Finite Group Databasebeta

← #6605616

Subgroups — #6605616

Order 512.

2185 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36161C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#33146C211yesmaximal, characteristic, complemented
4#32983C212yesmaximal, complemented
5#27950C211yesmaximal, characteristic, complemented
6#27680C211yesmaximal, characteristic, complemented
7#27602C212yesmaximal, complemented
8#26298C212yesmaximal, complemented
9#26148C214yesmaximal, complemented
10#25016C212yesmaximal, complemented
11#24953C214yesmaximal, complemented
12#24559C212yesmaximal, complemented
13#22665C212yesmaximal, complemented
14#22504C212yesmaximal, complemented
15#22401C211yesmaximal, characteristic
16C4×#483C211yesmaximal, characteristic, complemented
17#20844C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C22×C4×D4—21no—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C2×#484C2214yescomplemented
23C2×#483C2211yescharacteristic
24C2×#436C2212yes—
25C2×#428C2214yescomplemented
26C2×#407C2212yes—
27#2857C2211yescharacteristic
28#2848C2212yes—
29#2547C2211yescharacteristic
30#2538C2211yescharacteristic
31#2485C2212yes—
32#2480—22no—
33#2466—22no—
34#2447—21no—
35#2435—21no—
36#2415C2212yes—
37#2367—21no—
38#2360—21no—
39#2354C2211yescharacteristic
40D4×C4⋊C4—21no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—22no—
43D4×C4⋊C4C2214yes—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47C2×#389—21no—
48C2×#385—22no—
49C2×#384C2211yescharacteristic
50#2280C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2212yescomplemented
54#2096C2212yescomplemented
55#2083C2214yescomplemented
56#2077C2214yescomplemented
57#2077C2214yescomplemented
58#2076C2214yescomplemented
59#2036C2214yescomplemented
60#1990C2212yescomplemented
61#1988C2212yescomplemented
62#1954C2212yes—
63#1948C2212yescomplemented
64#1947C2212yescomplemented
65#1927C2212yescomplemented
66#1920C2212yescomplemented
67#1918C2214yescomplemented
68#1894C2214yescomplemented
69#1872C2212yes—
70#1872C2211yescharacteristic
71#1872C2214yescomplemented
72#1870C2214yes—
73#1854C2214yescomplemented
74#1853C2212yes—
75#1853C2211yescharacteristic
76#1853C2214yescomplemented
77#1852C2212yes—
78#1852C2212yes—
79#1852C2212yes—
80#1852C2214yescomplemented
81#1852C2214yescomplemented
82#1845C2212yescomplemented
83#1825C2214yescomplemented
84#1824C2214yescomplemented
85#1823C2211yescharacteristic
86#1823C2212yes—
87#1809C2214yescomplemented
88#1803C2212yescomplemented
89#1797C2211yescharacteristic
90#1779C2212yes—
91C4×#106C2212yescomplemented
92C4×#103C2212yescomplemented
93C4×C22⋊C8C2211yescharacteristic
94#1736C2214yescomplemented
95C23×D4D411yescharacteristic
96C24×C4—22no—
97C24×C4C2312yes—
98C22×M16C2314yes—
99#556—28no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#124—21no—
103C2×#124—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—28no—
111C2×C4×D4—24no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—48no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D412yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117C2312yes—
122C2×#117C2312yes—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—48no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C22⋊C4—22no—
136C22×C22⋊C4D412yes—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#484—48no—
140#483—24no—
141#436—28no—
142#428—48no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2312yes—
147C2×#106C2312yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153#407—28no—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161#397—22no—
162#397—22no—
163#397—22no—
164#396—22no—
165#394—22no—
166#394—22no—
167#391—21no—
168#391—21no—
169#391C2311yescharacteristic
170#390—21no—
171#390—21no—
172#389—24no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389C2312yes—
179#389C2311yescharacteristic
180#388—22no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—21no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—21no—
189#385—24no—
190#385—24no—
191#385—24no—
192#385—24no—
193#384—21no—
194#384—21no—
195#384C2312yes—
196#384C2312yes—
197#384C2311yescharacteristic
198#381—22no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2314yes—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2311yescharacteristic
210#379—22no—
211#379—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C2×#95—21no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25C2×D412yes—
238C22×Q8—21no—
239C22×Q8C2411yescharacteristic
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4C2×D412yes—
244C22×D4C2×D411yescharacteristic
245C22×D4C2×D411yescharacteristic
246C23×C4—21no—
247C23×C4—28no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—44no—
255C23×C4—48no—
256C23×C4—48no—
257C23×C4—84no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2412yes—
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C2×M16—28no—
267C2×M16—28no—
268C2×M16—44no—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273#126—48no—
274#124—44no—
275#124—44no—
276C22⋊Q8—28no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C4×Q8—24no—
282C4×Q8—44no—
283C4×D4—24no—
284C4×D4—28no—
285C4×D4—24no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—416no—
289C4×D4—48no—
290C4×D4—48no—
291C4×D4#4118yes—
292#117—28no—
293#117—44no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—48no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—28no—
369#107—44no—
370#107—44no—
371#106—24no—
372#106—24no—
373#106—44no—
374C4⋊C8—24no—
375#103—24no—
376#103—24no—
377#103—44no—
378C22⋊C8—24no—
379C22⋊C8—28no—
380C22⋊C8—22no—
381C22⋊C8—28no—
382C22⋊C8—44no—
383C22⋊C8#4114yes—
384C4×C8—24no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400C24—24no—
401C24—24no—
402C24—24no—
403C24—24no—
404C24—24no—
405C24—42no—
406C24—42no—
407C24C22×D412yes—
408C24C22×D412yes—
409C24C22×D411yescharacteristic
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×D4—232no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4#12314yes—
426C2×D4#13714yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—832no—
477C22×C4AGL(1,ℤ/8)14yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4118yes—
486C22×C4C2×#4112yes—
487C22×C42−1+412yes—
488C22×C4C2511yescharacteristic
489M16—416no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8C2×#4114yes—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—81no—
526C4⋊C4—81no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4#12314yes—
531C4⋊C4H(5,𝔽2)14yes—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4#12318yes—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—44no—
553C42—42no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—82no—
558C42C2×#4114yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—216no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—48no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—48no—
581C23—48no—
582C23C2×#12314yes—
583C23#54712yes—
584C23C2×AGL(1,ℤ/8)12yes—
585C23C2×#13714yes—
586C23#57812yes—
587C23C23×D411yescharacteristic
588Q8—42no—
589Q8—42no—
590Q8—84no—
591D4—216no—
592D4—28no—
593D4—28no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—816no—
648C2×C4—816no—
649C2×C4—88no—
650C2×C4—84no—
651C2×C4C2×#12312yes—
652C2×C4C2×#12314yes—
653C2×C4C2×#12312yes—
654C2×C4#54712yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4C2×#13712yes—
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4112yes—
660C2×C4C2×H(5,𝔽2)12yes—
661C2×C4C2×2−1+411yescharacteristic
662C8—24no—
663C8—44no—
664C8—42no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—432no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22#335314yes—
687C22C22×#12311yescharacteristic
688C22C2×#54711yescharacteristic
689C22#348411yescharacteristic
690C22C22×AGL(1,ℤ/8)11yescharacteristic
691C22C22×#13711yescharacteristic
692C22C2×#57811yescharacteristic
693C22#358511yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—24no—
698C4—41no—
699C4—42no—
700C4—44no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4—81no—
705C4—81no—
706C4—82no—
707C4—84no—
708C4—81no—
709C4#309214yes—
710C4#348112yes—
711C4#352012yes—
712C2—24no—
713C2—44no—
714C2—44no—
715C2#3623711yescharacteristic
716C2C2×#309211yescharacteristic
717C2#6052611yescharacteristic
718C2C2×#335311yescharacteristic
719C2#6125411yescharacteristic
720C2#6158211yescharacteristic
721C2#6160211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33146C2semidirect
4. #32983C2semidirect
5. #27950C2semidirect
6. #27680C2semidirect
7. #27602C2semidirect
8. #26298C2semidirect
9. #26148C2semidirect
10. #25016C2semidirect
11. #24953C2semidirect
12. #24559C2semidirect
13. #22665C2semidirect
14. #22504C2semidirect
16. C4×#483C2semidirect
17. #20844C2semidirect
22. C2×#484C22semidirect
25. C2×#428C22semidirect
53. #2099C22semidirect
54. #2096C22semidirect
55. #2083C22semidirect
56. #2077C22semidirect
57. #2077C22semidirect
58. #2076C22semidirect
59. #2036C22semidirect
60. #1990C22semidirect
61. #1988C22semidirect
63. #1948C22semidirect
64. #1947C22semidirect
65. #1927C22semidirect
66. #1920C22semidirect
67. #1918C22semidirect
68. #1894C22semidirect
71. #1872C22semidirect
73. #1854C22semidirect
76. #1853C22semidirect
80. #1852C22semidirect
81. #1852C22semidirect
82. #1845C22semidirect
83. #1825C22semidirect
84. #1824C22semidirect
87. #1809C22semidirect
88. #1803C22semidirect
91. C4×#106C22semidirect
92. C4×#103C22semidirect
94. #1736C22semidirect

Maximal subgroups

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

subgroupindex
#361612
C2×D4×C4⋊C42
#331462
#329832
#279502
#276802
#276022
#262982
#261482
#250162
#249532
#245592
#226652
#225042
#224012
C4×#4832
#208442