Finite Group Databasebeta

← #6434879

Subgroups — #6434879

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 638 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#30145C214yesmaximal, complemented
4#30091C212yesmaximal, complemented
5#30059C212yesmaximal, complemented
6#29966C212yesmaximal, complemented
7#29906C212yesmaximal, complemented
8#29754C212yesmaximal
9#29638C212yesmaximal, complemented
10#23835C212yesmaximal, complemented
11#23815C212yesmaximal, complemented
12#23774C214yesmaximal, complemented
13C4×#480C214yesmaximal, complemented
14#20777C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529—21no—
17C22×C4×Q8—21no—
18C2×#480C2214yes—
19#2861—21no—
20#2860—21no—
21#2860C2212yes—
22#2859C2211yescharacteristic
23#2858C2211yescharacteristic
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2854—21no—
27#2854C2212yes—
28#2850—21no—
29C4×C4⋊Q8—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8C2212yes—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×#127C2211yescharacteristic
39C4×#124—21no—
40C4×#121—21no—
41C4×#121C2212yes—
42C2×#391—21no—
43#2293—21no—
44#2293C2212yes—
45C2×#389—21no—
46C2×#389C2212yes—
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2212yes—
50C2×C43C2211yescharacteristic
51#2099C2214yes—
52#2099C2214yes—
53#2093C2214yes—
54#2086C2218yes—
55#2083C2214yes—
56#2083C2214yes—
57#2071C2214yes—
58#2067C2218yes—
59#2057C2214yes—
60#2057C2214yes—
61#2057C2214yes—
62#2041C2214yes—
63#2023C2214yes—
64#2023C2214yes—
65#1925C2218yes—
66#1918C2218yes—
67#1874C2214yes—
68#1852C2214yes—
69#1839C2212yes—
70#1831C2212yes—
71#1823C2214yes—
72#1817C2212yes—
73#1797C2212yes—
74C4×#107C2218yes—
75C4×C4⋊C8C2212yes—
76C4×#102C2218yes—
77C4×C22⋊C8C2212yes—
78#1744C2212yes—
79C23×Q8—21no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127C2311yescharacteristic
87#529—48no—
88C2×#126—22no—
89C2×#125—21no—
90C2×#124—21no—
91C2×#124—21no—
92C2×#123—22no—
93C2×C22⋊Q8—21no—
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—21no—
102C2×C4×Q8—48no—
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#480—216no—
123C2×#107C2318yes—
124C2×#107C2318yes—
125C2×C4⋊C8C2314yes—
126C2×#102C2318yes—
127C2×#102C2318yes—
128C2×C22⋊C8C2314yes—
129C2×C4×C8C2314yes—
130#391—22no—
131#391—22no—
132#391—22no—
133#391—22no—
134#391—21no—
135#391—21no—
136#390—21no—
137#390—21no—
138#390—21no—
139#390—21no—
140#390—21no—
141#390—21no—
142#390C2312yes—
143#390C2312yes—
144#390C2312yes—
145#390C2312yes—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389C2314yes—
157#389C2314yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#386—22no—
171#386—22no—
172#384—21no—
173#384—21no—
174#384C2311yescharacteristic
175#384C2312yes—
176#384C2312yes—
177#384C2311yescharacteristic
178#382—21no—
179#380—21no—
180#380C2312yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C2×#95—22no—
194C2×#95—22no—
195C43D411yescharacteristic
196C43D411yescharacteristic
197C43D411yescharacteristic
198C43D411yescharacteristic
199C43C2311yescharacteristic
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#340C2318yes—
204#340C2318yes—
205#340C2318yes—
206#340C2318yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211C22×Q8—28no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—21no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4C2412yes—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C22×C8C2418yes—
232C4⋊Q8—28no—
233C4⋊Q8—28no—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236C4⋊Q8—42no—
237#127—42no—
238#126—48no—
239#125—44no—
240#124—44no—
241#124—44no—
242#123—48no—
243C22⋊Q8—44no—
244#121—28no—
245#121—44no—
246#121—44no—
247C4×Q8—48no—
248C4×Q8—48no—
249C4×Q8—48no—
250C4×Q8—48no—
251C4×Q8—44no—
252C4×D4—28no—
253C4×D4—48no—
254C4×D4—44no—
255C4×D4—44no—
256#117—48no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4#4118yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2414yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
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×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—48no—
309C2×C42D818yes—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—216no—
339#107—48no—
340C4⋊C8—44no—
341#102—216no—
342#102—48no—
343C22⋊C8—28no—
344C4×C8—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355C24—22no—
356C24C22×D411yescharacteristic
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—432no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—42no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—42no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4C2×D814yes—
411C22×C4C2×D814yes—
412C22×C4C2×D814yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
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×C4C2×#4114yes—
427C22×C4C2511yescharacteristic
428C2×C8—28no—
429C2×C8—28no—
430C2×C8—28no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4C2×#4118yes—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—48no—
490C42—48no—
491C42—44no—
492C42C2×D814yes—
493C42C2×D814yes—
494C42C2×D814yes—
495C42C2×D814yes—
496C42#13714yes—
497C422−1+414yes—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23#52512yes—
510C23C22×D812yes—
511C23#57512yes—
512C23C23×D411yescharacteristic
513Q8—416no—
514Q8—416no—
515Q8—84no—
516D4—48no—
517D4—84no—
518D4—84no—
519D4—88no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
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—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—416no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—88no—
570C2×C4—88no—
571C2×C4#52512yes—
572C2×C4#53612yes—
573C2×C4#53612yes—
574C2×C4C22×D812yes—
575C2×C4C22×D812yes—
576C2×C4C22×D812yes—
577C2×C4C22×D812yes—
578C2×C4C22×D812yes—
579C2×C4C22×D812yes—
580C2×C4C2×#13712yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13712yes—
583C2×C4#57512yes—
584C2×C4C22×#4112yes—
585C2×C4C2×2−1+412yes—
586C2×C4#58512yes—
587C8—48no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22#310014yes—
602C22C2×#52511yescharacteristic
603C22C2×#53611yescharacteristic
604C22#344911yescharacteristic
605C22C23×D811yescharacteristic
606C22C22×#13711yescharacteristic
607C22C2×#57511yescharacteristic
608C22C2×#57511yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—41no—
620C4—48no—
621C4—41no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4#322614yes—
627C4#346114yes—
628C2—24no—
629C2—162no—
630C2#3460211yescharacteristic
631C2C2×#310011yescharacteristic
632C2#6056011yescharacteristic
633C2C2×#322611yescharacteristic
634C2#6085411yescharacteristic
635C2#6094011yescharacteristic
636C2#6095011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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. #30145C2semidirect
4. #30091C2semidirect
5. #30059C2semidirect
6. #29966C2semidirect
7. #29906C2semidirect
9. #29638C2semidirect
10. #23835C2semidirect
11. #23815C2semidirect
12. #23774C2semidirect
13. C4×#480C2semidirect
14. #20777C2semidirect

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
#301452
#300912
#300592
#299662
#299062
#297542
#296382
#238352
#238152
#237742
C4×#4802
#207772