Finite Group Databasebeta

← #6593938

Subgroups — #6593938

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#33005C212yesmaximal
4#32954C211yesmaximal, characteristic, complemented
5#27961C211yesmaximal, characteristic, complemented
6#27645C211yesmaximal, characteristic
7#27611C212yesmaximal, complemented
8#27554C214yesmaximal, complemented
9#27298C214yesmaximal, complemented
10#26108C212yesmaximal, complemented
11#24813C212yesmaximal, complemented
12#24603C212yesmaximal
13#22964C212yesmaximal, complemented
14#22767C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16C4×#448C211yesmaximal, characteristic
17#20734C212yesmaximal, complemented
18C2×#552—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C2×#481C2212yes—
22C2×#475C2212yes—
23C2×#448C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#431C2212yes—
26#2857C2211yescharacteristic
27#2854C2212yes—
28#2552C2211yescharacteristic
29#2542C2211yescharacteristic
30#2519—22no—
31#2512C2212yes—
32#2506C2212yes—
33#2498—22no—
34#2489—21no—
35#2417—21no—
36#2365C2211yescharacteristic
37#2362—21no—
38#2354C2211yescharacteristic
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4—22no—
42Q8×C4⋊C4C2214yes—
43Q8×C4⋊C4C2212yes—
44Q8×C4⋊C4C2212yes—
45C4×#121C2211yescharacteristic
46C2×#389—22no—
47C2×#389C2212yes—
48#2285C2212yes—
49C2×#384—21no—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2214yes—
54#2099C2214yes—
55#2088C2214yes—
56#2084C2212yes—
57#2084C2214yes—
58#2071C2212yes—
59#2065C2214yes—
60#1990C2212yes—
61#1948C2212yes—
62#1941C2214yes—
63#1939C2212yes—
64#1934C2214yes—
65#1920C2214yes—
66#1920C2214yes—
67#1916C2214yes—
68#1880C2212yes—
69#1880C2214yes—
70#1880C2214yes—
71#1874C2211yescharacteristic
72#1865C2212yes—
73#1865C2214yes—
74#1865C2214yes—
75#1865C2214yes—
76#1853C2212yes—
77#1853C2211yescharacteristic
78#1834C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1824C2212yes—
82#1817C2212yes—
83#1803C2214yes—
84#1803C2214yes—
85#1803C2212yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1787C2212yes—
89#1787C2211yescharacteristic
90C8×C4⋊C4C2212yes—
91C4×#103C2214yes—
92C4×#103C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×SD16C2312yes—
97#552—48no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#126—22no—
100C2×#125—21no—
101C2×#125—22no—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×#121C2311yescharacteristic
107C2×C4×Q8—28no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8—28no—
114C2×C4×Q8—44no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C42—21no—
133C22×C42—22no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#481—44no—
137#475—28no—
138#448—24no—
139C4×SD16—28no—
140#431—44no—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2311yescharacteristic
156C2×C4×C8C2312yes—
157C2×C4×C8C2314yes—
158#400—22no—
159#399—22no—
160#398—22no—
161#398—22no—
162#397—22no—
163#396—22no—
164#391—21no—
165#391—21no—
166#390C2312yes—
167#390C2311yescharacteristic
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—24no—
174#389—24no—
175#389—24no—
176#389—24no—
177#389C2314yes—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2314yes—
183#388—22no—
184#388—22no—
185#388—21no—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#386—22no—
191#386—21no—
192#384—22no—
193#384—22no—
194#384—24no—
195#384—21no—
196#384—21no—
197#384C2312yes—
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#382—22no—
203#380—21no—
204#380—22no—
205#380—21no—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—24no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8#4114yes—
243C22×Q8C2412yes—
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2411yescharacteristic
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—24no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262#126—48no—
263#125—48no—
264#125—44no—
265#123—48no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269#121—22no—
270#121—22no—
271C4×Q8—28no—
272C4×Q8—28no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—28no—
277C4×Q8—28no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—48no—
282C4×Q8—44no—
283C4×Q8#4118yes—
284C4×D4—24no—
285C4×D4—44no—
286#117—48no—
287#117—44no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4SD1618yescomplemented
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C42—28no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—48no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364#106—24no—
365#106—28no—
366#106—44no—
367C4⋊C8—24no—
368C4⋊C8—44no—
369#103—24no—
370#103—24no—
371#103—28no—
372#103—28no—
373#103—24no—
374#102—24no—
375#102—44no—
376C22⋊C8—22no—
377C22⋊C8—22no—
378C22⋊C8—24no—
379C4×C8—24no—
380C4×C8—44no—
381#95—24no—
382#95—24no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24C22×D411yescharacteristic
398C2×Q8—24no—
399C2×Q8—216no—
400C2×Q8—24no—
401C2×Q8—216no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8C2×#4114yes—
410C2×Q8C2×#4114yes—
411C2×Q82−1+414yes—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4C2×SD1618yes—
452C22×C4C2×SD1614yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4C2511yescharacteristic
465SD16—48no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—84no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4C2×SD1614yes—
511C4⋊C4C2×SD1614yes—
512C4⋊C4C2×SD1618yes—
513C4⋊C4C2×#4114yes—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—81no—
523C22⋊C4—82no—
524C42—28no—
525C42—28no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—48no—
536C42—48no—
537C42—44no—
538C42—44no—
539C42—42no—
540C42C2×#4114yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23#54612yes—
553C23C22×SD1612yes—
554C23#57812yes—
555C23C23×D411yescharacteristic
556Q8—216no—
557Q8—24no—
558Q8—24no—
559Q8—24no—
560Q8—84no—
561Q8#55218yescomplemented
562D4—42no—
563D4—42no—
564D4—84no—
565C2×C4—216no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—216no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4#52512yes—
619C2×C4#52512yes—
620C2×C4#54612yes—
621C2×C4C22×SD1614yes—
622C2×C4C22×SD1612yes—
623C2×C4C22×SD1614yes—
624C2×C4C22×SD1612yes—
625C2×C4#57512yes—
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4112yes—
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×2−1+412yes—
630C8—42no—
631C8—42no—
632C8—44no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22#329814yes—
647C22C2×#52511yescharacteristic
648C22C2×#54611yescharacteristic
649C22#348611yescharacteristic
650C22C23×SD1611yescharacteristic
651C22C2×#57511yescharacteristic
652C22C2×#57811yescharacteristic
653C22#358511yescharacteristic
654C4—28no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—24no—
659C4—24no—
660C4—41no—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4—81no—
669C4—84no—
670C4#310314yes—
671C4C2×#55212yes—
672C4#349212yes—
673C2—24no—
674C2—161no—
675C2#3616311yescharacteristic
676C2C2×#310311yescharacteristic
677C2#6058911yescharacteristic
678C2C2×#329811yescharacteristic
679C2#6116011yescharacteristic
680C2#6182011yescharacteristic
681C2#6184211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
4. #32954C2semidirect
5. #27961C2semidirect
7. #27611C2semidirect
8. #27554C2semidirect
9. #27298C2semidirect
10. #26108C2semidirect
11. #24813C2semidirect
13. #22964C2semidirect
14. #22767C2semidirect
15. #22599C2semidirect
17. #20734C2semidirect
320. C2×C4⋊C4SD16semidirect
561. Q8#552semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×Q8×C4⋊C42
#330052
#329542
#279612
#276452
#276112
#275542
#272982
#261082
#248132
#246032
#229642
#227672
#225992
C4×#4482
#207342