Finite Group Databasebeta

← #6434749

Subgroups — #6434749

Order 512.

1968 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#36639C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#30064C214yesmaximal, complemented
4#30063C212yesmaximal, complemented
5#30062C212yesmaximal, complemented
6#30061C214yesmaximal, complemented
7#30060C212yesmaximal, complemented
8#30059C212yesmaximal, complemented
9#23734C212yesmaximal, complemented
10#23554C212yesmaximal, complemented
11#23508C212yesmaximal
12#23452C212yesmaximal, complemented
13#21059C214yesmaximal, complemented
14#20772C211yesmaximal, characteristic, complemented
15C2×#533C2211yescharacteristic
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8—21no—
19C2×#428C2214yes—
20C2×#417C2214yes—
21C2×#417C2214yes—
22#2862C2211yescharacteristic
23#2862C2211yescharacteristic
24#2861C2212yes—
25#2861C2212yes—
26#2858C2211yescharacteristic
27#2858C2211yescharacteristic
28#2850—21no—
29#2850—21no—
30#2850—21no—
31#2850—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×#124—21no—
41C4×#124—21no—
42C4×#124—21no—
43C4×#124—21no—
44#2297C2212yes—
45#2297C2212yes—
46C2×#391—21no—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2212yes—
52C2×C43C2211yescharacteristic
53#2096C2214yes—
54#2095C2214yes—
55#2094C2214yes—
56#2093C2214yes—
57#2059C2218yes—
58#2058C2214yes—
59#2057C2214yes—
60#2043C2214yes—
61#2042C2218yes—
62#2041C2214yes—
63#2024C2214yes—
64#2024C2214yes—
65#2023C2214yes—
66#2023C2214yes—
67#1928C2218yes—
68#1927C2218yes—
69#1926C2218yes—
70#1925C2218yes—
71#1839C2212yes—
72#1839C2212yes—
73#1839C2212yes—
74#1839C2212yes—
75#1736C2214yes—
76#1728C2212yes—
77#1728C2212yes—
78#1728C2212yes—
79C22×#41C2312yes—
80C23×Q8—21no—
81C22×M16C2314yes—
82#533—44no—
83#532—44no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#125C2312yes—
94C2×#124—21no—
95C2×#124—21no—
96C2×#124—21no—
97C2×#124—21no—
98C2×#124C2312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×#121—21no—
102C2×#121—21no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—48no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×#117C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C42—22no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#428—48no—
128C2×#107C2318yes—
129C2×#106C2318yes—
130#417—216no—
131#417—48no—
132C2×#103C2318yes—
133C2×#103C2318yes—
134C2×#102C2318yes—
135C2×#102C2318yes—
136#391—22no—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—21no—
141#391—21no—
142#391—21no—
143#391—21no—
144#391C2312yes—
145#391C2312yes—
146#391C2312yes—
147#391C2312yes—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2314yes—
163#389C2314yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#386—22no—
176#384—21no—
177#384—21no—
178#384C2312yes—
179#384C2312yes—
180#382—21no—
181#382—21no—
182#382—21no—
183#382—21no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2314yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C2×#95—22no—
197C2×#95—22no—
198C43D411yescharacteristic
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206#340C2318yes—
207#340C2318yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214C2×#41—28no—
215C2×#41—28no—
216C2×#41—28no—
217C22×Q8—28no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×Q8C2412yes—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4C2412yes—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×M16—28no—
240C2×M16—28no—
241C2×M16—28no—
242C22×C8C2418yes—
243C4⋊Q8—28no—
244C4⋊Q8—28no—
245C4⋊Q8—28no—
246C4⋊Q8—28no—
247C4⋊Q8—42no—
248C4⋊Q8—42no—
249C4⋊Q8—42no—
250#127—42no—
251#125—44no—
252#124—44no—
253#124—44no—
254#124—44no—
255#124—44no—
256#124—44no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259#121—44no—
260#121—44no—
261C4×Q8—48no—
262C4×Q8—48no—
263C4×Q8—48no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8—48no—
267C4×D4—44no—
268C4×D4—44no—
269#117—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4#4118yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—48no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—48no—
348#106—48no—
349#103—216no—
350#103—48no—
351#102—216no—
352#102—48no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365C24—22no—
366C24C22×D411yescharacteristic
367#41—432no—
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—44no—
374C2×Q8—432no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—42no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C42−1+414yes—
444C22×C4C2511yescharacteristic
445M16—416no—
446C2×C8—28no—
447C2×C8—28no—
448C2×C8—28no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4C2×#4118yes—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—48no—
508C42—44no—
509C42—48no—
510C42AGL(1,ℤ/8)14yes—
511C42#13714yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23#52512yes—
524C23#53512yes—
525C23#53612yes—
526C23C23×D411yescharacteristic
527Q8—416no—
528Q8—416no—
529Q8—48no—
530Q8—84no—
531Q8—84no—
532D4—48no—
533D4—84no—
534D4—84no—
535D4—88no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—416no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—88no—
587C2×C4—88no—
588C2×C4#52512yes—
589C2×C4#53512yes—
590C2×C4#53612yes—
591C2×C4#542116yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4#57512yes—
599C2×C4#57512yes—
600C2×C4C22×#4112yes—
601C2×C4C2×H(5,𝔽2)12yes—
602C2×C4C2×2−1+412yes—
603C8—48no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22#263214yes—
618C22C2×#52511yescharacteristic
619C22C2×#53511yescharacteristic
620C22C2×#53611yescharacteristic
621C22C22×AGL(1,ℤ/8)11yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57511yescharacteristic
624C22C2×#57511yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—48no—
636C4—41no—
637C4—41no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4—82no—
642C4#263214yes—
643C4C2×#54214yes—
644C2—24no—
645C2—162no—
646C2C2×#263211yescharacteristic
647C2#6055811yescharacteristic
648C2#6056011yescharacteristic
649C2#6080311yescharacteristic
650C2#6080411yescharacteristic
651C2#6086011yescharacteristic
652C2#6086311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. #30064C2semidirect
4. #30063C2semidirect
5. #30062C2semidirect
6. #30061C2semidirect
7. #30060C2semidirect
8. #30059C2semidirect
9. #23734C2semidirect
10. #23554C2semidirect
12. #23452C2semidirect
13. #21059C2semidirect
14. #20772C2semidirect

Maximal subgroups

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

subgroupindex
#366392
C2×C4×C4⋊Q82
#300642
#300632
#300622
#300612
#300602
#300592
#237342
#235542
#235082
#234522
#210592
#207722