Finite Group Databasebeta

← #190222

Subgroups — #190222

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34123C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#8763C212yesmaximal, complemented
5#8762C212yesmaximal, complemented
6#8761C212yesmaximal, complemented
7#8760C212yesmaximal, complemented
8#8022C212yesmaximal, complemented
9#7984C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11C2×#432C2212yes—
12C2×#410—21no—
13#2356C411yescharacteristic, complemented
14#2356C411yescharacteristic, complemented
15#2353C411yescharacteristic, complemented
16#2353C411yescharacteristic, complemented
17#2353C2211yescharacteristic
18#2353C2211yescharacteristic
19#2349C2211yescharacteristic
20#2349C2211yescharacteristic
21#2346—21no—
22#2343—21no—
23#2343—21no—
24#2339—21no—
25#2295—21no—
26#2290—21no—
27#2290—21no—
28#2285C2212yes—
29#2283C411yescharacteristic, complemented
30#2283C411yescharacteristic, complemented
31#2283C2211yescharacteristic
32C42×D4—21no—
33C4×#117C411yescharacteristic, complemented
34C2×#376C411yescharacteristic, complemented
35C2×#376C2211yescharacteristic
36#1799—21no—
37#1799—21no—
38#1799C2212yes—
39#1799C2212yes—
40#1740—21no—
41#1467—21no—
42#1466—21no—
43#1464—21no—
44#1464—21no—
45#1459C2212yes—
46#1459C2212yes—
47#1294C2214yes—
48#1294C2214yes—
49#1289C2214yes—
50#1289C2214yes—
51C22×#41D411yescharacteristic
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117—21no—
67C2×#117—22no—
68C2×#117D411yescharacteristic
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78#432—22no—
79#432—22no—
80C2×C4⋊C8—22no—
81#410—44no—
82C2×#99—22no—
83C2×#99—22no—
84C2×C22⋊C8C2314yes—
85C2×#97—22no—
86#391—21no—
87#391—21no—
88#389—21no—
89#389—21no—
90#388—21no—
91#388—21no—
92#388—21no—
93#388—21no—
94#387—21no—
95#387—21no—
96#387—21no—
97#387—21no—
98#387—21no—
99#387—21no—
100#387—21no—
101#387—21no—
102#386—21no—
103#386—21no—
104#386—21no—
105#386—21no—
106#385—21no—
107#385—21no—
108#385—21no—
109#385—21no—
110#384—22no—
111#383—21no—
112#383C2×C411yescharacteristic
113#383C2×C411yescharacteristic
114#383C2×C411yescharacteristic
115#383C2×C411yescharacteristic
116#382—21no—
117#382—21no—
118#382—22no—
119#382C2×C411yescharacteristic
120#382C2×C411yescharacteristic
121#381—21no—
122#381—21no—
123#381C2×C411yescharacteristic
124#381C2×C411yescharacteristic
125#380—21no—
126#380—22no—
127#380C2×C411yescharacteristic
128#380C2×C411yescharacteristic
129#380C2314yes—
130#380C2311yescharacteristic
131#380C2311yescharacteristic
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4C2×C411yescharacteristic
151C4×C22⋊C4C2×C411yescharacteristic
152#376—21no—
153#376—21no—
154#376—21no—
155#376—21no—
156#376C2×C411yescharacteristic
157#376C2×C411yescharacteristic
158#376C2×C411yescharacteristic
159#376C2×C411yescharacteristic
160#376C2×C411yescharacteristic
161#376C2×C411yescharacteristic
162#376C2×C411yescharacteristic
163#376C2×C411yescharacteristic
164C2×#95—21no—
165C2×#95—21no—
166C2×#95C2×C411yescharacteristic
167C2×#95C2×C411yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C43—21no—
171#342—22no—
172#342—22no—
173#324—22no—
174#324—22no—
175#323—22no—
176#323—22no—
177C2×#41—22no—
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—42no—
182C22×Q8C22⋊C411yescharacteristic
183C22×D4C22⋊C411yescharacteristic
184C22×D4C2×D411yescharacteristic
185C22×D4C2×D411yescharacteristic
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2411yescharacteristic
199C22×C8—24no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—44no—
217C4×D4—42no—
218C4×D4—42no—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—22no—
225#117—44no—
226#117—44no—
227#117—44no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2×D411yescharacteristic
258C2×C22⋊C4C2×D411yescharacteristic
259C2×C22⋊C4C2×D411yescharacteristic
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—42no—
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42#4112yes—
294C2×C42#4112yes—
295C2×C42#4112yes—
296C2×C42#4112yes—
297C4⋊C8—42no—
298C4⋊C8—42no—
299#99—44no—
300#99—44no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303#97—42no—
304#97—42no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—24no—
330#95—42no—
331#95C22×C411yescharacteristic
332#95C22×C411yescharacteristic
333#95C22×C411yescharacteristic
334#95C22×C411yescharacteristic
335#95C22×C411yescharacteristic
336#95C22×C411yescharacteristic
337#95C22×C411yescharacteristic
338#95C22×C411yescharacteristic
339C24C2×C22⋊C411yescharacteristic
340C24C2×C22⋊C411yescharacteristic
341C24C22×D411yescharacteristic
342#41—44no—
343#41—44no—
344#41—44no—
345#41—44no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C2×D4AGL(1,ℤ/8)12yes—
359C2×D4#13712yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4#12312yes—
418C22×C4#12312yes—
419C22×C4#12312yes—
420C22×C4#12312yes—
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C2×C8—44no—
426C2×C8—84no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—84no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—81no—
460C22⋊C4—81no—
461C42—24no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—84no—
484C42—81no—
485C42—81no—
486C42#11711yescharacteristic
487C42#11711yescharacteristic
488C42#11711yescharacteristic
489C42#11711yescharacteristic
490C42#11712yes—
491C42#11712yes—
492C42#12312yes—
493C42#12312yes—
494C42#12312yes—
495C42#12312yes—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23#41712yes—
508C23C22×C22⋊C411yescharacteristic
509C23C2×#12312yes—
510C23C2×AGL(1,ℤ/8)11yescharacteristic
511C23C2×#13711yescharacteristic
512C23#57512yes—
513Q8—44no—
514D4—24no—
515D4—24no—
516D4—44no—
517D4—42no—
518D4—42no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4#38111yescharacteristic
593C2×C4#38111yescharacteristic
594C2×C4#38111yescharacteristic
595C2×C4#38112yes—
596C2×C4#38112yes—
597C2×C4#38111yescharacteristic
598C2×C4#41712yes—
599C2×C4C2×#11711yescharacteristic
600C2×C4C2×#11711yescharacteristic
601C2×C4C2×#12312yes—
602C8—84no—
603C22—21no—
604C22—22no—
605C22—21no—
606C22—22no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—81no—
619C22—81no—
620C22C2×#38111yescharacteristic
621C22C2×#41711yescharacteristic
622C22#289611yescharacteristic
623C22#309412yes—
624C22#309512yes—
625C4—21no—
626C4—21no—
627C4—22no—
628C4—22no—
629C4—21no—
630C4—22no—
631C4—22no—
632C4—21no—
633C4—21no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—164no—
648C4#182712yes—
649C2—21no—
650C2—21no—
651C2—22no—
652C2—22no—
653C2—42no—
654C2—81no—
655C2#2070211yescharacteristic
656C2C2×#182711yescharacteristic
657C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8763C2semidirect
5. #8762C2semidirect
6. #8761C2semidirect
7. #8760C2semidirect
8. #8022C2semidirect
9. #7984C2semidirect
13. #2356C4semidirect
14. #2356C4semidirect
15. #2353C4semidirect
16. #2353C4semidirect
29. #2283C4semidirect
30. #2283C4semidirect
33. C4×#117C4semidirect
34. C2×#376C4semidirect

Maximal subgroups

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

subgroupindex
#341232
#206322
#173632
#87632
#87622
#87612
#87602
#80222
#79842