Finite Group Databasebeta

← #6469395

Subgroups — #6469395

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29397C214yesmaximal, complemented
4#29277C211yesmaximal, characteristic, complemented
5#29276C211yesmaximal, characteristic, complemented
6#28202C211yesmaximal, characteristic, complemented
7#28201C211yesmaximal, characteristic, complemented
8#27646C212yesmaximal, complemented
9#27612C212yesmaximal, complemented
10#24300C212yesmaximal, complemented
11#24062C212yesmaximal
12#23838C212yesmaximal, complemented
13#23819C212yesmaximal, complemented
14C4×#476C212yesmaximal, complemented
15C4×#474C212yesmaximal, complemented
16C4×#482C214yesmaximal, complemented
17#20777C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×C4×#41—21no—
21C2×#482C2214yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2211yescharacteristic
30C4×#127C2211yescharacteristic
31C4×#127C2211yescharacteristic
32C4×#125C2212yes—
33C4×#125C2212yes—
34C4×#124—21no—
35C4×#124—21no—
36C4×C22⋊Q8—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8—21no—
40C4×#121—21no—
41C4×#121—21no—
42C4×#121C2212yes—
43C4×#121C2212yes—
44#2297—21no—
45#2296C2212yes—
46#2295—21no—
47#2295—21no—
48#2293—21no—
49#2293C2212yes—
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52C42×Q8—21no—
53C42×Q8—21no—
54C42×D4—21no—
55C42×D4C2212yes—
56C4×#117C2212yes—
57C2×C43C2211yescharacteristic
58#2100C2212yes—
59#2100C2212yes—
60#2085C2214yes—
61#2084C2212yes—
62#2083C2212yes—
63#2073C2214yes—
64#2059C2214yes—
65#2058C2212yes—
66#2057C2212yes—
67#2041C2212yes—
68#2041C2212yes—
69#2025C2214yes—
70#1988C2212yes—
71#1987C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1926C2214yes—
75#1925C2214yes—
76#1921C2214yes—
77#1920C2214yes—
78#1920C2214yes—
79#1918C2214yes—
80#1875C2214yes—
81#1864C2214yes—
82#1862C2214yes—
83#1854C2214yes—
84#1839C2212yes—
85#1831C2211yescharacteristic
86#1831C2211yescharacteristic
87#1825C2214yes—
88#1817C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×#107C2214yes—
93C4×#106C2214yes—
94C4×C4⋊C8C2212yes—
95C4×#102C2214yes—
96C4×#102C2214yes—
97C4×C22⋊C8C2212yes—
98#1744C2212yes—
99C22×#41—21no—
100#533—24no—
101#533—24no—
102#532—28no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125C2312yes—
108C2×#124—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
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×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
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—22no—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C22×C42—22no—
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#482—216no—
145#476—28no—
146#474—28no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C4×C8C2314yes—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#382—21no—
200#382C2312yes—
201#380—21no—
202#380—21no—
203#380—21no—
204#380C2312yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235#376—22no—
236C43—22no—
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#336C2314yes—
254#336C2312yes—
255#336C2312yes—
256#336C2314yes—
257#336C2312yes—
258#336C2312yes—
259C2×#41—28no—
260C2×#41—44no—
261C2×#41—44no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—21no—
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
279C22×C8C2414yes—
280C22×C8C2414yes—
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8#4114yescomplemented
285C4⋊Q8#4114yescomplemented
286#127—24no—
287#125—24no—
288#125—24no—
289#124—28no—
290C22⋊Q8—28no—
291C22⋊Q8—28no—
292#121—28no—
293#121—24no—
294#121#4118yes—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×D4—28no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—48no—
314C4×D4—48no—
315C4×D4—44no—
316#117—24no—
317#117—24no—
318#117—28no—
319#117—48no—
320#117—44no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
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×C4⋊C4C2412yes—
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×C22⋊C4—22no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—42no—
374C2×C42—42no—
375C2×C42—48no—
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×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404#107—28no—
405#107—28no—
406#106—28no—
407#106—28no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410#102—28no—
411#102—28no—
412#102—28no—
413#102—28no—
414C22⋊C8—28no—
415C4×C8—28no—
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—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432C24—22no—
433C24C22×D411yescharacteristic
434#41—432no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—28no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
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—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
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—22no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
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×C4C2×#4114yes—
510C22×C4C2511yescharacteristic
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4C2×#4114yes—
546C4⋊C4C2×#4114yes—
547C4⋊C4C2×#4114yes—
548C22⋊C4—28no—
549C22⋊C4—28no—
550C22⋊C4—28no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C42—28no—
560C42—22no—
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—21no—
578C42—21no—
579C42—22no—
580C42—22no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—44no—
586C42—44no—
587C42—44no—
588C42—44no—
589C42—44no—
590C42—44no—
591C42—44no—
592C42—44no—
593C42#13514yes—
594C42AGL(1,ℤ/8)14yes—
595C42C2×#4112yes—
596C42C2×#4112yes—
597C422−1+414yes—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—24no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23#52512yes—
612C23C23×D411yescharacteristic
613Q8—48no—
614Q8—48no—
615Q8—48no—
616D4—24no—
617D4—24no—
618D4—48no—
619D4—48no—
620D4—416no—
621D4—42no—
622D4—42no—
623C2×C4—28no—
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—24no—
639C2×C4—24no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—416no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—88no—
680C2×C4—88no—
681C2×C4#52512yes—
682C2×C4#52512yes—
683C2×C4#52512yes—
684C2×C4#53612yes—
685C2×C4#53612yes—
686C2×C4C2×#13512yes—
687C2×C4C2×#13512yes—
688C2×C4C2×#13512yes—
689C2×C4C2×AGL(1,ℤ/8)12yes—
690C2×C4C2×AGL(1,ℤ/8)12yes—
691C2×C4C2×AGL(1,ℤ/8)12yes—
692C2×C4#57512yes—
693C2×C4#57512yes—
694C2×C4C22×#4112yes—
695C2×C4C22×#4111yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4C2×2−1+412yes—
698C8—44no—
699C8—44no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22C2×#52511yescharacteristic
715C22C2×#52511yescharacteristic
716C22C2×#53611yescharacteristic
717C22C22×#13511yescharacteristic
718C22C22×AGL(1,ℤ/8)11yescharacteristic
719C22C2×#57511yescharacteristic
720C22C2×#57511yescharacteristic
721C4—22no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—22no—
728C4—41no—
729C4—44no—
730C4—41no—
731C4—44no—
732C4—41no—
733C4—44no—
734C4—41no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4#310614yes—
740C4#323514yes—
741C4#348212yes—
742C4#348212yes—
743C2—42no—
744C2—84no—
745C2#3497111yescharacteristic
746C2C2×#310611yescharacteristic
747C2#6053611yescharacteristic
748C2#6055811yescharacteristic
749C2#6055911yescharacteristic
750C2#6084611yescharacteristic
751C2C2×#323511yescharacteristic
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. #29397C2semidirect
4. #29277C2semidirect
5. #29276C2semidirect
6. #28202C2semidirect
7. #28201C2semidirect
8. #27646C2semidirect
9. #27612C2semidirect
10. #24300C2semidirect
12. #23838C2semidirect
13. #23819C2semidirect
14. C4×#476C2semidirect
15. C4×#474C2semidirect
16. C4×#482C2semidirect
17. #20777C2semidirect
284. C4⋊Q8#41semidirect
285. C4⋊Q8#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
#293972
#292772
#292762
#282022
#282012
#276462
#276122
#243002
#240622
#238382
#238192
C4×#4762
C4×#4742
C4×#4822
#207772