Finite Group Databasebeta

← #6491862

Subgroups — #6491862

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#33200C212yesmaximal, complemented
4#33073C212yesmaximal, complemented
5#29997C212yesmaximal, complemented
6#29988C212yesmaximal, complemented
7#26109C212yesmaximal, complemented
8#26089C212yesmaximal, complemented
9#26043C212yesmaximal, complemented
10#26023C212yesmaximal, complemented
11#26002C214yesmaximal
12#25965C214yesmaximal
13#21981C211yesmaximal, characteristic
14#21972C211yesmaximal, characteristic
15#21961C211yesmaximal, characteristic
16#20734C211yesmaximal, characteristic
17C2×#1803C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#479C2214yes—
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103C2212yes—
23#2859C2212yes—
24#2859C2212yes—
25#2647—22no—
26#2571—21no—
27#2571C2212yes—
28#2552—21no—
29#2552C2212yes—
30#2442—22no—
31#2442C2214yes—
32#2355—21no—
33#2355—21no—
34#2355C2211yescharacteristic
35#2355C2211yescharacteristic
36Q8×C4⋊C4C2211yescharacteristic
37Q8×C4⋊C4C2211yescharacteristic
38#2345—22no—
39#2345C2212yes—
40Q8×C22⋊C4C2212yes—
41#2317—21no—
42#2316—21no—
43#2295—22no—
44#2295C2212yes—
45#2285—21no—
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C42×Q8C2212yes—
50C4×#117—21no—
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2097C2214yes—
54#2088C2214yes—
55#2086C2214yes—
56#2068C2214yes—
57#2066C2214yes—
58#2059C2214yes—
59#2026C2214yes—
60#1989C2214yes—
61#1949C2214yes—
62#1948C2212yes—
63#1948C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1939C2212yes—
69#1939C2212yes—
70#1920C2214yes—
71#1918C2214yes—
72#1865C2214yes—
73#1863C2214yes—
74#1853C2214yes—
75#1850C2214yes—
76#1844C2212yes—
77#1842C2212yes—
78#1834C2212yes—
79#1831C2212yes—
80#1824C2212yes—
81#1808C2214yes—
82#1805C2214yes—
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1803C2211yescharacteristic
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2211yescharacteristic
91#1787C2212yes—
92C4×#103C2212yes—
93C4×#103C2212yes—
94C2×#340C2212yes—
95C23×Q8C2311yescharacteristic
96C23×C8C2312yes—
97C2×#125—22no—
98C2×#125C2314yes—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—24no—
102C2×C4×Q8Q818yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×#117—21no—
110C2×#117—21no—
111C2×#117D411yescharacteristic
112C2×#117D411yescharacteristic
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120#479—48no—
121#431—24no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8—24no—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103—44no—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C4×C8C2312yes—
142#400—24no—
143#400—24no—
144#398—24no—
145#398—24no—
146#391—22no—
147#391—22no—
148#391C2312yes—
149#391C2312yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2314yes—
162#389C2314yes—
163#389C2312yes—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#385—24no—
171#384—22no—
172#384—21no—
173#384—21no—
174#384—21no—
175#384—21no—
176#384C2312yes—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#382—22no—
183#382—21no—
184#382—21no—
185#382—22no—
186#382—21no—
187#382—21no—
188#382C2314yes—
189#380—21no—
190#380—21no—
191#380—24no—
192#380—22no—
193#380—22no—
194#380C2312yes—
195#380C2312yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219#376D412yes—
220C43—22no—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—41no—
239C22×Q8—41no—
240C22×Q8C2×Q814yes—
241C22×Q8C2411yescharacteristic
242C22×Q8C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C22×C8—24no—
250C22×C8—42no—
251C22×C8—42no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255#125—44no—
256#125—44no—
257#125—84no—
258C4×Q8—28no—
259C4×Q8—28no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—42no—
267C4×Q8—42no—
268C4×Q8C2×Q814yes—
269C4×Q8C2×Q814yes—
270C4×Q8C2×Q814yes—
271C4×Q8C2×Q814yes—
272#117—24no—
273#117—24no—
274#117—44no—
275#117—44no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×Q814yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—28no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×Q814yes—
342C2×C42#4114yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—44no—
348#107—44no—
349#106—44no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353C4⋊C8—22no—
354C4⋊C8—22no—
355C4⋊C8—44no—
356C4⋊C8—44no—
357#103—28no—
358#103—24no—
359#103—24no—
360#103—44no—
361#103—42no—
362#103—42no—
363C22⋊C8—24no—
364C4×C8—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—28no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8C22×Q812yes—
402C2×Q8C22×Q812yes—
403C2×Q8H(5,𝔽2)12yes—
404C2×Q82−1+412yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4C22⋊Q814yes—
430C22×C4C22⋊Q814yes—
431C22×C4#13514yes—
432C22×C4#13714yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q812yes—
443C22×C4C2×#4112yes—
444C22×C4C2511yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—84no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4C22×Q812yes—
498C4⋊C4C22×Q812yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—42no—
514C42—42no—
515C42—44no—
516C42—44no—
517C42—44no—
518C42—44no—
519C42—82no—
520C42—82no—
521C42C22⋊Q814yes—
522C42C22⋊Q814yes—
523C42C22×Q812yes—
524C42C22×Q812yes—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23C2×C22⋊Q812yes—
532C23C2×#13512yes—
533C23C2×#13712yes—
534C23C23×D411yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—22no—
540Q8—48no—
541Q8—41no—
542Q8—41no—
543Q8#54414yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—84no—
596C2×C4—164no—
597C2×C4#479116yescomplemented
598C2×C4C2×C22⋊Q812yes—
599C2×C4C2×C22⋊Q812yes—
600C2×C4C2×C22⋊Q812yes—
601C2×C4C2×C22⋊Q812yes—
602C2×C4C2×C22⋊Q812yes—
603C2×C4C2×#13512yes—
604C2×C4C2×#13512yes—
605C2×C4C2×#13712yes—
606C2×C4C2×#13712yes—
607C2×C4C23×Q811yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C2×C4#58511yescharacteristic
612C2×C4#58511yescharacteristic
613C8—42no—
614C8—44no—
615C8—42no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22C2×#47914yes—
623C22C22×C22⋊Q811yescharacteristic
624C22#348811yescharacteristic
625C22#348911yescharacteristic
626C22C22×#13511yescharacteristic
627C22C22×#13711yescharacteristic
628C22#358511yescharacteristic
629C22#358611yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—82no—
643C4—84no—
644C4—82no—
645C4—82no—
646C4C2×#47914yes—
647C4C2×#47914yes—
648C4C2×#54411yescharacteristic
649C4#345811yescharacteristic
650C4#349411yescharacteristic
651C4#349511yescharacteristic
652C2—24no—
653C2#3525311yescharacteristic
654C2C22×#47911yescharacteristic
655C2#6049311yescharacteristic
656C2#6195411yescharacteristic
657C2#6196511yescharacteristic
658C2#6210211yescharacteristic
659C2#6211311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33200C2semidirect
4. #33073C2semidirect
5. #29997C2semidirect
6. #29988C2semidirect
7. #26109C2semidirect
8. #26089C2semidirect
9. #26043C2semidirect
10. #26023C2semidirect
597. C2×C4#479semidirect

Maximal subgroups

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

subgroupindex
#352862
Q8×#1172
#332002
#330732
#299972
#299882
#261092
#260892
#260432
#260232
#260022
#259652
#219812
#219722
#219612
#207342
C2×#18032