Finite Group Databasebeta

← #6626294

Subgroups — #6626294

Order 512.

1763 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#32957C211yesmaximal, characteristic
4#32954C212yesmaximal
5#29301C212yesmaximal
6#29243C212yesmaximal
7#27948C211yesmaximal, characteristic
8#27680C211yesmaximal, characteristic
9#27313C214yesmaximal, complemented
10#26389C214yesmaximal, complemented
11#26042C212yesmaximal, complemented
12#24817C212yesmaximal, complemented
13#22971C212yesmaximal, complemented
14#22674C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16#21455C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#477C2212yes—
21C2×#475C2214yes—
22C2×#451C2211yescharacteristic
23C2×#441C2214yes—
24C2×C4×Q16C2212yes—
25#2857C2211yescharacteristic
26#2857C2212yes—
27#2586—22no—
28#2567C2211yescharacteristic
29#2551C2211yescharacteristic
30#2549—22no—
31#2507—21no—
32#2507C2212yes—
33#2451C2212yes—
34#2440—21no—
35#2369—21no—
36#2369C2211yescharacteristic
37Q8×C4⋊C4—22no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4C2214yes—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2211yescharacteristic
44#2345—21no—
45C4×C22⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47C2×#389—22no—
48C2×#389C2212yes—
49C2×#384—21no—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2212yes—
54#2096C2212yes—
55#2088C2214yes—
56#2084C2214yes—
57#2083C2214yes—
58#2068C2214yes—
59#2058C2214yes—
60#1990C2212yes—
61#1948C2212yes—
62#1943C2214yes—
63#1943C2214yes—
64#1939C2212yes—
65#1918C2214yes—
66#1917C2214yes—
67#1878C2212yes—
68#1878C2214yes—
69#1872C2212yes—
70#1872C2211yescharacteristic
71#1865C2214yes—
72#1865C2214yes—
73#1865C2214yes—
74#1865C2214yes—
75#1863C2214yes—
76#1863C2212yes—
77#1853C2212yes—
78#1853C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1852C2211yescharacteristic
82#1844C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2214yes—
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1787C2211yescharacteristic
89#1787C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2312yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C4×Q8—28no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—48no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C42—21no—
125C22×C42—22no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#477—28no—
129#475—48no—
130#451Q818yescomplemented
131#441—48no—
132C4×Q16—28no—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×C22⋊C8C2311yescharacteristic
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#398—22no—
153#398—22no—
154#398—22no—
155#395—22no—
156#395—22no—
157#395—22no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2311yescharacteristic
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—24no—
174#389—24no—
175#389—24no—
176#389—24no—
177#389—22no—
178#389C2314yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2314yes—
184#389C2311yescharacteristic
185#387—22no—
186#387—22no—
187#387—21no—
188#387—22no—
189#385—22no—
190#385—21no—
191#384—22no—
192#384—24no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#380—21no—
202#380—22no—
203#380C2312yes—
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⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2×Q814yes—
242C22×Q8C2411yescharacteristic
243C22×Q8C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16C2×Q818yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C22⋊Q8—22no—
262C22⋊Q8—84no—
263C22⋊Q8—82no—
264C22⋊Q8C2×Q814yes—
265C4×Q8—28no—
266C4×Q8—28no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—28no—
271C4×Q8—28no—
272C4×Q8—24no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—48no—
277C4×Q8—48no—
278C4×Q8#4118yes—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C42—28no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42—48no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—24no—
351#107—28no—
352#106—44no—
353#106—44no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#103—24no—
358#103—28no—
359#103—28no—
360#103—24no—
361#103—24no—
362#103—44no—
363#103C2×Q818yes—
364C22⋊C8—24no—
365C22⋊C8—22no—
366C22⋊C8C2×Q814yes—
367#97—44no—
368C4×C8—24no—
369#95—24no—
370#95—24no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—216no—
392C2×Q8—216no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—81no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8—81no—
406C2×Q8C22×Q812yes—
407C2×Q8C22×Q814yes—
408C2×Q8C2×#4114yes—
409C2×Q8H(5,𝔽2)14yes—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4#13714yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×Q812yes—
452C22×C42−1+412yes—
453C22×C4C2511yescharacteristic
454Q16—216no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8C22×Q814yes—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4#13714yes—
504C4⋊C4C22×Q812yes—
505C4⋊C4C2×#4114yes—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C42—28no—
510C42—28no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—24no—
515C42—24no—
516C42—42no—
517C42—42no—
518C42—48no—
519C42—48no—
520C42—42no—
521C42—42no—
522C42—82no—
523C42—82no—
524C42C2×#4114yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23D4×Q812yes—
530C23C2×#13712yes—
531C23#57712yes—
532C23C23×D411yescharacteristic
533Q8—216no—
534Q8—24no—
535Q8—24no—
536Q8—24no—
537Q8—24no—
538Q8—42no—
539Q8—164no—
540Q8—162no—
541Q8#55418yes—
542C2×C4—216no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—24no—
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—21no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—88no—
598C2×C4#52512yes—
599C2×C4#52512yes—
600C2×C4D4×Q812yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13714yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4C23×Q811yescharacteristic
606C2×C4C22×#4112yes—
607C2×C4C2×H(5,𝔽2)12yes—
608C2×C4C2×2−1+411yescharacteristic
609C8—24no—
610C8—42no—
611C8—44no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22#338714yes—
619C22C2×#52511yescharacteristic
620C22C2×D4×Q811yescharacteristic
621C22#348411yescharacteristic
622C22C22×#13711yescharacteristic
623C22C22×#13711yescharacteristic
624C22C2×#57711yescharacteristic
625C22#358511yescharacteristic
626C4—22no—
627C4—24no—
628C4—22no—
629C4—28no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—84no—
639C4—82no—
640C4—81no—
641C4—81no—
642C4—82no—
643C4#311314yes—
644C4C2×#55412yes—
645C4#349212yes—
646C2—24no—
647C2#3654611yescharacteristic
648C2C2×#311311yescharacteristic
649C2#6058811yescharacteristic
650C2C2×#338711yescharacteristic
651C2Q8×#13711yescharacteristic
652C2#6158811yescharacteristic
653C2#6159311yescharacteristic
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
9. #27313C2semidirect
10. #26389C2semidirect
11. #26042C2semidirect
12. #24817C2semidirect
13. #22971C2semidirect
14. #22674C2semidirect
130. #451Q8semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×Q8×C4⋊C42
#329572
#329542
#293012
#292432
#279482
#276802
#273132
#263892
#260422
#248172
#229712
#226742
#226292
#214552
C4×#4512