Finite Group Databasebeta

← #6471245

Subgroups — #6471245

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33094C212yesmaximal, complemented
4#32957C211yesmaximal, characteristic, complemented
5#32903C211yesmaximal, characteristic
6#29953C211yesmaximal, characteristic, complemented
7#29713C211yesmaximal, characteristic
8#29624C212yesmaximal, complemented
9#27083C214yesmaximal, complemented
10#26919C214yesmaximal, complemented
11#26773C214yesmaximal, complemented
12#26605C214yesmaximal, complemented
13#22047C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21454C212yesmaximal, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#477C2212yes—
21C2×#460C2212yes—
22C2×#448C2212yes—
23C2×#448C2212yes—
24C2×#440C2214yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2512C2212yes—
33#2506C2212yes—
34#2452C2212yes—
35#2445C2212yes—
36#2365—22no—
37#2365—22no—
38#2365C2214yes—
39#2365C2214yes—
40#2325—21no—
41C4×#121C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#389—21no—
45C2×#387—21no—
46C2×#387—21no—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2095C2212yes—
51#2072C2212yes—
52#2068C2214yes—
53#2055C2214yes—
54#2043C2214yes—
55#2043C2214yes—
56#2024C2212yes—
57#2014C2212yes—
58#2013C2214yes—
59#2013C2214yes—
60#1973C2214yes—
61#1935C2214yes—
62#1928C2214yes—
63#1925C2214yes—
64#1925C2214yes—
65#1918C2214yes—
66#1918C2214yes—
67#1917C2214yes—
68#1917C2214yes—
69#1880C2214yes—
70#1880C2214yes—
71#1877C2214yes—
72#1874C2212yes—
73#1874C2212yes—
74#1863C2214yes—
75#1863C2214yes—
76#1852C2212yes—
77#1852C2212yes—
78#1817C2212yes—
79#1803C2212yes—
80#1803C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1787C2212yes—
84#1787C2212yes—
85#1763C2214yes—
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C22×SD16C2314yes—
90C22×SD16C2314yes—
91#539—48no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8C2314yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#126—22no—
99C2×#125—22no—
100C2×#123—22no—
101C2×#123—22no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—44no—
111C2×C4×Q8—44no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C42—21no—
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#477—44no—
132#460—44no—
133#448—28no—
134#448—28no—
135#440—48no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×#97C2314yes—
151#400—22no—
152#398—22no—
153#397—22no—
154#396—22no—
155#396—22no—
156#394—22no—
157#394—22no—
158#393—22no—
159#391—21no—
160#391—22no—
161#391—21no—
162#391—21no—
163#391C2312yes—
164#391C2312yes—
165#391C2314yes—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—24no—
173#389—24no—
174#389—21no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#386—22no—
189#386—22no—
190#386—22no—
191#385—22no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4—24no—
197C4×C4⋊C4—24no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220#336C2314yes—
221#336C2314yes—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8—42no—
231C22×Q8#4114yes—
232C22×Q8#4114yes—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16—44no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—216no—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—42no—
263#127—42no—
264#126—48no—
265#125—48no—
266#123—48no—
267#123—48no—
268C22⋊Q8—48no—
269C22⋊Q8—44no—
270#121—24no—
271#121—24no—
272#121—44no—
273#121—44no—
274C4×Q8—28no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—48no—
283C4×D4—44no—
284C4×D4—48no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—22no—
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—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42—44no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349#107—28no—
350#107—44no—
351C4⋊C8—42no—
352C4⋊C8—42no—
353#103—28no—
354#103—28no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#102—28no—
360#102—44no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363#97—44no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—22no—
382C24C22×D411yescharacteristic
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—48no—
394C2×Q8—48no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—42no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8C2×#4114yes—
405C2×Q8C2×#4114yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4—81no—
418C2×D4—81no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4AGL(1,ℤ/8)14yes—
460C22×C4#13714yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473SD16—416no—
474SD16—88no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C4⋊C4—24no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4H(5,𝔽2)14yes—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—41no—
526C42—41no—
527C42—42no—
528C42—44no—
529C42—44no—
530C42—48no—
531C42—44no—
532C42—48no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23#52512yes—
545C23C2×AGL(1,ℤ/8)12yes—
546C23C2×#13712yes—
547C23C23×D411yescharacteristic
548Q8—28no—
549Q8—24no—
550Q8—24no—
551Q8—48no—
552Q8—48no—
553Q8—42no—
554Q8—42no—
555Q8—44no—
556Q8—162no—
557D4—44no—
558D4—44no—
559D4—88no—
560D4—162no—
561D4—162no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—84no—
618C2×C4—164no—
619C2×C4#52512yes—
620C2×C4C2×AGL(1,ℤ/8)14yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)12yes—
626C2×C4#58512yes—
627C8—44no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22#311214yes—
643C22C2×#52511yescharacteristic
644C22#348411yescharacteristic
645C22#348711yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22C22×#13711yescharacteristic
648C22#358611yescharacteristic
649C22#358611yescharacteristic
650C4—24no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—44no—
659C4—42no—
660C4—44no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4#345912yes—
667C4#350512yes—
668C2—24no—
669C2—162no—
670C2#3498711yescharacteristic
671C2C2×#311211yescharacteristic
672C2#6056611yescharacteristic
673C2#6157411yescharacteristic
674C2#6158511yescharacteristic
675C2#6188811yescharacteristic
676C2#6189311yescharacteristic
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×#2365C2semidirect
3. #33094C2semidirect
4. #32957C2semidirect
6. #29953C2semidirect
8. #29624C2semidirect
9. #27083C2semidirect
10. #26919C2semidirect
11. #26773C2semidirect
12. #26605C2semidirect
13. #22047C2semidirect
14. #21916C2semidirect
15. #21454C2semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#23652
#330942
#329572
#329032
#299532
#297132
#296242
#270832
#269192
#267732
#266052
#220472
#219162
#214542