Finite Group Databasebeta

← #6993207

Subgroups — #6993207

Order 512.

1703 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36274C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#32724C212yesmaximal, complemented
4#31190C214yesmaximal, complemented
5#31019C214yesmaximal, complemented
6#30358C212yesmaximal, complemented
7#29953C212yesmaximal, complemented
8#29664C211yesmaximal, characteristic, complemented
9#28304C212yesmaximal, complemented
10#28265C212yesmaximal, complemented
11#27910C211yesmaximal, characteristic, complemented
12#27681C211yesmaximal, characteristic, complemented
13#25294C212yesmaximal, complemented
14#25016C212yesmaximal, complemented
15#24243C212yesmaximal, complemented
16#23571C211yesmaximal, characteristic, complemented
17#22530C211yesmaximal, characteristic
18C2×D4×Q8C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#484C2214yes—
22C2×#484C2214yes—
23C2×#462C2212yes—
24C2×#451C2214yes—
25C2×C8×Q8C2212yes—
26C2×C4×Q16C2212yes—
27#2857C2212yes—
28#2857C2211yescharacteristic
29#2857C2212yes—
30#2855C2212yes—
31#2629—22no—
32#2582—21no—
33#2582C2211yescharacteristic
34#2582C2212yes—
35#2568C2211yescharacteristic
36#2538—21no—
37#2538C2212yes—
38#2451—21no—
39#2450—22no—
40#2439—21no—
41#2438—22no—
42#2417—22no—
43#2417C2214yes—
44#2417C2212yes—
45#2409C2212yes—
46#2349—21no—
47D4×C4⋊C4C2211yescharacteristic
48#2339—21no—
49#2316—21no—
50#2311—21no—
51C4×C22⋊Q8C2211yescharacteristic
52C2×#391C2212yes—
53C2×#384C2211yescharacteristic
54#2100C2212yes—
55#2095C2212yes—
56#2095C2212yes—
57#2095C2212yes—
58#2076C2214yes—
59#2068C2214yes—
60#2068C2214yes—
61#2068C2214yes—
62#2057C2211yescharacteristic
63#2056C2214yes—
64#2043C2211yescharacteristic
65#2036C2212yes—
66#2036C2214yes—
67#2036C2214yes—
68#2024C2211yescharacteristic
69#2024C2212yes—
70#2016C2211yescharacteristic
71#2016C2212yes—
72#2001C2212yes—
73#2001C2214yes—
74#2001C2214yes—
75#1969C2214yes—
76#1969C2214yes—
77#1956C2212yes—
78#1954C2214yes—
79#1954C2212yes—
80#1947C2212yes—
81#1942C2214yes—
82#1933C2214yes—
83#1872C2212yes—
84#1852C2214yes—
85#1852C2212yes—
86#1845C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1787C2214yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99D4×Q8—44no—
100#533—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C4×#41—44no—
115C2×C4×Q8—21no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D412yes—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117D411yescharacteristic
128C2×#117D412yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#484—48no—
136#484—48no—
137#462—44no—
138#451—48no—
139C8×Q8—44no—
140C4×Q16—44no—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2314yes—
162#400—22no—
163#398—24no—
164#398—24no—
165#398—22no—
166#398—22no—
167#396—24no—
168#396—24no—
169#396—22no—
170#395—22no—
171#393—22no—
172#391—22no—
173#391—21no—
174#391—21no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391C2312yes—
179#391C2314yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#391C2311yescharacteristic
184#390—22no—
185#389—22no—
186#389—22no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—21no—
194#389C2311yescharacteristic
195#389C2312yes—
196#389C2314yes—
197#387—22no—
198#387—22no—
199#387—21no—
200#386—24no—
201#386—22no—
202#386—22no—
203#385—22no—
204#385—21no—
205#385—22no—
206#385—22no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2312yes—
211#384C2312yes—
212#382—21no—
213#381—22no—
214#380—22no—
215#380—22no—
216#380—21no—
217#380—22no—
218#380—21no—
219#380—24no—
220#380—22no—
221#380C2312yes—
222#380C2314yes—
223#380C2311yescharacteristic
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—21no—
227C2×#95—22no—
228C2×#95—22no—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2311yescharacteristic
237C2×#41—42no—
238C2×#41—42no—
239C2×#41Q1618yescomplemented
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—44no—
244C22×Q8—42no—
245C22×Q8—44no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D412yes—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C23×C4C2412yes—
261C2×Q16—24no—
262C2×Q16—24no—
263C2×Q16—42no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×Q16—44no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273C4⋊Q8—44no—
274#125—44no—
275C22⋊Q8—22no—
276C22⋊Q8—48no—
277C22⋊Q8—41no—
278C22⋊Q8—42no—
279C22⋊Q8—42no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—84no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—42no—
291C4×Q8—44no—
292C4×D4—24no—
293C4×D4—42no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—44no—
298#117—44no—
299#117—42no—
300#117—44no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
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—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#107—42no—
363#107—44no—
364#107—44no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367#103—28no—
368#103—24no—
369#103—42no—
370#103—42no—
371#103—42no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376C22⋊C8—28no—
377C22⋊C8—24no—
378C22⋊C8—44no—
379C22⋊C8—42no—
380C4×C8—24no—
381C4×C8—44no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401C24C22×D412yes—
402C24C22×D411yescharacteristic
403#41—416no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—82no—
425C2×Q8—88no—
426C2×Q8—88no—
427C2×Q8C2×Q1614yes—
428C2×Q82−1+412yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4C2×Q1614yes—
434C2×D4C2×Q1618yes—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C2×Q1614yes—
472C22×C4C2×Q1618yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2511yescharacteristic
482Q16—48no—
483Q16—88no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8H(5,𝔽2)14yes—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—81no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C22⋊C4—24no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—81no—
537C22⋊C4—84no—
538C22⋊C4—82no—
539C22⋊C4—84no—
540C42—22no—
541C42—22no—
542C42—24no—
543C42—22no—
544C42—42no—
545C42—42no—
546C42—44no—
547C42—42no—
548C42—42no—
549C42H(5,𝔽2)14yes—
550C23—22no—
551C23—21no—
552C23—22no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—22no—
557C23—21no—
558C23—22no—
559C23—21no—
560C23—21no—
561C23—22no—
562C23C22×Q1612yes—
563C23C22×Q1614yes—
564C23C23×D411yescharacteristic
565Q8—28no—
566Q8—24no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—44no—
571Q8—42no—
572Q8—44no—
573Q8—42no—
574Q8—84no—
575Q8—88no—
576D4—28no—
577D4—48no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—84no—
630C2×C4—88no—
631C2×C4—164no—
632C2×C4—164no—
633C2×C4#53412yes—
634C2×C4#53412yes—
635C2×C4C22×Q1612yes—
636C2×C4C22×Q1614yes—
637C2×C4C22×Q1612yes—
638C2×C4#57512yes—
639C2×C4#57812yes—
640C2×C4C2×H(5,𝔽2)12yes—
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4C2×2−1+411yescharacteristic
643C2×C4#58511yescharacteristic
644C8—24no—
645C8—42no—
646C8—44no—
647C22—22no—
648C22—24no—
649C22—24no—
650C22—22no—
651C22—22no—
652C22—24no—
653C22—41no—
654C22—42no—
655C22—41no—
656C22—42no—
657C22—42no—
658C22—41no—
659C22—41no—
660C22—42no—
661C22C2×#53411yescharacteristic
662C22#348911yescharacteristic
663C22#349811yescharacteristic
664C22C23×Q1611yescharacteristic
665C22C2×#57511yescharacteristic
666C22C2×#57811yescharacteristic
667C22#358511yescharacteristic
668C4—22no—
669C4—24no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—81no—
677C4—82no—
678C4—81no—
679C4—82no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4—81no—
684C4#318914yes—
685C4#353612yes—
686C4#354612yes—
687C2—42no—
688C2—44no—
689C2#4298811yescharacteristic
690C2C2×#318911yescharacteristic
691C2#6076411yescharacteristic
692C2#6210011yescharacteristic
693C2#6214611yescharacteristic
694C2#6241611yescharacteristic
695C2#6245511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
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. #32724C2semidirect
4. #31190C2semidirect
5. #31019C2semidirect
6. #30358C2semidirect
7. #29953C2semidirect
8. #29664C2semidirect
9. #28304C2semidirect
10. #28265C2semidirect
11. #27910C2semidirect
12. #27681C2semidirect
13. #25294C2semidirect
14. #25016C2semidirect
15. #24243C2semidirect
16. #23571C2semidirect
239. C2×#41Q16semidirect

Maximal subgroups

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

subgroupindex
#362742
#352802
#327242
#311902
#310192
#303582
#299532
#296642
#283042
#282652
#279102
#276812
#252942
#250162
#242432
#235712
#225302