Finite Group Databasebeta

← #6492140

Subgroups — #6492140

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2C4⋊C4×#41C211yesmaximal, characteristic
3#33149C212yesmaximal, complemented
4#33109C212yesmaximal, complemented
5#29939C212yesmaximal, complemented
6#29923C212yesmaximal, complemented
7#26204C214yesmaximal, complemented
8#26132C214yesmaximal, complemented
9#25062C212yesmaximal, complemented
10#25016C212yesmaximal, complemented
11#24996C212yesmaximal, complemented
12#24971C212yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21885C211yesmaximal, characteristic, complemented
15#21847C211yesmaximal, characteristic
16#21385C211yesmaximal, characteristic, complemented
17C2×#1947C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×C4×#41—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#484C2214yes—
22C2×#431C2211yescharacteristic
23C22×#107C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2629—22no—
28#2582—21no—
29#2582C2212yes—
30#2538—21no—
31#2538C2212yes—
32#2417—22no—
33#2417C2214yes—
34#2351—21no—
35Q8×C4⋊C4C2211yescharacteristic
36Q8×C4⋊C4C2211yescharacteristic
37#2349—22no—
38#2349—21no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41D4×C4⋊C4C2211yescharacteristic
42D4×C4⋊C4C2211yescharacteristic
43#2316—21no—
44#2311—21no—
45#2296—21no—
46#2296—22no—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2083C2214yes—
55#2076C2214yes—
56#2076C2214yes—
57#2072C2214yes—
58#2036C2214yes—
59#2036C2214yes—
60#1987C2212yes—
61#1987C2211yescharacteristic
62#1987C2211yescharacteristic
63#1984C2212yes—
64#1981C2211yescharacteristic
65#1981C2212yes—
66#1981C2211yescharacteristic
67#1954C2212yes—
68#1947C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1928C2214yes—
75#1928C2214yes—
76#1921C2214yes—
77#1921C2214yes—
78#1892C2214yes—
79#1872C2212yes—
80#1872C2214yes—
81#1872C2212yes—
82#1869C2214yes—
83#1852C2214yes—
84#1852C2214yes—
85#1845C2212yes—
86#1834C2212yes—
87#1824C2214yes—
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91#1797C2212yes—
92C4×#103C2212yes—
93C4×#103C2212yes—
94C2×#340C2212yes—
95C22×#41D411yescharacteristic
96C23×C8C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2314yes—
106C4×#41—24no—
107C4×#41—24no—
108C4×#41—48no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4D412yes—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—22no—
125C2×#117—21no—
126C2×#117D411yescharacteristic
127C2×#117D412yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—22no—
136C22×C42—21no—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#484—48no—
140#431—24no—
141C2×#107—44no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161#398—24no—
162#398—24no—
163#396—24no—
164#396—24no—
165#391—22no—
166#391—22no—
167#391C2314yes—
168#390—22no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#386—24no—
184#385—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#382—21no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—21no—
205#380—24no—
206#380C2314yes—
207#380C2312yes—
208#380C2312yes—
209#380C2312yes—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C2×#95—22no—
224C2×#95—21no—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—24no—
238C2×#41—24no—
239C2×#41Q1618yes—
240C22×Q8—21no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×D4C2×D412yes—
244C22×D4C2×D411yescharacteristic
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C22×C8—24no—
257C22×C8—42no—
258C22×C8—42no—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266#125—44no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—84no—
273C4×Q8—24no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4—44no—
290C4×D4—44no—
291#117—24no—
292#117—28no—
293#117—48no—
294#117—44no—
295#117—44no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
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—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—41no—
325C2×C4⋊C4—41no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—48no—
364C2×C42—44no—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4114yes—
370C2×C42#4114yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—28no—
375#107—44no—
376#107—44no—
377#107—44no—
378C4⋊C8—24no—
379C4⋊C8—24no—
380C4⋊C8—22no—
381C4⋊C8—22no—
382#103—24no—
383#103—24no—
384#103—42no—
385#103—44no—
386#103—42no—
387#103—44no—
388C22⋊C8—24no—
389C22⋊C8—44no—
390C22⋊C8—44no—
391C4×C8—24no—
392#95—21no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406C24C22×D412yes—
407C24C22×D411yescharacteristic
408#41—232no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8C2×Q1614yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4C2×Q1618yes—
426C2×D4C2×Q1614yes—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—81no—
463C22×C4—82no—
464C22×C4—81no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4#12314yes—
469C22×C4#12314yes—
470C22×C4C2×Q1618yes—
471C22×C4C2×Q1614yes—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—48no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
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—44no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—81no—
529C4⋊C4—81no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4H(5,𝔽2)12yes—
537C4⋊C42−1+412yes—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42—42no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42—42no—
559C42#12314yes—
560C42#12314yes—
561C42C2×#4112yes—
562C42C2×#4112yes—
563C42C2×#4112yes—
564C42C2×#4112yes—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23C2×#12312yes—
576C23C22×Q1614yes—
577C23C22×Q1612yes—
578C23C2×AGL(1,ℤ/8)12yes—
579C23C23×D411yescharacteristic
580Q8—28no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—84no—
586D4—216no—
587D4—28no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—216no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4—164no—
645C2×C4#484116yescomplemented
646C2×C4C2×#12312yes—
647C2×C4C2×#12312yes—
648C2×C4C2×#12312yes—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4C22×Q1614yes—
652C2×C4C22×Q1612yes—
653C2×C4C22×Q1612yes—
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C22×#4111yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×2−1+411yescharacteristic
660C2×C4#58511yescharacteristic
661C2×C4#58511yescharacteristic
662C8—44no—
663C8—42no—
664C8—42no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22C2×#48414yes—
679C22C22×#12311yescharacteristic
680C22#348811yescharacteristic
681C22#348911yescharacteristic
682C22C23×Q1611yescharacteristic
683C22C22×AGL(1,ℤ/8)11yescharacteristic
684C22#358511yescharacteristic
685C22#358611yescharacteristic
686C4—24no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—81no—
695C4—82no—
696C4—84no—
697C4—82no—
698C4—81no—
699C4—82no—
700C4—82no—
701C4C2×#48414yes—
702C4C2×#48414yes—
703C4#346011yescharacteristic
704C4#346111yescharacteristic
705C4#351111yescharacteristic
706C4#351211yescharacteristic
707C2—24no—
708C2—44no—
709C2#3525511yescharacteristic
710C2C22×#48411yescharacteristic
711C2#6052011yescharacteristic
712C2#6195211yescharacteristic
713C2#6196111yescharacteristic
714C2#6210011yescharacteristic
715C2#6210911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33149C2semidirect
4. #33109C2semidirect
5. #29939C2semidirect
6. #29923C2semidirect
7. #26204C2semidirect
8. #26132C2semidirect
9. #25062C2semidirect
10. #25016C2semidirect
11. #24996C2semidirect
12. #24971C2semidirect
13. #21982C2semidirect
14. #21885C2semidirect
16. #21385C2semidirect
17. C2×#1947C2semidirect
645. C2×C4#484semidirect

Maximal subgroups

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

subgroupindex
#352802
C4⋊C4×#412
#331492
#331092
#299392
#299232
#262042
#261322
#250622
#250162
#249962
#249712
#219822
#218852
#218472
#213852
C2×#19472