Finite Group Databasebeta

← #6816635

Subgroups — #6816635

Order 512.

1728 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3237C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic
3#36132C211yesmaximal, characteristic
4#33316C211yesmaximal, characteristic, complemented
5#31107C214yesmaximal, complemented
6#31099C214yesmaximal, complemented
7#29883C212yesmaximal, complemented
8#28725C211yesmaximal, characteristic, complemented
9#28694C211yesmaximal, characteristic, complemented
10#26046C212yesmaximal, complemented
11#26042C212yesmaximal, complemented
12#25818C212yesmaximal, complemented
13#25677C212yesmaximal, complemented
14#25017C212yesmaximal
15#25015C212yesmaximal
16C4×#491C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#533C2212yes—
20#3237—216no—
21C2×#501C2212yes—
22C2×#501C2212yes—
23C2×#501C2212yes—
24C2×#491C2212yes—
25C2×#491C2211yescharacteristic
26C2×#485C2214yes—
27C2×#485C2214yes—
28C2×#477C2214yes—
29C2×#475C2214yes—
30C2×#432C2212yes—
31#2868C2211yescharacteristic
32#2859C2211yescharacteristic
33#2857C2212yes—
34#2851C2212yes—
35#2552—21no—
36#2551C2211yescharacteristic
37#2551C2211yescharacteristic
38#2542—21no—
39#2538C2212yes—
40#2465—22no—
41#2465—22no—
42#2452—21no—
43#2452—21no—
44#2441—22no—
45#2439C2212yes—
46#2439C2212yes—
47#2429—22no—
48#2423C2214yes—
49#2365—21no—
50#2365—21no—
51#2365—21no—
52#2354—21no—
53#2354C2211yescharacteristic
54#2354C2211yescharacteristic
55Q8×C4⋊C4C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×#125C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2088C2214yes—
60#2087C2214yes—
61#2084C2212yes—
62#2083C2212yes—
63#2077C2214yes—
64#2075C2214yes—
65#2074C2214yes—
66#2058C2214yes—
67#2057C2214yes—
68#2036C2214yes—
69#2035C2214yes—
70#1993C2212yes—
71#1992C2212yes—
72#1992C2211yescharacteristic
73#1960C2211yescharacteristic
74#1954C2212yes—
75#1948C2212yes—
76#1947C2212yes—
77#1939C2212yes—
78#1935C2212yes—
79#1935C2212yes—
80#1924C2214yes—
81#1865C2214yes—
82#1863C2214yes—
83#1854C2214yes—
84#1854C2214yes—
85#1847C2214yes—
86#1845C2212yes—
87#1844C2212yes—
88#1836C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1767C2211yescharacteristic
92C4×#103C2212yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C4×#97C2211yescharacteristic
96#533—28no—
97#533—28no—
98#533—44no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#125—22no—
109C2×#125C2312yes—
110C2×#125C2311yescharacteristic
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2314yes—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2314yes—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C22×C42C2312yes—
128#501—28no—
129#501—28no—
130#501—44no—
131#491—28no—
132#491D418yescomplemented
133#485—216no—
134#485—216no—
135#477—216no—
136#475—216no—
137#432—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×#97C2311yescharacteristic
157C2×#97C2311yescharacteristic
158C2×C4×C8C2312yes—
159#399—24no—
160#398—24no—
161#396—24no—
162#396—22no—
163#396—22no—
164#395—22no—
165#395—22no—
166#393—22no—
167#393—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2312yes—
172#390—22no—
173#390—21no—
174#390—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—24no—
186#389C2311yescharacteristic
187#389C2311yescharacteristic
188#389C2314yes—
189#389C2312yes—
190#389C2312yes—
191#388—24no—
192#388—22no—
193#388—22no—
194#387—21no—
195#387—21no—
196#387—22no—
197#387—22no—
198#387—24no—
199#387—22no—
200#387—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2312yes—
210#384C2312yes—
211#382C2311yescharacteristic
212#382C2312yes—
213#380—22no—
214#380C2312yes—
215#380C2314yes—
216#380C2312yes—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4D412yes—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4C2312yes—
232C43C2311yescharacteristic
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#336C2314yes—
242#336C2312yes—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×D4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265C4⋊Q8C2×D414yes—
266#125—24no—
267#125—22no—
268#125—24no—
269#125—48no—
270#125C2×D414yes—
271C22⋊Q8—28no—
272C22⋊Q8—44no—
273C22⋊Q8—48no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276C22⋊Q8—84no—
277C4×Q8—28no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—48no—
282C4×Q8—48no—
283C4×Q8—48no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—42no—
287#117—24no—
288#117—24no—
289#117—42no—
290#117—42no—
291#117—48no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—28no—
358#107—28no—
359#107—44no—
360#106—28no—
361#106—28no—
362#106—44no—
363C4⋊C8—28no—
364C4⋊C8—24no—
365#103—28no—
366#103—28no—
367#103—24no—
368#103—24no—
369#103C2×D418yes—
370#103C2×D418yes—
371C22⋊C8—28no—
372C22⋊C8—28no—
373#97—24no—
374#97—24no—
375#97—22no—
376#97C2×D414yes—
377C4×C8—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—21no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24C22×D412yes—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8C22×D414yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8C22×D414yes—
453C2×C82−1+414yes—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4#13714yes—
496C4⋊C4C22×D412yes—
497C4⋊C4C22×D412yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—84no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—41no—
515C42—41no—
516C42—44no—
517C42—44no—
518C42—48no—
519C42—44no—
520C42—42no—
521C42—42no—
522C42C22×D412yes—
523C42H(5,𝔽2)12yes—
524C422−1+414yes—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23C23×D411yescharacteristic
533Q8—24no—
534Q8—24no—
535Q8—42no—
536Q8—42no—
537Q8—88no—
538Q8—88no—
539Q8—88no—
540D4—44no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—48no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—84no—
592C2×C4#53612yes—
593C2×C4#53612yes—
594C2×C4D4212yes—
595C2×C4D4212yes—
596C2×C4C2×#13714yes—
597C2×C4C2×#13712yes—
598C2×C4#57512yes—
599C2×C4C23×D411yescharacteristic
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+412yes—
602C2×C4C2×2−1+412yes—
603C8—24no—
604C8—24no—
605C8—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C2×#53611yescharacteristic
614C22C2×D4211yescharacteristic
615C22#349911yescharacteristic
616C22C22×#13711yescharacteristic
617C22C2×#57511yescharacteristic
618C22#358511yescharacteristic
619C22#358611yescharacteristic
620C4—22no—
621C4—24no—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—44no—
631C4—81no—
632C4—81no—
633C4—82no—
634C4—84no—
635C4—82no—
636C4—82no—
637C4#323714yescomplemented
638C4#349912yes—
639C4#355212yes—
640C2—82no—
641C2#3936011yescharacteristic
642C2C2×#323711yescharacteristic
643C2#6088611yescharacteristic
644C2D4×#13711yescharacteristic
645C2#6107011yescharacteristic
646C2#6250711yescharacteristic
647C2#6251011yescharacteristic
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
1. C2×#3237C2semidirect
4. #33316C2semidirect
5. #31107C2semidirect
6. #31099C2semidirect
7. #29883C2semidirect
8. #28725C2semidirect
9. #28694C2semidirect
10. #26046C2semidirect
11. #26042C2semidirect
12. #25818C2semidirect
13. #25677C2semidirect
16. C4×#491C2semidirect
132. #491D4semidirect
637. C4#3237semidirect

Maximal subgroups

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

subgroupindex
C2×#32372
#361342
#361322
#333162
#311072
#310992
#298832
#287252
#286942
#260462
#260422
#258182
#256772
#250172
#250152
C4×#4912
#212702