Finite Group Databasebeta

← #6856700

Subgroups — #6856700

Order 512.

1645 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36595C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic, complemented
3#33161C211yesmaximal, characteristic
4#32088C214yesmaximal, complemented
5#30884C214yesmaximal, complemented
6#29797C212yesmaximal, complemented
7#29256C212yesmaximal, complemented
8#28752C212yesmaximal, complemented
9#28749C211yesmaximal, characteristic
10#28402C212yesmaximal, complemented
11#28283C211yesmaximal, characteristic, complemented
12#27770C212yesmaximal, complemented
13#25652C212yesmaximal, complemented
14#24190C212yesmaximal, complemented
15#22821C211yesmaximal, characteristic, complemented
16#22147C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#559C2212yes—
19C2×#488C2214yes—
20C2×#486C2212yes—
21C2×#475C2212yes—
22C2×#462C2211yescharacteristic
23C2×#460C2212yes—
24C2×#460C2212yes—
25C2×#460C2212yes—
26C2×C4×Q16C2212yes—
27#2867C2211yescharacteristic
28#2858C2212yes—
29#2585—22no—
30#2582C2211yescharacteristic
31#2568C2211yescharacteristic
32#2552C2211yescharacteristic
33#2548—21no—
34#2548—21no—
35#2548C2212yes—
36#2546C2211yescharacteristic
37#2544C2214yes—
38#2511—21no—
39#2509C2212yes—
40#2453C2212yes—
41#2446—21no—
42#2445—22no—
43#2445—22no—
44#2402—22no—
45#2366—21no—
46#2366—21no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#127C2212yes—
51C4×#125C2212yes—
52#2295—21no—
53#2295C2212yes—
54C42×Q8C2211yescharacteristic
55C42×Q8C2211yescharacteristic
56#2092C2214yes—
57#2066C2214yes—
58#2042C2214yes—
59#2042C2214yes—
60#2042C2214yes—
61#2042C2214yes—
62#2041C2214yes—
63#2025C2214yes—
64#2015C2214yes—
65#2015C2214yes—
66#1984C2212yes—
67#1982C2212yes—
68#1982C2212yes—
69#1982C2212yes—
70#1982C2212yes—
71#1982C2212yes—
72#1981C2211yescharacteristic
73#1981C2212yes—
74#1968C2214yes—
75#1958C2214yes—
76#1957C2212yes—
77#1948C2212yes—
78#1947C2211yescharacteristic
79#1934C2214yes—
80#1928C2212yes—
81#1916C2214yes—
82#1878C2212yes—
83#1877C2214yes—
84#1864C2214yes—
85#1864C2214yes—
86#1862C2214yes—
87#1842C2211yescharacteristic
88#1831C2212yes—
89#1803C2212yes—
90#1803C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×SD16C2314yes—
97#559—44no—
98#559—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127—21no—
103C2×#127—21no—
104C2×#127—21no—
105C2×#127—21no—
106C2×#127C2312yes—
107C2×#127C2312yes—
108C2×#127C2312yes—
109C2×#125C2312yes—
110C2×#124—22no—
111C2×#124C2314yes—
112C2×#124C2314yes—
113C2×#123—24no—
114C2×C4×Q8—21no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4C2314yes—
126#488—48no—
127#486—44no—
128#475—44no—
129#462—24no—
130#460—28no—
131#460—44no—
132#460—44no—
133C4×Q16—44no—
134C2×#107C2312yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156#400—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#394—24no—
165#394—24no—
166#394—24no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390—22no—
183#390C2314yes—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#386—24no—
203#386—24no—
204#384—21no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#382C2312yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227#376—22no—
228C43—21no—
229C43C2312yes—
230C43C2311yescharacteristic
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×Q8C2411yescharacteristic
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4C2414yes—
249C23×C4—24no—
250C2×Q16—24no—
251C2×Q16—42no—
252C2×Q16—42no—
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—44no—
264#127—24no—
265#127—24no—
266#127—24no—
267#127—24no—
268#127—42no—
269#127—42no—
270#127—44no—
271#127—44no—
272#127—82no—
273#127—82no—
274#125—24no—
275#125—42no—
276#124—44no—
277#124—44no—
278#124—44no—
279#124—44no—
280#124—48no—
281#123—416no—
282C4×Q8—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—42no—
292C4×Q8—41no—
293C4×Q8—44no—
294C4×Q8#4114yes—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—48no—
298C4×D4—84no—
299C4×D4—84no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—42no—
359#106—44no—
360C4⋊C8—24no—
361C4⋊C8—22no—
362C4⋊C8—24no—
363C4⋊C8—22no—
364C4⋊C8—42no—
365#103—24no—
366#103—24no—
367#103—42no—
368#103—42no—
369#103—44no—
370#103—44no—
371#102—28no—
372#102—28no—
373#102—44no—
374#102—44no—
375#102—44no—
376#102—44no—
377#102—44no—
378C4×C8—24no—
379C4×C8—44no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8C2×#4112yes—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—48no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2511yescharacteristic
465Q16—48no—
466SD16—88no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4C2×#4112yes—
511C4⋊C4H(5,𝔽2)12yes—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—22no—
530C42—42no—
531C42—42no—
532C42—41no—
533C42—42no—
534C42—44no—
535C42—42no—
536C42—44no—
537C42—42no—
538C42—42no—
539C42—81no—
540C42#13514yes—
541C42C2×#4112yes—
542C42H(5,𝔽2)12yes—
543C42H(5,𝔽2)12yes—
544C422−1+412yes—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23C23×D411yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—44no—
556Q8—41no—
557Q8—41no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—84no—
563D4—44no—
564D4—44no—
565D4—44no—
566D4—44no—
567D4—88no—
568D4—88no—
569D4—88no—
570D4—164no—
571D4—164no—
572D4—164no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—168no—
619C2×C4#53512yes—
620C2×C4#53512yes—
621C2×C4#54812yes—
622C2×C4#54812yes—
623C2×C4C2×#13512yes—
624C2×C4C2×#13512yes—
625C2×C4C2×#13512yes—
626C2×C4#57512yes—
627C2×C4#57712yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4#58511yescharacteristic
634C8—42no—
635C8—44no—
636C8—42no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22C2×#53511yescharacteristic
645C22C2×#54811yescharacteristic
646C22#348811yescharacteristic
647C22C22×#13511yescharacteristic
648C22C2×#57511yescharacteristic
649C22C2×#57711yescharacteristic
650C22#358511yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—81no—
666C4—82no—
667C4#321214yes—
668C4#336514yes—
669C4#350611yescharacteristic
670C4#350811yescharacteristic
671C4#354811yescharacteristic
672C4#355411yescharacteristic
673C2—164no—
674C2#3972511yescharacteristic
675C2C2×#321211yescharacteristic
676C2#6083311yescharacteristic
677C2C2×#336511yescharacteristic
678C2#6132411yescharacteristic
679C2#6195311yescharacteristic
680C2#6198611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2867C22
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
2. #34951C2semidirect
4. #32088C2semidirect
5. #30884C2semidirect
6. #29797C2semidirect
7. #29256C2semidirect
8. #28752C2semidirect
10. #28402C2semidirect
11. #28283C2semidirect
12. #27770C2semidirect
13. #25652C2semidirect
14. #24190C2semidirect
15. #22821C2semidirect
16. #22147C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#365952
#349512
#331612
#320882
#308842
#297972
#292562
#287522
#287492
#284022
#282832
#277702
#256522
#241902
#228212
#221472
C4×#4622