Finite Group Databasebeta

← #6853741

Subgroups — #6853741

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33256C211yesmaximal, characteristic, complemented
4#32568C214yesmaximal, complemented
5#31870C214yesmaximal, complemented
6#30305C211yesmaximal, characteristic, complemented
7#30305C211yesmaximal, characteristic, complemented
8#29219C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic
10#29190C211yesmaximal, characteristic
11#29177C211yesmaximal, characteristic, complemented
12#29177C211yesmaximal, characteristic, complemented
13#29139C211yesmaximal, characteristic
14#27296C214yesmaximal, complemented
15#27002C214yesmaximal, complemented
16#22219C211yesmaximal, characteristic
17#22219C211yesmaximal, characteristic
18C4×#486C211yesmaximal, characteristic
19#21339C211yesmaximal, characteristic, complemented
20C2×#554—21no—
21C2×#486C2211yescharacteristic
22#2868C2211yescharacteristic
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2582—21no—
29#2567C2212yes—
30#2566—21no—
31#2566—21no—
32#2563—21no—
33#2552—21no—
34#2550—21no—
35#2550—21no—
36#2538—21no—
37#2538C2212yes—
38#2507C2212yes—
39#2507C2212yes—
40#2452—21no—
41#2452—21no—
42#2452C2212yes—
43#2452C2212yes—
44#2451—21no—
45#2451—21no—
46#2369C2212yes—
47#2355—21no—
48Q8×C4⋊C4—21no—
49D4×C4⋊C4—21no—
50D4×C4⋊C4C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C42×Q8C2211yescharacteristic
58#2099C2214yes—
59#2088C2214yes—
60#2088C2214yes—
61#2084C2214yes—
62#2084C2214yes—
63#2065C2214yes—
64#2065C2214yes—
65#2058C2214yes—
66#2058C2214yes—
67#2023C2214yes—
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1988C2214yes—
72#1984C2212yes—
73#1957C2211yescharacteristic
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1941C2214yes—
77#1941C2214yes—
78#1939C2212yes—
79#1934C2214yes—
80#1926C2214yes—
81#1926C2214yes—
82#1919C2214yes—
83#1919C2214yes—
84#1916C2212yes—
85#1916C2212yes—
86#1864C2214yes—
87#1864C2214yes—
88#1864C2214yes—
89#1864C2214yes—
90#1844C2211yescharacteristic
91#1844C2211yescharacteristic
92#1844C2211yescharacteristic
93#1844C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1802C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×#102C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#554—48no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#125—21no—
117C2×#125—21no—
118C2×#125—21no—
119C2×#125—21no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—22no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8D411yescharacteristic
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C22×C4⋊C4—22no—
139#486Q818yescomplemented
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2312yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2311yescharacteristic
157#398—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#396—22no—
162#396—22no—
163#396—22no—
164#396—22no—
165#395—22no—
166#395—22no—
167#393—22no—
168#393—22no—
169#391—22no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—22no—
174#391—22no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—22no—
197#389—22no—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#387—22no—
210#387—22no—
211#385—22no—
212#385—22no—
213#385—22no—
214#385—22no—
215#384—22no—
216#384—21no—
217#384—21no—
218#384—21no—
219#384—21no—
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2312yes—
227#384C2312yes—
228#382—21no—
229#382—21no—
230#382—21no—
231#382—21no—
232#380—22no—
233#380—22no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2311yescharacteristic
250C43C2311yescharacteristic
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#336C2312yes—
262#336C2312yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268C22×Q8—21no—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8C2×D411yescharacteristic
272C22×D4—21no—
273C22×D4—21no—
274C22×D4C2412yes—
275C23×C4—22no—
276C23×C4—22no—
277C23×C4—22no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—22no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C4⋊Q8—82no—
287C4⋊Q8—82no—
288C4⋊Q8—82no—
289C4⋊Q8C2×Q814yes—
290#127—22no—
291#127C2×Q814yes—
292#125—44no—
293#125—44no—
294#125—82no—
295#125—82no—
296C22⋊Q8—48no—
297C22⋊Q8—48no—
298C22⋊Q8—48no—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×Q8—44no—
302C4×Q8—82no—
303C4×Q8—82no—
304C4×Q8SD1618yes—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—44no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—82no—
313C4×D4—82no—
314C2×C4⋊C4—21no—
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—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—48no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
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×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#106—44no—
393#106—44no—
394#106—44no—
395#106—44no—
396C4⋊C8—42no—
397C4⋊C8—42no—
398C4⋊C8—42no—
399#102—44no—
400#102—44no—
401#102—44no—
402#102C2×Q8116yes—
403C4×C8—22no—
404C4×C8—22no—
405C4×C8—22no—
406C4×C8—42no—
407C4×C8—42no—
408C4×C8C2×Q814yes—
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#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429C24—22no—
430C2×Q8—24no—
431C2×Q8—41no—
432C2×Q8—41no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—82no—
438C2×Q8—82no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×Q8C2×SD1614yes—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4C22×Q814yes—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8C22×Q814yes—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4C2×SD1614yes—
539C4⋊C4C2×SD1614yes—
540C4⋊C4C2×SD1614yes—
541C4⋊C4C22×Q814yes—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—44no—
567C42—82no—
568C42—82no—
569C42C2×SD1614yes—
570C42C2×SD1614yes—
571C42C2×SD1614yes—
572C42C22×Q812yes—
573C422−1+412yes—
574C23—24no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23C23×D411yescharacteristic
582Q8—44no—
583Q8—48no—
584Q8—164no—
585Q8—164no—
586D4—28no—
587D4—48no—
588D4—48no—
589D4—44no—
590C2×C4—21no—
591C2×C4—21no—
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—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
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—88no—
642C2×C4—164no—
643C2×C4—164no—
644C2×C4D4×Q812yes—
645C2×C4D4×Q812yes—
646C2×C4C22×SD1612yes—
647C2×C4C22×SD1612yes—
648C2×C4C22×SD1612yes—
649C2×C4C22×SD1612yes—
650C2×C4C22×SD1612yes—
651C2×C4C22×SD1612yes—
652C2×C4C22×SD1612yes—
653C2×C4#57512yes—
654C2×C4#57612yes—
655C2×C4#57712yes—
656C2×C4C23×Q811yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C2×C4#58512yes—
659C2×C4#58512yes—
660C8—24no—
661C8—42no—
662C8—42no—
663C8—42no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22#344911yescharacteristic
672C22C2×D4×Q811yescharacteristic
673C22#350011yescharacteristic
674C22C23×SD1611yescharacteristic
675C22C2×#57511yescharacteristic
676C22C2×#57611yescharacteristic
677C22C2×#57711yescharacteristic
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—44no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4Q8×SD1614yes—
698C4#338914yes—
699C4#350412yes—
700C4#356512yes—
701C2—84no—
702C2#3961311yescharacteristic
703C2#6094111yescharacteristic
704C2#6100611yescharacteristic
705C2C2×Q8×SD1611yescharacteristic
706C2C2×#338911yescharacteristic
707C2#6252511yescharacteristic
708C2#6254311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36563C2semidirect
3. #33256C2semidirect
4. #32568C2semidirect
5. #31870C2semidirect
6. #30305C2semidirect
7. #30305C2semidirect
8. #29219C2semidirect
11. #29177C2semidirect
12. #29177C2semidirect
14. #27296C2semidirect
15. #27002C2semidirect
19. #21339C2semidirect
139. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#365592
#332562
#325682
#318702
#303052
#292192
#291902
#291772
#291392
#272962
#270022
#222192
C4×#4862
#213392