Finite Group Databasebeta

← #6852323

Subgroups — #6852323

Order 512.

1683 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3#32954C211yesmaximal, characteristic, complemented
4#32367C214yesmaximal, complemented
5#31543C214yesmaximal, complemented
6#29953C212yesmaximal, complemented
7#28540C212yesmaximal, complemented
8#27956C212yesmaximal, complemented
9#27823C212yesmaximal, complemented
10#27684C212yesmaximal
11#25813C211yesmaximal, characteristic, complemented
12#25456C212yesmaximal, complemented
13#24816C212yesmaximal, complemented
14#24298C211yesmaximal, characteristic
15#22629C211yesmaximal, characteristic, complemented
16#22047C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#500C2212yes—
22C2×#475C2212yes—
23C2×#474C2212yes—
24C2×#448C2211yescharacteristic
25C2×C4×SD16C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2568—21no—
29#2568—21no—
30#2568—22no—
31#2568C2214yes—
32#2568C2212yes—
33#2568C2212yes—
34#2564—22no—
35#2556—22no—
36#2551C2211yescharacteristic
37#2542C2211yescharacteristic
38#2506C2212yes—
39#2492—21no—
40#2452C2212yes—
41#2417—21no—
42C2×#400—22no—
43#2365C2211yescharacteristic
44#2360—21no—
45D4×C4⋊C4C2211yescharacteristic
46C4×#121C2212yes—
47C4×#121C2211yescharacteristic
48C2×#391—21no—
49C2×#391—21no—
50C2×#391C2212yes—
51C2×#382C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2095C2214yes—
55#2092C2214yes—
56#2085C2214yes—
57#2085C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2057C2214yes—
61#2057C2212yes—
62#2024C2214yes—
63#2023C2212yes—
64#1993C2214yes—
65#1990C2212yes—
66#1988C2212yes—
67#1973C2214yes—
68#1949C2214yes—
69#1939C2212yes—
70#1935C2214yes—
71#1928C2214yes—
72#1917C2214yes—
73#1880C2212yes—
74#1877C2214yes—
75#1874C2212yes—
76#1874C2211yescharacteristic
77#1865C2214yes—
78#1865C2212yes—
79#1853C2212yes—
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1847C2214yes—
85#1804C2214yes—
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2211yescharacteristic
90#1787C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98#554—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—21no—
113C2×#121C2311yescharacteristic
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—44no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#500—44no—
135#475—44no—
136#474—44no—
137#448Q818yescomplemented
138C4×SD16—44no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#400—24no—
160#400—24no—
161#400—24no—
162#400—24no—
163#399—22no—
164#398—22no—
165#396—22no—
166#395—22no—
167#393—22no—
168#391—22no—
169#391—24no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—24no—
174#391—21no—
175#391—22no—
176#391—22no—
177#391—21no—
178#391—22no—
179#391C2314yes—
180#391C2314yes—
181#390C2312yes—
182#390C2311yescharacteristic
183#389—21no—
184#389—22no—
185#389—21no—
186#389C2312yes—
187#389C2311yescharacteristic
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#388—22no—
192#388—21no—
193#387—22no—
194#387—22no—
195#386—21no—
196#386—22no—
197#385—22no—
198#385—22no—
199#384C2311yescharacteristic
200#384C2312yes—
201#382—22no—
202#382—22no—
203#382C2314yes—
204#382C2314yes—
205#380—21no—
206#380—22no—
207#380—21no—
208#380C2312yes—
209#380C2311yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—42no—
243C22×Q8—42no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2×Q814yes—
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×SD16—42no—
262C2×SD16—42no—
263C2×SD16C2×Q818yescomplemented
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275#125—48no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—44no—
281#121—22no—
282#121C2×Q814yes—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—42no—
286C4×Q8—42no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—48no—
293C4×D4—44no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355#107—44no—
356#107—44no—
357#107—44no—
358#106—42no—
359#106—44no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—44no—
363#103—42no—
364#103—42no—
365#103—44no—
366#103C2×Q818yescomplemented
367#102—42no—
368#102—44no—
369C22⋊C8—22no—
370C22⋊C8—24no—
371C22⋊C8C2×Q814yes—
372#97—44no—
373C4×C8—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24C22×D411yescharacteristic
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—48no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—48no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8—816no—
420C2×Q8C22×Q814yes—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C22×Q812yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4AGL(1,ℤ/8)14yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×Q812yes—
475C22×C42−1+412yes—
476C22×C4C2511yescharacteristic
477SD16—48no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8C22×Q814yes—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4C22×Q812yes—
520C22⋊C4—24no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—82no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—44no—
538C42—44no—
539C42—42no—
540C42H(5,𝔽2)14yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23D4×Q812yes—
553C23C2×AGL(1,ℤ/8)12yes—
554C23#57812yes—
555C23C23×D411yescharacteristic
556Q8—24no—
557Q8—24no—
558Q8—48no—
559Q8—42no—
560Q8—42no—
561Q8—88no—
562Q8—84no—
563Q8—88no—
564Q8—164no—
565D4—24no—
566D4—48no—
567D4—42no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4—164no—
624C2×C4#53512yes—
625C2×C4#53512yes—
626C2×C4D4×Q812yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4#57512yes—
630C2×C4C23×Q811yescharacteristic
631C2×C4C2×H(5,𝔽2)12yes—
632C2×C4C2×2−1+411yescharacteristic
633C2×C4#58512yes—
634C8—24no—
635C8—42no—
636C8—44no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22#338614yes—
651C22C2×#53511yescharacteristic
652C22C2×D4×Q811yescharacteristic
653C22#348511yescharacteristic
654C22C22×AGL(1,ℤ/8)11yescharacteristic
655C22C2×#57511yescharacteristic
656C22C2×#57811yescharacteristic
657C22#358611yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—44no—
667C4—42no—
668C4—82no—
669C4—81no—
670C4—81no—
671C4—84no—
672C4—82no—
673C4—82no—
674C4#321314yes—
675C4#350112yes—
676C4#354112yes—
677C2—24no—
678C2—82no—
679C2#3959411yescharacteristic
680C2C2×#321311yescharacteristic
681C2#6081411yescharacteristic
682C2C2×#338611yescharacteristic
683C2#6136911yescharacteristic
684C2#6172211yescharacteristic
685C2#6173011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2568C2semidirect
3. #32954C2semidirect
4. #32367C2semidirect
5. #31543C2semidirect
6. #29953C2semidirect
7. #28540C2semidirect
8. #27956C2semidirect
9. #27823C2semidirect
11. #25813C2semidirect
12. #25456C2semidirect
13. #24816C2semidirect
15. #22629C2semidirect
137. #448Q8semidirect
263. C2×SD16C2×Q8semidirect
366. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#25682
#329542
#323672
#315432
#299532
#285402
#279562
#278232
#276842
#258132
#254562
#248162
#242982
#226292
#220472
C4×#4482