Finite Group Databasebeta

← #6537113

Subgroups — #6537113

Order 512.

1733 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic, complemented
2#34322C211yesmaximal, characteristic, complemented
3#30082C212yesmaximal, complemented
4#29981C211yesmaximal, characteristic, complemented
5#29573C212yesmaximal, complemented
6#29367C212yesmaximal, complemented
7#28511C211yesmaximal, characteristic, complemented
8#28157C211yesmaximal, characteristic, complemented
9#26890C214yesmaximal, complemented
10#26377C214yesmaximal, complemented
11#25057C212yesmaximal, complemented
12#24185C212yesmaximal, complemented
13#23796C212yesmaximal
14#23676C211yesmaximal, characteristic
15#22996C212yesmaximal, complemented
16#22828C211yesmaximal, characteristic, complemented
17C4⋊C4×D8C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#554—21no—
20C2×#477C2214yes—
21C2×#461C2212yes—
22C2×C4×SD16C2214yes—
23C2×C4×D8C2212yes—
24C2×C4×D8C2214yes—
25#2868C2211yescharacteristic
26#2859C2212yes—
27#2566—22no—
28#2563—21no—
29#2538—21no—
30#2538C2212yes—
31#2452—22no—
32#2452C2214yes—
33#2373—22no—
34#2367C2212yes—
35#2365C2211yescharacteristic
36#2360—21no—
37#2355C2211yescharacteristic
38#2351C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2211yescharacteristic
41D4×C4⋊C4C2212yes—
42#2346—21no—
43#2334—22no—
44#2323—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#127C2212yes—
48C4×#124C2212yes—
49#2295C2212yes—
50#2293C2211yescharacteristic
51#2274—21no—
52C42×Q8C2211yescharacteristic
53#2100C2212yes—
54#2095C2212yes—
55#2095C2212yes—
56#2094C2212yes—
57#2092C2214yes—
58#2085C2214yes—
59#2083C2211yescharacteristic
60#2072C2212yes—
61#2071C2211yescharacteristic
62#2067C2212yes—
63#2058C2212yes—
64#2054C2212yes—
65#2042C2214yes—
66#2041C2211yescharacteristic
67#2023C2212yes—
68#2014C2212yes—
69#2012C2214yes—
70#2011C2211yescharacteristic
71#1984C2212yes—
72#1956C2212yes—
73#1948C2212yes—
74#1942C2214yes—
75#1933C2214yes—
76#1925C2214yes—
77#1921C2214yes—
78#1916C2214yes—
79#1880C2214yes—
80#1880C2214yes—
81#1877C2214yes—
82#1865C2214yes—
83#1863C2214yes—
84#1862C2214yes—
85#1862C2214yes—
86#1845C2212yes—
87#1839C2212yes—
88#1831C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1744C2212yes—
95C22×SD16C2314yes—
96C22×D8C2312yes—
97#556—24no—
98#556—24no—
99#554—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#126—22no—
106C2×#125—22no—
107C2×#124C2312yes—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2314yes—
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129#477—48no—
130#461—28no—
131C4×SD16—48no—
132C4×D8—28no—
133C4×D8—48no—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#396—24no—
152#396—24no—
153#393—24no—
154#391—21no—
155#391C2312yes—
156#391C2312yes—
157#390—22no—
158#390—21no—
159#390—21no—
160#390—22no—
161#390C2312yes—
162#390C2314yes—
163#390C2311yescharacteristic
164#390C2311yescharacteristic
165#389—21no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2314yes—
175#389C2314yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—21no—
180#388—22no—
181#388—21no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2312yes—
197#382—22no—
198#382—22no—
199#382C2312yes—
200#380—22no—
201#380—22no—
202#380—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221#376—22no—
222C43C2311yescharacteristic
223C43C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—22no—
247C2×SD16—28no—
248C2×SD16—28no—
249C2×SD16—44no—
250C2×D8—24no—
251C2×D8—24no—
252C2×D8—24no—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—22no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yes—
261#127—24no—
262#126—216no—
263#125—48no—
264#124—24no—
265#124—24no—
266C22⋊Q8—28no—
267C22⋊Q8—416no—
268C22⋊Q8—48no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×D4—22no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—84no—
287C4×D4#4114yes—
288#117—28no—
289#117—44no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—21no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—28no—
356#107—24no—
357#107—44no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361#103—28no—
362#103—28no—
363#103—44no—
364#102—24no—
365#102—24no—
366#102—24no—
367#102—24no—
368#102—44no—
369#102—44no—
370C4×C8—28no—
371C4×C8—24no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—24no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C24—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—44no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8C2×#4114yes—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×#4112yes—
415C2×D42−1+414yes—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—82no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2511yescharacteristic
446SD16—416no—
447D8—216no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8H(5,𝔽2)14yes—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4AGL(1,ℤ/8)14yes—
490C4⋊C4C2×#4112yes—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—81no—
502C22⋊C4—82no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C42—28no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—42no—
517C42—41no—
518C42—42no—
519C42—82no—
520C42—82no—
521C42—82no—
522C42#13514yes—
523C42C2×#4112yes—
524C42C2×#4112yes—
525C23—22no—
526C23—24no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23C23×D411yescharacteristic
540Q8—28no—
541Q8—44no—
542Q8—44no—
543Q8—164no—
544D4—24no—
545D4—24no—
546D4—24no—
547D4—48no—
548D4—48no—
549D4—42no—
550D4—42no—
551D4—42no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—88no—
597C2×C4—164no—
598C2×C4—168no—
599C2×C4#52512yes—
600C2×C4#52512yes—
601C2×C4C2×#13512yes—
602C2×C4C2×#13512yes—
603C2×C4C2×#13512yes—
604C2×C4C2×AGL(1,ℤ/8)14yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4#57812yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4C2×2−1+412yes—
611C8—24no—
612C8—42no—
613C8—44no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—82no—
621C22—81no—
622C22—81no—
623C22—82no—
624C22—81no—
625C22—82no—
626C22—82no—
627C22—81no—
628C22C2×#52511yescharacteristic
629C22#348611yescharacteristic
630C22#349811yescharacteristic
631C22C22×#13511yescharacteristic
632C22C22×AGL(1,ℤ/8)11yescharacteristic
633C22C2×#57811yescharacteristic
634C22#358511yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—24no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—82no—
647C4—81no—
648C4—81no—
649C4—82no—
650C4—84no—
651C4#310614yes—
652C4#349212yes—
653C4#352012yes—
654C2—82no—
655C2—84no—
656C2#3570111yescharacteristic
657C2C2×#310611yescharacteristic
658C2#6060011yescharacteristic
659C2#6181911yescharacteristic
660C2#6183711yescharacteristic
661C2#6241811yescharacteristic
662C2#6244711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2011C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36559C2semidirect
2. #34322C2semidirect
3. #30082C2semidirect
4. #29981C2semidirect
5. #29573C2semidirect
6. #29367C2semidirect
7. #28511C2semidirect
8. #28157C2semidirect
9. #26890C2semidirect
10. #26377C2semidirect
11. #25057C2semidirect
12. #24185C2semidirect
15. #22996C2semidirect
16. #22828C2semidirect
17. C4⋊C4×D8C2semidirect

Maximal subgroups

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

subgroupindex
#365592
#343222
#300822
#299812
#295732
#293672
#285112
#281572
#268902
#263772
#250572
#241852
#237962
#236762
#229962
#228282
C4⋊C4×D82