Finite Group Databasebeta

← #6566749

Subgroups — #6566749

Order 512.

1769 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#34334C211yesmaximal, characteristic, complemented
3#30115C212yesmaximal, complemented
4#29984C211yesmaximal, characteristic, complemented
5#28511C211yesmaximal, characteristic, complemented
6#28314C212yesmaximal, complemented
7#28157C211yesmaximal, characteristic, complemented
8#27586C212yesmaximal, complemented
9#27239C214yesmaximal, complemented
10#26805C214yesmaximal, complemented
11#25010C212yesmaximal, complemented
12#24239C212yesmaximal, complemented
13#23762C212yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15#23100C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22811C212yesmaximal, complemented
18C2×#555—21no—
19C2×D4×Q8C2211yescharacteristic
20C2×#480C2214yes—
21C2×#477C2214yes—
22C2×#461C2212yes—
23C2×#448C2214yes—
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2856C2212yes—
27#2563—21no—
28#2537—21no—
29#2537C2212yes—
30#2500—22no—
31#2491—22no—
32#2491C2214yes—
33#2369—21no—
34#2368—22no—
35#2365C2211yescharacteristic
36#2360C2212yes—
37#2355C2211yescharacteristic
38#2354—21no—
39#2354C2212yes—
40#2349—21no—
41D4×C4⋊C4C2211yescharacteristic
42Q8×C22⋊C4C2212yes—
43#2331—21no—
44#2329—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2295C2211yescharacteristic
49C2×#391C2212yes—
50#2285C2212yes—
51#2275—21no—
52C42×D4C2211yescharacteristic
53#2100C2212yes—
54#2095C2212yes—
55#2095C2212yes—
56#2095C2212yes—
57#2084C2211yescharacteristic
58#2075C2214yes—
59#2072C2211yescharacteristic
60#2072C2212yes—
61#2065C2212yes—
62#2057C2212yes—
63#2055C2212yes—
64#2041C2211yescharacteristic
65#2035C2214yes—
66#2033C2214yes—
67#2024C2212yes—
68#2014C2211yescharacteristic
69#2014C2212yes—
70#2000C2214yes—
71#1973C2214yes—
72#1973C2214yes—
73#1956C2212yes—
74#1954C2212yes—
75#1947C2212yes—
76#1928C2214yes—
77#1928C2214yes—
78#1925C2214yes—
79#1924C2214yes—
80#1880C2214yes—
81#1880C2214yes—
82#1852C2214yes—
83#1845C2212yes—
84#1831C2211yescharacteristic
85#1817C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1787C2214yes—
89#1776C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C23×Q8C2312yes—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98#555—48no—
99D4×Q8—24no—
100D4×Q8—24no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#124—24no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121C2314yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C2×#117—22no—
127C22×C4⋊C4C2312yes—
128C22×C42C2312yes—
129#480—48no—
130#477—48no—
131#461—28no—
132#448—48no—
133C4×SD16—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#394—24no—
154#394—24no—
155#393—24no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391C2312yes—
161#391C2311yescharacteristic
162#391C2311yescharacteristic
163#391C2312yes—
164#391C2312yes—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#390C2314yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#388—24no—
179#388—24no—
180#387—21no—
181#387—22no—
182#387—21no—
183#387—22no—
184#387—22no—
185#387—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—21no—
196#382—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376—21no—
223C2×#95—22no—
224C2×#95—22no—
225C43C2311yescharacteristic
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2314yes—
235#336C2311yescharacteristic
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C2×SD16—24no—
256C2×SD16—28no—
257C2×SD16—24no—
258C2×SD16—42no—
259C2×SD16—44no—
260C2×SD16—44no—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—28no—
265C4⋊Q8—22no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—82no—
269C4⋊Q8#4114yes—
270#127—42no—
271#124—416no—
272C22⋊Q8—216no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275#121—48no—
276#121—48no—
277#121—44no—
278#121—44no—
279#121—84no—
280C4×Q8—22no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8#4114yes—
284C4×D4—28no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—44no—
294#117—48no—
295#117—48no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#107—44no—
355#107—44no—
356#106—24no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359#103—24no—
360#103—28no—
361#103—28no—
362#103—44no—
363#103—44no—
364#102—24no—
365#102—24no—
366#102—42no—
367#102—44no—
368C22⋊C8—28no—
369C22⋊C8—44no—
370#97—44no—
371C4×C8—24no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—24no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95—21no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—22no—
393C24C22×D412yes—
394C2×Q8—28no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—88no—
407C2×Q8—88no—
408C2×Q8C2×#4112yes—
409C2×Q8C2×#4114yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4AGL(1,ℤ/8)14yes—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2511yescharacteristic
465SD16—48no—
466SD16—88no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8H(5,𝔽2)14yes—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C2×#4112yes—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—42no—
525C42—42no—
526C42—41no—
527C42—48no—
528C42—42no—
529C42—82no—
530C42—82no—
531C42—82no—
532C42#13714yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23C2×AGL(1,ℤ/8)14yes—
549C23C23×D411yescharacteristic
550Q8—24no—
551Q8—28no—
552Q8—44no—
553Q8—42no—
554Q8—88no—
555Q8—164no—
556D4—28no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—84no—
562D4—84no—
563D4—168no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—164no—
615C2×C4#52512yes—
616C2×C4#52512yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4#57812yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4#58512yes—
626C8—24no—
627C8—42no—
628C8—44no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C2×#52511yescharacteristic
644C22#348511yescharacteristic
645C22#349811yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57811yescharacteristic
649C22#358511yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—42no—
660C4—41no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—81no—
665C4—84no—
666C4#311214yes—
667C4#349412yes—
668C4#351412yes—
669C2—44no—
670C2—162no—
671C2#3602211yescharacteristic
672C2C2×#311211yescharacteristic
673C2#6060011yescharacteristic
674C2#6172111yescharacteristic
675C2#6173711yescharacteristic
676C2#6242411yescharacteristic
677C2#6244711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
2. #34334C2semidirect
3. #30115C2semidirect
4. #29984C2semidirect
5. #28511C2semidirect
6. #28314C2semidirect
7. #28157C2semidirect
8. #27586C2semidirect
9. #27239C2semidirect
10. #26805C2semidirect
11. #25010C2semidirect
12. #24239C2semidirect
13. #23762C2semidirect
14. #23678C2semidirect
15. #23100C2semidirect
16. #22839C2semidirect
17. #22811C2semidirect

Maximal subgroups

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

subgroupindex
#361872
#343342
#301152
#299842
#285112
#283142
#281572
#275862
#272392
#268052
#250102
#242392
#237622
#236782
#231002
#228392
#228112