Finite Group Databasebeta

← #228195

Subgroups — #228195

Order 512.

801 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23113C211yesmaximal, characteristic, complemented
2D4×#104C211yesmaximal, characteristic, complemented
3#18516C211yesmaximal, characteristic
4#17994C211yesmaximal, characteristic, complemented
5#17511C211yesmaximal, characteristic, complemented
6#9852C211yesmaximal, characteristic, complemented
7#9590C211yesmaximal, characteristic, complemented
8#9577C211yesmaximal, characteristic, complemented
9#9233C211yesmaximal, characteristic, complemented
10#9210C211yesmaximal, characteristic, complemented
11#9053C211yesmaximal, characteristic, complemented
12#8908C211yesmaximal, characteristic, complemented
13#8905C211yesmaximal, characteristic, complemented
14#8197C211yesmaximal, characteristic, complemented
15#8170C211yesmaximal, characteristic, complemented
16D4×#41C2211yescharacteristic
17#3169C2211yescharacteristic
18#3100C2211yescharacteristic
19#3035C2211yescharacteristic
20#2965C2211yescharacteristic
21#2935C2211yescharacteristic, complemented
22D4×M16C2211yescharacteristic
23#2925C2211yescharacteristic
24C22×#104—21no—
25C42×D4C2211yescharacteristic
26#1980C2211yescharacteristic
27#1974—21no—
28#1962C2211yescharacteristic
29#1962C2211yescharacteristic, complemented
30#1950C2211yescharacteristic, complemented
31#1929—21no—
32#1896—21no—
33#1876—21no—
34#1871—21no—
35#1806C2211yescharacteristic
36#1789—21no—
37C4×C4⋊C8C2211yescharacteristic, complemented
38#1764—21no—
39C4×#104C2211yescharacteristic
40#1744C2211yescharacteristic, complemented
41#1649C411yescharacteristic, complemented
42#1647C411yescharacteristic, complemented
43#1641C411yescharacteristic, complemented
44#1639C411yescharacteristic, complemented
45#1627C411yescharacteristic, complemented
46#1625C411yescharacteristic, complemented
47#1621C411yescharacteristic, complemented
48#1619C411yescharacteristic, complemented
49#1589—21no—
50#1581—21no—
51#1581—21no—
52C8×D8—21no—
53#1571C2211yescharacteristic, complemented
54#1566C2211yescharacteristic, complemented
55#1540C2211yescharacteristic
56#1502C2211yescharacteristic, complemented
57#1496—21no—
58#1492C2211yescharacteristic
59#1492C2211yescharacteristic, complemented
60#1490—21no—
61#1486C2211yescharacteristic
62#1484—21no—
63#1480C2211yescharacteristic
64#1478—21no—
65#1476C2211yescharacteristic
66C2×D4⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68C8×M16C2211yescharacteristic, complemented
69#1303C2211yescharacteristic, complemented
70#1301C2211yescharacteristic, complemented
71#1279C2211yescharacteristic, complemented
72#1279C2211yescharacteristic, complemented
73#1278C2211yescharacteristic, complemented
74#1276C2211yescharacteristic, complemented
75C22×#41—21no—
76C2×#135—21no—
77C2×#131—21no—
78C22×M16—21no—
79#550C2×C411yescharacteristic
80D4×Q8C2×C411yescharacteristic
81#548—21no—
82#547—21no—
83#547C2×C411yescharacteristic
84#546—21no—
85#546—41no—
86D42—21no—
87D42C2×C411yescharacteristic
88#532—21no—
89#532C2311yescharacteristic
90#532C2311yescharacteristic
91#525—21no—
92#525—41no—
93C4×#41C2311yescharacteristic
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2311yescharacteristic
97C22×C42—21no—
98#500C2×C411yescharacteristic, complemented
99#493C2×C411yescharacteristic, complemented
100#486C2×C411yescharacteristic, complemented
101#486C2×C411yescharacteristic, complemented
102#480—21no—
103#474C2×C411yescharacteristic
104#474C2×C411yescharacteristic
105#466—41no—
106#464—21no—
107#463—21no—
108#462C2×C411yescharacteristic, complemented
109#461C2×C411yescharacteristic, complemented
110#460C2×C411yescharacteristic, complemented
111#459C2×C411yescharacteristic
112#459C2×C411yescharacteristic
113#459C2×C411yescharacteristic, complemented
114#447—21no—
115#446—21no—
116C4×D8—21no—
117C4×D8—21no—
118C4×D8—41no—
119#436—21no—
120#436—21no—
121#436—21no—
122#435—41no—
123#435—41no—
124#435—41no—
125C8×D4—21no—
126C8×D4—21no—
127#433—41no—
128#432—21no—
129#431—21no—
130#431—21no—
131#431C2311yescharacteristic
132#431C2311yescharacteristic
133C2×#108—21no—
134#423C2311yescharacteristic
135#423C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×#104—21no—
139C2×#104—21no—
140C2×#104—21no—
141C2×#104—21no—
142C2×#104—21no—
143C2×#104—21no—
144C2×#104—21no—
145C2×#104—44no—
146C2×#104D411yescharacteristic
147C2×#104D411yescharacteristic
148#416—21no—
149#408—21no—
150#408—21no—
151#407—21no—
152#405—21no—
153C4×M16—21no—
154C4×M16C2311yescharacteristic
155C2×C4×C8—21no—
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2311yescharacteristic
160C4×C22⋊C4—21no—
161C4×C22⋊C4—41no—
162C43C2311yescharacteristic
163#341—21no—
164#339—21no—
165#339D411yescharacteristic
166#339D411yescharacteristic
167#336—21no—
168#336—41no—
169#335—21no—
170#335—21no—
171#334—41no—
172#332C2×C411yescharacteristic, complemented
173#332C2×C411yescharacteristic, complemented
174#331C2×C411yescharacteristic, complemented
175#331C2×C411yescharacteristic, complemented
176#327—41no—
177Q8⋊C8C2×C411yescharacteristic, complemented
178Q8⋊C8C2×C411yescharacteristic, complemented
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8C2×C411yescharacteristic
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8C2×C411yescharacteristic
190D4⋊C8C2×C411yescharacteristic
191D4⋊C8C2×C411yescharacteristic, complemented
192D4⋊C8C2×C411yescharacteristic, complemented
193D4⋊C8D411yescharacteristic
194D4⋊C8D411yescharacteristic
195D4⋊C8D411yescharacteristic
196D4⋊C8D411yescharacteristic
197#322—21no—
198C82—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—41no—
204C2×#41—41no—
205C2×#41—42no—
206C2×#41—42no—
207C2×#41—81no—
208C2×#41—81no—
209C2×#41C2×D411yescharacteristic
210C22×Q8—21no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—41no—
214C22×D4#4111yescharacteristic
215C23×C4—21no—
216C23×C4—41no—
217C23×C4—41no—
218#135—81no—
219#135—81no—
220C2×Q16—21no—
221C2×SD16—21no—
222C2×SD16—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—41no—
227#131—42no—
228#131—81no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—41no—
232C2×M16—41no—
233C2×M16—41no—
234C2×M16—42no—
235C2×M16—81no—
236C2×M16C2×D411yescharacteristic
237C2×M16C2×D411yescharacteristic
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—41no—
243C22×C8—41no—
244C4⋊Q8—21no—
245C4⋊Q8C22×C411yescharacteristic
246C4⋊Q8C22×C411yescharacteristic
247#127—21no—
248#127C22×C411yescharacteristic
249#127C22×C411yescharacteristic
250#124—21no—
251#124—21no—
252#124—41no—
253#123—41no—
254#123—41no—
255#123—41no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—41no—
259C22⋊Q8—41no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264#121—21no—
265#121—21no—
266#121—21no—
267#121—21no—
268#121—41no—
269#121—41no—
270#121—41no—
271#120—41no—
272#120—41no—
273#120—41no—
274C4×Q8C22×C411yescharacteristic
275C4×Q8#4111yescharacteristic
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—24no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—41no—
288C4×D4—41no—
289C4×D4—41no—
290C4×D4—42no—
291C4×D4C22×C411yescharacteristic
292C4×D4C22×C411yescharacteristic
293C4×D4C22×C411yescharacteristic
294C4×D4C2×D411yescharacteristic
295C4×D4C2×D411yescharacteristic
296C4×D4#4111yescharacteristic
297#117—21no—
298#117C2×D411yescharacteristic
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4#4111yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—41no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—42no—
310C2×C42—41no—
311C2×C42—81no—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2411yescharacteristic
315#108—81no—
316#107—21no—
317#107—21no—
318#107—41no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—41no—
327C4⋊C8C22×C411yescharacteristic
328C4⋊C8C22×C411yescharacteristic
329C4⋊C8C22×C411yescharacteristic
330C4⋊C8C22×C411yescharacteristic
331C4⋊C8C2×D411yescharacteristic
332C4⋊C8C2×D411yescharacteristic
333#104—22no—
334#104—41no—
335#104—42no—
336#104—42no—
337#104—42no—
338#104—42no—
339#104—81no—
340#104—81no—
341#104—81no—
342#103—21no—
343#103—21no—
344#102—21no—
345#102—21no—
346#102—21no—
347#102—21no—
348#102—21no—
349#102—21no—
350#102—41no—
351#102—41no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—41no—
355C22⋊C8—41no—
356C22⋊C8—41no—
357C22⋊C8—41no—
358C22⋊C8—41no—
359C22⋊C8—41no—
360C22⋊C8—41no—
361#97—21no—
362#97—41no—
363#97—41no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8—21no—
367C4×C8—21no—
368C4×C8—21no—
369C4×C8—41no—
370C4×C8—81no—
371C4×C8C22×C411yescharacteristic
372C4×C8C22×C411yescharacteristic
373C4×C8C2×D411yescharacteristic
374C4×C8C2×D411yescharacteristic
375#95—21no—
376#95—41no—
377C24—21no—
378C24—41no—
379#41—21no—
380#41—21no—
381#41—21no—
382#41—21no—
383#41—42no—
384#41—42no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—82no—
394#41—82no—
395#41—81no—
396#41—81no—
397#41—82no—
398#41—82no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—41no—
402C2×Q8—42no—
403C2×Q8—81no—
404C2×Q8C4×D411yescharacteristic
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—81no—
425C2×D4—81no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4C4×D411yescharacteristic
429C2×D4C4×D411yescharacteristic
430C2×D4C4×D411yescharacteristic
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—44no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4H(5,𝔽2)11yescharacteristic
459Q16—81no—
460SD16—81no—
461SD16—81no—
462D8—42no—
463D8—81no—
464D8—81no—
465D8—82no—
466M16—21no—
467M16—21no—
468M16—21no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—82no—
474M16—82no—
475M16—82no—
476M16AGL(1,ℤ/8)12yes—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—82no—
497C2×C8—82no—
498C2×C8—81no—
499C2×C8—81no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4C4×D411yescharacteristic
512C4⋊C4C4×D411yescharacteristic
513C4⋊C4C4×D411yescharacteristic
514C4⋊C4C4×D411yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—44no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—41no—
541C42—41no—
542C42—81no—
543C42—82no—
544C42C23×C411yescharacteristic
545C42C22×D411yescharacteristic
546C42C2×#4111yescharacteristic
547C42H(5,𝔽2)11yescharacteristic
548C23—21no—
549C23—21no—
550C23—21no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—81no—
557C23—82no—
558C23—82no—
559C23—161no—
560C23#54611yescharacteristic
561Q8—21no—
562Q8—21no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—82no—
568D4—21no—
569D4—21no—
570D4—21no—
571D4—21no—
572D4—21no—
573D4—42no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—81no—
585D4—81no—
586D4—82no—
587D4—82no—
588D4—82no—
589D4—82no—
590D4—81no—
591D4—81no—
592D4—81no—
593D4—81no—
594D4—161no—
595D4#44312yes—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—81no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—88no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4C2×C4×D411yescharacteristic
639C2×C4C2×C4×D411yescharacteristic
640C2×C4#51811yescharacteristic
641C2×C4D4211yescharacteristic
642C2×C4#54611yescharacteristic
643C2×C4#55111yescharacteristic
644C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
645C2×C4#57511yescharacteristic
646C2×C4#57611yescharacteristic
647C2×C4#57611yescharacteristic
648C8—22no—
649C8—41no—
650C8—41no—
651C8—41no—
652C8—41no—
653C8—41no—
654C8—41no—
655C8—41no—
656C8—41no—
657C8—41no—
658C8—41no—
659C8—41no—
660C8—81no—
661C8—81no—
662C8—81no—
663C22—21no—
664C22—21no—
665C22—22no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—82no—
675C22—82no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—164no—
683C22#235811yescharacteristic
684C22#332211yescharacteristic
685C22#332511yescharacteristic
686C4—21no—
687C4—21no—
688C4—21no—
689C4—42no—
690C4—42no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—82no—
697C4—81no—
698C4—81no—
699C4C4×AGL(1,ℤ/8)11yescharacteristic
700C4#294811yescharacteristic
701C4C2×#44311yescharacteristic
702C4#295611yescharacteristic
703C4#326411yescharacteristic
704C4#326811yescharacteristic
705C2—21no—
706C2—21no—
707C2—42no—
708C2—81no—
709C2—81no—
710C2—161no—
711C2—161no—
712C2#2288011yescharacteristic
713C2#2288811yescharacteristic
714C2#2289811yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radical#18516C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23113C2semidirect
2. D4×#104C2semidirect
4. #17994C2semidirect
5. #17511C2semidirect
6. #9852C2semidirect
7. #9590C2semidirect
8. #9577C2semidirect
9. #9233C2semidirect
10. #9210C2semidirect
11. #9053C2semidirect
12. #8908C2semidirect
13. #8905C2semidirect
14. #8197C2semidirect
15. #8170C2semidirect
21. #2935C22semidirect
29. #1962C22semidirect
30. #1950C22semidirect
37. C4×C4⋊C8C22semidirect
40. #1744C22semidirect
41. #1649C4semidirect
42. #1647C4semidirect
43. #1641C4semidirect
44. #1639C4semidirect
45. #1627C4semidirect
46. #1625C4semidirect
47. #1621C4semidirect
48. #1619C4semidirect
53. #1571C22semidirect
54. #1566C22semidirect
56. #1502C22semidirect
59. #1492C22semidirect
68. C8×M16C22semidirect
69. #1303C22semidirect
70. #1301C22semidirect
71. #1279C22semidirect
72. #1279C22semidirect
73. #1278C22semidirect
74. #1276C22semidirect
98. #500C2×C4semidirect
99. #493C2×C4semidirect
100. #486C2×C4semidirect
101. #486C2×C4semidirect
108. #462C2×C4semidirect
109. #461C2×C4semidirect
110. #460C2×C4semidirect
113. #459C2×C4semidirect
172. #332C2×C4semidirect
173. #332C2×C4semidirect
174. #331C2×C4semidirect
175. #331C2×C4semidirect
177. Q8⋊C8C2×C4semidirect
178. Q8⋊C8C2×C4semidirect
191. D4⋊C8C2×C4semidirect
192. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#231132
D4×#1042
#185162
#179942
#175112
#98522
#95902
#95772
#92332
#92102
#90532
#89082
#89052
#81972
#81702