Finite Group Databasebeta

← #6625124

Subgroups — #6625124

Order 512.

1819 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33109C211yesmaximal, characteristic
4#32906C212yesmaximal
5#28304C212yesmaximal
6#28121C212yesmaximal
7#27068C214yesmaximal, complemented
8#26590C214yesmaximal, complemented
9#26532C212yesmaximal, complemented
10#25722C211yesmaximal, characteristic
11#25370C212yesmaximal, complemented
12#24212C211yesmaximal, characteristic
13#23184C212yesmaximal, complemented
14#23096C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic
16C4×#484C211yesmaximal, characteristic
17C4×#451C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#494C2212yes—
21C2×#484C2211yescharacteristic
22C2×#484C2212yes—
23C2×#462C2212yes—
24C2×#462C2214yes—
25C2×#451C2212yes—
26C2×#451C2212yes—
27C2×#451C2212yes—
28C2×C4×Q16C2214yes—
29#2857C2211yescharacteristic
30#2857C2212yes—
31#2585—22no—
32#2569—22no—
33#2552C2211yescharacteristic
34#2546C2211yescharacteristic
35#2511C2212yes—
36#2509—21no—
37#2446C2212yes—
38#2442—21no—
39#2370—21no—
40#2365—22no—
41#2365—21no—
42#2365—21no—
43#2365C2212yes—
44#2365C2212yes—
45#2365C2214yes—
46#2365C2211yescharacteristic
47#2355C2211yescharacteristic
48#2345—21no—
49C4×C22⋊Q8C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51C2×#391C2212yes—
52C2×#389—21no—
53C2×#387—22no—
54C2×C4×C4⋊C4—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#2095C2214yes—
58#2068C2214yes—
59#2056C2214yes—
60#2043C2212yes—
61#2016C2214yes—
62#2016C2214yes—
63#2016C2212yes—
64#1981C2212yes—
65#1981C2212yes—
66#1969C2214yes—
67#1969C2214yes—
68#1956C2212yes—
69#1947C2212yes—
70#1939C2212yes—
71#1928C2214yes—
72#1928C2212yes—
73#1928C2214yes—
74#1921C2212yes—
75#1916C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1863C2214yes—
83#1852C2211yescharacteristic
84#1824C2211yescharacteristic
85#1803C2214yes—
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1787C2212yes—
89C4×#107C2212yes—
90C8×C4⋊C4C2212yes—
91C4×#103C2212yes—
92C4×#103C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2311yescharacteristic
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—48no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C42—22no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#494—28no—
128#484—24no—
129#484—44no—
130#462—28no—
131#462—48no—
132#451—28no—
133#451—44no—
134#451—44no—
135C4×Q16—48no—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×C4×C8C2312yes—
154C2×C4×C8C2314yes—
155#400—22no—
156#400—22no—
157#400—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#391—22no—
162#391—22no—
163#391C2312yes—
164#391C2314yes—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#391C2314yes—
169#391C2311yescharacteristic
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—24no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—21no—
181#389C2312yes—
182#389C2311yescharacteristic
183#387—22no—
184#387—22no—
185#387—21no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#385—21no—
196#385—22no—
197#384C2312yes—
198#384C2311yescharacteristic
199#382—21no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380—21no—
204#380C2311yescharacteristic
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—24no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C2×#95—22no—
227C2×#95—21no—
228C2×#95—22no—
229C2×#95D412yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—28no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8—44no—
244C22×Q8C2411yescharacteristic
245C22×Q8C2412yes—
246C22×Q8C2414yes—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2411yescharacteristic
255C2×Q16—28no—
256C2×Q16—28no—
257C2×Q16—28no—
258C2×Q16—28no—
259C2×Q16—44no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×Q16—44no—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—28no—
268C4⋊Q8—28no—
269C4⋊Q8—28no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—44no—
273C4⋊Q8#4118yes—
274#125—44no—
275C22⋊Q8—24no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278C22⋊Q8—24no—
279C4×Q8—28no—
280C4×Q8—28no—
281C4×Q8—24no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—48no—
286C4×Q8—48no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
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—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4Q1618yescomplemented
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42—48no—
343C2×C42—44no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—24no—
353#107—24no—
354#107—44no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#103—28no—
358#103—28no—
359#103—24no—
360#103—28no—
361#103—24no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366C22⋊C8—22no—
367C22⋊C8—24no—
368C22⋊C8#4114yes—
369C4×C8—24no—
370C4×C8—44no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24C22×D411yescharacteristic
397C2×Q8—28no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—416no—
412C2×Q8—416no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8C2×#4118yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4C2×Q1614yes—
447C22×C4C2×Q1618yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4C2511yescharacteristic
460Q16—416no—
461Q16—88no—
462Q16—88no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8C2×#4114yes—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—42no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4C2×Q1618yes—
507C4⋊C4C2×Q1614yes—
508C4⋊C4C2×Q1614yes—
509C4⋊C4H(5,𝔽2)14yes—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—24no—
517C42—24no—
518C42—42no—
519C42—42no—
520C42—48no—
521C42—48no—
522C42—44no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42—82no—
527C42—82no—
528C42C2×#4114yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23#55012yes—
534C23C22×Q1612yes—
535C23#57812yes—
536C23C23×D411yescharacteristic
537Q8—28no—
538Q8—28no—
539Q8—416no—
540Q8—44no—
541Q8—44no—
542Q8—44no—
543Q8—42no—
544Q8—42no—
545Q8—44no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—416no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4#52512yes—
602C2×C4#52512yes—
603C2×C4#55012yes—
604C2×C4C22×Q1612yes—
605C2×C4C22×Q1614yes—
606C2×C4C22×Q1614yes—
607C2×C4C22×Q1612yes—
608C2×C4#57512yes—
609C2×C4C22×#4111yescharacteristic
610C2×C4C22×#4112yes—
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C8—24no—
614C8—44no—
615C8—42no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22#339514yes—
623C22C2×#52511yescharacteristic
624C22C2×#55011yescharacteristic
625C22#348411yescharacteristic
626C22C23×Q1611yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57811yescharacteristic
629C22#358611yescharacteristic
630C4—22no—
631C4—24no—
632C4—24no—
633C4—22no—
634C4—22no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—41no—
639C4—48no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4—82no—
645C4—81no—
646C4—82no—
647C4#310414yes—
648C4#348112yes—
649C4#351912yes—
650C2—24no—
651C2#3652911yescharacteristic
652C2C2×#310411yescharacteristic
653C2#6058311yescharacteristic
654C2C2×#339511yescharacteristic
655C2#6140311yescharacteristic
656C2#6156311yescharacteristic
657C2#6159511yescharacteristic
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
7. #27068C2semidirect
8. #26590C2semidirect
9. #26532C2semidirect
11. #25370C2semidirect
13. #23184C2semidirect
14. #23096C2semidirect
316. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×#23652
#331092
#329062
#283042
#281212
#270682
#265902
#265322
#257222
#253702
#242122
#231842
#230962
#226252
C4×#4842
C4×#4512