Finite Group Databasebeta

← #6627755

Subgroups — #6627755

Order 512.

1745 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2C4×D4×Q8C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic
4#33026C212yesmaximal
5#29347C212yesmaximal, complemented
6#29287C211yesmaximal, characteristic, complemented
7#29255C212yesmaximal, complemented
8#28748C211yesmaximal, characteristic, complemented
9#26690C214yesmaximal, complemented
10#26320C212yesmaximal, complemented
11#26166C214yesmaximal
12#25929C212yesmaximal
13#23214C212yesmaximal, complemented
14#22996C212yesmaximal
15#22821C211yesmaximal, characteristic, complemented
16C4×#476C211yesmaximal, characteristic
17C4×#461C212yesmaximal
18C2×D4×Q8C2211yescharacteristic
19C22×C4×Q8—21no—
20C2×#487C2212yes—
21C2×#476C2211yescharacteristic
22C2×#461C2212yes—
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×C4×SD16C2214yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2582—21no—
29#2571—21no—
30#2570—22no—
31#2568C2212yes—
32#2511—22no—
33#2446C2214yes—
34#2365C2212yes—
35#2365C2211yescharacteristic
36#2365C2212yes—
37#2360—21no—
38#2360—22no—
39#2355—21no—
40#2355—21no—
41Q8×C4⋊C4C2211yescharacteristic
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2211yescharacteristic
44Q8×C22⋊C4—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2212yes—
49C4×C22⋊Q8—22no—
50C4×C22⋊Q8—21no—
51C42×Q8C2211yescharacteristic
52C42×D4C2212yes—
53C42×D4C2211yescharacteristic
54#2096C2214yes—
55#2088C2214yes—
56#2072C2214yes—
57#2043C2214yes—
58#2024C2214yes—
59#2024C2214yes—
60#1988C2212yes—
61#1987C2212yes—
62#1987C2211yescharacteristic
63#1984C2212yes—
64#1981C2212yes—
65#1981C2212yes—
66#1981C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1956C2212yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1933C2214yes—
74#1933C2214yes—
75#1927C2214yes—
76#1927C2214yes—
77#1927C2214yes—
78#1920C2212yes—
79#1916C2214yes—
80#1877C2214yes—
81#1877C2214yes—
82#1877C2214yes—
83#1865C2214yes—
84#1865C2214yes—
85#1864C2212yes—
86#1842C2211yescharacteristic
87#1831C2212yes—
88#1802C2211yescharacteristic
89C4×#106C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×C4⋊C8C2212yes—
93C4×#103C2214yes—
94C4×#102C2212yes—
95C23×Q8—21no—
96C22×SD16C2314yes—
97D4×Q8—24no—
98D4×Q8—24no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—48no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C42—22no—
126C22×C42—21no—
127#487—28no—
128#476—24no—
129#461—28no—
130#461—28no—
131#461—44no—
132C4×SD16—48no—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#400—24no—
153#400—24no—
154#398—24no—
155#391—22no—
156#391—21no—
157#391—22no—
158#391—21no—
159#391—22no—
160#391—21no—
161#391—22no—
162#391—22no—
163#391C2314yes—
164#391C2314yes—
165#391C2314yes—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—22no—
182#387—22no—
183#387—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2311yescharacteristic
194#382—22no—
195#382—22no—
196#382—22no—
197#380—21no—
198#380—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C43C2312yes—
225C43C2311yescharacteristic
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—28no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2414yes—
245C22×D4C2411yescharacteristic
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—21no—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—28no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8#4114yes—
269C22⋊Q8—28no—
270C22⋊Q8—216no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—42no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—48no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—44no—
282C4×Q8SD1618yescomplemented
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—42no—
290C4×D4—41no—
291C4×D4#4114yes—
292C4×D4#4114yes—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—48no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#106—24no—
357#106—42no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361C4⋊C8—22no—
362C4⋊C8—22no—
363C4⋊C8—42no—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—44no—
369#103—44no—
370#103—44no—
371#102—24no—
372#102—24no—
373C4×C8—24no—
374C4×C8—44no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—24no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C2×Q8—28no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—432no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8C2×SD1614yes—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×#4112yes—
419C2×D4C2×#4112yes—
420C2×D4H(5,𝔽2)12yes—
421C22×C4—28no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—81no—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455SD16—416no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4C2×SD1618yes—
501C4⋊C4C2×SD1614yes—
502C4⋊C4C2×#4112yes—
503C4⋊C4C2×#4114yes—
504C4⋊C4C2×#4112yes—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—22no—
518C42—21no—
519C42—41no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42—44no—
527C42—42no—
528C42—44no—
529C42—84no—
530C42C2×SD1618yes—
531C42C2×SD1614yes—
532C42C2×#4112yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C422−1+412yes—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23C23×D411yescharacteristic
544Q8—28no—
545Q8—416no—
546Q8—48no—
547Q8—44no—
548Q8—44no—
549Q8—44no—
550Q8—44no—
551Q8—84no—
552D4—22no—
553D4—22no—
554D4—22no—
555D4#55914yes—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—416no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—88no—
603C2×C4—84no—
604C2×C4#52512yes—
605C2×C4#52512yes—
606C2×C4#54812yes—
607C2×C4#54812yes—
608C2×C4C22×SD1614yes—
609C2×C4C22×SD1612yes—
610C2×C4C22×SD1614yes—
611C2×C4C22×SD1612yes—
612C2×C4C22×SD1612yes—
613C2×C4#57512yes—
614C2×C4#57812yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58511yescharacteristic
621C8—44no—
622C8—42no—
623C8—42no—
624C22—24no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22C2×#52511yescharacteristic
632C22C2×#54811yescharacteristic
633C22#348811yescharacteristic
634C22C23×SD1611yescharacteristic
635C22C2×#57511yescharacteristic
636C22C2×#57811yescharacteristic
637C22#358611yescharacteristic
638C4—24no—
639C4—22no—
640C4—22no—
641C4—24no—
642C4—22no—
643C4—41no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—82no—
651C4—84no—
652C4—81no—
653C4—82no—
654C4#310114yes—
655C4#335914yes—
656C4C2×#55911yescharacteristic
657C4#348011yescharacteristic
658C4#350711yescharacteristic
659C4#352211yescharacteristic
660C2—44no—
661C2#3658011yescharacteristic
662C2C2×#310111yescharacteristic
663C2#6059411yescharacteristic
664C2C2×#335911yescharacteristic
665C2#6132311yescharacteristic
666C2#6194811yescharacteristic
667C2#6198511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
5. #29347C2semidirect
6. #29287C2semidirect
7. #29255C2semidirect
8. #28748C2semidirect
9. #26690C2semidirect
10. #26320C2semidirect
13. #23214C2semidirect
15. #22821C2semidirect
282. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×D4×Q82
#331622
#330262
#293472
#292872
#292552
#287482
#266902
#263202
#261662
#259292
#232142
#229962
#228212
C4×#4762
C4×#4612