Finite Group Databasebeta

← #6435658

Subgroups — #6435658

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34060C211yesmaximal, characteristic
3#29947C214yesmaximal, complemented
4#29945C214yesmaximal, complemented
5#29941C212yesmaximal, complemented
6#29941C212yesmaximal, complemented
7#29937C212yesmaximal, complemented
8#29936C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23662C212yesmaximal, complemented
11#23643C212yesmaximal
12#23641C212yesmaximal
13#21313C211yesmaximal, characteristic, complemented
14C2×#1988C211yesmaximal, characteristic, complemented
15#21170C211yesmaximal, characteristic, complemented
16#21169C211yesmaximal, characteristic, complemented
17#20629C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#428C2212yes—
21C2×#427C2212yes—
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#408C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2628C2214yes—
28#2568—21no—
29#2567—21no—
30#2563—21no—
31#2542—21no—
32#2487—22no—
33#2409—22no—
34#2320—21no—
35#2319—21no—
36#2318—22no—
37#2317C2212yes—
38#2317C2211yescharacteristic
39#2317C2211yescharacteristic
40#2314C2212yes—
41#2314C2212yes—
42#2311C2211yescharacteristic
43#2311C2211yescharacteristic
44#2311C2212yes—
45#2295—21no—
46#2293—21no—
47#2290—22no—
48#2285C2212yes—
49#2285C2211yescharacteristic
50#2283C2212yes—
51C4×#117C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2085C2214yes—
58#2085C2214yes—
59#2084C2214yes—
60#2083C2214yes—
61#2077C2214yes—
62#2077C2214yes—
63#2077C2214yes—
64#2077C2214yes—
65#2076C2214yes—
66#2076C2214yes—
67#2075C2214yes—
68#2075C2214yes—
69#1988C2212yes—
70#1988C2212yes—
71#1988C2214yes—
72#1988C2212yes—
73#1988C2212yes—
74#1987C2212yes—
75#1987C2212yes—
76#1949C2212yes—
77#1949C2212yes—
78#1949C2214yes—
79#1948C2211yescharacteristic
80#1948C2211yescharacteristic
81#1948C2212yes—
82#1947C2211yescharacteristic
83#1947C2211yescharacteristic
84#1947C2212yes—
85#1798C2214yes—
86#1797C2212yes—
87#1797C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C4×#41—24no—
108C4×#41—24no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D412yes—
113C2×C4×D4D412yes—
114C2×C4×D4D411yescharacteristic
115C2×#117—22no—
116C2×#117C2311yescharacteristic
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C42—21no—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#428—44no—
129#427—44no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×#106C2312yes—
136#423—24no—
137C2×C4⋊C8—24no—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143#408—44no—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146#400—24no—
147#399—24no—
148#395—24no—
149#393—24no—
150#391—22no—
151#391—21no—
152#391—21no—
153#391—22no—
154#391—22no—
155#390—22no—
156#390—21no—
157#390—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#386—22no—
175#386—22no—
176#386—24no—
177#385—22no—
178#385—22no—
179#385—24no—
180#384—22no—
181#384C2312yes—
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#383—22no—
186#382—22no—
187#382—22no—
188#382C2312yes—
189#382C2312yes—
190#382C2312yes—
191#381—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—21no—
196#380—21no—
197#380C2312yes—
198#380C2312yes—
199#380C2312yes—
200#380C2312yes—
201#380C2314yes—
202#380C2314yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217#376—21no—
218#376—22no—
219C2×#95—21no—
220C43—21no—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—44no—
246C22×Q8—82no—
247C22×Q8—82no—
248C22×Q8C2×D411yescharacteristic
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C22⋊Q8—84no—
276C22⋊Q8—84no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—24no—
283#117—22no—
284#117—22no—
285#117—24no—
286#117—24no—
287#117—22no—
288#117—24no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—24no—
293#117—84no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—82no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42#4114yes—
358C2×C42#4114yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—42no—
363#107—42no—
364#107—44no—
365#106—42no—
366#106—42no—
367#106—44no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373C22⋊C8—44no—
374C22⋊C8—44no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—816no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4#12314yes—
452C22×C4#12314yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C22×C4C2511yescharacteristic
468M16—48no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—48no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—81no—
539C42—81no—
540C42—82no—
541C42#12314yes—
542C42#12314yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23C2×#12312yes—
559C23#53612yes—
560C23#53712yes—
561C23C23×D411yescharacteristic
562Q8—44no—
563Q8—88no—
564Q8—88no—
565D4—48no—
566D4—44no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—88no—
620C2×C4C2×#12312yes—
621C2×C4C2×#12312yes—
622C2×C4C2×#12312yes—
623C2×C4C2×#12312yes—
624C2×C4C2×#12312yes—
625C2×C4#53612yes—
626C2×C4#53712yes—
627C2×C4#54114yes—
628C2×C4#54114yes—
629C2×C4#54214yes—
630C2×C4#54214yes—
631C2×C4#57612yes—
632C2×C4#57712yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C8—42no—
640C8—42no—
641C8—44no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22#262514yes—
656C22C22×#12311yescharacteristic
657C22C2×#53611yescharacteristic
658C22C2×#53711yescharacteristic
659C22C2×#57611yescharacteristic
660C22C2×#57711yescharacteristic
661C22#358611yescharacteristic
662C22#358611yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—81no—
670C4—82no—
671C4—82no—
672C4—81no—
673C4—82no—
674C4—84no—
675C4—81no—
676C4—82no—
677C4—81no—
678C4#262514yes—
679C4#309614yes—
680C4C2×#54111yescharacteristic
681C4C2×#54111yescharacteristic
682C4C2×#54211yescharacteristic
683C4C2×#54211yescharacteristic
684C2—24no—
685C2—82no—
686C2C2×#262511yescharacteristic
687C2C2×#309611yescharacteristic
688C2#6051911yescharacteristic
689C2#6088811yescharacteristic
690C2#6088911yescharacteristic
691C2#6092811yescharacteristic
692C2#6093111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
3. #29947C2semidirect
4. #29945C2semidirect
5. #29941C2semidirect
6. #29941C2semidirect
7. #29937C2semidirect
8. #29936C2semidirect
9. #23748C2semidirect
10. #23662C2semidirect
13. #21313C2semidirect
14. C2×#1988C2semidirect
15. #21170C2semidirect
16. #21169C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#340602
#299472
#299452
#299412
#299372
#299362
#237482
#236622
#236432
#236412
#213132
C2×#19882
#211702
#211692
#206292