Finite Group Databasebeta

← #6487867

Subgroups — #6487867

Order 512.

2058 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#32959C212yesmaximal
4#32956C212yesmaximal
5#29973C212yesmaximal, complemented
6#29967C212yesmaximal, complemented
7#26118C214yesmaximal, complemented
8#26107C212yesmaximal, complemented
9#26106C212yesmaximal, complemented
10#26053C214yesmaximal, complemented
11#26045C212yesmaximal, complemented
12#26044C212yesmaximal, complemented
13#21987C211yesmaximal, characteristic, complemented
14#21981C211yesmaximal, characteristic, complemented
15#21953C211yesmaximal, characteristic, complemented
16#20738C211yesmaximal, characteristic
17#20733C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#478C2214yes—
22C2×#476C2214yes—
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2857C2212yes—
26#2851C2212yes—
27#2666—22no—
28#2571C2211yescharacteristic, complemented
29#2570C2212yescomplemented
30#2552C2211yescharacteristic, complemented
31#2552C2211yescharacteristic, complemented
32#2551C2211yescharacteristic, complemented
33#2550C2212yescomplemented
34#2442—21no—
35#2440—21no—
36#2438—22no—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4—22no—
40D4×C4⋊C4C2214yes—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2212yes—
43#2331—21no—
44#2330—21no—
45C2×#389C2212yes—
46C2×#385—22no—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49#2283C2211yescharacteristic
50C2×#380—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2214yescomplemented
53#2099C2214yescomplemented
54#2088C2214yescomplemented
55#2088C2214yescomplemented
56#2088C2214yescomplemented
57#2086C2214yescomplemented
58#2059C2214yescomplemented
59#2058C2214yescomplemented
60#1948C2214yescomplemented
61#1948C2212yescomplemented
62#1947C2212yescomplemented
63#1940C2214yescomplemented
64#1939C2212yescomplemented
65#1939C2212yescomplemented
66#1864C2214yes—
67#1864C2214yes—
68#1864C2214yes—
69#1862C2214yes—
70#1853C2212yes—
71#1853C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1847C2212yescomplemented
75#1844C2212yescomplemented
76#1836C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1824C2211yescharacteristic
80#1824C2211yescharacteristic
81#1804C2212yes—
82#1804C2214yes—
83#1804C2212yes—
84#1802C2212yes—
85#1802C2212yes—
86#1802C2214yes—
87#1797C2212yes—
88#1797C2212yes—
89#1786C2212yes—
90#1786C2211yescharacteristic
91#1786C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94C4×#102C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×#125—21no—
101C2×#125C2312yes—
102C2×C22⋊Q8—22no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—24no—
108C2×C4×D4—48no—
109C2×C4×D4Q818yescomplemented
110C2×C4×D4C2314yes—
111C2×C4×D4C2314yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—48no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—22no—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#478—48no—
131#476—48no—
132#432—24no—
133#431—24no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×C22⋊C8C2311yescharacteristic
151C2×#97C2312yes—
152C2×C4×C8C2312yes—
153#400—22no—
154#400—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#398—22no—
159#395—22no—
160#395—22no—
161#391—22no—
162#391—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#389C2312yes—
176#389C2312yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—22no—
183#385—24no—
184#385—24no—
185#385—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#383—21no—
196#382—21no—
197#382—21no—
198#382C2311yescharacteristic
199#382C2312yes—
200#382C2311yescharacteristic
201#381—21no—
202#380—22no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380—24no—
207#380C2311yescharacteristic
208#380C2312yes—
209#380C2311yescharacteristic
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221#376—21no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236C25—21no—
237C22×Q8—22no—
238C22×D4—24no—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4C2×Q814yes—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—28no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—48no—
254C23×C4—48no—
255C23×C4—84no—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266#125—44no—
267#125—44no—
268C22⋊Q8—84no—
269C4×D4—28no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4C2×Q818yes—
280C4×D4C2×Q818yes—
281#117—82no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—416no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×Q814yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—48no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×Q814yes—
346C2×C42#4114yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—44no—
352#106—44no—
353#106—44no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358C4⋊C8—44no—
359#102—24no—
360#102—28no—
361#102—24no—
362#102—28no—
363#102—24no—
364#102—24no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C22⋊C8—24no—
368#97—24no—
369C4×C8—24no—
370#95—24no—
371#95—24no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—24no—
393C24—24no—
394C24—24no—
395C24—22no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24C22×D411yescharacteristic
401C2×Q8—44no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—216no—
407C2×D4—216no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4C22×Q814yes—
413C2×D4H(5,𝔽2)14yes—
414C2×D42−1+414yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—432no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—832no—
455C22×C4C22⋊Q814yes—
456C22×C4C22⋊Q814yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×Q812yes—
467C22×C4C2×#4112yes—
468C22×C4C2511yescharacteristic
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4AGL(1,ℤ/8)14yes—
514C4⋊C4C22×Q814yes—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C42—24no—
523C42—24no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—82no—
529C42C22⋊Q814yes—
530C42C22⋊Q814yes—
531C42C22×Q812yes—
532C42C22×Q812yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—216no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—48no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—48no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C2×C22⋊Q812yes—
554C23#57712yes—
555C23#57712yes—
556C23C23×D411yescharacteristic
557Q8—164no—
558D4—216no—
559D4—24no—
560D4—24no—
561D4—24no—
562D4#54418yes—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—416no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—416no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—816no—
615C2×C4—84no—
616C2×C4—816no—
617C2×C4—88no—
618C2×C4C2×C22⋊Q812yes—
619C2×C4C2×C22⋊Q812yes—
620C2×C4C2×C22⋊Q812yes—
621C2×C4C2×C22⋊Q812yes—
622C2×C4C2×C22⋊Q812yes—
623C2×C4C2×AGL(1,ℤ/8)14yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4#57512yes—
626C2×C4C23×Q811yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×2−1+412yes—
630C8—42no—
631C8—42no—
632C8—44no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—28no—
640C22—432no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22#308414yes—
648C22C22×C22⋊Q811yescharacteristic
649C22#348411yescharacteristic
650C22#348611yescharacteristic
651C22C22×AGL(1,ℤ/8)11yescharacteristic
652C22C2×#57511yescharacteristic
653C22C2×#57711yescharacteristic
654C22C2×#57711yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—24no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—82no—
667C4—81no—
668C4—84no—
669C4—82no—
670C4—81no—
671C4#307714yes—
672C4C2×#54412yes—
673C4#349212yes—
674C2—24no—
675C2—48no—
676C2#3522411yescharacteristic
677C2C2×#307711yescharacteristic
678C2C2×#308411yescharacteristic
679C2#6160011yescharacteristic
680C2#6160411yescharacteristic
681C2#6184111yescharacteristic
682C2#6184311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35239C2semidirect
5. #29973C2semidirect
6. #29967C2semidirect
7. #26118C2semidirect
8. #26107C2semidirect
9. #26106C2semidirect
10. #26053C2semidirect
11. #26045C2semidirect
12. #26044C2semidirect
13. #21987C2semidirect
14. #21981C2semidirect
15. #21953C2semidirect
28. #2571C22semidirect
29. #2570C22semidirect
30. #2552C22semidirect
31. #2552C22semidirect
32. #2551C22semidirect
33. #2550C22semidirect
52. #2101C22semidirect
53. #2099C22semidirect
54. #2088C22semidirect
55. #2088C22semidirect
56. #2088C22semidirect
57. #2086C22semidirect
58. #2059C22semidirect
59. #2058C22semidirect
60. #1948C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1940C22semidirect
64. #1939C22semidirect
65. #1939C22semidirect
74. #1847C22semidirect
75. #1844C22semidirect
76. #1836C22semidirect
77. #1834C22semidirect
109. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#352392
C2×D4×C4⋊C42
#329592
#329562
#299732
#299672
#261182
#261072
#261062
#260532
#260452
#260442
#219872
#219812
#219532
#207382
#207332