Finite Group Databasebeta

← #6485007

Subgroups — #6485007

Order 512.

1618 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#35249C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33095C212yesmaximal, complemented
4#33094C212yesmaximal, complemented
5#29912C212yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#26323C212yesmaximal, complemented
8#26321C212yesmaximal, complemented
9#26279C212yesmaximal, complemented
10#26277C212yesmaximal, complemented
11#25069C214yesmaximal, complemented
12#25003C214yesmaximal, complemented
13#21987C211yesmaximal, characteristic, complemented
14#21859C211yesmaximal, characteristic
15#21820C211yesmaximal, characteristic, complemented
16#21395C211yesmaximal, characteristic
17#21361C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2854C2212yes—
23#2854C2212yes—
24#2632—22no—
25#2580C2212yes—
26#2580C2212yes—
27#2567C2211yescharacteristic
28#2567C2211yescharacteristic
29#2548C2211yescharacteristic
30#2548C2211yescharacteristic
31#2501—21no—
32#2436—21no—
33#2406—22no—
34#2355—21no—
35#2355—21no—
36#2355—22no—
37#2355C2212yes—
38#2355C2212yes—
39#2355C2214yes—
40#2326—21no—
41#2326—21no—
42C2×#391—21no—
43C2×#387—22no—
44C2×#384C2211yescharacteristic
45#2283C2211yescharacteristic
46C2×#382C2212yes—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2092C2214yes—
53#2092C2214yes—
54#2085C2214yes—
55#2085C2214yes—
56#2073C2214yes—
57#2073C2214yes—
58#1990C2212yes—
59#1987C2212yes—
60#1985C2214yes—
61#1984C2212yes—
62#1984C2212yes—
63#1982C2212yes—
64#1957C2212yes—
65#1949C2212yes—
66#1949C2212yes—
67#1948C2212yes—
68#1947C2212yes—
69#1926C2214yes—
70#1926C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1919C2214yes—
74#1919C2214yes—
75#1918C2214yes—
76#1918C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1874C2212yes—
81#1853C2211yescharacteristic
82#1853C2212yes—
83#1853C2211yescharacteristic
84#1852C2211yescharacteristic
85#1852C2212yes—
86#1852C2211yescharacteristic
87#1847C2212yes—
88#1836C2212yes—
89#1817C2212yes—
90#1817C2212yes—
91#1802C2214yes—
92C4×#107C2212yes—
93C4×#106C2212yes—
94#1762C2214yes—
95C22×#41—21no—
96C23×Q8D411yescharacteristic
97#532—24no—
98#532—24no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#124—21no—
102C2×#124—21no—
103C2×#124—21no—
104C2×#123—22no—
105C2×#123—22no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D412yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D412yes—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C42—21no—
126C22×C42—22no—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#432—24no—
130#431—24no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2311yescharacteristic
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×#97C2312yes—
147C2×C4×C8C2312yes—
148#399—22no—
149#399—22no—
150#395—22no—
151#395—22no—
152#394—22no—
153#394—22no—
154#393—22no—
155#393—22no—
156#391—21no—
157#391—21no—
158#391—22no—
159#391—22no—
160#391—24no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#387—24no—
176#387—24no—
177#387—24no—
178#387—24no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#383—21no—
192#382—22no—
193#382—22no—
194#382C2311yescharacteristic
195#382C2311yescharacteristic
196#382C2314yes—
197#382C2314yes—
198#381—21no—
199#380C2311yescharacteristic
200#380C2311yescharacteristic
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—24no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215#376—21no—
216C2×#95—22no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234C2×#41—44no—
235C2×#41—82no—
236C2×#41—82no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—42no—
240C22×Q8—42no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2×D411yescharacteristic
243C22×Q8C2×D412yes—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262#127—22no—
263#127—22no—
264#124—44no—
265#124—44no—
266#124—44no—
267#123—84no—
268#123—84no—
269#121—24no—
270#121—24no—
271#121—44no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—28no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—44no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
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×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
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×C22⋊C4—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42—84no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42#4114yes—
345C2×C42#4114yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—24no—
351#107—44no—
352#107—42no—
353#106—24no—
354#106—44no—
355#106—42no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358#102—28no—
359#102—44no—
360#102—44no—
361#102—44no—
362C22⋊C8—22no—
363C22⋊C8—22no—
364C22⋊C8—24no—
365#97—24no—
366C4×C8—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387C24—22no—
388C24C22×D411yescharacteristic
389#41—816no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—416no—
400C2×Q8—42no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4#12314yes—
449C22×C4#12314yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2511yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C42−1+414yes—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—44no—
521C42—44no—
522C42—42no—
523C42—42no—
524C42—84no—
525C42—84no—
526C42#12314yes—
527C42#12314yes—
528C42C2×#4112yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C2×#12312yes—
544C23#57612yes—
545C23#57712yes—
546C23C23×D411yescharacteristic
547Q8—48no—
548Q8—44no—
549Q8—44no—
550Q8—84no—
551D4—44no—
552D4—44no—
553D4—84no—
554D4—88no—
555D4—84no—
556C2×C4—28no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—88no—
608C2×C4—88no—
609C2×C4—84no—
610C2×C4C2×#12312yes—
611C2×C4C2×#12312yes—
612C2×C4C2×#12312yes—
613C2×C4C2×#12312yes—
614C2×C4C2×#12312yes—
615C2×C4#57512yes—
616C2×C4#57512yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×2−1+412yes—
620C2×C4#58512yes—
621C8—42no—
622C8—42no—
623C8—44no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22#309614yes—
638C22C22×#12311yescharacteristic
639C22#348611yescharacteristic
640C22#348711yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57511yescharacteristic
643C22C2×#57611yescharacteristic
644C22C2×#57711yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—82no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—84no—
659C4—81no—
660C4—82no—
661C4#309314yes—
662C4#346112yes—
663C4#350712yes—
664C2—24no—
665C2—162no—
666C2#3518211yescharacteristic
667C2C2×#309311yescharacteristic
668C2C2×#309611yescharacteristic
669C2#6183511yescharacteristic
670C2#6183611yescharacteristic
671C2#6189811yescharacteristic
672C2#6189911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2355C2semidirect
3. #33095C2semidirect
4. #33094C2semidirect
5. #29912C2semidirect
6. #29912C2semidirect
7. #26323C2semidirect
8. #26321C2semidirect
9. #26279C2semidirect
10. #26277C2semidirect
11. #25069C2semidirect
12. #25003C2semidirect
13. #21987C2semidirect
15. #21820C2semidirect
17. #21361C2semidirect

Maximal subgroups

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

subgroupindex
#352492
C2×#23552
#330952
#330942
#299122
#263232
#263212
#262792
#262772
#250692
#250032
#219872
#218592
#218202
#213952
#213612