Finite Group Databasebeta

← #6854495

Subgroups — #6854495

Order 512.

1796 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#127C211yesmaximal, characteristic, complemented
2#36555C211yesmaximal, characteristic, complemented
3#33294C211yesmaximal, characteristic, complemented
4#31540C214yesmaximal, complemented
5#31538C214yesmaximal, complemented
6#30309C212yesmaximal, complemented
7#29195C212yesmaximal, complemented
8#29149C211yesmaximal, characteristic, complemented
9#29148C212yesmaximal, complemented
10#28520C211yesmaximal, characteristic, complemented
11#24813C212yesmaximal, complemented
12#24811C212yesmaximal, complemented
13#24330C212yesmaximal, complemented
14#24326C212yesmaximal, complemented
15#22272C212yesmaximal
16C4×#496C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21C2×#496C2211yescharacteristic
22C2×#475C2212yes—
23C2×#474C2212yes—
24C2×C4×SD16C2212yes—
25C2×C4×D8C2212yes—
26#2868C2211yescharacteristic
27#2868C2212yes—
28#2864C2212yes—
29#2859C2211yescharacteristic
30#2587—22no—
31#2582C2211yescharacteristic
32#2542—22no—
33#2542—21no—
34#2542—21no—
35#2542—22no—
36#2542C2214yes—
37#2542C2212yes—
38#2538C2211yescharacteristic
39#2522—22no—
40#2497—21no—
41#2454C2212yes—
42#2453C2212yes—
43#2423—21no—
44#2360—21no—
45Q8×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47#2345—21no—
48C4×#127C2212yes—
49C4×#127C2211yescharacteristic
50C4×#124C2212yes—
51C4×#124C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2095C2212yes—
54#2094C2212yes—
55#2084C2214yes—
56#2083C2214yes—
57#2067C2214yes—
58#2065C2214yes—
59#2058C2214yes—
60#2057C2214yes—
61#2025C2214yes—
62#2024C2214yes—
63#2023C2214yes—
64#1992C2214yes—
65#1990C2212yes—
66#1990C2212yes—
67#1984C2212yes—
68#1973C2212yes—
69#1972C2212yes—
70#1961C2212yes—
71#1961C2211yescharacteristic
72#1959C2212yes—
73#1959C2211yescharacteristic
74#1939C2212yes—
75#1917C2214yes—
76#1880C2214yes—
77#1879C2214yes—
78#1879C2214yes—
79#1877C2214yes—
80#1864C2214yes—
81#1862C2214yes—
82#1847C2212yes—
83#1847C2211yescharacteristic
84#1836C2212yes—
85#1836C2212yes—
86#1836C2212yes—
87#1803C2214yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2214yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C23×Q8—22no—
96C22×SD16C2312yes—
97C22×D8C2312yes—
98D4×Q8—44no—
99D4×Q8—44no—
100D4×Q8—48no—
101D4×Q8—48no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×#127C2311yescharacteristic
109C2×#125C2312yes—
110C2×#124C2311yescharacteristic
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—22no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2314yes—
130C2×C4×D4C2312yes—
131C22×C4⋊C4—21no—
132#496Q818yescomplemented
133#475—44no—
134#474—44no—
135C4×SD16—44no—
136C4×D8—44no—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#97C2311yescharacteristic
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×#97C2311yescharacteristic
155C2×C4×C8C2312yes—
156#400—22no—
157#399—22no—
158#399—24no—
159#399—24no—
160#399—24no—
161#398—22no—
162#398—22no—
163#396—22no—
164#396—22no—
165#391—21no—
166#391C2312yes—
167#391C2311yescharacteristic
168#390—22no—
169#390—22no—
170#390—21no—
171#390—21no—
172#390C2314yes—
173#390C2312yes—
174#390C2311yescharacteristic
175#390C2312yes—
176#389—21no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#388—21no—
188#388—22no—
189#388—22no—
190#388—21no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#385—22no—
197#385—22no—
198#384—21no—
199#384—22no—
200#384—22no—
201#384—21no—
202#384—21no—
203#384C2312yes—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2314yes—
208#384C2312yes—
209#382C2312yes—
210#382C2311yescharacteristic
211#380—21no—
212#380—21no—
213#380—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C43C2311yescharacteristic
226C43C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8—48no—
240C22×Q8—48no—
241C22×Q8—48no—
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2411yescharacteristic
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2411yescharacteristic
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—22no—
252C2×SD16—42no—
253C2×SD16—42no—
254C2×SD16C2×Q818yescomplemented
255C2×D8—42no—
256C2×D8—42no—
257C2×D8C2×Q818yescomplemented
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269#127—22no—
270#127C2×Q814yes—
271#125—44no—
272#124—22no—
273#124C2×Q814yescomplemented
274C22⋊Q8—48no—
275C22⋊Q8—416no—
276C22⋊Q8—48no—
277C22⋊Q8—416no—
278C22⋊Q8—416no—
279C22⋊Q8—44no—
280C22⋊Q8—48no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—44no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—48no—
295C4×D4—44no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
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⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—42no—
361#107—44no—
362#106—42no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—42no—
367#103—44no—
368#102—44no—
369#102—44no—
370#102—42no—
371#102—42no—
372#102—42no—
373#102—44no—
374#97—22no—
375#97—24no—
376#97—42no—
377#97C2×Q814yescomplemented
378C4×C8—24no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—21no—
395C24—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×Q8—48no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—832no—
408C2×Q8AGL(1,ℤ/8)14yes—
409C2×Q8C22×Q812yes—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C22×Q812yes—
425C2×D4C22×Q814yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456SD16—48no—
457D8—48no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8C22×Q814yes—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4AGL(1,ℤ/8)14yes—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—82no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—82no—
517C42—24no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—22no—
523C42—24no—
524C42—41no—
525C42—41no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—82no—
530C42C22×Q812yes—
531C42H(5,𝔽2)14yes—
532C422−1+412yes—
533C23—22no—
534C23—24no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23C23×D411yescharacteristic
548Q8—28no—
549Q8—24no—
550Q8—42no—
551Q8—816no—
552Q8—88no—
553D4—24no—
554D4—24no—
555D4—24no—
556D4—48no—
557D4—48no—
558D4—42no—
559D4—42no—
560D4—42no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—816no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4—164no—
608C2×C4#53512yes—
609C2×C4#53512yes—
610C2×C4D4×Q812yes—
611C2×C4D4×Q812yes—
612C2×C4C2×AGL(1,ℤ/8)14yes—
613C2×C4C2×AGL(1,ℤ/8)14yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4C23×Q811yescharacteristic
617C2×C4C2×H(5,𝔽2)12yes—
618C2×C4C2×2−1+411yescharacteristic
619C2×C4#58512yes—
620C8—24no—
621C8—42no—
622C8—44no—
623C22—44no—
624C22—44no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—82no—
630C22—81no—
631C22—81no—
632C22—82no—
633C22—81no—
634C22—82no—
635C22—81no—
636C22—82no—
637C22C2×#53511yescharacteristic
638C22C2×D4×Q811yescharacteristic
639C22#350011yescharacteristic
640C22C22×AGL(1,ℤ/8)11yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22#358611yescharacteristic
643C22#358611yescharacteristic
644C4—24no—
645C4—24no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—41no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—81no—
660C4#321714yes—
661C4#350112yes—
662C4#354712yes—
663C2—84no—
664C2—82no—
665C2#3961811yescharacteristic
666C2C2×#321711yescharacteristic
667C2#6083111yescharacteristic
668C2Q8×AGL(1,ℤ/8)11yescharacteristic
669C2Q8×AGL(1,ℤ/8)11yescharacteristic
670C2#6258311yescharacteristic
671C2#6259511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#496C22
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. Q8×#127C2semidirect
2. #36555C2semidirect
3. #33294C2semidirect
4. #31540C2semidirect
5. #31538C2semidirect
6. #30309C2semidirect
7. #29195C2semidirect
8. #29149C2semidirect
9. #29148C2semidirect
10. #28520C2semidirect
11. #24813C2semidirect
12. #24811C2semidirect
13. #24330C2semidirect
14. #24326C2semidirect
17. #21345C2semidirect
132. #496Q8semidirect
254. C2×SD16C2×Q8semidirect
257. C2×D8C2×Q8semidirect
273. #124C2×Q8semidirect
377. #97C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×#1272
#365552
#332942
#315402
#315382
#303092
#291952
#291492
#291482
#285202
#248132
#248112
#243302
#243262
#222722
C4×#4962
#213452