Finite Group Databasebeta

← #6529806

Subgroups — #6529806

Order 512.

1994 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3112C211yesmaximal, characteristic
2C2×#2983C211yesmaximal, characteristic, complemented
3#35178C211yesmaximal, characteristic
4C2×#2581C211yesmaximal, characteristic, complemented
5#28275C214yesmaximal, complemented
6#28199C214yesmaximal, complemented
7#25483C214yesmaximal, complemented
8#25310C214yesmaximal, complemented
9#24608C212yesmaximal, complemented
10#24600C212yesmaximal, complemented
11#24509C212yesmaximal, complemented
12#24503C212yesmaximal, complemented
13#21986C211yesmaximal, characteristic, complemented
14#21865C211yesmaximal, characteristic
15#20740C211yesmaximal, characteristic, complemented
16C2×#536—21no—
17C2×#532C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#3112—28no—
21C2×#480C2212yes—
22C2×#477C2212yes—
23C2×#460C2212yes—
24C2×#448C2212yes—
25#2983—28no—
26C2×#446C2214yes—
27C2×#446C2214yes—
28C2×#440C2214yes—
29C2×#432C2211yescharacteristic
30C2×#432C2212yes—
31C2×#432C2211yescharacteristic
32C2×#431C2212yes—
33#2854C2212yes—
34#2854C2212yes—
35#2618—21no—
36#2618—21no—
37#2581—22no—
38#2581—22no—
39#2581C2214yes—
40#2581C2214yes—
41#2548C2212yes—
42#2546C2212yes—
43#2545—22no—
44#2538C2212yes—
45#2538C2212yes—
46#2536—22no—
47C2×#400—21no—
48C2×#395—21no—
49#2311—21no—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#387—21no—
55C2×#387—21no—
56#2285C2211yescharacteristic
57C2×#382C2211yescharacteristic
58#2087C2214yes—
59#2086C2214yes—
60#2068C2214yes—
61#2067C2214yes—
62#2066C2214yes—
63#2065C2214yes—
64#2055C2214yes—
65#2055C2214yes—
66#1993C2212yes—
67#1973C2214yes—
68#1968C2214yes—
69#1961C2212yes—
70#1949C2214yes—
71#1944C2214yes—
72#1942C2214yes—
73#1936C2214yes—
74#1935C2214yes—
75#1934C2214yes—
76#1933C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1847C2212yes—
82#1836C2212yes—
83#1823C2212yes—
84#1804C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1787C2212yes—
88#1763C2214yes—
89C22×#41—21no—
90C23×Q8D411yescharacteristic
91C23×Q8C2311yescharacteristic
92C22×SD16C2314yes—
93#536—48no—
94#532—24no—
95#532—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#124—22no—
100C2×#124—21no—
101C2×#123—22no—
102C2×#123—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—24no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C2×C4×Q8D412yes—
114C2×C4×Q8D412yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#480—28no—
136#477—28no—
137#460—28no—
138#448—28no—
139#446—216no—
140#446—216no—
141#440—216no—
142#432—24no—
143#432—28no—
144#432D418yes—
145#431—28no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×#97C2314yes—
162C2×#97C2312yes—
163C2×#97C2312yes—
164C2×C4×C8C2314yes—
165#400—24no—
166#400—24no—
167#398—22no—
168#398—22no—
169#396—22no—
170#396—22no—
171#396—22no—
172#396—22no—
173#395—24no—
174#395—24no—
175#394—22no—
176#394—22no—
177#391—22no—
178#391—22no—
179#391—22no—
180#391C2314yes—
181#391C2314yes—
182#390—22no—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#390C2312yes—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2312yes—
191#389C2314yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—24no—
200#387—24no—
201#387—24no—
202#387—24no—
203#386—22no—
204#386—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#382—21no—
208#382—21no—
209#382—21no—
210#382C2312yes—
211#382C2312yes—
212#380—21no—
213#380—22no—
214#380—22no—
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2312yes—
218C2×#95—21no—
219C2×#95—21no—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95—21no—
223C2×#95—21no—
224C2×#95—21no—
225C2×#95—21no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232C2×#41—44no—
233C2×#41—44no—
234C2×#41—44no—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—42no—
243C22×Q8—42no—
244C22×Q8C2×D412yes—
245C22×Q8C2×D412yes—
246C22×Q8C2412yes—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×SD16—28no—
264C2×SD16—28no—
265C2×SD16—44no—
266C22×C8C2414yes—
267C22×C8C2414yes—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—48no—
271#127—22no—
272#127—22no—
273#124—48no—
274#124—44no—
275#123—48no—
276#123—48no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279#121—24no—
280#121—24no—
281#121—44no—
282C4×Q8—28no—
283C4×Q8—28no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—44no—
297#117—44no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—44no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D414yes—
360C2×C42#4114yes—
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—28no—
366#107—44no—
367#106—44no—
368C4⋊C8—28no—
369C4⋊C8—28no—
370C4⋊C8—24no—
371C4⋊C8C2×D418yes—
372#103—28no—
373#103—28no—
374#103—44no—
375#102—28no—
376#102—44no—
377#102—44no—
378C22⋊C8—24no—
379C22⋊C8C2×D418yes—
380#97—28no—
381#97—24no—
382#97C2×D418yes—
383C4×C8—28no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400C24—22no—
401C24C22×D411yescharacteristic
402#41—816no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—416no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q82−1+414yes—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—81no—
436C2×D4—81no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4#12114yes—
477C22×C4#12114yes—
478C22×C4AGL(1,ℤ/8)14yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490C22×C4C2511yescharacteristic
491SD16—416no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8C22×D418yes—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—44no—
546C42—44no—
547C42—44no—
548C42#12114yes—
549C42#12114yes—
550C42C22×D412yes—
551C42C22×D412yes—
552C42C2×#4112yes—
553C42C2×#4112yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23C2×#12112yes—
566C23C2×AGL(1,ℤ/8)12yes—
567C23#57512yes—
568C23C23×D411yescharacteristic
569Q8—24no—
570Q8—24no—
571Q8—48no—
572Q8—48no—
573Q8—48no—
574Q8—42no—
575Q8—42no—
576Q8—84no—
577D4—44no—
578D4—44no—
579D4—84no—
580D4—84no—
581D4—88no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—88no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4C2×#12112yes—
638C2×C4C2×#12112yes—
639C2×C4C2×#12112yes—
640C2×C4C2×#12112yes—
641C2×C4C2×#12112yes—
642C2×C4C2×AGL(1,ℤ/8)14yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×2−1+412yes—
646C2×C4#58512yes—
647C8—28no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#305514yes—
663C22C22×#12111yescharacteristic
664C22#348611yescharacteristic
665C22#348711yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C2×#57511yescharacteristic
668C22#358611yescharacteristic
669C22#358611yescharacteristic
670C4—22no—
671C4—22no—
672C4—24no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4#348912yes—
687C4#349512yes—
688C2—24no—
689C2—162no—
690C2#3566111yescharacteristic
691C2C2×#305511yescharacteristic
692C2#6045511yescharacteristic
693C2#6182811yescharacteristic
694C2#6183211yescharacteristic
695C2#6188811yescharacteristic
696C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2983C2semidirect
4. C2×#2581C2semidirect
5. #28275C2semidirect
6. #28199C2semidirect
7. #25483C2semidirect
8. #25310C2semidirect
9. #24608C2semidirect
10. #24600C2semidirect
11. #24509C2semidirect
12. #24503C2semidirect
13. #21986C2semidirect
15. #20740C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31122
C2×#29832
#351782
C2×#25812
#282752
#281992
#254832
#253102
#246082
#246002
#245092
#245032
#219862
#218652
#207402