Finite Group Databasebeta

← #228255

Subgroups — #228255

Order 512.

813 conjugacy classes of subgroups.

The subgroup lattice has 717 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#18001C211yesmaximal, characteristic, complemented
5#17511C211yesmaximal, characteristic, complemented
6#9844C211yesmaximal, characteristic, complemented
7#9592C211yesmaximal, characteristic, complemented
8#9578C211yesmaximal, characteristic, complemented
9#9233C211yesmaximal, characteristic, complemented
10#9210C211yesmaximal, characteristic, complemented
11#9053C211yesmaximal, characteristic, complemented
12#8906C211yesmaximal, characteristic, complemented
13#8903C211yesmaximal, characteristic, complemented
14#8194C211yesmaximal, characteristic, complemented
15C4×D4⋊C8C211yesmaximal, 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
40C42×C8C2211yescharacteristic, 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#1594—21no—
50#1586—21no—
51#1586—21no—
52#1580—21no—
53#1570C2211yescharacteristic, complemented
54#1567C2211yescharacteristic, 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
68#1451C2211yescharacteristic, 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×C8—41no—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2311yescharacteristic
161C4×C22⋊C4—21no—
162C4×C22⋊C4—41no—
163C43C2311yescharacteristic
164#341—21no—
165#339—21no—
166#339D411yescharacteristic
167#339D411yescharacteristic
168#336—21no—
169#335—41no—
170#334—21no—
171#334—21no—
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—
198#322—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—44no—
371C4×C8—81no—
372C4×C8C22×C411yescharacteristic
373C4×C8C22×C411yescharacteristic
374C4×C8C2×D411yescharacteristic
375C4×C8C2×D411yescharacteristic
376#95—21no—
377#95—41no—
378C24—21no—
379C24—41no—
380#41—21no—
381#41—21no—
382#41—21no—
383#41—21no—
384#41—42no—
385#41—42no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—82no—
395#41—82no—
396#41—81no—
397#41—81no—
398#41—82no—
399#41—82no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—41no—
403C2×Q8—42no—
404C2×Q8—81no—
405C2×Q8C4×D411yescharacteristic
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4C4×D411yescharacteristic
430C2×D4C4×D411yescharacteristic
431C2×D4C4×D411yescharacteristic
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—21no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—44no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4111yescharacteristic
459C22×C4H(5,𝔽2)11yescharacteristic
460Q16—81no—
461SD16—81no—
462SD16—81no—
463D8—42no—
464D8—82no—
465D8—81no—
466D8—81no—
467M16—21no—
468M16—21no—
469M16—21no—
470M16—21no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—82no—
476M16—82no—
477M16—82no—
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—21no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—82no—
500C2×C8—82no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4C4×D411yescharacteristic
513C4⋊C4C4×D411yescharacteristic
514C4⋊C4C4×D411yescharacteristic
515C4⋊C4C4×D411yescharacteristic
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—44no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—41no—
542C42—41no—
543C42—82no—
544C42AGL(1,ℤ/8)12yes—
545C42C23×C411yescharacteristic
546C42C22×D411yescharacteristic
547C42C2×#4111yescharacteristic
548C42H(5,𝔽2)11yescharacteristic
549C23—21no—
550C23—21no—
551C23—21no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—81no—
558C23—82no—
559C23—82no—
560C23—161no—
561C23#54611yescharacteristic
562Q8—21no—
563Q8—21no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—41no—
568Q8—82no—
569D4—21no—
570D4—21no—
571D4—21no—
572D4—21no—
573D4—21no—
574D4—42no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4—81no—
586D4—81no—
587D4—82no—
588D4—82no—
589D4—82no—
590D4—82no—
591D4—81no—
592D4—81no—
593D4—81no—
594D4—81no—
595D4—161no—
596D4#44312yes—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
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—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—81no—
629C2×C4—88no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4C2×C4×D411yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4#51811yescharacteristic
642C2×C4D4211yescharacteristic
643C2×C4#54611yescharacteristic
644C2×C4#55111yescharacteristic
645C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
646C2×C4#57511yescharacteristic
647C2×C4#57611yescharacteristic
648C2×C4#57611yescharacteristic
649C8—22no—
650C8—22no—
651C8—41no—
652C8—41no—
653C8—41no—
654C8—41no—
655C8—41no—
656C8—41no—
657C8—41no—
658C8—41no—
659C8—41no—
660C8—41no—
661C8—81no—
662C8—81no—
663C8—81no—
664C22—21no—
665C22—21no—
666C22—22no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—82no—
676C22—82no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—164no—
684C22#235811yescharacteristic
685C22#331811yescharacteristic
686C22#331911yescharacteristic
687C4—24no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—42no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—82no—
698C4—81no—
699C4—81no—
700C4C4×AGL(1,ℤ/8)11yescharacteristic
701C4#294811yescharacteristic
702C4C2×#44311yescharacteristic
703C4#295611yescharacteristic
704C4#327311yescharacteristic
705C4#327411yescharacteristic
706C2—21no—
707C2—21no—
708C2—42no—
709C2—81no—
710C2—81no—
711C2—161no—
712C2—161no—
713C2#2286811yescharacteristic
714C2#2288811yescharacteristic
715C2#2289911yescharacteristic
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. #18001C2semidirect
5. #17511C2semidirect
6. #9844C2semidirect
7. #9592C2semidirect
8. #9578C2semidirect
9. #9233C2semidirect
10. #9210C2semidirect
11. #9053C2semidirect
12. #8906C2semidirect
13. #8903C2semidirect
14. #8194C2semidirect
15. C4×D4⋊C8C2semidirect
21. #2935C22semidirect
29. #1962C22semidirect
30. #1950C22semidirect
37. C4×C4⋊C8C22semidirect
40. C42×C8C22semidirect
41. #1649C4semidirect
42. #1647C4semidirect
43. #1641C4semidirect
44. #1639C4semidirect
45. #1627C4semidirect
46. #1625C4semidirect
47. #1621C4semidirect
48. #1619C4semidirect
53. #1570C22semidirect
54. #1567C22semidirect
56. #1502C22semidirect
59. #1492C22semidirect
68. #1451C22semidirect
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
#180012
#175112
#98442
#95922
#95782
#92332
#92102
#90532
#89062
#89032
#81942
C4×D4⋊C82