Finite Group Databasebeta

← #6563524

Subgroups — #6563524

Order 512.

1952 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36259C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29936C211yesmaximal, characteristic, complemented
4#29906C212yesmaximal, complemented
5#27950C211yesmaximal, characteristic, complemented
6#27856C212yesmaximal, complemented
7#27644C211yesmaximal, characteristic, complemented
8#27610C212yesmaximal, complemented
9#26276C212yesmaximal, complemented
10#26126C214yesmaximal, complemented
11#25056C212yesmaximal, complemented
12#24933C214yesmaximal, complemented
13#23754C211yesmaximal, characteristic, complemented
14C2×#2083C212yesmaximal, complemented
15D4×#107C212yesmaximal, complemented
16#22485C212yesmaximal, complemented
17#22469C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547C2211yescharacteristic
20C2×#520—21no—
21C2×C4×#41C2211yescharacteristic
22C2×#480C2212yes—
23C2×C8×D4C2212yes—
24C22×#107C2214yes—
25#2857C2211yescharacteristic
26#2854C2212yes—
27#2546C2211yescharacteristic
28#2538C2211yescharacteristic
29#2486C2212yes—
30#2429—21no—
31#2425—22no—
32#2416C2212yes—
33#2415—21no—
34#2413—22no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4C2212yes—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—22no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2211yescharacteristic
41D4×C4⋊C4C2212yes—
42#2339—21no—
43#2317—22no—
44#2317—21no—
45C4×#121C2211yescharacteristic
46C2×#389C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51#2099C2214yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2093C2214yes—
55#2093C2211yescharacteristic
56#2083C2214yes—
57#2083C2212yes—
58#2083C2212yes—
59#2083C2211yescharacteristic
60#2083C2214yes—
61#2083C2212yes—
62#2083C2212yes—
63#2075C2214yes—
64#2075C2212yes—
65#2071C2211yescharacteristic
66#2071C2212yes—
67#2033C2212yes—
68#1990C2212yes—
69#1988C2212yes—
70#1984C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1925C2214yes—
74#1918C2214yes—
75#1918C2214yes—
76#1918C2214yes—
77#1893C2214yes—
78#1874C2214yes—
79#1869C2214yes—
80#1852C2214yes—
81#1852C2214yes—
82#1852C2214yes—
83#1845C2212yes—
84#1823C2212yes—
85#1823C2214yes—
86#1817C2212yes—
87#1808C2214yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C4×#102C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93C2×#340C2214yes—
94C2×#336C2212yes—
95C22×#41—21no—
96C22×#41D411yescharacteristic
97C23×C8C2314yes—
98#547—28no—
99#547—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#123—22no—
103C2×#123—22no—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×#121—22no—
109C2×#121C2312yes—
110C2×#121C2311yescharacteristic
111#520—48no—
112C4×#41—28no—
113C2×C4×Q8—22no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—24no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D412yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C2×#117—21no—
127C2×#117D412yes—
128C2×#117D411yescharacteristic
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2314yes—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140#480—44no—
141C8×D4—28no—
142C2×#107—48no—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×C4⋊C8C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C4×C8C2312yes—
157#398—22no—
158#398—22no—
159#397—22no—
160#396—22no—
161#396—22no—
162#394—22no—
163#391—21no—
164#391—21no—
165#390C2311yescharacteristic
166#390C2312yes—
167#389—22no—
168#389—24no—
169#389—24no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#388—22no—
181#388—22no—
182#388—21no—
183#386—22no—
184#386—21no—
185#386—22no—
186#385—24no—
187#385—24no—
188#385—24no—
189#385—22no—
190#384—21no—
191#384—22no—
192#384—21no—
193#384—22no—
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2314yes—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382—21no—
201#382—22no—
202#381—22no—
203#380—21no—
204#380—22no—
205#380—21no—
206#380—21no—
207#380C2312yes—
208#380C2314yes—
209#380C2312yes—
210#380C2311yescharacteristic
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C2×#95—22no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2311yescharacteristic
234#336C2314yes—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—44no—
238C2×#41—44no—
239C2×#41—44no—
240C2×#41D818yes—
241C22×Q8—21no—
242C22×Q8C2×D411yescharacteristic
243C22×D4—22no—
244C22×D4—21no—
245C22×D4C2×D412yes—
246C22×D4C2×D411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—24no—
252C23×C4—21no—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2412yes—
259C22×C8—28no—
260C22×C8—28no—
261C22×C8—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267#123—48no—
268#123—48no—
269#123—48no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273#121—28no—
274#121—22no—
275#121—41no—
276#121—48no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—48no—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—416no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4#4118yes—
289#117—24no—
290#117—28no—
291#117—48no—
292#117—44no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4—84no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—28no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—24no—
367#107—24no—
368#107—28no—
369#107—44no—
370#107—44no—
371#107—44no—
372C4⋊C8—24no—
373#102—24no—
374#102—42no—
375#102—44no—
376C22⋊C8—28no—
377C22⋊C8—24no—
378C22⋊C8—42no—
379C4×C8—24no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393C24—21no—
394C24C22×D411yescharacteristic
395C24C22×D412yes—
396#41—232no—
397#41—816no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8C2×D814yes—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—81no—
416C2×D4#12314yes—
417C2×D4C2×D818yes—
418C2×D4C2×D814yes—
419C2×D4#13714yes—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—82no—
456C22×C4—84no—
457C22×C4—82no—
458C22×C4—84no—
459C22×C4—82no—
460C22×C4C2×D818yes—
461C22×C4C2×D814yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4118yes—
471C22×C4C2511yescharacteristic
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—416no—
481C2×C82−1+414yes—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—84no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4#12314yes—
520C4⋊C4H(5,𝔽2)12yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—82no—
535C22⋊C4#12318yes—
536C42—24no—
537C42—22no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—48no—
543C42—44no—
544C42—42no—
545C42C2×#4114yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×#12314yes—
564C23C22×D812yes—
565C23C22×D814yes—
566C23C2×#13714yes—
567C23C23×D411yescharacteristic
568Q8—28no—
569Q8—84no—
570D4—216no—
571D4—42no—
572D4—42no—
573D4—84no—
574D4—88no—
575D4#480116yescomplemented
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—216no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—88no—
625C2×C4—88no—
626C2×C4—84no—
627C2×C4—168no—
628C2×C4—164no—
629C2×C4C2×#12314yes—
630C2×C4C2×#12312yes—
631C2×C4C2×#12312yes—
632C2×C4C22×D814yes—
633C2×C4C22×D812yes—
634C2×C4C22×D812yes—
635C2×C4C2×#13712yes—
636C2×C4#57812yes—
637C2×C4C22×#4112yes—
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+412yes—
640C2×C4#58511yescharacteristic
641C8—24no—
642C8—42no—
643C8—44no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22C2×#48018yes—
665C22C22×#12311yescharacteristic
666C22#348811yescharacteristic
667C22#349911yescharacteristic
668C22C23×D811yescharacteristic
669C22C22×#13711yescharacteristic
670C22C2×#57811yescharacteristic
671C22#358511yescharacteristic
672C4—22no—
673C4—24no—
674C4—22no—
675C4—22no—
676C4—24no—
677C4—42no—
678C4—41no—
679C4—42no—
680C4—82no—
681C4—81no—
682C4—81no—
683C4—82no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4C2×#48014yes—
688C4#351512yes—
689C4#352012yes—
690C2—28no—
691C2—42no—
692C2—161no—
693C2#3599811yescharacteristic
694C2C22×#48011yescharacteristic
695C2#6052611yescharacteristic
696C2#6194711yescharacteristic
697C2#6198411yescharacteristic
698C2#6247111yescharacteristic
699C2#6250411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36259C2
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. #34183C2semidirect
3. #29936C2semidirect
4. #29906C2semidirect
5. #27950C2semidirect
6. #27856C2semidirect
7. #27644C2semidirect
8. #27610C2semidirect
9. #26276C2semidirect
10. #26126C2semidirect
11. #25056C2semidirect
12. #24933C2semidirect
13. #23754C2semidirect
14. C2×#2083C2semidirect
15. D4×#107C2semidirect
16. #22485C2semidirect
17. #22469C2semidirect
575. D4#480semidirect

Maximal subgroups

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

subgroupindex
#362592
#341832
#299362
#299062
#279502
#278562
#276442
#276102
#262762
#261262
#250562
#249332
#237542
C2×#20832
D4×#1072
#224852
#224692