Finite Group Databasebeta

← #6563276

Subgroups — #6563276

Order 512.

1627 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36492C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29943C212yesmaximal, complemented
4#29938C211yesmaximal, characteristic, complemented
5#27951C212yesmaximal, complemented
6#27950C211yesmaximal, characteristic, complemented
7#27650C212yesmaximal, complemented
8#27645C211yesmaximal, characteristic, complemented
9#26306C214yesmaximal, complemented
10#26297C212yesmaximal, complemented
11#25018C214yesmaximal, complemented
12#25011C212yesmaximal, complemented
13#23754C211yesmaximal, characteristic, complemented
14#23754C212yesmaximal, complemented
15#22727C212yesmaximal, complemented
16#22530C211yesmaximal, characteristic, complemented
17#22519C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#481C2212yes—
21C2×#482C2214yes—
22C2×C8×Q8C2212yes—
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2563C2211yescharacteristic
26#2562C2212yes—
27#2537C2211yescharacteristic
28#2535C2212yes—
29#2477—22no—
30#2463—22no—
31#2421—21no—
32#2407—21no—
33#2361—21no—
34#2355—21no—
35#2355C2212yes—
36#2355C2211yescharacteristic
37#2354—21no—
38#2354C2212yes—
39#2352—22no—
40#2352—21no—
41#2352C2214yes—
42#2314—22no—
43#2311—21no—
44C4×#121C2212yes—
45C4×#121C2211yescharacteristic
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2099C2212yes—
52#2099C2211yescharacteristic
53#2093C2211yescharacteristic
54#2084C2212yes—
55#2084C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#2083C2211yescharacteristic
59#2077C2214yes—
60#2077C2214yes—
61#2076C2214yes—
62#2075C2214yes—
63#2075C2212yes—
64#2071C2211yescharacteristic
65#2071C2212yes—
66#2033C2214yes—
67#2033C2212yes—
68#1991C2214yes—
69#1990C2212yes—
70#1989C2214yes—
71#1987C2212yes—
72#1954C2212yes—
73#1954C2214yes—
74#1949C2214yes—
75#1949C2214yes—
76#1948C2212yes—
77#1948C2212yes—
78#1854C2214yes—
79#1854C2214yes—
80#1852C2214yes—
81#1846C2214yes—
82#1845C2212yes—
83#1825C2214yes—
84#1824C2212yes—
85#1804C2214yes—
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90C4×#107C2214yes—
91C4×#102C2212yes—
92C4×#102C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96#554—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125C2312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×#121C2311yescharacteristic
105C4×#41—28no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4—22no—
110C2×C4×D4D412yes—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2311yescharacteristic
116C2×#117—21no—
117C2×#117—22no—
118C2×#117D412yes—
119C2×#117D411yescharacteristic
120C2×#117C2314yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#482—48no—
128#481—44no—
129C8×Q8—28no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—22no—
148#397—22no—
149#394—22no—
150#394—22no—
151#393—22no—
152#392—22no—
153#391—22no—
154#391—21no—
155#391—21no—
156#390—21no—
157#390—22no—
158#390C2311yescharacteristic
159#389—21no—
160#389C2312yes—
161#389C2311yescharacteristic
162#388—22no—
163#388—22no—
164#388—24no—
165#388—24no—
166#388—22no—
167#388—21no—
168#387—24no—
169#387—24no—
170#387—22no—
171#387—22no—
172#387—22no—
173#386—24no—
174#386—22no—
175#386—21no—
176#386—22no—
177#386—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2312yes—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#382—21no—
188#382—21no—
189#382—22no—
190#382C2314yes—
191#382C2312yes—
192#380—22no—
193#380—22no—
194#380—21no—
195#380—21no—
196#380—22no—
197#380—21no—
198#380C2311yescharacteristic
199#380C2312yes—
200#380C2312yes—
201#380C2312yes—
202#380C2312yes—
203#380C2314yes—
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—24no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—42no—
236C22×Q8—21no—
237C22×Q8C2×D411yescharacteristic
238C22×D4—22no—
239C22×D4—21no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2×D412yes—
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256#125—28no—
257C22⋊Q8—44no—
258C22⋊Q8—48no—
259C22⋊Q8—82no—
260#121—22no—
261#121—24no—
262#121—41no—
263#121—42no—
264#121—44no—
265C4×Q8—24no—
266C4×Q8#4118yes—
267C4×D4—24no—
268C4×D4—28no—
269C4×D4—28no—
270C4×D4—24no—
271C4×D4—28no—
272C4×D4—24no—
273C4×D4—44no—
274C4×D4—84no—
275#117—24no—
276#117—28no—
277#117—44no—
278#117—44no—
279#117—44no—
280#117—84no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—28no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345#107—24no—
346#107—44no—
347#107—42no—
348#107—44no—
349#106—44no—
350#106—44no—
351C4⋊C8—28no—
352C4⋊C8—24no—
353#102—28no—
354#102—24no—
355#102—24no—
356C22⋊C8—24no—
357C22⋊C8—28no—
358C22⋊C8—42no—
359C22⋊C8—44no—
360C4×C8—28no—
361C4×C8—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379C24—21no—
380C24C22×D412yes—
381C24C22×D411yescharacteristic
382#41—416no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8#12314yes—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—82no—
403C2×D4—81no—
404C2×D4H(5,𝔽2)12yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4#13514yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2511yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C82−1+414yes—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—81no—
491C4⋊C4#12314yes—
492C4⋊C4#12318yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—82no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—22no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—44no—
516C42—42no—
517C42—84no—
518C42—84no—
519C42—82no—
520C42C2×#4118yes—
521C42C2×#4114yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C2×#13512yes—
543C23C23×D411yescharacteristic
544Q8—28no—
545Q8—162no—
546D4—24no—
547D4—48no—
548D4—44no—
549D4—42no—
550D4—84no—
551D4—88no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—84no—
601C2×C4—164no—
602C2×C4—164no—
603C2×C4C2×#12312yes—
604C2×C4C2×#12314yes—
605C2×C4C2×#12314yes—
606C2×C4C2×#12312yes—
607C2×C4C2×#13512yes—
608C2×C4C2×#13512yes—
609C2×C4#57512yes—
610C2×C4#57612yes—
611C2×C4C22×#4112yes—
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C2×C4C2×2−1+412yes—
614C2×C4#58511yescharacteristic
615C8—24no—
616C8—44no—
617C8—42no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22C22×#12311yescharacteristic
640C22#348811yescharacteristic
641C22#349911yescharacteristic
642C22C22×#13511yescharacteristic
643C22C2×#57511yescharacteristic
644C22C2×#57611yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—24no—
650C4—22no—
651C4—42no—
652C4—41no—
653C4—81no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4—81no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4#309114yes—
662C4#351612yes—
663C4#352212yes—
664C2—42no—
665C2—82no—
666C2—82no—
667C2#3599611yescharacteristic
668C2C2×#309111yescharacteristic
669C2#6052111yescharacteristic
670C2#6195011yescharacteristic
671C2#6199511yescharacteristic
672C2#6247411yescharacteristic
673C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36492C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #29943C2semidirect
4. #29938C2semidirect
5. #27951C2semidirect
6. #27950C2semidirect
7. #27650C2semidirect
8. #27645C2semidirect
9. #26306C2semidirect
10. #26297C2semidirect
11. #25018C2semidirect
12. #25011C2semidirect
13. #23754C2semidirect
14. #23754C2semidirect
15. #22727C2semidirect
16. #22530C2semidirect
17. #22519C2semidirect

Maximal subgroups

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

subgroupindex
#364922
#341882
#299432
#299382
#279512
#279502
#276502
#276452
#263062
#262972
#250182
#250112
#237542
#227272
#225302
#225192