Finite Group Databasebeta

← #7567523

Subgroups — #7567523

Order 512.

1789 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#32584C214yesmaximal, complemented
4#29206C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#28799C212yesmaximal, complemented
7#27821C212yesmaximal, complemented
8#26464C212yesmaximal, complemented
9#26107C212yesmaximal, complemented
10#26044C212yesmaximal, complemented
11#24812C212yesmaximal, complemented
12#24338C214yesmaximal, complemented
13#23250C211yesmaximal, characteristic, complemented
14#23001C212yesmaximal
15#22670C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×D4×Q8C2212yes—
19C22×C4⋊Q8—21no—
20C2×#476C2211yescharacteristic
21C2×#476C2214yes—
22C2×#459C2211yescharacteristic
23C2×#442C2214yes—
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2857—21no—
27#2804—22no—
28#2580—22no—
29#2580C2214yes—
30#2571C2212yes—
31#2563—21no—
32#2563C2212yes—
33#2551C2212yes—
34#2551C2212yes—
35#2546C2212yes—
36#2487—22no—
37#2440—22no—
38#2440—22no—
39#2434—22no—
40#2366C2212yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2088C2214yes—
51#2085C2214yes—
52#2065C2214yes—
53#2058C2214yes—
54#2058C2214yes—
55#2023C2214yes—
56#2023C2214yes—
57#1993C2214yes—
58#1990C2212yes—
59#1987C2211yescharacteristic
60#1985C2214yes—
61#1982C2211yescharacteristic
62#1957C2212yes—
63#1948C2212yes—
64#1948C2212yes—
65#1948C2211yescharacteristic
66#1947C2211yescharacteristic
67#1943C2214yes—
68#1939C2212yes—
69#1939C2212yes—
70#1936C2214yes—
71#1925C2212yes—
72#1920C2212yes—
73#1879C2212yes—
74#1879C2214yes—
75#1864C2212yes—
76#1864C2214yes—
77#1864C2214yes—
78#1864C2214yes—
79#1844C2212yes—
80#1834C2212yes—
81#1804C2214yes—
82#1802C2211yescharacteristic
83#1802C2212yes—
84#1802C2212yes—
85#1802C2211yescharacteristic
86C4×#106C2212yes—
87#1770C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2212yes—
92C4×#102C2212yes—
93C23×Q8—22no—
94C22×D8C2312yes—
95D4×Q8—44no—
96D4×Q8—44no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8D411yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2312yes—
107C2×#127C2311yescharacteristic
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2314yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2314yes—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C42—21no—
125#476—24no—
126#476—48no—
127#459Q818yescomplemented
128#442—48no—
129C4×D8—28no—
130C2×#107C2312yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2312yes—
151#400—24no—
152#399—24no—
153#398—24no—
154#395—24no—
155#395—24no—
156#393—24no—
157#393—24no—
158#391—22no—
159#391—21no—
160#391D412yes—
161#390—22no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2314yes—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389D412yes—
171#389D412yes—
172#389C2312yes—
173#389C2314yes—
174#389C2314yes—
175#389C2314yes—
176#386—24no—
177#385—24no—
178#385—24no—
179#385—24no—
180#385—24no—
181#385—24no—
182#384—21no—
183#384—21no—
184#384D411yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—22no—
190#382C2314yes—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2311yescharacteristic
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—21no—
206C43C2311yescharacteristic
207#340C2314yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#336C2312yes—
218#336C2314yes—
219#336C2314yes—
220#336C2312yes—
221C22×Q8—22no—
222C22×Q8—24no—
223C22×Q8—21no—
224C22×Q8—48no—
225C22×Q8—48no—
226C22×Q8—84no—
227C22×Q8C2×D412yes—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4C2412yes—
231C22×D4C2411yescharacteristic
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—24no—
238C2×D8—24no—
239C2×D8—24no—
240C2×D8C2×Q818yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C4⋊Q8—28no—
246C4⋊Q8—24no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—44no—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C4⋊Q8—48no—
253C4⋊Q8—48no—
254C4⋊Q8—48no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257#127—22no—
258#127C2×Q814yescomplemented
259C22⋊Q8—416no—
260C22⋊Q8—48no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×Q8—82no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—28no—
267C4×D4—24no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—24no—
271C4×D4—44no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—82no—
276C4×D4—84no—
277C4×D4C2×Q814yes—
278C4×D4#4114yes—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4—84no—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—44no—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2414yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—24no—
334#106—24no—
335#106—24no—
336#106—44no—
337#106—44no—
338C4⋊C8—22no—
339C4⋊C8—22no—
340C4⋊C8—22no—
341C4⋊C8—44no—
342C4⋊C8C2×Q814yescomplemented
343#102—24no—
344#102—24no—
345#102—24no—
346#102—24no—
347#102—24no—
348#102—44no—
349#102—44no—
350#102C2×Q818yescomplemented
351#97—44no—
352C4×C8—24no—
353C4×C8—24no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95C2×D414yes—
365#95C2×D414yes—
366C24—21no—
367C24—22no—
368C2×Q8—24no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×Q8—44no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—84no—
380C2×Q8—832no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—81no—
394C2×D4—81no—
395C2×D4C22×Q814yes—
396C2×D4C22×Q812yes—
397C2×D4C2×#4112yes—
398C2×D4H(5,𝔽2)12yes—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—42no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433D8—216no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8C22×Q814yes—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—84no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4C22×Q812yes—
479C4⋊C4C2×#4112yes—
480C4⋊C4H(5,𝔽2)12yes—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—82no—
490C22⋊C4—84no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—21no—
496C42—21no—
497C42—41no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—44no—
502C42AGL(1,ℤ/8)14yes—
503C42C22×Q812yes—
504C42C2×#4112yes—
505C422−1+412yes—
506C422−1+412yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C23×D411yescharacteristic
522Q8—48no—
523Q8—88no—
524Q8—84no—
525Q8—816no—
526D4—24no—
527D4—22no—
528D4—24no—
529D4—22no—
530D4—22no—
531D4—48no—
532D4—42no—
533D4—42no—
534D4—84no—
535D4#55414yes—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—48no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—816no—
576C2×C4—88no—
577C2×C4—168no—
578C2×C4#54812yes—
579C2×C4#54812yes—
580C2×C4D4×Q812yes—
581C2×C4D4×Q812yes—
582C2×C4C2×AGL(1,ℤ/8)12yes—
583C2×C4C2×AGL(1,ℤ/8)12yes—
584C2×C4C2×AGL(1,ℤ/8)12yes—
585C2×C4#57712yes—
586C2×C4#57712yes—
587C2×C4C23×Q811yescharacteristic
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×H(5,𝔽2)11yescharacteristic
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4C2×2−1+411yescharacteristic
592C2×C4C2×2−1+411yescharacteristic
593C8—24no—
594C8—42no—
595C8—42no—
596C8—42no—
597C22—24no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22C2×#54811yescharacteristic
612C22C2×D4×Q811yescharacteristic
613C22#348411yescharacteristic
614C22C22×AGL(1,ℤ/8)11yescharacteristic
615C22C2×#57711yescharacteristic
616C22C2×#57711yescharacteristic
617C22#358511yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—44no—
627C4—82no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4#337314yes—
632C4#338414yes—
633C4C2×#55411yescharacteristic
634C4#350311yescharacteristic
635C4#351111yescharacteristic
636C4#354411yescharacteristic
637C2—44no—
638C2—84no—
639C2#5913611yescharacteristic
640C2C2×#337311yescharacteristic
641C2#6134711yescharacteristic
642C2C2×#338411yescharacteristic
643C2#6138211yescharacteristic
644C2#6158211yescharacteristic
645C2#6166411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59864C2semidirect
2. D4×C4⋊Q8C2semidirect
3. #32584C2semidirect
4. #29206C2semidirect
5. #29139C2semidirect
6. #28799C2semidirect
7. #27821C2semidirect
8. #26464C2semidirect
9. #26107C2semidirect
10. #26044C2semidirect
11. #24812C2semidirect
12. #24338C2semidirect
13. #23250C2semidirect
15. #22670C2semidirect
16. C4×#476C2semidirect
127. #459Q8semidirect
258. #127C2×Q8semidirect
342. C4⋊C8C2×Q8semidirect
350. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598642
D4×C4⋊Q82
#325842
#292062
#291392
#287992
#278212
#264642
#261072
#260442
#248122
#243382
#232502
#230012
#226702
C4×#4762
C4×#4592