Finite Group Databasebeta

← #6855920

Subgroups — #6855920

Order 512.

1733 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic, complemented
3#33106C211yesmaximal, characteristic
4#32246C214yesmaximal, complemented
5#30829C214yesmaximal, complemented
6#29760C212yesmaximal, complemented
7#28799C212yesmaximal, complemented
8#27973C211yesmaximal, characteristic
9#27778C212yesmaximal, complemented
10#27680C211yesmaximal, characteristic, complemented
11#26464C212yesmaximal, complemented
12#26026C212yesmaximal, complemented
13#25908C212yesmaximal, complemented
14#24299C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal, complemented
17C4×#483C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2212yes—
20C2×#489C2214yes—
21C2×#483C2211yescharacteristic
22C2×#480C2212yes—
23C2×#476C2214yes—
24#2856C2211yescharacteristic
25#2850C2212yes—
26#2597—22no—
27#2571C2211yescharacteristic
28#2551—22no—
29#2551—21no—
30#2551—21no—
31#2551C2212yes—
32#2551C2214yes—
33#2551C2212yes—
34#2548C2211yescharacteristic
35#2513—21no—
36#2513C2212yes—
37#2504—21no—
38#2447C2212yes—
39#2444—22no—
40C2×#395—22no—
41#2366C2211yescharacteristic
42#2361—21no—
43#2357—21no—
44#2355C2211yescharacteristic
45C4×C22⋊Q8C2211yescharacteristic
46C4×#121C2212yes—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#384C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2212yes—
53#2096C2212yes—
54#2085C2214yes—
55#2071C2214yes—
56#2068C2214yes—
57#2065C2214yes—
58#2058C2214yes—
59#2042C2214yes—
60#2024C2214yes—
61#2023C2214yes—
62#2023C2214yes—
63#1993C2214yes—
64#1987C2214yes—
65#1982C2212yes—
66#1957C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1943C2214yes—
70#1939C2212yes—
71#1936C2214yes—
72#1927C2212yes—
73#1925C2214yes—
74#1921C2214yes—
75#1920C2212yes—
76#1917C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1864C2214yes—
83#1853C2212yes—
84#1853C2211yescharacteristic
85#1834C2212yes—
86#1823C2211yescharacteristic
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1797C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—48no—
102C2×C4⋊Q8—48no—
103C2×C4⋊Q8C2314yes—
104C2×C4⋊Q8C2314yes—
105C2×#127C2311yescharacteristic
106C2×#124C2312yes—
107C2×C22⋊Q8C2311yescharacteristic
108C2×#121—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4C2312yes—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#489—48no—
128#483—24no—
129#480—28no—
130#476—48no—
131C2×#107C2314yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#397—22no—
151#397—22no—
152#397—22no—
153#397—22no—
154#395—24no—
155#395—24no—
156#395—24no—
157#395—24no—
158#394—22no—
159#391C2312yes—
160#391C2311yescharacteristic
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—24no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2314yes—
177#389C2314yes—
178#389C2311yescharacteristic
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—21no—
185#387—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—21no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2312yes—
197#382—21no—
198#382—21no—
199#382C2312yes—
200#381—22no—
201#380C2311yescharacteristic
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C2×#95—21no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—48no—
237C22×Q8—48no—
238C22×Q8—84no—
239C22×Q8C2411yescharacteristic
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261C4⋊Q8—48no—
262C4⋊Q8—416no—
263C4⋊Q8—48no—
264C4⋊Q8—48no—
265#127—24no—
266#124—44no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269#121—24no—
270#121—24no—
271#121—44no—
272#121—82no—
273C4×Q8—24no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—84no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4#4114yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C42—24no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—28no—
346#106—24no—
347#106—24no—
348#106—44no—
349C4⋊C8—24no—
350C4⋊C8—44no—
351#103—24no—
352#103—24no—
353#103—44no—
354#102—28no—
355#102—28no—
356#102—44no—
357#102—44no—
358C22⋊C8—24no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361#97—44no—
362C4×C8—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—22no—
383C24C22×D411yescharacteristic
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—84no—
393C2×Q8—84no—
394C2×Q8—832no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—81no—
407C2×D4—81no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4AGL(1,ℤ/8)14yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C42−1+412yes—
453C22×C4C2511yescharacteristic
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—81no—
495C4⋊C4#13714yes—
496C4⋊C4C2×#4114yes—
497C4⋊C4H(5,𝔽2)14yes—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—44no—
519C42—44no—
520C42—42no—
521C42—82no—
522C422−1+414yes—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#54812yes—
535C23C2×AGL(1,ℤ/8)12yes—
536C23#57712yes—
537C23C23×D411yescharacteristic
538Q8—42no—
539Q8—42no—
540Q8—816no—
541Q8—816no—
542D4—44no—
543D4—44no—
544D4—88no—
545D4—84no—
546D4—162no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—88no—
600C2×C4—164no—
601C2×C4#53612yes—
602C2×C4#53612yes—
603C2×C4#54812yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×#13714yes—
607C2×C4C2×#13712yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4C2×2−1+412yes—
611C2×C4C2×2−1+411yescharacteristic
612C8—44no—
613C8—42no—
614C8—42no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22#337314yes—
629C22C2×#53611yescharacteristic
630C22C2×#54811yescharacteristic
631C22#348411yescharacteristic
632C22C22×AGL(1,ℤ/8)11yescharacteristic
633C22C22×#13711yescharacteristic
634C22C2×#57711yescharacteristic
635C22#358511yescharacteristic
636C4—22no—
637C4—24no—
638C4—22no—
639C4—22no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—41no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4—84no—
650C4—81no—
651C4—81no—
652C4#323814yes—
653C4#351112yes—
654C4#355212yes—
655C2—24no—
656C2—162no—
657C2#3972011yescharacteristic
658C2C2×#323811yescharacteristic
659C2#6087311yescharacteristic
660C2C2×#337311yescharacteristic
661C2#6131911yescharacteristic
662C2#6158211yescharacteristic
663C2#6159311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2551C2semidirect
4. #32246C2semidirect
5. #30829C2semidirect
6. #29760C2semidirect
7. #28799C2semidirect
9. #27778C2semidirect
10. #27680C2semidirect
11. #26464C2semidirect
12. #26026C2semidirect
13. #25908C2semidirect
14. #24299C2semidirect
15. #22629C2semidirect
16. #22081C2semidirect
17. C4×#483C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#25512
#331062
#322462
#308292
#297602
#287992
#279732
#277782
#276802
#264642
#260262
#259082
#242992
#226292
#220812
C4×#4832