Finite Group Databasebeta

← #6800084

Subgroups — #6800084

Order 512.

1735 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic
4#32100C214yesmaximal, complemented
5#30896C214yesmaximal, complemented
6#29957C212yesmaximal
7#28417C212yesmaximal, complemented
8#27777C212yesmaximal, complemented
9#25861C212yesmaximal
10#25856C211yesmaximal, characteristic
11#25276C212yesmaximal, complemented
12#24732C212yesmaximal, complemented
13#24212C212yesmaximal
14#24212C211yesmaximal, characteristic
15#22629C211yesmaximal, characteristic
16#22050C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C2×#491C2214yes—
22C2×#487C2212yes—
23C2×#475C2212yes—
24C2×#462C2212yes—
25C2×#462C2214yes—
26C2×#451C2212yes—
27C2×#451C2211yescharacteristic
28C2×C4×Q16C2212yes—
29#2857C2211yescharacteristic
30#2851C2212yes—
31#2590—22no—
32#2568C2211yescharacteristic
33#2552C2211yescharacteristic
34#2546—22no—
35#2546—21no—
36#2546—21no—
37#2546C2212yes—
38#2546C2214yes—
39#2546C2212yes—
40#2511—21no—
41#2508C2212yes—
42#2505—22no—
43#2502—21no—
44#2453C2212yes—
45C2×#398—22no—
46#2365C2211yescharacteristic
47#2360—21no—
48#2355—21no—
49Q8×C4⋊C4C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51C4×C22⋊Q8C2211yescharacteristic
52C2×#391C2212yes—
53C2×#389—21no—
54C2×#389—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2097C2214yes—
57#2068C2214yes—
58#2066C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2043C2214yes—
62#2043C2212yes—
63#2043C2214yes—
64#2026C2214yes—
65#2016C2214yes—
66#2016C2212yes—
67#1981C2214yes—
68#1981C2212yes—
69#1971C2214yes—
70#1960C2214yes—
71#1956C2212yes—
72#1939C2212yes—
73#1935C2214yes—
74#1927C2214yes—
75#1917C2214yes—
76#1878C2214yes—
77#1878C2212yes—
78#1872C2211yescharacteristic
79#1865C2214yes—
80#1863C2212yes—
81#1853C2212yes—
82#1852C2211yescharacteristic
83#1803C2212yes—
84#1797C2212yes—
85#1797C2211yescharacteristic
86#1787C2212yes—
87#1787C2212yes—
88#1787C2211yescharacteristic
89#1787C2212yes—
90#1787C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8—48no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125C2312yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—44no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#491—48no—
133#487—44no—
134#475—44no—
135#462—48no—
136#462—44no—
137#451—24no—
138#451—28no—
139C4×Q16—44no—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162#400—22no—
163#400—22no—
164#400—22no—
165#398—22no—
166#398—22no—
167#398—22no—
168#398—24no—
169#398—24no—
170#398—24no—
171#398—24no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—22no—
177#391—22no—
178#391C2312yes—
179#391C2314yes—
180#391C2314yes—
181#391C2311yescharacteristic
182#389—22no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389—24no—
187#389—22no—
188#389—24no—
189#389—22no—
190#389—21no—
191#389—22no—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2311yescharacteristic
196#389C2312yes—
197#387—22no—
198#387—21no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#385—21no—
208#384—21no—
209#384—21no—
210#384C2311yescharacteristic
211#382—22no—
212#382C2312yes—
213#380—21no—
214#380C2311yescharacteristic
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D412yes—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—22no—
229C2×#95—22no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—42no—
248C22×Q8—44no—
249C22×Q8—44no—
250C22×Q8C2414yes—
251C22×Q8C2411yescharacteristic
252C22×Q8C2412yes—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×Q16—24no—
262C2×Q16—24no—
263C2×Q16—28no—
264C2×Q16—42no—
265C2×Q16—44no—
266C2×Q16—44no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—28no—
271C4⋊Q8—28no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—48no—
281C4⋊Q8—48no—
282#125—44no—
283C22⋊Q8—22no—
284C22⋊Q8—44no—
285C22⋊Q8#4114yes—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—42no—
290C4×Q8—42no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—44no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359#107—42no—
360#106—44no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363#103—28no—
364#103—24no—
365#103—24no—
366#103—28no—
367#103—42no—
368#103—42no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376C22⋊C8—22no—
377C22⋊C8—24no—
378C22⋊C8—22no—
379#97—44no—
380C4×C8—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—21no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24C22×D411yescharacteristic
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—88no—
431C2×Q8—88no—
432C2×Q8C2×#4112yes—
433C2×Q8H(5,𝔽2)14yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4#13714yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C42−1+412yes—
477C22×C4C2511yescharacteristic
478Q16—48no—
479Q16—88no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—42no—
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—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4#13714yes—
526C4⋊C4C2×#4112yes—
527C22⋊C4—24no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—44no—
539C42—42no—
540C42—42no—
541C42—44no—
542C42—82no—
543C42H(5,𝔽2)14yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23#54812yes—
549C23C2×#13712yes—
550C23#57812yes—
551C23C23×D411yescharacteristic
552Q8—24no—
553Q8—24no—
554Q8—24no—
555Q8—48no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—44no—
560Q8—44no—
561Q8—44no—
562Q8—44no—
563Q8—88no—
564Q8—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4#53512yes—
622C2×C4#53512yes—
623C2×C4#54812yes—
624C2×C4C2×#13714yes—
625C2×C4C2×#13712yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)12yes—
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4C2×2−1+411yescharacteristic
632C8—42no—
633C8—44no—
634C8—42no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22#337414yes—
642C22C2×#53511yescharacteristic
643C22C2×#54811yescharacteristic
644C22#348411yescharacteristic
645C22C22×#13711yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57811yescharacteristic
648C22#358611yescharacteristic
649C4—22no—
650C4—22no—
651C4—24no—
652C4—22no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—42no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—81no—
665C4—84no—
666C4#321814yes—
667C4#351912yes—
668C4#353612yes—
669C2—24no—
670C2#3912011yescharacteristic
671C2C2×#321811yescharacteristic
672C2#6081411yescharacteristic
673C2C2×#337411yescharacteristic
674C2#6132211yescharacteristic
675C2#6158411yescharacteristic
676C2#6160211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
4. #32100C2semidirect
5. #30896C2semidirect
7. #28417C2semidirect
8. #27777C2semidirect
11. #25276C2semidirect
12. #24732C2semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×#25462
#329542
#321002
#308962
#299572
#284172
#277772
#258612
#258562
#252762
#247322
#242122
#226292
#220502
C4×#4512