Finite Group Databasebeta

← #7533201

Subgroups — #7533201

Order 512.

1961 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59815C211yesmaximal, characteristic
2#36559C211yesmaximal, characteristic, complemented
3#31386C214yesmaximal, complemented
4#29218C211yesmaximal, characteristic, complemented
5#28716C211yesmaximal, characteristic, complemented
6#28599C212yesmaximal, complemented
7#28141C212yesmaximal, complemented
8#26300C212yesmaximal, complemented
9#26202C214yesmaximal, complemented
10#25569C212yesmaximal, complemented
11#24967C212yesmaximal, complemented
12#24120C212yesmaximal, complemented
13#23173C211yesmaximal, characteristic, complemented
14#22487C212yesmaximal, complemented
15#22485C212yesmaximal, complemented
16C4×#475C211yesmaximal, characteristic, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#550—21no—
20C2×#547C2212yes—
21C2×#486C2212yes—
22C2×#475C2211yescharacteristic
23C2×#460C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×C8×D4C2212yes—
26C2×C8×D4C2212yes—
27#2868C2211yescharacteristic
28#2859—21no—
29#2682—22no—
30#2566—22no—
31#2563C2212yes—
32#2541C2212yes—
33#2538—21no—
34#2538C2212yes—
35#2538C2212yes—
36#2537C2212yes—
37#2452—22no—
38#2452—22no—
39#2445—22no—
40#2416C2214yes—
41#2407—22no—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127—21no—
47C4×#127C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C42×Q8C2211yescharacteristic
50#2100C2214yes—
51#2077C2214yes—
52#2077C2214yes—
53#2071C2214yes—
54#2034C2214yes—
55#2034C2214yes—
56#2000C2214yes—
57#2000C2214yes—
58#1990C2212yes—
59#1990C2212yes—
60#1988C2211yescharacteristic
61#1987C2212yes—
62#1987C2212yes—
63#1982C2211yescharacteristic
64#1954C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1948C2211yescharacteristic
68#1926C2212yes—
69#1919C2212yes—
70#1893C2214yes—
71#1893C2214yes—
72#1892C2214yes—
73#1880C2212yes—
74#1874C2214yes—
75#1874C2214yes—
76#1872C2214yes—
77#1869C2214yes—
78#1865C2212yes—
79#1853C2214yes—
80#1834C2212yes—
81#1824C2214yes—
82#1823C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85#1802C2212yes—
86#1802C2212yes—
87C4×#106C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#102C2212yes—
92C2×#340C2214yes—
93C22×#41—22no—
94C22×SD16C2312yes—
95C23×C8C2314yes—
96#554—28no—
97#554—44no—
98#550—48no—
99#547—28no—
100#547—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127—22no—
106C2×#127D411yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#125—22no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×#121—22no—
114C2×#121—24no—
115C2×#121C2314yes—
116C2×#121C2314yes—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2314yes—
133#486—44no—
134#475—24no—
135#460—24no—
136C4×SD16—44no—
137C8×D4—44no—
138C8×D4—44no—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2312yes—
159#396—24no—
160#396—24no—
161#396—24no—
162#394—24no—
163#394—24no—
164#393—24no—
165#392—24no—
166#391—21no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390—21no—
172#390—21no—
173#390D412yes—
174#390D412yes—
175#390D412yes—
176#390C2312yes—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389C2312yes—
184#386—24no—
185#385—24no—
186#385—24no—
187#385—24no—
188#384—21no—
189#384—21no—
190#384D411yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#380—22no—
197#380C2314yes—
198#380C2314yes—
199#380C2314yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C43—21no—
209C43C2311yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2314yes—
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#336C2312yes—
223#336C2312yes—
224C2×#41—48no—
225C2×#41—48no—
226C2×#41—84no—
227C22×Q8—21no—
228C22×Q8—22no—
229C22×Q8C2411yescharacteristic
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2×D412yes—
235C22×D4C2×D412yes—
236C22×D4C2×D412yes—
237C22×D4C2412yes—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—22no—
241C23×C4C2414yes—
242C2×SD16—24no—
243C2×SD16—42no—
244C2×SD16—42no—
245C22×C8—44no—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—82no—
256C4⋊Q8—82no—
257C4⋊Q8#4114yes—
258#127—22no—
259#127—22no—
260#127—48no—
261#127SD1618yescomplemented
262#125—48no—
263#123—416no—
264C22⋊Q8—28no—
265C22⋊Q8—416no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268#121—28no—
269#121—44no—
270#121—48no—
271#121—416no—
272#121—44no—
273#121—44no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—41no—
278C4×Q8—41no—
279C4×Q8—44no—
280C4×Q8#4114yescomplemented
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—42no—
289C4×D4—48no—
290C4×D4—48no—
291C4×D4—44no—
292C4×D4—82no—
293C4×D4—84no—
294C4×D4—84no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4C2×D414yes—
336C2×C22⋊C4C2×D414yes—
337C2×C22⋊C4C2×D414yes—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#106—24no—
353#106—42no—
354#106—42no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359#103—24no—
360#103—42no—
361#103—42no—
362#102—24no—
363#102—42no—
364#102—42no—
365#102—44no—
366#102—44no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369C22⋊C8—44no—
370C4×C8—24no—
371C4×C8—24no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380C24—22no—
381C24C22×D414yes—
382#41—832no—
383C2×Q8—24no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—82no—
393C2×Q8—82no—
394C2×Q8C2×#4112yes—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—84no—
419C2×D4C2×SD1618yes—
420C2×D4C2×SD1618yes—
421C2×D4C2×SD1618yes—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D414yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2511yescharacteristic
453SD16—48no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—88no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C2×#4112yes—
500C4⋊C4C2×#4114yes—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—41no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42—42no—
524C42—42no—
525C42C2×SD1614yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C42H(5,𝔽2)12yes—
529C42H(5,𝔽2)12yes—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C22×SD1618yes—
544C23C23×D411yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—41no—
548Q8—41no—
549Q8—48no—
550Q8—164no—
551D4—216no—
552D4—216no—
553D4—48no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—84no—
559D4—816no—
560D4—84no—
561D4—168no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—816no—
603C2×C4—88no—
604C2×C4—168no—
605C2×C4#54612yes—
606C2×C4#54612yes—
607C2×C4#54612yes—
608C2×C4#54612yes—
609C2×C4C22×SD1612yes—
610C2×C4C22×SD1612yes—
611C2×C4C22×SD1612yes—
612C2×C4#57612yes—
613C2×C4#57812yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4#58511yescharacteristic
619C2×C4#58511yescharacteristic
620C8—42no—
621C8—42no—
622C8—42no—
623C8—42no—
624C22—28no—
625C22—28no—
626C22—28no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22C2×#54611yescharacteristic
639C22C2×#54611yescharacteristic
640C22#348511yescharacteristic
641C22C23×SD1611yescharacteristic
642C22C2×#57611yescharacteristic
643C22C2×#57811yescharacteristic
644C22#358511yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—41no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—82no—
654C4—81no—
655C4—84no—
656C4—81no—
657C4—82no—
658C4#329514yes—
659C4#329814yes—
660C4#348011yescharacteristic
661C4#350611yescharacteristic
662C4#351711yescharacteristic
663C4#354611yescharacteristic
664C2—48no—
665C2—162no—
666C2#5903311yescharacteristic
667C2C2×#329511yescharacteristic
668C2C2×#329811yescharacteristic
669C2#6120311yescharacteristic
670C2#6120511yescharacteristic
671C2#6169711yescharacteristic
672C2#6179111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
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
2. #36559C2semidirect
3. #31386C2semidirect
4. #29218C2semidirect
5. #28716C2semidirect
6. #28599C2semidirect
7. #28141C2semidirect
8. #26300C2semidirect
9. #26202C2semidirect
10. #25569C2semidirect
11. #24967C2semidirect
12. #24120C2semidirect
13. #23173C2semidirect
14. #22487C2semidirect
15. #22485C2semidirect
16. C4×#475C2semidirect
17. C4×#460C2semidirect
261. #127SD16semidirect
280. C4×Q8#41semidirect

Maximal subgroups

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

subgroupindex
#598152
#365592
#313862
#292182
#287162
#285992
#281412
#263002
#262022
#255692
#249672
#241202
#231732
#224872
#224852
C4×#4752
C4×#4602