Finite Group Databasebeta

← #6986974

Subgroups — #6986974

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36504C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic, complemented
3#32772C212yesmaximal, complemented
4#32032C214yesmaximal, complemented
5#32006C214yesmaximal, complemented
6#30188C212yesmaximal, complemented
7#30176C212yesmaximal, complemented
8#29923C211yesmaximal, characteristic, complemented
9#29160C212yesmaximal, complemented
10#28600C212yesmaximal, complemented
11#26281C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic, complemented
13#25650C212yesmaximal, complemented
14#25338C212yesmaximal, complemented
15#24298C211yesmaximal, characteristic, complemented
16#23755C211yesmaximal, characteristic, complemented
17#22469C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#536—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#487C2212yes—
23C2×#474C2212yes—
24C2×C8×D4C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2850C2212yes—
29#2621—22no—
30#2547—21no—
31#2547C2212yes—
32#2546—21no—
33#2546C2212yes—
34#2542C2211yescharacteristic
35#2540C2212yes—
36#2537C2211yescharacteristic
37#2536C2212yes—
38#2486—22no—
39#2486C2214yes—
40#2479—22no—
41#2467—22no—
42#2423—21no—
43#2407—21no—
44#2363—21no—
45#2355C2211yescharacteristic
46#2354—21no—
47#2310—21no—
48C4×C22⋊Q8C2212yes—
49C4×#121C2211yescharacteristic
50#2285C2211yescharacteristic
51#2101C2214yes—
52#2100C2211yescharacteristic
53#2095C2212yes—
54#2094C2211yescharacteristic
55#2085C2214yes—
56#2084C2211yescharacteristic
57#2077C2212yes—
58#2075C2214yes—
59#2072C2212yes—
60#2071C2211yescharacteristic
61#2057C2212yes—
62#2042C2214yes—
63#2042C2214yes—
64#2036C2214yes—
65#2035C2214yes—
66#2035C2214yes—
67#2034C2212yes—
68#2033C2214yes—
69#2024C2212yes—
70#2024C2212yes—
71#2023C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1984C2212yes—
75#1981C2212yes—
76#1947C2212yes—
77#1921C2214yes—
78#1920C2214yes—
79#1893C2214yes—
80#1892C2214yes—
81#1892C2214yes—
82#1874C2212yes—
83#1872C2214yes—
84#1869C2214yes—
85#1863C2214yes—
86#1863C2214yes—
87#1852C2212yes—
88#1817C2212yes—
89#1808C2214yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×C8C2314yes—
98#556—44no—
99#536—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#126—22no—
108C2×#124—22no—
109C2×#124C2312yes—
110C2×#123—24no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121C2311yescharacteristic
119C2×#121C2312yes—
120C4×#41—44no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D412yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117—21no—
132C2×#117D411yescharacteristic
133C2×#117D412yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#487—44no—
139#474—44no—
140C8×D4—44no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C4×C8C2312yes—
157#399—22no—
158#399—22no—
159#398—24no—
160#398—24no—
161#398—22no—
162#397—24no—
163#397—24no—
164#394—22no—
165#394—22no—
166#392—22no—
167#391—21no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391C2312yes—
173#390—21no—
174#390—22no—
175#390—22no—
176#390C2311yescharacteristic
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389C2311yescharacteristic
182#389C2314yes—
183#389C2314yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—21no—
189#387—22no—
190#387—22no—
191#386—22no—
192#386—22no—
193#386—21no—
194#385—22no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#382—21no—
200#382C2312yes—
201#381—24no—
202#380—21no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380—21no—
207#380—21no—
208#380C2312yes—
209#380C2314yes—
210#380C2314yes—
211#380C2311yescharacteristic
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—42no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2412yes—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4C2×D412yes—
248C22×D4C2×D411yescharacteristic
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8—44no—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271#126—48no—
272#124—44no—
273#124—48no—
274#123—416no—
275C22⋊Q8—24no—
276C22⋊Q8—42no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282#121—22no—
283#121—41no—
284#121—44no—
285#121—44no—
286C4×Q8—24no—
287C4×Q8—42no—
288C4×Q8—44no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295#117—44no—
296#117—42no—
297#117—44no—
298#117—44no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—44no—
362#107—44no—
363#106—24no—
364#106—42no—
365C4⋊C8—24no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#102—42no—
371#102—42no—
372#102—44no—
373C22⋊C8—42no—
374C22⋊C8—42no—
375C22⋊C8—44no—
376C4×C8—24no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—22no—
392C24—21no—
393C24C22×D411yescharacteristic
394C24C22×D412yes—
395#41—416no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—816no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4AGL(1,ℤ/8)14yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4#13514yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
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—
478C2×C8—88no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C42−1+412yes—
512C22⋊C4—24no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—42no—
538C42—44no—
539C42—82no—
540C42H(5,𝔽2)14yes—
541C23—22no—
542C23—21no—
543C23—22no—
544C23—21no—
545C23—22no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23C2×#13512yes—
561C23C2×AGL(1,ℤ/8)14yes—
562C23C23×D411yescharacteristic
563Q8—44no—
564Q8—44no—
565Q8—44no—
566Q8—88no—
567Q8—88no—
568D4—28no—
569D4—48no—
570D4—42no—
571D4—42no—
572D4—84no—
573D4—164no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4#53412yes—
627C2×C4#53412yes—
628C2×C4C2×#13512yes—
629C2×C4C2×#13512yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57812yes—
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4C2×2−1+411yescharacteristic
634C2×C4#58512yes—
635C2×C4#58511yescharacteristic
636C8—42no—
637C8—44no—
638C8—42no—
639C22—22no—
640C22—24no—
641C22—22no—
642C22—24no—
643C22—22no—
644C22—24no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—41no—
649C22—42no—
650C22—42no—
651C22—41no—
652C22—41no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22C2×#53411yescharacteristic
661C22#348911yescharacteristic
662C22#350011yescharacteristic
663C22C22×#13511yescharacteristic
664C22C22×AGL(1,ℤ/8)11yescharacteristic
665C22C2×#57811yescharacteristic
666C22#358611yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—42no—
675C4—81no—
676C4—84no—
677C4—81no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#318814yes—
683C4#355412yes—
684C4#356212yes—
685C2—42no—
686C2—44no—
687C2—161no—
688C2#4291311yescharacteristic
689C2C2×#318811yescharacteristic
690C2#6075911yescharacteristic
691C2#6209711yescharacteristic
692C2#6213211yescharacteristic
693C2#6252811yescharacteristic
694C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36504C2
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
2. #35316C2semidirect
3. #32772C2semidirect
4. #32032C2semidirect
5. #32006C2semidirect
6. #30188C2semidirect
7. #30176C2semidirect
8. #29923C2semidirect
9. #29160C2semidirect
10. #28600C2semidirect
11. #26281C2semidirect
12. #25902C2semidirect
13. #25650C2semidirect
14. #25338C2semidirect
15. #24298C2semidirect
16. #23755C2semidirect
17. #22469C2semidirect

Maximal subgroups

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

subgroupindex
#365042
#353162
#327722
#320322
#320062
#301882
#301762
#299232
#291602
#286002
#262812
#259022
#256502
#253382
#242982
#237552
#224692