Finite Group Databasebeta

← #6795516

Subgroups — #6795516

Order 512.

1557 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#34941C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#32284C214yesmaximal, complemented
5#31936C214yesmaximal, complemented
6#29923C212yesmaximal, complemented
7#28689C211yesmaximal, characteristic, complemented
8#28531C212yesmaximal, complemented
9#28284C211yesmaximal, characteristic, complemented
10#27729C212yesmaximal, complemented
11#27681C212yesmaximal, complemented
12#25706C212yesmaximal
13#24652C212yesmaximal, complemented
14#24122C212yesmaximal
15#22795C211yesmaximal, characteristic
16#21982C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#500C2212yes—
20C2×#477C2212yes—
21C2×#477C2212yes—
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2574—22no—
28#2567C2212yes—
29#2567C2211yescharacteristic
30#2563C2211yescharacteristic
31#2552—21no—
32#2552—21no—
33#2552C2211yescharacteristic
34#2538C2211yescharacteristic
35#2508—21no—
36#2502C2214yes—
37#2497C2212yes—
38#2454—21no—
39#2440—22no—
40#2440—22no—
41#2423C2212yes—
42#2396—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125—21no—
48C4×C22⋊Q8C2212yes—
49#2293—21no—
50#2285C2212yes—
51C2×#384C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2100C2214yes—
55#2095C2214yes—
56#2095C2214yes—
57#2083C2214yes—
58#2072C2214yes—
59#2068C2214yes—
60#2036C2214yes—
61#2036C2214yes—
62#2036C2214yes—
63#2001C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1981C2212yes—
67#1981C2211yescharacteristic
68#1969C2214yes—
69#1954C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2211yescharacteristic
76#1933C2214yes—
77#1928C2212yes—
78#1923C2214yes—
79#1878C2212yes—
80#1872C2214yes—
81#1852C2214yes—
82#1852C2214yes—
83#1852C2214yes—
84#1845C2214yes—
85#1842C2211yescharacteristic
86#1834C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96#554—44no—
97#554—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—21no—
104C2×#125—22no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2314yes—
108C2×C4×Q8—21no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2314yes—
121C22×C42C2312yes—
122#500—44no—
123#477—44no—
124#477—44no—
125#462—24no—
126C4×Q16—44no—
127#431—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#399—22no—
151#398—22no—
152#398—22no—
153#398—24no—
154#398—24no—
155#396—22no—
156#396—22no—
157#395—22no—
158#395—24no—
159#393—22no—
160#391—21no—
161#391C2312yes—
162#391C2314yes—
163#391C2312yes—
164#390—21no—
165#390—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384—22no—
190#384—22no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2314yes—
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#382—21no—
199#382—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380C2312yes—
204#380C2312yes—
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C43—21no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C23×C4—24no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C2×Q16—24no—
246C2×Q16—24no—
247C2×Q16—42no—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—82no—
258C4⋊Q8—82no—
259#125—48no—
260#125—44no—
261C22⋊Q8—24no—
262C22⋊Q8—416no—
263C22⋊Q8—42no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—41no—
274C4×Q8—82no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—48no—
279C4×D4Q1618yescomplemented
280#117—84no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—42no—
351#107—44no—
352#107—44no—
353#107—44no—
354#107—44no—
355C4⋊C8—24no—
356C4⋊C8—24no—
357C4⋊C8—22no—
358C4⋊C8—42no—
359C4⋊C8#4114yes—
360#103—24no—
361#103—42no—
362#103—44no—
363#103—42no—
364#103—42no—
365#103—44no—
366#103—44no—
367C22⋊C8—28no—
368C22⋊C8—44no—
369C4×C8—28no—
370C4×C8—24no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24C22×D412yes—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q82−1+412yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4C2×Q1614yes—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4C2×Q1618yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445Q16—48no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8C2×#4114yes—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—82no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4C2×Q1614yes—
498C4⋊C4#13714yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4C2×Q1618yes—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—42no—
517C42—44no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42—81no—
524C42C2×Q1614yes—
525C42C2×#4112yes—
526C42H(5,𝔽2)12yes—
527C42H(5,𝔽2)12yes—
528C422−1+412yes—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23C22×Q1614yes—
536C23C23×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—84no—
546Q8—164no—
547D4—28no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—216no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—48no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—168no—
598C2×C4#53512yes—
599C2×C4#53512yes—
600C2×C4#54712yes—
601C2×C4#54712yes—
602C2×C4C22×Q1612yes—
603C2×C4C22×Q1614yes—
604C2×C4C22×Q1612yes—
605C2×C4C22×Q1612yes—
606C2×C4C2×#13714yes—
607C2×C4C2×#13712yes—
608C2×C4#57612yes—
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C2×C4#58511yescharacteristic
615C8—24no—
616C8—44no—
617C8—42no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22C2×#53511yescharacteristic
626C22C2×#54711yescharacteristic
627C22#348911yescharacteristic
628C22C23×Q1611yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57611yescharacteristic
631C22#358511yescharacteristic
632C4—24no—
633C4—22no—
634C4—24no—
635C4—22no—
636C4—22no—
637C4—42no—
638C4—42no—
639C4—41no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4—84no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4#320914yes—
649C4#333814yes—
650C4#351711yescharacteristic
651C4#352011yescharacteristic
652C4#354711yescharacteristic
653C4#355211yescharacteristic
654C2—44no—
655C2#3892011yescharacteristic
656C2C2×#320911yescharacteristic
657C2#6083211yescharacteristic
658C2C2×#333811yescharacteristic
659C2#6125411yescharacteristic
660C2#6210011yescharacteristic
661C2#6214911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33198C2semidirect
4. #32284C2semidirect
5. #31936C2semidirect
6. #29923C2semidirect
7. #28689C2semidirect
8. #28531C2semidirect
9. #28284C2semidirect
10. #27729C2semidirect
11. #27681C2semidirect
13. #24652C2semidirect
16. #21982C2semidirect
17. C4×#462C2semidirect
279. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#361802
#349412
#331982
#322842
#319362
#299232
#286892
#285312
#282842
#277292
#276812
#257062
#246522
#241222
#227952
#219822
C4×#4622