Finite Group Databasebeta

← #6806104

Subgroups — #6806104

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36591C211yesmaximal, characteristic, complemented
2Q8×#127C211yesmaximal, characteristic, complemented
3#33238C211yesmaximal, characteristic, complemented
4#31656C214yesmaximal, complemented
5#31654C214yesmaximal, complemented
6#30216C212yesmaximal, complemented
7#29225C211yesmaximal, characteristic, complemented
8#29148C212yesmaximal, complemented
9#28618C212yesmaximal, complemented
10#28450C211yesmaximal, characteristic
11#25933C212yesmaximal, complemented
12#25929C212yesmaximal, complemented
13#25320C212yesmaximal, complemented
14#25318C212yesmaximal, complemented
15#22272C212yesmaximal, complemented
16C4×#490C211yesmaximal, characteristic, complemented
17#21345C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21C2×#496C2212yes—
22C2×#490C2211yescharacteristic
23C2×#487C2212yes—
24C2×#486C2212yes—
25C2×#461C2212yes—
26C2×#459C2212yes—
27#2868C2212yes—
28#2868C2211yescharacteristic
29#2861C2212yes—
30#2858C2211yescharacteristic
31#2595—22no—
32#2568C2211yescharacteristic
33#2548C2211yescharacteristic
34#2542—22no—
35#2542—21no—
36#2542—21no—
37#2542—22no—
38#2542C2214yes—
39#2542C2212yes—
40#2516—22no—
41#2513—21no—
42#2448—21no—
43#2448C2212yes—
44#2447C2212yes—
45#2367—21no—
46#2366C2211yescharacteristic
47#2365C2211yescharacteristic
48#2357—21no—
49C4×#127C2212yes—
50C4×#127C2211yescharacteristic
51C4×#125C2211yescharacteristic
52C4×#124C2212yes—
53C42×Q8C2211yescharacteristic
54#2100C2212yes—
55#2100C2212yes—
56#2072C2214yes—
57#2071C2214yes—
58#2055C2214yes—
59#2054C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#2023C2214yes—
63#2023C2214yes—
64#2012C2214yes—
65#1992C2211yescharacteristic
66#1992C2212yes—
67#1988C2212yes—
68#1988C2212yes—
69#1984C2212yes—
70#1961C2211yescharacteristic
71#1959C2212yes—
72#1959C2212yes—
73#1959C2214yes—
74#1959C2212yes—
75#1939C2212yes—
76#1936C2212yes—
77#1936C2212yes—
78#1917C2214yes—
79#1880C2214yes—
80#1879C2214yes—
81#1879C2214yes—
82#1879C2214yes—
83#1877C2214yes—
84#1877C2214yes—
85#1836C2212yes—
86#1836C2211yescharacteristic
87#1803C2214yes—
88#1802C2214yes—
89#1767C2211yescharacteristic
90#1767C2212yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C23×Q8—22no—
96C22×SD16C2314yes—
97C22×D8C2314yes—
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⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×#127C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#125C2311yescharacteristic
112C2×#124—21no—
113C2×#124C2312yes—
114C2×#124C2312yes—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—24no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—22no—
130C2×C4×D4C2312yes—
131C2×C4×D4C2314yes—
132C2×C4×D4C2312yes—
133#496—28no—
134#490—24no—
135#487—44no—
136#486—44no—
137#461—44no—
138#459—44no—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#97C2311yescharacteristic
152C2×#97C2312yes—
153C2×#97C2311yescharacteristic
154C2×#97C2312yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#399—22no—
158#399—24no—
159#399—24no—
160#399—24no—
161#397—22no—
162#397—22no—
163#397—22no—
164#394—22no—
165#391—21no—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—21no—
173#390—21no—
174#390—22no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390C2314yes—
180#390C2312yes—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#389—21no—
185#389C2311yescharacteristic
186#389C2311yescharacteristic
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—21no—
196#387—21no—
197#387—22no—
198#386—22no—
199#385—22no—
200#384—22no—
201#384—21no—
202#384—21no—
203#384—22no—
204#384C2312yes—
205#384C2311yescharacteristic
206#384C2314yes—
207#384C2312yes—
208#382—21no—
209#382C2311yescharacteristic
210#381—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C43C2311yescharacteristic
225C43C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8—48no—
241C22×Q8—48no—
242C22×Q8—48no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2412yes—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2412yes—
250C22×D4C2412yes—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—22no—
254C2×SD16—28no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×D8—28no—
258C2×D8—44no—
259C2×D8—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272#127—24no—
273#127—22no—
274#127#4114yes—
275#125—22no—
276#125—22no—
277#124—24no—
278#124—24no—
279#124—44no—
280#124—44no—
281C22⋊Q8—48no—
282C22⋊Q8—416no—
283C22⋊Q8—48no—
284C22⋊Q8—416no—
285C22⋊Q8—416no—
286C22⋊Q8—48no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—48no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—44no—
300C4×D4—44no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
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—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#102—24no—
369#102—24no—
370#102—42no—
371#102—42no—
372#102—44no—
373#102—44no—
374#102—44no—
375#97—22no—
376#97—24no—
377#97—22no—
378#97—42no—
379C4×C8—24no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—22no—
396C24—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—44no—
406C2×Q8—48no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—832no—
410C2×Q8AGL(1,ℤ/8)14yes—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4C2×#4114yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460SD16—88no—
461D8—88no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4AGL(1,ℤ/8)14yes—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—24no—
528C42—42no—
529C42—41no—
530C42—41no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—82no—
535C42C2×#4112yes—
536C42H(5,𝔽2)14yes—
537C422−1+412yes—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23C23×D411yescharacteristic
553Q8—28no—
554Q8—44no—
555Q8—44no—
556Q8—816no—
557Q8—88no—
558D4—24no—
559D4—24no—
560D4—48no—
561D4—48no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4—44no—
566D4—84no—
567D4—162no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—816no—
612C2×C4—88no—
613C2×C4—164no—
614C2×C4#53512yes—
615C2×C4#53512yes—
616C2×C4#54812yes—
617C2×C4#54812yes—
618C2×C4C2×AGL(1,ℤ/8)14yes—
619C2×C4C2×AGL(1,ℤ/8)14yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4C2×2−1+411yescharacteristic
625C2×C4#58512yes—
626C8—42no—
627C8—44no—
628C8—42no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C2×#53511yescharacteristic
644C22C2×#54811yescharacteristic
645C22#350011yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22#358611yescharacteristic
649C22#358611yescharacteristic
650C4—24no—
651C4—24no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—82no—
660C4—84no—
661C4—82no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4#321714yes—
667C4#351412yes—
668C4#355412yes—
669C2—84no—
670C2—162no—
671C2#3917111yescharacteristic
672C2C2×#321711yescharacteristic
673C2#6082811yescharacteristic
674C2#6131711yescharacteristic
675C2#6131711yescharacteristic
676C2#6258311yescharacteristic
677C2#6259511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28450C2
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. #36591C2semidirect
2. Q8×#127C2semidirect
3. #33238C2semidirect
4. #31656C2semidirect
5. #31654C2semidirect
6. #30216C2semidirect
7. #29225C2semidirect
8. #29148C2semidirect
9. #28618C2semidirect
11. #25933C2semidirect
12. #25929C2semidirect
13. #25320C2semidirect
14. #25318C2semidirect
15. #22272C2semidirect
16. C4×#490C2semidirect
17. #21345C2semidirect

Maximal subgroups

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

subgroupindex
#365912
Q8×#1272
#332382
#316562
#316542
#302162
#292252
#291482
#286182
#284502
#259332
#259292
#253202
#253182
#222722
C4×#4902
#213452