Finite Group Databasebeta

← #6432729

Subgroups — #6432729

Order 512.

2044 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#23839C211yesmaximal, characteristic, complemented
4#23823C211yesmaximal, characteristic, complemented
5#23822C211yesmaximal, characteristic, complemented
6#23803C211yesmaximal, characteristic, complemented
7#23782C212yesmaximal, complemented
8#23774C212yesmaximal, complemented
9#23761C212yesmaximal, complemented
10#23757C212yesmaximal, complemented
11#23711C214yesmaximal, complemented
12#23684C212yesmaximal, complemented
13#23677C212yesmaximal, complemented
14#23574C212yesmaximal, complemented
15#23570C212yesmaximal
16C4×#440C214yesmaximal, complemented
17C4×#432C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×C4×#41—21no—
21C2×#440C2214yes—
22C2×#440C2214yes—
23C2×C4×SD16C2214yes—
24C2×#432C2212yes—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2211yescharacteristic
30C4×C4⋊Q8C2211yescharacteristic
31C4×#127C2211yescharacteristic
32C4×#127C2211yescharacteristic
33C4×#125—21no—
34C4×#125—21no—
35C4×#124—21no—
36C4×#124—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C4×#121—21no—
42C4×#121—21no—
43C4×#121C2212yes—
44C4×#121C2212yes—
45#2297—21no—
46#2296—21no—
47#2295—21no—
48#2295C2212yes—
49#2293—21no—
50#2293C2212yes—
51#2285C2211yescharacteristic
52#2285C2211yescharacteristic
53C42×Q8—21no—
54C42×Q8C2212yes—
55C42×D4—21no—
56C42×D4C2212yes—
57C4×#117—21no—
58C2×C43C2211yescharacteristic
59#2100C2212yes—
60#2099C2212yes—
61#2095C2212yes—
62#2093C2212yes—
63#2086C2214yes—
64#2084C2212yes—
65#2083C2212yes—
66#2076C2214yes—
67#2072C2212yes—
68#2071C2212yes—
69#2067C2214yes—
70#2066C2214yes—
71#2058C2212yes—
72#2057C2212yes—
73#2055C2214yes—
74#2043C2212yes—
75#2041C2212yes—
76#2036C2214yes—
77#2034C2214yes—
78#2024C2212yes—
79#2023C2212yes—
80#2014C2212yes—
81#2013C2212yes—
82#2000C2214yes—
83#1776C2214yes—
84#1776C2214yes—
85C4×#107C2214yes—
86#1767C2212yes—
87C4×C4⋊C8C2212yes—
88#1763C2214yes—
89#1763C2214yes—
90#1762C2214yes—
91#1762C2214yes—
92C4×#103C2214yes—
93C4×#102C2214yes—
94#1749C2212yes—
95C4×C22⋊C8C2212yes—
96#1745C2211yescharacteristic
97#1745C2211yescharacteristic
98C4×#97C2212yes—
99C22×#41—21no—
100C22×SD16C2314yes—
101#533—28no—
102#532—28no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125—21no—
108C2×#124—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121C2312yes—
113C4×#41—48no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×#117—21no—
133C2×#117—21no—
134C2×#117—21no—
135C2×#117—22no—
136C22×C42—22no—
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#440—216no—
145#440—216no—
146C4×SD16—216no—
147#432—28no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2314yes—
159C2×#97C2312yes—
160C2×#97C2314yes—
161C2×#97C2312yes—
162C2×C4×C8C2314yes—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#386—22no—
196#386—22no—
197#385—22no—
198#385—22no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382—21no—
204#382—21no—
205#380—21no—
206#380—21no—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239#376—22no—
240C43—22no—
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43D411yescharacteristic
244C43D411yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247C43C2311yescharacteristic
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#336C2312yes—
256#336C2312yes—
257#336C2312yes—
258#336C2312yes—
259C2×#41—28no—
260C2×#41—44no—
261C2×#41—44no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8C2412yes—
265C22×D4—22no—
266C22×D4—21no—
267C22×D4—21no—
268C22×D4C2412yes—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2411yescharacteristic
279C2×SD16—28no—
280C2×SD16—28no—
281C2×SD16—28no—
282C22×C8C2414yes—
283C22×C8C2414yes—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287#127—24no—
288#125—28no—
289#124—28no—
290C22⋊Q8—28no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293#121—28no—
294#121—24no—
295#121—24no—
296C4×Q8—28no—
297C4×Q8—24no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8#4118yescomplemented
305C4×D4—28no—
306C4×D4—24no—
307C4×D4—28no—
308C4×D4—28no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—48no—
314C4×D4—48no—
315C4×D4#4118yes—
316#117—28no—
317#117—44no—
318#117—44no—
319#117—48no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—42no—
372C2×C42—42no—
373C2×C42—48no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402#107—28no—
403#107—28no—
404#106—28no—
405C4⋊C8—28no—
406#103—28no—
407#103—28no—
408#103—28no—
409#102—28no—
410#102—28no—
411#102—28no—
412C22⋊C8—28no—
413#97—24no—
414#97—28no—
415#97#4118yescomplemented
416C4×C8—28no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433C24—22no—
434C24C22×D411yescharacteristic
435#41—432no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8C2×#4114yes—
446C2×D4—24no—
447C2×D4—28no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×#4114yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4#13714yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C2511yescharacteristic
510SD16—232no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8C2×#4118yes—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—28no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4C2×#4114yes—
544C4⋊C4C2×#4114yes—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—28no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—28no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—44no—
582C42—44no—
583C42—44no—
584C42—44no—
585C42—44no—
586C42—44no—
587C42—44no—
588C42—44no—
589C42AGL(1,ℤ/8)14yes—
590C42C2×#4114yes—
591C42C2×#4114yes—
592C42C2×#4114yes—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—24no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23C2×#13712yes—
607C23C23×D411yescharacteristic
608Q8—28no—
609Q8—48no—
610Q8—48no—
611Q8—44no—
612D4—28no—
613D4—48no—
614D4—48no—
615D4—416no—
616D4—44no—
617C2×C4—28no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—28no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—416no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—88no—
673C2×C4—88no—
674C2×C4#52512yes—
675C2×C4#52512yes—
676C2×C4#52512yes—
677C2×C4#52512yes—
678C2×C4#53212yes—
679C2×C4#53212yes—
680C2×C4C2×AGL(1,ℤ/8)12yes—
681C2×C4C2×AGL(1,ℤ/8)12yes—
682C2×C4C2×AGL(1,ℤ/8)12yes—
683C2×C4C2×#13712yes—
684C2×C4C2×#13712yes—
685C2×C4#57512yes—
686C2×C4#57512yes—
687C2×C4C22×#4112yes—
688C2×C4C22×#4112yes—
689C2×C4C22×#4112yes—
690C8—28no—
691C8—44no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22C2×#52511yescharacteristic
707C22C2×#52511yescharacteristic
708C22C2×#53211yescharacteristic
709C22C22×AGL(1,ℤ/8)11yescharacteristic
710C22C22×#13711yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57511yescharacteristic
713C4—22no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—41no—
724C4—44no—
725C4—44no—
726C4—41no—
727C4—44no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4#311014yes—
733C4#344414yes—
734C2—42no—
735C2—84no—
736C2#3454711yescharacteristic
737C2C2×#311011yescharacteristic
738C2#6054611yescharacteristic
739C2#6056011yescharacteristic
740C2#6056111yescharacteristic
741C2#6070011yescharacteristic
742C2#6070111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#533C2semidirect
3. #23839C2semidirect
4. #23823C2semidirect
5. #23822C2semidirect
6. #23803C2semidirect
7. #23782C2semidirect
8. #23774C2semidirect
9. #23761C2semidirect
10. #23757C2semidirect
11. #23711C2semidirect
12. #23684C2semidirect
13. #23677C2semidirect
14. #23574C2semidirect
16. C4×#440C2semidirect
17. C4×#432C2semidirect
304. C4×Q8#41semidirect
415. #97#41semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C4×#5322
#238392
#238232
#238222
#238032
#237822
#237742
#237612
#237572
#237112
#236842
#236772
#235742
#235702
C4×#4402
C4×#4322