Finite Group Databasebeta

← #6464348

Subgroups — #6464348

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic
3#29417C214yesmaximal
4#29395C212yesmaximal, complemented
5#29371C212yesmaximal, complemented
6#29273C211yesmaximal, characteristic, complemented
7#29255C211yesmaximal, characteristic, complemented
8#28206C211yesmaximal, characteristic, complemented
9#28161C211yesmaximal, characteristic, complemented
10#27628C212yesmaximal
11#24155C212yesmaximal
12#23763C212yesmaximal
13#23577C212yesmaximal
14#21616C212yesmaximal, complemented
15#21506C212yesmaximal, complemented
16#21123C214yesmaximal, complemented
17#20778C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#477C2212yes—
21C2×#462C2212yes—
22C2×#441C2214yes—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2211yescharacteristic
30C4×C4⋊Q8C2211yescharacteristic
31C4×#125—21no—
32C4×#125—21no—
33C4×#125—21no—
34C4×#125—21no—
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C4×C22⋊Q8C2212yes—
42C4×C22⋊Q8C2212yes—
43#2296—21no—
44#2296—21no—
45#2295—21no—
46#2295—21no—
47#2295C2212yes—
48#2295C2212yes—
49#2285C2211yescharacteristic
50#2285C2211yescharacteristic
51C42×Q8—21no—
52C42×Q8—21no—
53C42×Q8C2212yes—
54C42×Q8C2212yes—
55C4×#117—21no—
56C4×#117—21no—
57C2×C43C2211yescharacteristic
58#2099C2212yes—
59#2096C2212yes—
60#2083C2212yes—
61#2077C2214yes—
62#2072C2212yes—
63#2058C2212yes—
64#2043C2212yes—
65#2036C2214yes—
66#2035C2214yes—
67#2024C2212yes—
68#2016C2212yes—
69#2001C2214yes—
70#1987C2212yes—
71#1981C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1928C2214yes—
75#1927C2214yes—
76#1927C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1918C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1865C2214yes—
84#1865C2214yes—
85#1863C2214yes—
86#1839C2212yes—
87#1831C2211yescharacteristic
88#1831C2211yescharacteristic
89#1817C2212yes—
90#1803C2212yes—
91#1803C2212yes—
92#1797C2212yes—
93#1776C2214yes—
94#1767C2212yes—
95#1763C2214yes—
96#1763C2214yes—
97#1749C2212yes—
98#1745C2212yes—
99C22×Q16C2314yes—
100#533—24no—
101#533—24no—
102#533—28no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—21no—
108C2×#125—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
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×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—21no—
128C2×#117—21no—
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#477—28no—
137#462—28no—
138#441—216no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×C22⋊C8C2314yes—
151C2×#97C2314yes—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#382—21no—
193#382—21no—
194#380—21no—
195#380—21no—
196#380C2312yes—
197#380C2312yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234#376—22no—
235#376—22no—
236C43—22no—
237C43—22no—
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#336C2314yes—
253#336C2312yes—
254#336C2312yes—
255#336C2312yes—
256#336C2314yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8C2412yes—
261C22×Q8C2412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×Q16—28no—
270C2×Q16—28no—
271C2×Q16—44no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8#4114yes—
279C4⋊Q8#4114yes—
280#125—28no—
281#125—28no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C4×Q8—28no—
289C4×Q8—28no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—28no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—42no—
297C4×Q8—44no—
298C4×Q8—42no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8#4118yes—
304#117—28no—
305#117—28no—
306#117—44no—
307#117—44no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—42no—
362C2×C42—42no—
363C2×C42—42no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—28no—
393#106—28no—
394#106—44no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—28no—
401#103—44no—
402#103—44no—
403C22⋊C8—44no—
404#97—28no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425C24C22×D411yescharacteristic
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—24no—
435C2×Q8—22no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8C2×#4114yes—
441C2×Q8C2×#4114yes—
442C22×C4—22no—
443C22×C4—22no—
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—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4#13714yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2511yescharacteristic
486Q16—416no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4C2×#4114yes—
528C4⋊C4C2×#4114yes—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C42—28no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—28no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—21no—
551C42—21no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—44no—
562C42—44no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42—44no—
567C42—44no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42#13714yes—
572C42C2×#4114yes—
573C42C2×#4112yes—
574C42C2×#4112yes—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23C2×#13712yes—
582C23C23×D411yescharacteristic
583Q8—24no—
584Q8—28no—
585Q8—24no—
586Q8—48no—
587Q8—48no—
588Q8—42no—
589Q8—42no—
590Q8—44no—
591C2×C4—28no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—28no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4#52512yes—
652C2×C4#52512yes—
653C2×C4#52512yes—
654C2×C4#52512yes—
655C2×C4C2×#13712yes—
656C2×C4C2×#13712yes—
657C2×C4C2×#13712yes—
658C2×C4C2×#13712yes—
659C2×C4C2×#13712yes—
660C2×C4#57512yes—
661C2×C4#57512yes—
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4112yes—
664C2×C4C22×#4111yescharacteristic
665C2×C4#58512yes—
666C8—44no—
667C8—44no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22C2×#52511yescharacteristic
676C22C2×#52511yescharacteristic
677C22#344911yescharacteristic
678C22C22×#13711yescharacteristic
679C22C22×#13711yescharacteristic
680C22C2×#57511yescharacteristic
681C22C2×#57511yescharacteristic
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—44no—
694C4—44no—
695C4—41no—
696C4—44no—
697C4—44no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#311314yes—
703C4#347512yes—
704C4#348012yes—
705C2—42no—
706C2#3490911yescharacteristic
707C2C2×#311311yescharacteristic
708C2#6055911yescharacteristic
709C2#6056011yescharacteristic
710C2#6056211yescharacteristic
711C2#6096311yescharacteristic
712C2#6096611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #29395C2semidirect
5. #29371C2semidirect
6. #29273C2semidirect
7. #29255C2semidirect
8. #28206C2semidirect
9. #28161C2semidirect
14. #21616C2semidirect
15. #21506C2semidirect
16. #21123C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
#294172
#293952
#293712
#292732
#292552
#282062
#281612
#276282
#241552
#237632
#235772
#216162
#215062
#211232
#207782