Finite Group Databasebeta

← #6490767

Subgroups — #6490767

Order 512.

1904 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#33108C212yesmaximal, complemented
4#33106C212yesmaximal
5#30061C214yesmaximal, complemented
6#29259C212yesmaximal, complemented
7#29258C212yesmaximal, complemented
8#29212C212yesmaximal, complemented
9#28754C212yesmaximal, complemented
10#27914C212yesmaximal, complemented
11#25908C212yesmaximal, complemented
12#24066C214yesmaximal, complemented
13#22224C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529C2212yes—
17C22×C4×Q8—21no—
18C2×#489C2214yes—
19C2×#483C2214yes—
20C2×#480C2214yes—
21#2861C2212yes—
22#2860—21no—
23#2860—21no—
24#2859C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2854—21no—
29#2854—21no—
30#2850C2212yes—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2211yescharacteristic
40C4×#127C2211yescharacteristic
41C4×#124C2212yes—
42C4×#121—21no—
43C4×#121—21no—
44C2×#391—21no—
45#2293—21no—
46#2293—21no—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2212yes—
52C2×C43C2211yescharacteristic
53#2095C2214yes—
54#2094C2214yes—
55#2059C2218yes—
56#2042C2218yes—
57#2024C2214yes—
58#2023C2214yes—
59#1993C2214yes—
60#1982C2214yes—
61#1981C2214yes—
62#1959C2214yes—
63#1948C2214yes—
64#1947C2214yes—
65#1927C2218yes—
66#1925C2218yes—
67#1874C2214yes—
68#1872C2214yes—
69#1864C2218yes—
70#1863C2218yes—
71#1847C2214yes—
72#1839C2212yes—
73#1831C2212yes—
74#1817C2212yes—
75#1797C2212yes—
76#1767C2212yes—
77#1749C2212yes—
78#1745C2212yes—
79C23×Q8—21no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87#529—48no—
88C2×#126—22no—
89C2×#125C2312yes—
90C2×#124C2312yes—
91C2×#124C2312yes—
92C2×#123—22no—
93C2×C22⋊Q8C2312yes—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121C2312yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—48no—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4C2312yes—
107C2×#117—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C42—22no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#489—216no—
123#483—48no—
124#480—48no—
125C2×#107C2318yes—
126C2×#106C2318yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2318yes—
129C2×#103C2318yes—
130C2×#102C2318yes—
131C2×#102C2318yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2314yes—
134#391—22no—
135#391—22no—
136#391—22no—
137#391—22no—
138#391C2312yes—
139#391C2312yes—
140#390—21no—
141#390—21no—
142#390—21no—
143#390—21no—
144#390—21no—
145#390—21no—
146#390—21no—
147#390—21no—
148#390C2312yes—
149#390C2312yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389C2314yes—
165#389C2314yes—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#386—22no—
175#386—22no—
176#384—21no—
177#384—21no—
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2312yes—
181#384C2311yescharacteristic
182#382C2312yes—
183#380—21no—
184#380—21no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4C2314yes—
194C4×C4⋊C4C2314yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C2×#95—22no—
198C2×#95—22no—
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C22×Q8—28no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8C2412yes—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C2412yes—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2411yescharacteristic
233C22×C8C2418yes—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237C4⋊Q8—24no—
238C4⋊Q8—24no—
239C4⋊Q8#4118yes—
240#127—24no—
241#126—48no—
242#125—44no—
243#124—28no—
244#124—44no—
245#123—48no—
246C22⋊Q8—28no—
247#121—28no—
248#121—44no—
249#121—44no—
250C4×Q8—44no—
251C4×Q8—48no—
252C4×Q8—48no—
253C4×Q8—48no—
254C4×Q8—48no—
255C4×D4—44no—
256C4×D4—44no—
257C4×D4—48no—
258C4×D4—44no—
259#117—48no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2414yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42—48no—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—48no—
342#106—48no—
343C4⋊C8—28no—
344#103—216no—
345#103—48no—
346#102—216no—
347#102—48no—
348C22⋊C8—28no—
349#97—28no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360C24—22no—
361C24C22×D411yescharacteristic
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—432no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4AGL(1,ℤ/8)14yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4H(5,𝔽2)14yes—
432C22×C4C2511yescharacteristic
433C2×C8—28no—
434C2×C8—28no—
435C2×C8—28no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4C2×#4118yes—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—44no—
499C42—48no—
500C42—48no—
501C42#13714yes—
502C42C2×#4114yes—
503C422−1+414yes—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23#53512yes—
516C23C2×AGL(1,ℤ/8)12yes—
517C23#57512yes—
518C23C23×D411yescharacteristic
519Q8—416no—
520Q8—416no—
521Q8—48no—
522D4—48no—
523D4—84no—
524D4—88no—
525D4—84no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
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—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—416no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—88no—
578C2×C4—88no—
579C2×C4#52512yes—
580C2×C4#52512yes—
581C2×C4#53512yes—
582C2×C4#53612yes—
583C2×C4#53612yes—
584C2×C4C2×AGL(1,ℤ/8)12yes—
585C2×C4C2×AGL(1,ℤ/8)12yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4C2×#13712yes—
589C2×C4#57512yes—
590C2×C4C22×#4112yes—
591C2×C4C2×H(5,𝔽2)12yes—
592C2×C4C2×2−1+412yes—
593C8—48no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22#321414yes—
608C22C2×#52511yescharacteristic
609C22C2×#53511yescharacteristic
610C22C2×#53611yescharacteristic
611C22C22×AGL(1,ℤ/8)11yescharacteristic
612C22C22×#13711yescharacteristic
613C22C2×#57511yescharacteristic
614C22C2×#57511yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—48no—
627C4—41no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4#323814yes—
633C4#352214yes—
634C2—24no—
635C2—162no—
636C2#3524911yescharacteristic
637C2#6055811yescharacteristic
638C2#6056011yescharacteristic
639C2C2×#321411yescharacteristic
640C2#6080411yescharacteristic
641C2C2×#323811yescharacteristic
642C2#6086611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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. #33108C2semidirect
5. #30061C2semidirect
6. #29259C2semidirect
7. #29258C2semidirect
8. #29212C2semidirect
9. #28754C2semidirect
10. #27914C2semidirect
11. #25908C2semidirect
12. #24066C2semidirect
13. #22224C2semidirect
14. #20778C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×C4⋊Q82
#331082
#331062
#300612
#292592
#292582
#292122
#287542
#279142
#259082
#240662
#222242
#207782