Finite Group Databasebeta

← #7567415

Subgroups — #7567415

Order 512.

1767 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#59863C211yesmaximal, characteristic, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#32639C214yesmaximal, complemented
4#29141C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#27778C212yesmaximal, complemented
7#27776C212yesmaximal, complemented
8#26026C212yesmaximal, complemented
9#26024C212yesmaximal, complemented
10#25939C214yesmaximal, complemented
11#24731C212yesmaximal
12#24727C212yesmaximal, complemented
13#23250C211yesmaximal, characteristic, complemented
14#22762C212yesmaximal, complemented
15#22758C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×D4×Q8C2212yes—
19C22×C4⋊Q8—21no—
20C2×#489C2214yes—
21C2×#463C2214yes—
22C2×#461C2211yescharacteristic
23C2×#459C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×C4×D8C2212yes—
26#2868C2211yescharacteristic
27#2857—21no—
28#2808—22no—
29#2592C2214yes—
30#2563—21no—
31#2551C2212yes—
32#2551C2212yes—
33#2548C2212yes—
34#2547C2212yes—
35#2546C2212yes—
36#2513—22no—
37#2485—22no—
38#2440—22no—
39#2440—22no—
40#2434—22no—
41#2355C2212yes—
42#2354—21no—
43#2354C2212yes—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C42×D4C2211yescharacteristic
50#2068C2214yes—
51#2065C2214yes—
52#2042C2214yes—
53#2042C2214yes—
54#2024C2214yes—
55#2024C2214yes—
56#2023C2214yes—
57#2023C2214yes—
58#1987C2214yes—
59#1985C2214yes—
60#1982C2211yescharacteristic
61#1981C2211yescharacteristic
62#1957C2212yes—
63#1956C2212yes—
64#1948C2211yescharacteristic
65#1947C2211yescharacteristic
66#1939C2212yes—
67#1939C2212yes—
68#1936C2214yes—
69#1935C2214yes—
70#1927C2212yes—
71#1925C2212yes—
72#1921C2214yes—
73#1920C2214yes—
74#1879C2212yes—
75#1879C2214yes—
76#1877C2212yes—
77#1877C2214yes—
78#1834C2212yes—
79#1834C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1802C2212yes—
83#1802C2212yes—
84#1802C2211yescharacteristic
85#1802C2211yescharacteristic
86#1770C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93C23×Q8—22no—
94C22×SD16C2312yes—
95C22×D8C2312yes—
96D4×Q8—44no—
97D4×Q8—44no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—48no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#124C2314yes—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2314yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C42—21no—
125#489—48no—
126#463—48no—
127#461—24no—
128#459—24no—
129C4×SD16—28no—
130C4×D8—28no—
131C2×#107C2312yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2312yes—
153#398—24no—
154#397—24no—
155#397—24no—
156#395—24no—
157#395—24no—
158#394—24no—
159#393—24no—
160#391—22no—
161#391D412yes—
162#391C2312yes—
163#391C2314yes—
164#390—22no—
165#390—22no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2314yes—
170#389—22no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389D412yes—
175#389D412yes—
176#389C2314yes—
177#389C2314yes—
178#388—24no—
179#388—24no—
180#388—24no—
181#387—24no—
182#385—24no—
183#385—24no—
184#384—21no—
185#384—21no—
186#384D411yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209C43C2311yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2312yes—
218#336C2314yes—
219#336C2312yes—
220#336C2314yes—
221#336C2312yes—
222C22×Q8—22no—
223C22×Q8—24no—
224C22×Q8—48no—
225C22×Q8—48no—
226C22×Q8—84no—
227C22×Q8C2×D412yes—
228C22×Q8C2412yes—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C22×D4C2411yescharacteristic
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—24no—
238C2×SD16—24no—
239C2×SD16—24no—
240C2×SD16—24no—
241C2×D8—24no—
242C2×D8—24no—
243C2×D8—24no—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C4⋊Q8—28no—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—44no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—48no—
256C4⋊Q8—48no—
257C4⋊Q8—48no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260#127—22no—
261#127—22no—
262#124—44no—
263#124—44no—
264C22⋊Q8—416no—
265C22⋊Q8—48no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—42no—
269C4×Q8—42no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—28no—
273C4×D4—24no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—82no—
280C4×D4#4114yes—
281C4×D4#4114yes—
282#117—84no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2414yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—24no—
338#106—24no—
339C4⋊C8—22no—
340C4⋊C8—22no—
341C4⋊C8—22no—
342C4⋊C8—22no—
343C4⋊C8—44no—
344#103—24no—
345#103—24no—
346#103—24no—
347#103—44no—
348#103—44no—
349#102—24no—
350#102—24no—
351#102—24no—
352#102—24no—
353#102—24no—
354#102—44no—
355#102—44no—
356#97—44no—
357C4×C8—24no—
358C4×C8—24no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95C2×D414yes—
370#95C2×D414yes—
371C24—21no—
372C24—22no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—84no—
388C2×Q8—832no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4C2×#4112yes—
405C2×D4C2×#4112yes—
406C2×D4H(5,𝔽2)12yes—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—84no—
429C22×C4C22×D414yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D414yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2511yescharacteristic
438SD16—48no—
439D8—48no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4C2×#4112yes—
483C4⋊C4C2×#4112yes—
484C4⋊C4H(5,𝔽2)12yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—82no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—24no—
500C42—21no—
501C42—21no—
502C42—41no—
503C42—44no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—84no—
508C42AGL(1,ℤ/8)14yes—
509C42C2×#4112yes—
510C42C2×#4112yes—
511C422−1+412yes—
512C422−1+412yes—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23C23×D411yescharacteristic
528Q8—48no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—42no—
533Q8—88no—
534Q8—816no—
535D4—22no—
536D4—22no—
537D4—22no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—84no—
543D4—84no—
544D4#55514yes—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—48no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—816no—
585C2×C4—88no—
586C2×C4#54812yes—
587C2×C4#54812yes—
588C2×C4#54812yes—
589C2×C4#54812yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4#57712yes—
594C2×C4#57812yes—
595C2×C4C22×#4111yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×H(5,𝔽2)11yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C8—42no—
602C8—42no—
603C8—42no—
604C8—42no—
605C22—24no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C2×#54811yescharacteristic
620C22C2×#54811yescharacteristic
621C22#348411yescharacteristic
622C22C22×AGL(1,ℤ/8)11yescharacteristic
623C22C2×#57711yescharacteristic
624C22C2×#57811yescharacteristic
625C22#358511yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—41no—
634C4—44no—
635C4—82no—
636C4—84no—
637C4—82no—
638C4—82no—
639C4#336714yes—
640C4#337314yes—
641C4C2×#55511yescharacteristic
642C4#351111yescharacteristic
643C4#351111yescharacteristic
644C4#354411yescharacteristic
645C2—44no—
646C2—162no—
647C2#5913511yescharacteristic
648C2C2×#336711yescharacteristic
649C2C2×#337311yescharacteristic
650C2#6134711yescharacteristic
651C2#6134811yescharacteristic
652C2#6158211yescharacteristic
653C2#6166711yescharacteristic
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. #59863C2semidirect
2. D4×C4⋊Q8C2semidirect
3. #32639C2semidirect
4. #29141C2semidirect
5. #29139C2semidirect
6. #27778C2semidirect
7. #27776C2semidirect
8. #26026C2semidirect
9. #26024C2semidirect
10. #25939C2semidirect
12. #24727C2semidirect
13. #23250C2semidirect
14. #22762C2semidirect
15. #22758C2semidirect
16. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#598632
D4×C4⋊Q82
#326392
#291412
#291392
#277782
#277762
#260262
#260242
#259392
#247312
#247272
#232502
#227622
#227582
C4×#4612
C4×#4592