Finite Group Databasebeta

← #6557118

Subgroups — #6557118

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36290C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic, complemented
4#30190C212yesmaximal, complemented
5#29335C212yesmaximal, complemented
6#29132C211yesmaximal, characteristic, complemented
7#28755C211yesmaximal, characteristic, complemented
8#27777C212yesmaximal, complemented
9#27757C212yesmaximal, complemented
10#27268C214yesmaximal, complemented
11#27268C214yesmaximal, complemented
12#25861C212yesmaximal, complemented
13#23820C211yesmaximal, characteristic, complemented
14C2×#2043C212yesmaximal, complemented
15D4×#103C212yesmaximal
16D4×#103C212yesmaximal
17#20558C211yesmaximal, characteristic
18C2×D4×Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C2×#491C2214yes—
21C2×#491C2212yes—
22C2×C8×D4C2212yes—
23C22×#103C2214yes—
24C22×#103C2214yes—
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2212yes—
28#2851C2212yes—
29#2571C2211yescharacteristic
30#2552C2211yescharacteristic
31#2551—21no—
32#2551—21no—
33#2546C2212yes—
34#2509—21no—
35#2507—21no—
36#2505—22no—
37#2473—22no—
38#2442C2212yes—
39#2442C2212yes—
40#2440—22no—
41#2440—22no—
42#2434C2214yes—
43#2349—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4—21no—
47D4×C4⋊C4C2211yescharacteristic
48D4×C4⋊C4C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#125C2211yescharacteristic
51C2×#389C2212yes—
52C42×D4C2211yescharacteristic
53#2099C2211yescharacteristic
54#2099C2211yescharacteristic
55#2099C2212yes—
56#2074C2212yes—
57#2074C2214yes—
58#2043C2211yescharacteristic
59#2043C2212yes—
60#2043C2212yes—
61#2043C2212yes—
62#2043C2211yescharacteristic
63#2043C2214yes—
64#2043C2212yes—
65#2043C2212yes—
66#2043C2214yes—
67#2043C2214yes—
68#2026C2212yes—
69#2026C2214yes—
70#1981C2212yes—
71#1981C2212yes—
72#1960C2214yes—
73#1960C2212yes—
74#1956C2212yes—
75#1956C2212yes—
76#1935C2214yes—
77#1935C2214yes—
78#1935C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1872C2214yes—
82#1872C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1787C2214yes—
86#1787C2214yes—
87#1770C2212yes—
88#1767C2211yescharacteristic
89C4×#103C2212yes—
90C4×#103C2212yes—
91#1752C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C23×Q8C2314yes—
96C23×C8C2314yes—
97D4×Q8—28no—
98D4×Q8—28no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2314yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#125—21no—
109C2×#125C2311yescharacteristic
110C2×#125C2312yes—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C42C2312yes—
126#491—44no—
127#491—48no—
128C8×D4—28no—
129C2×C4⋊C8C2312yes—
130C2×#103—48no—
131C2×#103—48no—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#400—22no—
152#400—22no—
153#398—24no—
154#398—22no—
155#398—22no—
156#398—22no—
157#395—24no—
158#395—24no—
159#391C2312yes—
160#391C2314yes—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2311yescharacteristic
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#387—21no—
182#387—21no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—24no—
188#385—24no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#382—21no—
202#382—21no—
203#382C2311yescharacteristic
204#382C2312yes—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380C2312yes—
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C2×#95—22no—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2311yescharacteristic
236C22×Q8—24no—
237C22×Q8—28no—
238C22×Q8—28no—
239C22×Q8—28no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8C2414yes—
244C22×Q8C2412yes—
245C22×D4C2×D411yescharacteristic
246C23×C4—24no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C22×C8—28no—
253C22×C8—44no—
254C22×C8—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—28no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268C4⋊Q8—48no—
269#125—22no—
270#125—41no—
271#125—44no—
272#125—82no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—416no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4#4118yes—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—44no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C4⋊C8—24no—
351#103—24no—
352#103—28no—
353#103—24no—
354#103—24no—
355#103—28no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—42no—
362#103—44no—
363#103—44no—
364#103—44no—
365C22⋊C8—28no—
366#97—42no—
367#97—44no—
368C4×C8—24no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24C22×D412yes—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—48no—
403C2×Q8—416no—
404C2×Q8—416no—
405C2×Q8—48no—
406C2×Q8—84no—
407C2×Q8—88no—
408C2×Q8—88no—
409C2×Q8H(5,𝔽2)14yes—
410C2×D4#12414yes—
411C2×D4#13714yes—
412C2×D4#13714yes—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—82no—
437C22×C4—84no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4118yes—
450C22×C4C2511yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—88no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4#12414yes—
504C4⋊C4H(5,𝔽2)12yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4#12418yes—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—24no—
517C42—22no—
518C42—21no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—42no—
523C42—42no—
524C42C2×#4114yes—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23C2×#12414yes—
530C23C2×#13714yes—
531C23C2×#13714yes—
532C23C23×D411yescharacteristic
533Q8—24no—
534Q8—24no—
535Q8—42no—
536Q8—42no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—44no—
541Q8—816no—
542D4#491116yescomplemented
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—88no—
595C2×C4—816no—
596C2×C4—164no—
597C2×C4C2×#12412yes—
598C2×C4C2×#12414yes—
599C2×C4C2×#12412yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4#57712yes—
603C2×C4#57812yes—
604C2×C4C22×#4112yes—
605C2×C4C2×H(5,𝔽2)12yes—
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4#58511yescharacteristic
608C8—42no—
609C8—44no—
610C8—42no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22C2×#49118yes—
618C22C22×#12411yescharacteristic
619C22#348411yescharacteristic
620C22#348811yescharacteristic
621C22C22×#13711yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57711yescharacteristic
624C22C2×#57811yescharacteristic
625C4—24no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—81no—
637C4—84no—
638C4—82no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4C2×#49114yes—
643C4#351312yes—
644C4#351912yes—
645C2—28no—
646C2#3590311yescharacteristic
647C2C22×#49111yescharacteristic
648C2#6062311yescharacteristic
649C2#6159311yescharacteristic
650C2#6160211yescharacteristic
651C2#6198011yescharacteristic
652C2#6198411yescharacteristic
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. #30220C2semidirect
4. #30190C2semidirect
5. #29335C2semidirect
6. #29132C2semidirect
7. #28755C2semidirect
8. #27777C2semidirect
9. #27757C2semidirect
10. #27268C2semidirect
11. #27268C2semidirect
12. #25861C2semidirect
13. #23820C2semidirect
14. C2×#2043C2semidirect
542. D4#491semidirect

Maximal subgroups

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

subgroupindex
#362902
D4×C4⋊Q82
#302202
#301902
#293352
#291322
#287552
#277772
#277572
#272682
#258612
#238202
C2×#20432
D4×#1032
#205582