Finite Group Databasebeta

← #6537283

Subgroups — #6537283

Order 512.

1885 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2#34334C211yesmaximal, characteristic
3#29983C211yesmaximal, characteristic, complemented
4#29951C212yesmaximal, complemented
5#29255C211yesmaximal, characteristic, complemented
6#29159C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#26462C212yesmaximal, complemented
9#26412C214yesmaximal, complemented
10#26022C212yesmaximal, complemented
11#25955C214yesmaximal, complemented
12#24301C212yesmaximal, complemented
13#23776C211yesmaximal, characteristic, complemented
14C2×#2024C212yesmaximal, complemented
15#22703C212yesmaximal
16#22696C211yesmaximal, characteristic
17D4×#103C212yesmaximal
18C2×D4×Q8—21no—
19C2×D4×Q8C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C2×#475C2214yes—
22C2×C8×D4C2212yes—
23C22×#103C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2551—21no—
27#2551C2212yes—
28#2546—21no—
29#2440—22no—
30#2440C2214yes—
31#2434—22no—
32#2369—21no—
33#2368—22no—
34#2365C2211yescharacteristic
35#2360C2212yes—
36#2355C2211yescharacteristic
37#2349—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2211yescharacteristic
41Q8×C22⋊C4C2212yes—
42#2331—21no—
43#2329—22no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47#2295C2211yescharacteristic
48C2×#391C2212yes—
49#2275—21no—
50C42×D4C2211yescharacteristic
51#2099C2212yes—
52#2099C2211yescharacteristic
53#2096C2212yes—
54#2096C2212yes—
55#2096C2211yescharacteristic
56#2088C2212yes—
57#2066C2214yes—
58#2066C2212yes—
59#2066C2214yes—
60#2058C2212yes—
61#2058C2211yescharacteristic
62#2024C2212yes—
63#2024C2214yes—
64#2024C2212yes—
65#2024C2214yes—
66#2024C2211yescharacteristic
67#2024C2212yes—
68#2024C2212yes—
69#1988C2214yes—
70#1981C2212yes—
71#1956C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1939C2212yes—
75#1919C2214yes—
76#1892C2214yes—
77#1872C2214yes—
78#1865C2214yes—
79#1865C2214yes—
80#1853C2214yes—
81#1853C2214yes—
82#1850C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1807C2214yes—
86#1803C2212yes—
87#1803C2214yes—
88#1797C2212yes—
89#1787C2214yes—
90C4×#106C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C2×#340C2214yes—
94C2×#336C2212yes—
95C23×Q8—22no—
96C23×Q8C2312yes—
97C23×C8C2314yes—
98D4×Q8—24no—
99D4×Q8—48no—
100D4×Q8Q818yescomplemented
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2312yes—
129C22×C42C2312yes—
130#475—48no—
131C8×D4—28no—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2312yes—
139C2×#103—48no—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8C2312yes—
149#398—24no—
150#395—24no—
151#395—24no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2311yescharacteristic
160#391C2312yes—
161#391C2312yes—
162#391C2311yescharacteristic
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2311yescharacteristic
174#389C2314yes—
175#389C2311yescharacteristic
176#387—21no—
177#387—22no—
178#387—21no—
179#387—22no—
180#387—22no—
181#387—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#382—21no—
194#380—22no—
195#380—22no—
196#380C2312yes—
197#380C2314yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217#376—21no—
218C2×#95—22no—
219C2×#95—22no—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8—48no—
243C22×Q8—48no—
244C22×Q8—84no—
245C22×Q8C2×Q818yescomplemented
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C23×C4—22no—
250C23×C4—24no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C22×C8—28no—
256C22×C8—44no—
257C22×C8—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—28no—
262C4⋊Q8—24no—
263C4⋊Q8—28no—
264C4⋊Q8—22no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—48no—
268C4⋊Q8—48no—
269C4⋊Q8—48no—
270C4⋊Q8—44no—
271C4⋊Q8—82no—
272C4⋊Q8C2×Q814yescomplemented
273C22⋊Q8—24no—
274C22⋊Q8—216no—
275C22⋊Q8—48no—
276C22⋊Q8—416no—
277C22⋊Q8C2×Q818yescomplemented
278C4×Q8—22no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8C2×Q814yescomplemented
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4SD1618yes—
288C4×D4C2×Q814yes—
289C4×D4#4114yes—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—44no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#106—24no—
351#106—42no—
352#106—44no—
353#106—44no—
354C4⋊C8—24no—
355C4⋊C8—42no—
356C4⋊C8—44no—
357#103—24no—
358#103—28no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—42no—
363#103—44no—
364C22⋊C8—28no—
365C4×C8—24no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24C22×D412yes—
389C2×Q8—24no—
390C2×Q8—28no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—48no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—416no—
402C2×Q8—42no—
403C2×Q8—48no—
404C2×Q8—44no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—84no—
408C2×Q8—88no—
409C2×Q8—832no—
410C2×Q8C22×Q812yes—
411C2×Q8C22×Q814yes—
412C2×D4C22⋊Q814yes—
413C2×D4C2×SD1614yes—
414C2×D4#13714yes—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—84no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C2×SD1618yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×Q814yes—
451C22×C4C2×#4114yes—
452C22×C4C2511yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—88no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C22⋊Q814yes—
497C4⋊C4C2×SD1614yes—
498C4⋊C4C22×Q812yes—
499C4⋊C4H(5,𝔽2)14yes—
500C22⋊C4—28no—
501C22⋊C4—24no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4C22⋊Q818yes—
505C22⋊C4C2×SD1618yes—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—42no—
513C42—42no—
514C42—41no—
515C42—44no—
516C42—42no—
517C42—82no—
518C42—82no—
519C42—82no—
520C42C2×SD1614yes—
521C42C22×Q812yes—
522C42C2×#4112yes—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23C2×C22⋊Q814yes—
528C23C22×SD1614yes—
529C23C2×#13714yes—
530C23C23×D411yescharacteristic
531Q8—24no—
532Q8—28no—
533Q8—48no—
534Q8—44no—
535Q8—42no—
536Q8—88no—
537Q8—816no—
538Q8—88no—
539D4#475116yescomplemented
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—88no—
590C2×C4—816no—
591C2×C4—164no—
592C2×C4C2×C22⋊Q812yes—
593C2×C4C2×C22⋊Q814yes—
594C2×C4C2×C22⋊Q812yes—
595C2×C4C22×SD1612yes—
596C2×C4C22×SD1612yes—
597C2×C4C22×SD1614yes—
598C2×C4C22×SD1612yes—
599C2×C4C2×#13712yes—
600C2×C4#57812yes—
601C2×C4C23×Q811yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×H(5,𝔽2)12yes—
604C2×C4#58512yes—
605C8—42no—
606C8—44no—
607C8—42no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22C2×#47518yes—
615C22C22×C22⋊Q811yescharacteristic
616C22#348411yescharacteristic
617C22#350011yescharacteristic
618C22C23×SD1611yescharacteristic
619C22C22×#13711yescharacteristic
620C22C2×#57811yescharacteristic
621C22#358511yescharacteristic
622C4—22no—
623C4—22no—
624C4—24no—
625C4—22no—
626C4—24no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—44no—
631C4—41no—
632C4—44no—
633C4—42no—
634C4—82no—
635C4—82no—
636C4—81no—
637C4—84no—
638C4—81no—
639C4C2×#47514yes—
640C4#350112yes—
641C4#351412yes—
642C2—28no—
643C2#3570211yescharacteristic
644C2C22×#47511yescharacteristic
645C2#6049811yescharacteristic
646C2#6156011yescharacteristic
647C2#6160211yescharacteristic
648C2#6252611yescharacteristic
649C2#6258211yescharacteristic
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. #29983C2semidirect
4. #29951C2semidirect
5. #29255C2semidirect
6. #29159C2semidirect
7. #29141C2semidirect
8. #26462C2semidirect
9. #26412C2semidirect
10. #26022C2semidirect
11. #25955C2semidirect
12. #24301C2semidirect
13. #23776C2semidirect
14. C2×#2024C2semidirect
100. D4×Q8Q8semidirect
245. C22×Q8C2×Q8semidirect
272. C4⋊Q8C2×Q8semidirect
277. C22⋊Q8C2×Q8semidirect
282. C4×Q8C2×Q8semidirect
539. D4#475semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#343342
#299832
#299512
#292552
#291592
#291412
#264622
#264122
#260222
#259552
#243012
#237762
C2×#20242
#227032
#226962
D4×#1032