yesmaximal, complemented
10#25056C212yesmaximal, complemented
11#24811C212yesmaximal, complemented
12#24184C212yesmaximal, complemented
13#23149C211yesmaximal, characteristic, complemented
14#22962C212yesmaximal, complemented
15#22540C212yesmaximal, complemented
16C4×#459C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic, complemented
18C2×#550C2212yes—
19C2×#547—21no—
20C2×#533—21no—
21C2×#486C2212yes—
22C2×#474C2212yes—
23C2×#474C2212yes—
24C2×#459C2211yescharacteristic
25C2×#459C2211yescharacteristic
26C2×C8×Q8C2212yes—
27C2×C4×D8C2212yes—
28C2×C4×D8C2212yes—
29#2868C2211yescharacteristic
30#2857—21no—
31#2682—22no—
32#2552C2212yes—
33#2542C2212yes—
34#2541C2212yes—
35#2538C2212yes—
36#2538C2212yes—
37#2537—21no—
38#2445C2214yes—
39#2441—22no—
40#2429—22no—
41#2423—22no—
42#2416—22no—
43#2407—22no—
44#2365C2212yes—
45#2354C2212yes—
46Q8×C4⋊C4C2212yes—
47D4×C4⋊C4C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2093C2214yes—
53#2093C2214yes—
54#2083C2214yes—
55#2071C2214yes—
56#2067C2214yes—
57#2067C2214yes—
58#2041C2214yes—
59#2011C2214yes—
60#1990C2212yes—
61#1984C2212yes—
62#1982C2211yescharacteristic
63#1982C2214yes—
64#1982C2211yescharacteristic
65#1970C2214yes—
66#1957C2214yes—
67#1957C2212yes—
68#1947C2211yescharacteristic
69#1947C2211yescharacteristic
70#1942C2214yes—
71#1939C2212yes—
72#1925C2214yes—
73#1925C2212yes—
74#1925C2212yes—
75#1918C2214yes—
76#1916C2214yes—
77#1879C2212yes—
78#1879C2212yes—
79#1879C2214yes—
80#1862C2214yes—
81#1845C2212yes—
82#1802C2212yes—
83#1802C2214yes—
84#1802C2212yes—
85#1802C2211yescharacteristic
86#1802C2211yescharacteristic
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#102C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×#41—22no—
94C22×D8C2312yes—
95C22×D8C2312yes—
96#550—28no—
97#550—44no—
98#547—48no—
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8D411yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#127C2314yes—
106C2×#127C2311yescharacteristic
107C2×#125—22no—
108C2×#123—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×#121—22no—
114C2×#121—24no—
115C2×#121—22no—
116C2×#121—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2314yes—
130C2×#117—22no—
131C22×C4⋊C4—22no—
132C22×C42—21no—
133#486—44no—
134#474—28no—
135#474—44no—
136#459—24no—
137#459—24no—
138C8×Q8—44no—
139C4×D8—44no—
140C4×D8—44no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2311yescharacteristic
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2312yes—
164C2×C4×C8C2314yes—
165#399—24no—
166#398—24no—
167#396—24no—
168#396—24no—
169#394—24no—
170#394—24no—
171#392—24no—
172#391—22no—
173#391D412yes—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#390C2312yes—
180#390C2314yes—
181#390C2314yes—
182#389—22no—
183#389—22no—
184#389—24no—
185#389D412yes—
186#389D412yes—
187#389C2314yes—
188#389C2314yes—
189#388—24no—
190#387—24no—
191#386—24no—
192#385—24no—
193#384—21no—
194#384—21no—
195#384D411yescharacteristic
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2314yes—
200#384C2312yes—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C43C2311yescharacteristic
215#340C2312yes—
216#340C2314yes—
217#340C2312yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#336C2312yes—
222#336C2314yes—
223#336C2312yes—
224#336C2314yes—
225C2×#41—48no—
226C2×#41—48no—
227C2×#41—84no—
228C22×Q8—22no—
229C22×Q8C2×D412yes—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—24no—
234C22×D4—21no—
235C22×D4—24no—
236C22×D4C2412yes—
237C22×D4C2412yes—
238C22×D4C2411yescharacteristic
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C2×D8—24no—
246C2×D8—24no—
247C2×D8—24no—
248C2×D8—24no—
249C2×D8—42no—
250C2×D8—42no—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8D818yescomplemented
260#127—28no—
261#127—22no—
262#127—22no—
263#127—22no—
264#127—44no—
265#127#4114yes—
266#125—48no—
267#123—416no—
268C22⋊Q8—48no—
269C22⋊Q8—48no—
270C22⋊Q8—48no—
271C22⋊Q8—84no—
272#121—48no—
273#121—416no—
274#121—48no—
275#121—84no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—48no—
279C4×Q8—48no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—82no—
295C4×D4—82no—
296#117—48no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2414yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—28no—
351#107—42no—
352#107—42no—
353#107—44no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—44no—
358C4⋊C8#4114yescomplemented
359#102—24no—
360#102—24no—
361#102—28no—
362#102—24no—
363#102—24no—
364#102—42no—
365#102—42no—
366#102—42no—
367#102—42no—
368#102—44no—
369#102—44no—
370#102—44no—
371C4×C8—24no—
372C4×C8—24no—
373C4×C8—44no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D414yes—
381#95C2×D414yes—
382C24—21no—
383C24—22no—
384C24—22no—
385#41—832no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8C2×D818yes—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—48no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—81no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4C2×#4114yes—
427C2×D42−1+412yes—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—84no—
452C22×C4C22×D414yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D414yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461D8—48no—
462D8—48no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8C2×#4114yes—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—84no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C2×D818yes—
504C4⋊C4C2×D818yes—
505C4⋊C42−1+412yes—
506C22⋊C4—24no—
507C22⋊C4—48no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C42—22no—
521C42—22no—
522C42—24no—
523C42—24no—
524C42—21no—
525C42—21no—
526C42—41no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—48no—
531C42C2×D814yes—
532C42C2×#4112yes—
533C42C2×#4112yes—
534C42H(5,𝔽2)12yes—
535C42H(5,𝔽2)12yes—
536C23—24no—
537C23—22no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23C23×D411yescharacteristic
558Q8—216no—
559Q8—88no—
560D4—24no—
561D4—22no—
562D4—22no—
563D4—24no—
564D4—41no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—41no—
571D4—42no—
572D4—48no—
573D4—84no—
574D4—816no—
575D4—84no—
576D4—84no—
577D4—168no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—48no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—816no—
612C2×C4—84no—
613C2×C4—88no—
614C2×C4—84no—
615C2×C4—88no—
616C2×C4#54612yes—
617C2×C4#54612yes—
618C2×C4#55012yes—
619C2×C4#55012yes—
620C2×C4C22×D812yes—
621C2×C4C22×D818yes—
622C2×C4C22×D812yes—
623C2×C4C22×D812yes—
624C2×C4#57612yes—
625C2×C4#57712yes—
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C8—24no—
633C8—42no—
634C8—42no—
635C8—42no—
636C22—44no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—81no—
643C22—82no—
644C22—82no—
645C22—81no—
646C22—81no—
647C22—82no—
648C22—81no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22C2×#54611yescharacteristic
658C22C2×#55011yescharacteristic
659C22#348611yescharacteristic
660C22C23×D811yescharacteristic
661C22C2×#57611yescharacteristic
662C22C2×#57711yescharacteristic
663C22#358611yescharacteristic
664C4—22no—
665C4—22no—
666C4—28no—
667C4—22no—
668C4—44no—
669C4—42no—
670C4—41no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—84no—
676C4#329414yes—
677C4#339114yes—
678C4#348211yescharacteristic
679C4#352011yescharacteristic
680C4#352011yescharacteristic
681C4#354811yescharacteristic
682C2—84no—
683C2—82no—
684C2—162no—
685C2#5904011yescharacteristic
686C2C2×#329411yescharacteristic
687C2#6119911yescharacteristic
688C2C2×#339111yescharacteristic
689C2#6142011yescharacteristic
690C2#6181711yescharacteristic
691C2#6186811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59815C2semidirect
2. #36134C2semidirect
3. #31570C2semidirect
4. #28716C2semidirect
5. #28508C2semidirect
6. #28397C2semidirect
7. #28263C2semidirect
8. #27190C2semidirect
9. #25272C2semidirect
10. #25056C2semidirect
11. #24811C2semidirect
12. #24184C2semidirect
13. #23149C2semidirect
14. #22962C2semidirect
15. #22540C2semidirect
16. C4×#459C2semidirect
17. C4×#459C2semidirect
259. C4⋊Q8D8semidirect
358. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#598152
#361342
#315702
#287162
#285082
#283972
#282632
#271902
#252722
#250562
#248112
#241842
#231492
#229622
#225402
C4×#4592