Finite Group Databasebeta

← #6857031

Subgroups — #6857031

Order 512.

1541 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, complemented
2#34941C211yesmaximal, characteristic, complemented
3#33197C211yesmaximal, characteristic, complemented
4#32484C214yesmaximal, complemented
5#31814C214yesmaximal, complemented
6#29825C212yesmaximal
7#29348C212yesmaximal, complemented
8#29289C212yesmaximal, complemented
9#29218C211yesmaximal, characteristic, complemented
10#28280C211yesmaximal, characteristic, complemented
11#28185C212yesmaximal, complemented
12#26021C212yesmaximal, complemented
13#25470C212yesmaximal, complemented
14#24677C212yesmaximal, complemented
15Q8×C4⋊C8C211yesmaximal, characteristic, complemented
16#22111C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556—21no—
19C2×#476C2212yes—
20C2×#476C2212yes—
21C2×#459C2211yescharacteristic
22C2×C8×Q8C2212yes—
23C2×C4×D8C2212yes—
24#2868C2211yescharacteristic
25#2858C2212yes—
26#2574—22no—
27#2567C2211yescharacteristic
28#2563C2211yescharacteristic
29#2552C2211yescharacteristic
30#2547—21no—
31#2545—22no—
32#2538—21no—
33#2538C2211yescharacteristic
34#2538C2212yes—
35#2512—22no—
36#2512C2214yes—
37#2508C2212yes—
38#2497—21no—
39#2454C2212yes—
40#2423—21no—
41#2396—22no—
42#2354—21no—
43#2354C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×#127C2211yescharacteristic
47C4×#125C2212yes—
48C4×C22⋊Q8—21no—
49#2293C2212yes—
50#2285—21no—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2100C2214yes—
54#2088C2214yes—
55#2086C2214yes—
56#2073C2214yes—
57#2065C2214yes—
58#2065C2214yes—
59#2065C2214yes—
60#2054C2214yes—
61#2054C2214yes—
62#2025C2214yes—
63#1990C2212yes—
64#1989C2214yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1982C2212yes—
69#1982C2211yescharacteristic
70#1949C2214yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2211yescharacteristic
75#1941C2214yes—
76#1939C2214yes—
77#1939C2212yes—
78#1934C2214yes—
79#1934C2214yes—
80#1934C2214yes—
81#1926C2214yes—
82#1925C2212yes—
83#1919C2214yes—
84#1879C2212yes—
85#1842C2212yes—
86#1842C2211yescharacteristic
87#1834C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C22×D8C2312yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#126—24no—
103C2×#125C2312yes—
104C2×#124—22no—
105C2×#124—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C2×#117—22no—
121C22×C42—21no—
122#476—28no—
123#476—44no—
124#459—24no—
125C8×Q8—44no—
126C4×D8—44no—
127C2×#107C2312yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2314yes—
146#400—22no—
147#399—22no—
148#398—22no—
149#398—22no—
150#397—24no—
151#396—22no—
152#396—24no—
153#396—24no—
154#396—22no—
155#395—22no—
156#393—22no—
157#391—21no—
158#391—21no—
159#391—22no—
160#390—21no—
161#390—21no—
162#390—22no—
163#390—22no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2314yes—
168#390C2314yes—
169#389—21no—
170#389—21no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—24no—
181#388—24no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#382C2312yes—
196#380—21no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C43C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8C2×D411yescharacteristic
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2412yes—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C2×D8—24no—
246C2×D8—42no—
247C2×D8—42no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257#127—22no—
258#127—22no—
259#126—416no—
260#125—24no—
261#125—42no—
262#124—48no—
263#124—84no—
264C22⋊Q8—44no—
265C22⋊Q8—48no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—48no—
271C4×Q8—82no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—24no—
278C4×D4—41no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4#4114yes—
282#117—48no—
283#117—84no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—42no—
349#106—28no—
350#106—44no—
351C4⋊C8—22no—
352C4⋊C8—24no—
353C4⋊C8—24no—
354C4⋊C8—22no—
355C4⋊C8—42no—
356C4⋊C8—44no—
357C4⋊C8—44no—
358#102—28no—
359#102—24no—
360#102—24no—
361#102—44no—
362#102—42no—
363#102—42no—
364#102—44no—
365C4×C8—24no—
366C4×C8—44no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C24—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8AGL(1,ℤ/8)14yes—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C2×#4112yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442D8—48no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4C2×#4112yes—
489C4⋊C42−1+412yes—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—24no—
505C42—22no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—21no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—22no—
515C42—42no—
516C42—41no—
517C42—48no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42#13514yes—
523C42C2×#4112yes—
524C42H(5,𝔽2)12yes—
525C422−1+412yes—
526C422−1+412yes—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23C23×D411yescharacteristic
542Q8—28no—
543Q8—88no—
544Q8—84no—
545D4—22no—
546D4—22no—
547D4—42no—
548D4—42no—
549D4—41no—
550D4—42no—
551D4—41no—
552D4—42no—
553D4—84no—
554D4—84no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4#53712yes—
601C2×C4#53712yes—
602C2×C4#54812yes—
603C2×C4#54812yes—
604C2×C4C2×#13512yes—
605C2×C4C2×#13512yes—
606C2×C4C2×#13512yes—
607C2×C4C2×AGL(1,ℤ/8)14yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4#57712yes—
610C2×C4C22×#4111yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C2×C4C2×2−1+411yescharacteristic
615C2×C4#58511yescharacteristic
616C8—44no—
617C8—42no—
618C8—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C2×#53711yescharacteristic
634C22C2×#54811yescharacteristic
635C22#348911yescharacteristic
636C22C22×#13511yescharacteristic
637C22C22×AGL(1,ℤ/8)11yescharacteristic
638C22C2×#57711yescharacteristic
639C22#358511yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—82no—
651C4—84no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4#324714yes—
656C4#336014yes—
657C4#350611yescharacteristic
658C4#350911yescharacteristic
659C4#355011yescharacteristic
660C4#355511yescharacteristic
661C2—82no—
662C2—162no—
663C2#3972711yescharacteristic
664C2C2×#324711yescharacteristic
665C2#6092911yescharacteristic
666C2C2×#336011yescharacteristic
667C2#6132111yescharacteristic
668C2#6209811yescharacteristic
669C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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
1. #36594C2semidirect
2. #34941C2semidirect
3. #33197C2semidirect
4. #32484C2semidirect
5. #31814C2semidirect
7. #29348C2semidirect
8. #29289C2semidirect
9. #29218C2semidirect
10. #28280C2semidirect
11. #28185C2semidirect
12. #26021C2semidirect
13. #25470C2semidirect
14. #24677C2semidirect
15. Q8×C4⋊C8C2semidirect
16. #22111C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#349412
#331972
#324842
#318142
#298252
#293482
#292892
#292182
#282802
#281852
#260212
#254702
#246772
Q8×C4⋊C82
#221112
C4×#4592