Finite Group Databasebeta

← #6856865

Subgroups — #6856865

Order 512.

1655 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
1#36595C211yesmaximal, characteristic
2#34950C211yesmaximal, characteristic, complemented
3#33161C211yesmaximal, characteristic
4#31966C214yesmaximal, complemented
5#31156C214yesmaximal, complemented
6#29826C212yesmaximal, complemented
7#29346C212yesmaximal, complemented
8#29287C211yesmaximal, characteristic, complemented
9#28749C211yesmaximal, characteristic
10#28640C212yesmaximal, complemented
11#28280C212yesmaximal, complemented
12#26464C212yesmaximal, complemented
13#25322C212yesmaximal, complemented
14#25295C212yesmaximal, complemented
15Q8×C4⋊C8C211yesmaximal, characteristic, complemented
16#22147C212yesmaximal, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#486C2212yes—
20C2×#475C2211yescharacteristic
21C2×#476C2214yes—
22C2×#460C2212yes—
23C2×#460C2212yes—
24C2×#460C2212yes—
25C2×C8×Q8C2212yes—
26#2867C2211yescharacteristic
27#2858C2212yes—
28#2586—22no—
29#2582C2211yescharacteristic
30#2567C2211yescharacteristic
31#2567C2211yescharacteristic
32#2551C2211yescharacteristic
33#2548—21no—
34#2548—21no—
35#2548C2212yes—
36#2544—22no—
37#2507—21no—
38#2507C2212yes—
39#2451—21no—
40#2451C2212yes—
41#2445—22no—
42#2445C2214yes—
43#2399—22no—
44#2366—21no—
45#2366C2212yes—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2212yes—
50C4×#127C2212yes—
51#2295—21no—
52#2295C2212yes—
53C42×Q8C2211yescharacteristic
54C42×Q8C2211yescharacteristic
55#2088C2214yes—
56#2071C2214yes—
57#2065C2214yes—
58#2065C2214yes—
59#2065C2214yes—
60#2065C2214yes—
61#2055C2214yes—
62#2055C2214yes—
63#2041C2214yes—
64#2023C2214yes—
65#1988C2212yes—
66#1988C2211yescharacteristic
67#1987C2212yes—
68#1982C2212yes—
69#1982C2212yes—
70#1982C2212yes—
71#1982C2212yes—
72#1982C2212yes—
73#1968C2214yes—
74#1957C2212yes—
75#1948C2212yes—
76#1948C2212yes—
77#1948C2211yescharacteristic
78#1939C2212yes—
79#1939C2214yes—
80#1934C2214yes—
81#1934C2214yes—
82#1934C2214yes—
83#1919C2212yes—
84#1877C2214yes—
85#1865C2212yes—
86#1864C2214yes—
87#1842C2211yescharacteristic
88#1842C2212yes—
89#1803C2211yescharacteristic
90#1802C2214yes—
91C4×#106C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C22×SD16C2314yes—
96#559—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127—21no—
104C2×#127—21no—
105C2×#127C2312yes—
106C2×#127C2312yes—
107C2×#127C2312yes—
108C2×#124—22no—
109C2×#124—22no—
110C2×#124—22no—
111C2×#123—24no—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4C2314yes—
125#486—44no—
126#476—48no—
127#475—24no—
128#460—28no—
129#460—44no—
130#460—44no—
131C8×Q8—44no—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#398—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#395—22no—
158#395—22no—
159#395—22no—
160#395—22no—
161#394—24no—
162#394—24no—
163#394—24no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391—21no—
169#391—22no—
170#391C2312yes—
171#391C2312yes—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#390C2314yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#386—24no—
200#386—24no—
201#384—21no—
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2311yescharacteristic
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223#376—22no—
224C43—21no—
225C43C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2411yescharacteristic
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4C2414yes—
248C23×C4—24no—
249C2×SD16—28no—
250C2×SD16—44no—
251C2×SD16—44no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262#127—24no—
263#127—24no—
264#127—24no—
265#127—24no—
266#127—42no—
267#127—42no—
268#127—44no—
269#127—44no—
270#127—82no—
271#127—82no—
272#124—48no—
273#124—48no—
274#124—84no—
275#123—416no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—42no—
283C4×Q8—41no—
284C4×Q8—44no—
285C4×Q8—84no—
286C4×Q8—82no—
287C4×Q8SD1618yescomplemented
288C4×Q8#4114yes—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—48no—
293C4×D4—84no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#106—24no—
354#106—24no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—22no—
358C4⋊C8—24no—
359C4⋊C8—22no—
360C4⋊C8—44no—
361C4⋊C8—42no—
362C4⋊C8—44no—
363#103—24no—
364#103—42no—
365#102—28no—
366#102—28no—
367#102—44no—
368#102—44no—
369#102—44no—
370#102—44no—
371#102—44no—
372C4×C8—24no—
373C4×C8—44no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—81no—
407C2×Q8—82no—
408C2×Q8—81no—
409C2×Q8C2×SD1614yes—
410C2×Q8C2×#4112yes—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—82no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—48no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2511yescharacteristic
457SD16—88no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4C2×SD1614yes—
502C4⋊C4C2×SD1618yes—
503C4⋊C4C2×#4112yes—
504C4⋊C4H(5,𝔽2)12yes—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C42—24no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—21no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—42no—
523C42—42no—
524C42—41no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42—42no—
529C42—42no—
530C42—44no—
531C42—81no—
532C42C2×SD1618yes—
533C42C2×SD1614yes—
534C42C2×#4112yes—
535C42H(5,𝔽2)12yes—
536C422−1+412yes—
537C422−1+412yes—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23C23×D411yescharacteristic
546Q8—28no—
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—84no—
552Q8—88no—
553Q8—84no—
554D4—44no—
555D4—44no—
556D4—44no—
557D4—44no—
558D4—88no—
559D4—88no—
560D4—88no—
561D4—164no—
562D4—164no—
563D4—164no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—168no—
608C2×C4#53712yes—
609C2×C4#53712yes—
610C2×C4#54812yes—
611C2×C4#54812yes—
612C2×C4C22×SD1614yes—
613C2×C4C22×SD1614yes—
614C2×C4C22×SD1612yes—
615C2×C4C22×SD1612yes—
616C2×C4C22×SD1612yes—
617C2×C4#57512yes—
618C2×C4#57612yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C2×C4#58511yescharacteristic
625C8—42no—
626C8—44no—
627C8—42no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22C2×#53711yescharacteristic
636C22C2×#54811yescharacteristic
637C22#348811yescharacteristic
638C22C23×SD1611yescharacteristic
639C22C2×#57511yescharacteristic
640C22C2×#57611yescharacteristic
641C22#358511yescharacteristic
642C4—22no—
643C4—24no—
644C4—22no—
645C4—24no—
646C4—22no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—82no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4#324614yes—
659C4#336214yes—
660C4#350711yescharacteristic
661C4#350811yescharacteristic
662C4#355011yescharacteristic
663C4#355411yescharacteristic
664C2—164no—
665C2#3972611yescharacteristic
666C2C2×#324611yescharacteristic
667C2#6092211yescharacteristic
668C2C2×#336211yescharacteristic
669C2#6132411yescharacteristic
670C2#6195111yescharacteristic
671C2#6199511yescharacteristic
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. #34950C2semidirect
4. #31966C2semidirect
5. #31156C2semidirect
6. #29826C2semidirect
7. #29346C2semidirect
8. #29287C2semidirect
10. #28640C2semidirect
11. #28280C2semidirect
12. #26464C2semidirect
13. #25322C2semidirect
14. #25295C2semidirect
15. Q8×C4⋊C8C2semidirect
16. #22147C2semidirect
17. C4×#475C2semidirect
287. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365952
#349502
#331612
#319662
#311562
#298262
#293462
#292872
#287492
#286402
#282802
#264642
#253222
#252952
Q8×C4⋊C82
#221472
C4×#4752