Finite Group Databasebeta

← #6623899

Subgroups — #6623899

Order 512.

1667 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
1#36594C211yesmaximal, characteristic
2C4×#557C211yesmaximal, characteristic, complemented
3#33198C211yesmaximal, characteristic, complemented
4#33031C212yesmaximal
5#28511C211yesmaximal, characteristic, complemented
6#28386C212yesmaximal
7#28252C211yesmaximal, characteristic
8#28206C212yesmaximal, complemented
9#27503C214yesmaximal, complemented
10#27083C214yesmaximal, complemented
11#25056C212yesmaximal, complemented
12#24187C212yesmaximal, complemented
13#23100C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22811C212yesmaximal, complemented
16#21616C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#556—21no—
20C2×#477C2212yes—
21C2×#477C2212yes—
22C2×#477C2212yes—
23C2×#460C2211yescharacteristic
24C2×#460C2212yes—
25C2×#461C2212yes—
26C2×C4×SD16C2212yes—
27#2868C2211yescharacteristic
28#2858C2212yes—
29#2547—21no—
30#2545—22no—
31#2538—21no—
32#2538C2212yes—
33#2512—22no—
34#2512C2214yes—
35#2370—22no—
36#2370—21no—
37#2365C2211yescharacteristic
38#2365C2212yes—
39#2355C2211yescharacteristic
40#2355C2212yes—
41#2354—21no—
42#2354C2212yes—
43#2351—21no—
44Q8×C4⋊C4C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C4×#125—21no—
50C4×#125—22no—
51C42×Q8—21no—
52C42×Q8C2212yes—
53C42×Q8C2211yescharacteristic
54C42×Q8C2211yescharacteristic
55#2095C2214yes—
56#2093C2214yes—
57#2083C2214yes—
58#2055C2214yes—
59#2043C2214yes—
60#2013C2214yes—
61#1987C2212yes—
62#1984C2212yes—
63#1982C2211yescharacteristic
64#1973C2214yes—
65#1973C2214yes—
66#1956C2212yes—
67#1948C2211yescharacteristic
68#1947C2212yes—
69#1947C2212yes—
70#1928C2214yes—
71#1928C2214yes—
72#1926C2212yes—
73#1925C2214yes—
74#1925C2214yes—
75#1918C2214yes—
76#1917C2214yes—
77#1880C2214yes—
78#1880C2214yes—
79#1880C2212yes—
80#1865C2214yes—
81#1845C2212yes—
82#1842C2211yescharacteristic
83#1831C2212yes—
84#1803C2212yes—
85#1803C2211yescharacteristic
86#1803C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#102C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2314yes—
97#557—24no—
98#557—24no—
99#556—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#126—24no—
107C2×#125—21no—
108C2×#125—22no—
109C2×#124—22no—
110C2×#124—22no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×#117—22no—
128#477—28no—
129#477—44no—
130#477—44no—
131#461—28no—
132#460—24no—
133#460—44no—
134C4×SD16—28no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#397—24no—
156#396—24no—
157#396—24no—
158#391—22no—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#390—21no—
163#390—21no—
164#390—22no—
165#390—22no—
166#390C2312yes—
167#390C2314yes—
168#390C2314yes—
169#389—21no—
170#389—22no—
171#389—21no—
172#389—22no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—22no—
178#389C2314yes—
179#388—24no—
180#388—24no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#384C2312yes—
194#382—21no—
195#382—22no—
196#380—22no—
197#380—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C43—22no—
226C43—21no—
227C43C2311yescharacteristic
228C43C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—22no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×Q8C2411yescharacteristic
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2412yes—
245C23×C4—22no—
246C2×SD16—24no—
247C2×SD16—28no—
248C2×SD16—24no—
249C2×SD16—42no—
250C2×SD16—44no—
251C2×SD16—44no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262C4⋊Q8#4114yes—
263#127—22no—
264#127—22no—
265#126—416no—
266#125—216no—
267#125—28no—
268#124—48no—
269#124—84no—
270C22⋊Q8—48no—
271C4×Q8—28no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—24no—
281C4×Q8—22no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8—42no—
286C4×Q8—41no—
287C4×Q8—48no—
288C4×Q8—44no—
289C4×Q8#4114yes—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—44no—
293C4×D4—44no—
294#117—48no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#107—44no—
363#107—44no—
364#106—24no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—22no—
368C4⋊C8—42no—
369C4⋊C8#4114yes—
370#103—24no—
371#103—28no—
372#103—28no—
373#103—44no—
374#103—44no—
375#102—24no—
376#102—24no—
377#102—24no—
378#102—44no—
379#97—44no—
380C4×C8—24no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8AGL(1,ℤ/8)14yes—
414C2×Q8C2×#4114yes—
415C2×Q8C2×#4112yes—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2511yescharacteristic
450SD16—48no—
451SD16—88no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8C2×#4114yes—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—48no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C2×#4112yes—
500C4⋊C42−1+412yes—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—28no—
509C42—24no—
510C42—22no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—22no—
519C42—22no—
520C42—21no—
521C42—42no—
522C42—41no—
523C42—42no—
524C42—44no—
525C42—48no—
526C42—42no—
527C42—44no—
528C42—44no—
529C42—42no—
530C42—42no—
531C42—81no—
532C42—82no—
533C42#13714yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C42H(5,𝔽2)12yes—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C23×D411yescharacteristic
546Q8—28no—
547Q8—22no—
548Q8—24no—
549Q8—22no—
550Q8—24no—
551Q8—42no—
552Q8—41no—
553Q8—42no—
554Q8—41no—
555Q8—88no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—84no—
561D4—84no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—88no—
607C2×C4#52512yes—
608C2×C4#52512yes—
609C2×C4#55012yes—
610C2×C4#55012yes—
611C2×C4C2×AGL(1,ℤ/8)14yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×#13712yes—
614C2×C4C2×#13712yes—
615C2×C4C2×#13712yes—
616C2×C4#57812yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C2×C4#58511yescharacteristic
623C8—24no—
624C8—42no—
625C8—44no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C2×#52511yescharacteristic
634C22C2×#55011yescharacteristic
635C22#348911yescharacteristic
636C22C22×AGL(1,ℤ/8)11yescharacteristic
637C22C22×#13711yescharacteristic
638C22C2×#57811yescharacteristic
639C22#358511yescharacteristic
640C4—24no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4—84no—
656C4#311214yes—
657C4#339814yes—
658C4#347511yescharacteristic
659C4#348211yescharacteristic
660C4#351111yescharacteristic
661C4#351411yescharacteristic
662C2—162no—
663C2#3650411yescharacteristic
664C2C2×#311211yescharacteristic
665C2#6060011yescharacteristic
666C2C2×#339811yescharacteristic
667C2#6140211yescharacteristic
668C2#6211411yescharacteristic
669C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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
2. C4×#557C2semidirect
3. #33198C2semidirect
5. #28511C2semidirect
8. #28206C2semidirect
9. #27503C2semidirect
10. #27083C2semidirect
11. #25056C2semidirect
12. #24187C2semidirect
13. #23100C2semidirect
14. #22825C2semidirect
15. #22811C2semidirect
16. #21616C2semidirect

Maximal subgroups

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

subgroupindex
#365942
C4×#5572
#331982
#330312
#285112
#283862
#282522
#282062
#275032
#270832
#250562
#241872
#231002
#228252
#228112
#216162
C4×#4602