Finite Group Databasebeta

← #6802694

Subgroups — #6802694

Order 512.

1653 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36577C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4#32667C214yesmaximal, complemented
5#32625C214yesmaximal, complemented
6#29756C212yesmaximal
7#28313C212yesmaximal
8#28274C212yesmaximal, complemented
9#28124C212yesmaximal
10#27905C211yesmaximal, characteristic
11#27647C211yesmaximal, characteristic
12#25302C212yesmaximal, complemented
13#25060C212yesmaximal, complemented
14#24191C212yesmaximal, complemented
15#22510C211yesmaximal, characteristic
16#22084C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#483C2212yes—
22C2×#483C2212yes—
23C2×#483C2212yes—
24C2×#462C2212yes—
25C2×#451C2211yescharacteristic
26C2×#451C2212yes—
27C2×#446C2212yes—
28C2×C4×Q16C2212yes—
29#2857C2211yescharacteristic
30#2857C2212yes—
31#2582—21no—
32#2582—21no—
33#2582—22no—
34#2582C2214yes—
35#2582C2211yescharacteristic
36#2582C2212yes—
37#2582C2212yes—
38#2582C2212yes—
39#2568C2212yes—
40#2568C2211yescharacteristic
41#2453—21no—
42#2453—22no—
43#2442—22no—
44#2442—21no—
45C2×#398—22no—
46#2351—21no—
47Q8×C4⋊C4C2211yescharacteristic
48Q8×C22⋊C4—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2211yescharacteristic
51C2×#391—21no—
52C2×#391C2212yes—
53C2×#389—21no—
54C2×#389C2212yes—
55C2×#384C2211yescharacteristic
56#2096C2214yes—
57#2096C2214yes—
58#2084C2214yes—
59#2084C2212yes—
60#2072C2212yes—
61#2066C2214yes—
62#2066C2214yes—
63#2066C2214yes—
64#2056C2214yes—
65#2043C2214yes—
66#2016C2214yes—
67#1985C2214yes—
68#1984C2212yes—
69#1971C2214yes—
70#1971C2214yes—
71#1956C2212yes—
72#1948C2212yes—
73#1944C2214yes—
74#1935C2214yes—
75#1927C2214yes—
76#1920C2214yes—
77#1878C2212yes—
78#1878C2214yes—
79#1878C2214yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1863C2214yes—
83#1863C2212yes—
84#1852C2211yescharacteristic
85#1844C2212yes—
86#1823C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99#557—44no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125—21no—
103C2×#125—22no—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—44no—
111C2×C4×Q8—44no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D412yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D412yes—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#483—28no—
134#483—44no—
135#483—44no—
136#462—44no—
137#451—24no—
138#451—44no—
139#446—44no—
140C4×Q16—44no—
141C2×#107C2312yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161#400—22no—
162#400—22no—
163#400—22no—
164#398—24no—
165#398—24no—
166#398—24no—
167#398—24no—
168#398—22no—
169#398—22no—
170#398—22no—
171#391—21no—
172#391—22no—
173#391—22no—
174#391—24no—
175#391—22no—
176#391—21no—
177#391—22no—
178#391—22no—
179#391—22no—
180#391C2312yes—
181#391C2314yes—
182#391C2311yescharacteristic
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2314yes—
187#389—22no—
188#389—22no—
189#389—24no—
190#389—21no—
191#389—22no—
192#389—21no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—21no—
197#389—22no—
198#389—21no—
199#389—22no—
200#389C2314yes—
201#389C2312yes—
202#389C2311yescharacteristic
203#389C2314yes—
204#387—21no—
205#387—22no—
206#385—21no—
207#385—22no—
208#384—21no—
209#384—21no—
210#384C2311yescharacteristic
211#384C2312yes—
212#384C2312yes—
213#384C2312yes—
214#382—21no—
215#382—22no—
216#380—22no—
217#380—21no—
218#380—21no—
219#380—22no—
220#380C2311yescharacteristic
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—22no—
226C4×C22⋊C4—21no—
227C2×#95—22no—
228C2×#95—22no—
229C2×#95—22no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2312yes—
238#336C2314yes—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2×D412yes—
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2412yes—
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—28no—
263C2×Q16—24no—
264C2×Q16—42no—
265C2×Q16—44no—
266C2×Q16—44no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—44no—
272#125—48no—
273#125—44no—
274C22⋊Q8—22no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—42no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×Q8—42no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4—84no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—44no—
351C2×C42—44no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360#107—42no—
361#106—28no—
362#106—44no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—24no—
367#103—24no—
368#103—28no—
369#103—42no—
370#103—42no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375C22⋊C8—24no—
376C22⋊C8—22no—
377C22⋊C8#4114yes—
378#97—44no—
379C4×C8—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—21no—
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#95—22no—
401C24C22×D411yescharacteristic
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—42no—
424C2×Q8—416no—
425C2×Q8—48no—
426C2×Q8—48no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8#13714yes—
430C2×Q82−1+414yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4#13714yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4C2511yescharacteristic
475Q16—48no—
476Q16—88no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8C2×#4114yes—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—22no—
534C42—24no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—44no—
539C42—44no—
540C42—42no—
541C42—44no—
542C42—42no—
543C42H(5,𝔽2)14yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23#55012yes—
549C23C2×#13712yes—
550C23#57712yes—
551C23C23×D411yescharacteristic
552Q8—28no—
553Q8—24no—
554Q8—24no—
555Q8—48no—
556Q8—48no—
557Q8—44no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—44no—
562Q8—44no—
563Q8—42no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—88no—
620C2×C4#53412yes—
621C2×C4#53412yes—
622C2×C4#55012yes—
623C2×C4C2×#13714yes—
624C2×C4C2×#13712yes—
625C2×C4C2×#13712yes—
626C2×C4C2×#13712yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+412yes—
631C8—24no—
632C8—42no—
633C8—44no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22#340014yes—
641C22C2×#53411yescharacteristic
642C22C2×#55011yescharacteristic
643C22#348611yescharacteristic
644C22C22×#13711yescharacteristic
645C22C22×#13711yescharacteristic
646C22C2×#57711yescharacteristic
647C22#358611yescharacteristic
648C4—22no—
649C4—24no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—81no—
658C4—82no—
659C4—81no—
660C4—84no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4#319514yes—
666C4#352012yes—
667C4#353612yes—
668C2—24no—
669C2#3913311yescharacteristic
670C2C2×#319511yescharacteristic
671C2#6076311yescharacteristic
672C2C2×#340011yescharacteristic
673C2#6140411yescharacteristic
674C2#6182911yescharacteristic
675C2#6183811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #32667C2semidirect
5. #32625C2semidirect
8. #28274C2semidirect
12. #25302C2semidirect
13. #25060C2semidirect
14. #24191C2semidirect

Maximal subgroups

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

subgroupindex
#365772
C2×#25822
#329062
#326672
#326252
#297562
#283132
#282742
#281242
#279052
#276472
#253022
#250602
#241912
#225102
#220842
C4×#4512