Finite Group Databasebeta

← #7533581

Subgroups — #7533581

Order 512.

2049 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3328C212yesmaximal, complemented
2#59817C211yesmaximal, characteristic
3#36594C211yesmaximal, characteristic
4#30423C214yesmaximal, complemented
5#28717C211yesmaximal, characteristic, complemented
6#28512C211yesmaximal, characteristic, complemented
7#28232C212yesmaximal, complemented
8#25884C212yesmaximal, complemented
9#25348C212yesmaximal, complemented
10#25015C212yesmaximal, complemented
11#24892C214yesmaximal, complemented
12#24243C212yesmaximal, complemented
13#24123C212yesmaximal, complemented
14#23176C211yesmaximal, characteristic
15#22430C212yesmaximal, complemented
16C4×#462C211yesmaximal, characteristic, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×D4×Q8C2212yes—
20C2×D4×Q8C2212yes—
21#3328—216no—
22C2×#487C2212yes—
23C2×#485C2214yes—
24C2×#475C2212yes—
25C2×#475C2212yes—
26C2×#470C2214yes—
27C2×#462C2211yescharacteristic
28C2×#462C2211yescharacteristic
29C2×#451C2214yes—
30C2×C4×Q16C2212yes—
31C2×C4×Q16C2212yes—
32C2×#436C2212yes—
33C2×#436C2212yes—
34C2×#419C2214yes—
35#2868C2211yescharacteristic
36#2858—21no—
37#2683—22no—
38#2551—21no—
39#2547C2212yes—
40#2545—22no—
41#2538C2212yes—
42#2538C2212yes—
43#2538C2212yes—
44#2538C2212yes—
45#2512—22no—
46#2512—22no—
47#2452—22no—
48#2452—22no—
49#2409C2214yes—
50#2366—21no—
51#2354C2212yes—
52#2354C2212yes—
53C4×C4⋊Q8—21no—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C42×Q8C2211yescharacteristic
58#2096C2214yes—
59#2077C2214yes—
60#2072C2214yes—
61#2036C2214yes—
62#2035C2214yes—
63#2035C2214yes—
64#2035C2214yes—
65#2001C2214yes—
66#1990C2212yes—
67#1981C2211yescharacteristic
68#1981C2211yescharacteristic
69#1956C2212yes—
70#1954C2212yes—
71#1954C2212yes—
72#1947C2211yescharacteristic
73#1947C2211yescharacteristic
74#1928C2212yes—
75#1928C2212yes—
76#1894C2214yes—
77#1878C2212yes—
78#1878C2212yes—
79#1873C2214yes—
80#1870C2214yes—
81#1854C2214yes—
82#1844C2212yes—
83#1809C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2211yescharacteristic
88#1787C2214yes—
89#1787C2214yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#103C2212yes—
94C23×Q8C2314yes—
95C22×Q16C2312yes—
96C22×Q16C2312yes—
97C22×M16C2314yes—
98#556—28no—
99#556—44no—
100D4×Q8—28no—
101D4×Q8—28no—
102D4×Q8—44no—
103D4×Q8—44no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×#127D411yescharacteristic
112C2×#126—24no—
113C2×#124—22no—
114C2×#124—22no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8C2314yes—
118C2×C22⋊Q8C2314yes—
119C2×C22⋊Q8C2314yes—
120C2×C4×Q8—22no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×#117C2314yes—
132#487—28no—
133#485—216no—
134#475—28no—
135#475—44no—
136#470—216no—
137#462—24no—
138#462D418yescomplemented
139#451—216no—
140C4×Q16—28no—
141C4×Q16—44no—
142#436—28no—
143#436—44no—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2311yescharacteristic
152#419—216no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C2×C22⋊C8C2314yes—
168C2×C4×C8C2312yes—
169C2×C4×C8C2312yes—
170#397—24no—
171#396—24no—
172#396—24no—
173#396—24no—
174#396—24no—
175#395—24no—
176#393—24no—
177#391—22no—
178#391C2312yes—
179#391C2312yes—
180#390—22no—
181#390—22no—
182#390—22no—
183#390—22no—
184#390—22no—
185#390—22no—
186#390D412yes—
187#390D412yes—
188#390D412yes—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#388—24no—
198#388—24no—
199#386—24no—
200#385—24no—
201#384—21no—
202#384—21no—
203#384D411yescharacteristic
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#384C2312yes—
208#380C2314yes—
209#380C2314yes—
210#380C2314yes—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C43—21no—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2312yes—
233C22×Q8—22no—
234C22×Q8—24no—
235C22×Q8—28no—
236C22×Q8—28no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2412yes—
244C22×D4C2×D412yes—
245C22×D4C2×D412yes—
246C22×D4C2×D412yes—
247C23×C4—24no—
248C23×C4C2414yes—
249C2×Q16—24no—
250C2×Q16—24no—
251C2×Q16—42no—
252C2×Q16—42no—
253C2×Q16—42no—
254C2×Q16C2×D418yes—
255C2×M16—28no—
256C2×M16—44no—
257C2×M16—44no—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268C4⋊Q8—44no—
269C4⋊Q8—82no—
270C4⋊Q8C2×D414yes—
271C4⋊Q8#4114yes—
272#127—24no—
273#126—416no—
274#124—48no—
275#124—84no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278C22⋊Q8—28no—
279C22⋊Q8—416no—
280C22⋊Q8—416no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—22no—
291C4×Q8—41no—
292C4×Q8—41no—
293C4×Q8—48no—
294C4×Q8C2×D414yes—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—48no—
302C4×D4—48no—
303#117—28no—
304#117—44no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4C2×D414yes—
344C2×C22⋊C4C2×D414yes—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—42no—
363#106—28no—
364#106—44no—
365C4⋊C8—22no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368C4⋊C8C2×D414yes—
369#103—28no—
370#103—24no—
371#103—28no—
372#103—24no—
373#103—28no—
374#103—24no—
375#103—42no—
376#103—42no—
377#103—42no—
378#103—42no—
379#103—44no—
380#103C2×D418yes—
381C22⋊C8—28no—
382C22⋊C8—44no—
383C22⋊C8—44no—
384C4×C8—24no—
385C4×C8—24no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—24no—
396C24C22×D414yes—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—416no—
418C2×Q8—84no—
419C2×Q8—88no—
420C2×Q8—88no—
421C2×Q8—88no—
422C2×Q8C22×D414yes—
423C2×Q8C22×D412yes—
424C2×Q8C2×#4114yes—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—42no—
433C2×D4—42no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—84no—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D414yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460Q16—48no—
461Q16—48no—
462M16—88no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8C22×D414yes—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4C22×D412yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—21no—
526C42—41no—
527C42—48no—
528C42—42no—
529C42—42no—
530C42—44no—
531C42—44no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42#13714yes—
536C42C22×D412yes—
537C42C2×#4112yes—
538C42H(5,𝔽2)12yes—
539C42H(5,𝔽2)12yes—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23C23×D411yescharacteristic
548Q8—22no—
549Q8—24no—
550Q8—24no—
551Q8—22no—
552Q8—24no—
553Q8—24no—
554Q8—42no—
555Q8—41no—
556Q8—42no—
557Q8—41no—
558Q8—42no—
559Q8—42no—
560Q8—816no—
561Q8—88no—
562D4—48no—
563D4—48no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—24no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—816no—
609C2×C4D4212yes—
610C2×C4D4212yes—
611C2×C4#54612yes—
612C2×C4#54612yes—
613C2×C4C2×#13712yes—
614C2×C4C2×#13712yes—
615C2×C4C2×#13712yes—
616C2×C4#57712yes—
617C2×C4#57812yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4#58511yescharacteristic
623C2×C4#58511yescharacteristic
624C8—24no—
625C8—42no—
626C8—42no—
627C8—42no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22C2×D4211yescharacteristic
636C22C2×#54611yescharacteristic
637C22#348511yescharacteristic
638C22C22×#13711yescharacteristic
639C22C2×#57711yescharacteristic
640C22C2×#57811yescharacteristic
641C22#358611yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4—84no—
655C4—82no—
656C4#327814yes—
657C4#332814yescomplemented
658C4D4×#4111yescharacteristic
659C4#349611yescharacteristic
660C4#351411yescharacteristic
661C4#354611yescharacteristic
662C2—84no—
663C2#5903511yescharacteristic
664C2C2×#327811yescharacteristic
665C2#6109711yescharacteristic
666C2C2×#332811yescharacteristic
667C2#6120211yescharacteristic
668C2#6172311yescharacteristic
669C2#6178911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3328C2semidirect
4. #30423C2semidirect
5. #28717C2semidirect
6. #28512C2semidirect
7. #28232C2semidirect
8. #25884C2semidirect
9. #25348C2semidirect
10. #25015C2semidirect
11. #24892C2semidirect
12. #24243C2semidirect
13. #24123C2semidirect
15. #22430C2semidirect
16. C4×#462C2semidirect
17. C4×#462C2semidirect
138. #462D4semidirect
657. C4#3328semidirect

Maximal subgroups

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

subgroupindex
C2×#33282
#598172
#365942
#304232
#287172
#285122
#282322
#258842
#253482
#250152
#248922
#242432
#241232
#231762
#224302
C4×#4622