Finite Group Databasebeta

← #6793569

Subgroups — #6793569

Order 512.

1615 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#31446C214yesmaximal, complemented
5#31378C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29287C211yesmaximal, characteristic, complemented
8#29218C211yesmaximal, characteristic, complemented
9#28797C212yesmaximal, complemented
10#27823C212yesmaximal, complemented
11#26296C212yesmaximal
12#24969C212yesmaximal
13#24711C212yesmaximal, complemented
14#24302C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic
16#21840C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#477C2211yescharacteristic
21C2×#477C2212yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2573—22no—
26#2571C2211yescharacteristic
27#2568C2211yescharacteristic
28#2551—21no—
29#2551C2212yes—
30#2547C2211yescharacteristic
31#2546—21no—
32#2538C2211yescharacteristic
33#2514—21no—
34#2494C2212yes—
35#2447—21no—
36#2440—22no—
37#2440C2214yes—
38#2434—22no—
39#2421C2212yes—
40#2401—22no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#124—21no—
46C4×C22⋊Q8C2212yes—
47#2295—21no—
48#2283C2212yes—
49C42×Q8C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2100C2214yes—
53#2088C2214yes—
54#2085C2214yes—
55#2077C2214yes—
56#2075C2214yes—
57#2075C2214yes—
58#2057C2214yes—
59#2035C2214yes—
60#2024C2214yes—
61#1993C2214yes—
62#1993C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2211yescharacteristic
68#1987C2212yes—
69#1987C2212yes—
70#1954C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2211yescharacteristic
74#1947C2212yes—
75#1947C2212yes—
76#1944C2214yes—
77#1935C2214yes—
78#1924C2214yes—
79#1921C2212yes—
80#1872C2214yes—
81#1863C2212yes—
82#1852C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1823C2214yes—
86#1823C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C23×Q8—22no—
96D4×Q8—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#124—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C22×C4⋊C4—22no—
122C22×C4⋊C4C2314yes—
123C22×C42C2312yes—
124#477—24no—
125#477—44no—
126#432—44no—
127C2×#107C2312yes—
128C2×#107C2314yes—
129C2×#107C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143#401—22no—
144#400—22no—
145#400—22no—
146#400—22no—
147#398—24no—
148#397—22no—
149#397—22no—
150#397—22no—
151#396—22no—
152#395—24no—
153#395—24no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391C2312yes—
161#390—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2314yes—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—22no—
179#385—24no—
180#385—24no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#383—22no—
188#382—21no—
189#382C2312yes—
190#382C2312yes—
191#381—22no—
192#380—22no—
193#380C2314yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376—22no—
217C43—21no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—24no—
238C22×Q8—48no—
239C22×Q8—48no—
240C22×Q8—84no—
241C22×Q8C2411yescharacteristic
242C22×D4C2×D411yescharacteristic
243C23×C4—24no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C4⋊Q8—28no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—44no—
257C4⋊Q8—48no—
258C4⋊Q8—48no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261C4⋊Q8—82no—
262C4⋊Q8#4114yes—
263#124—44no—
264C22⋊Q8—24no—
265C22⋊Q8—416no—
266C22⋊Q8—42no—
267C22⋊Q8—48no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—41no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—48no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—48no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—24no—
345#107—42no—
346#107—44no—
347#106—44no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350C4⋊C8—24no—
351C4⋊C8—24no—
352C4⋊C8—42no—
353#103—24no—
354#103—24no—
355#103—44no—
356C22⋊C8—44no—
357C22⋊C8—44no—
358#97—44no—
359C4×C8—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24C22×D412yes—
383C2×Q8—24no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—48no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—48no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—84no—
400C2×Q8—832no—
401C2×Q82−1+412yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4AGL(1,ℤ/8)14yes—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—84no—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2511yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4C2×#4114yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—82no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—21no—
503C42—21no—
504C42—41no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—44no—
509C42—42no—
510C42—44no—
511C42—42no—
512C42—81no—
513C42#13714yes—
514C42C2×#4112yes—
515C42H(5,𝔽2)12yes—
516C422−1+412yes—
517C422−1+412yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23C2×AGL(1,ℤ/8)14yes—
525C23C23×D411yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—48no—
529Q8—41no—
530Q8—41no—
531Q8—88no—
532Q8—816no—
533Q8—84no—
534D4—28no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—816no—
587C2×C4#53612yes—
588C2×C4#53612yes—
589C2×C4#54612yes—
590C2×C4#54612yes—
591C2×C4C2×AGL(1,ℤ/8)12yes—
592C2×C4C2×#13712yes—
593C2×C4C2×#13712yes—
594C2×C4C2×#13712yes—
595C2×C4#57812yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C2×C4#58511yescharacteristic
602C8—42no—
603C8—42no—
604C8—44no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22C2×#53611yescharacteristic
613C22C2×#54611yescharacteristic
614C22#348911yescharacteristic
615C22C22×AGL(1,ℤ/8)11yescharacteristic
616C22C22×#13711yescharacteristic
617C22C2×#57811yescharacteristic
618C22#358511yescharacteristic
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—42no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—81no—
631C4—82no—
632C4—81no—
633C4—81no—
634C4—84no—
635C4#323814yes—
636C4#332814yes—
637C4#351811yescharacteristic
638C4#352311yescharacteristic
639C4#355211yescharacteristic
640C4#355411yescharacteristic
641C2—44no—
642C2#3891111yescharacteristic
643C2C2×#323811yescharacteristic
644C2#6089311yescharacteristic
645C2C2×#332811yescharacteristic
646C2#6115511yescharacteristic
647C2#6211211yescharacteristic
648C2#6213211yescharacteristic
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. #33198C2semidirect
4. #31446C2semidirect
5. #31378C2semidirect
6. #29942C2semidirect
7. #29287C2semidirect
8. #29218C2semidirect
9. #28797C2semidirect
10. #27823C2semidirect
13. #24711C2semidirect
14. #24302C2semidirect
16. #21840C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#349452
#331982
#314462
#313782
#299422
#292872
#292182
#287972
#278232
#262962
#249692
#247112
#243022
#227992
#218402
C4×#4772