Finite Group Databasebeta

← #6627924

Subgroups — #6627924

Order 512.

1988 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36596C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#32903C211yesmaximal, characteristic, complemented
4#32903C212yesmaximal, complemented
5#29328C212yesmaximal, complemented
6#29235C212yesmaximal, complemented
7#27973C211yesmaximal, characteristic, complemented
8#27678C211yesmaximal, characteristic, complemented
9#27554C214yesmaximal, complemented
10#26163C214yesmaximal, complemented
11#26108C212yesmaximal, complemented
12#24728C212yesmaximal, complemented
13#22997C212yesmaximal, complemented
14#22767C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16C4×#448C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic, complemented
18C2×#553—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C2×#460C2212yes—
22C2×#460C2214yes—
23C2×#448C2212yes—
24C2×#448C2211yescharacteristic
25C2×C4×SD16C2212yes—
26C2×C4×SD16C2214yes—
27#2856C2211yescharacteristic
28#2856C2212yes—
29#2554—22no—
30#2552C2211yescharacteristic
31#2543—22no—
32#2541C2211yescharacteristic
33#2512C2212yes—
34#2490—21no—
35#2445C2212yes—
36#2416—21no—
37#2366C2211yescharacteristic
38#2363—21no—
39#2354C2211yescharacteristic
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4—22no—
43Q8×C4⋊C4C2214yes—
44Q8×C4⋊C4C2212yes—
45Q8×C4⋊C4C2212yes—
46C4×#121C2211yescharacteristic
47C4×#121C2212yes—
48C2×#389—22no—
49C2×#389C2212yes—
50C2×#384—21no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2212yes—
55#2099C2214yes—
56#2094C2212yes—
57#2088C2214yes—
58#2071C2214yes—
59#2065C2214yes—
60#2041C2214yes—
61#1957C2212yes—
62#1948C2212yes—
63#1939C2212yes—
64#1934C2214yes—
65#1934C2214yes—
66#1920C2214yes—
67#1916C2214yes—
68#1880C2212yes—
69#1880C2214yes—
70#1880C2214yes—
71#1880C2214yes—
72#1880C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1874C2212yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1865C2214yes—
79#1865C2212yes—
80#1864C2214yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1834C2212yes—
84#1803C2214yes—
85#1803C2212yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1787C2211yescharacteristic
90C8×C4⋊C4C2212yes—
91C4×#103C2214yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8C2311yescharacteristic
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98#553—48no—
99C2×#127C2311yescharacteristic
100C2×#127C2312yes—
101C2×#126—22no—
102C2×#125—21no—
103C2×#123—22no—
104C2×#121—21no—
105C2×#121—22no—
106C2×#121—22no—
107C2×#121—21no—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×#121C2311yescharacteristic
112C2×C4×Q8—28no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—48no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×#117—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4D411yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2312yes—
137C22×C42—21no—
138C22×C42—22no—
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141#460—28no—
142#460—48no—
143#448—28no—
144#448—24no—
145C4×SD16—28no—
146C4×SD16—48no—
147C2×#106C2312yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2314yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2311yescharacteristic
163C2×C4×C8C2312yes—
164C2×C4×C8C2314yes—
165#398—22no—
166#397—22no—
167#396—22no—
168#394—22no—
169#394—22no—
170#392—22no—
171#390—21no—
172#390—21no—
173#390—22no—
174#390—22no—
175#390—21no—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#390C2311yescharacteristic
181#389—21no—
182#389—22no—
183#389—22no—
184#389—24no—
185#389—24no—
186#389—24no—
187#389—24no—
188#389C2314yes—
189#389C2312yes—
190#389C2314yes—
191#389C2311yescharacteristic
192#388—22no—
193#388—21no—
194#388—22no—
195#388—22no—
196#386—22no—
197#386—21no—
198#386—22no—
199#386—22no—
200#384—22no—
201#384—22no—
202#384—24no—
203#384—21no—
204#384—21no—
205#384C2312yes—
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2312yes—
209#380—21no—
210#380C2311yescharacteristic
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C2×#95—22no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8#4114yes—
242C22×Q8C2412yes—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—28no—
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—24no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270#127—24no—
271#127—44no—
272#126—48no—
273#125—44no—
274#123—48no—
275#121—24no—
276#121—22no—
277#121—24no—
278#121—22no—
279#121—44no—
280#121—48no—
281#121—44no—
282#121—44no—
283#121—82no—
284#121—84no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—28no—
291C4×Q8—28no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—48no—
295C4×Q8—48no—
296C4×Q8#4118yes—
297C4×D4—24no—
298C4×D4—44no—
299C4×D4—48no—
300C4×D4—44no—
301C4×D4—44no—
302#117—44no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4SD1618yescomplemented
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C42—28no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—44no—
366C2×C42—48no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#106—24no—
376#106—44no—
377C4⋊C8—24no—
378C4⋊C8—44no—
379#103—24no—
380#103—24no—
381#103—28no—
382#103—24no—
383#102—24no—
384#102—28no—
385#102—44no—
386#102—44no—
387#102—44no—
388C22⋊C8—24no—
389C22⋊C8—22no—
390C22⋊C8—22no—
391C4×C8—24no—
392C4×C8—44no—
393#95—24no—
394#95—24no—
395#95—21no—
396#95—22no—
397#95—21no—
398#95—22no—
399#95—21no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—21no—
405C24—22no—
406C24C22×D411yescharacteristic
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—216no—
410C2×Q8—216no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8C2×#4114yes—
416C2×Q8C2×#4114yes—
417C2×Q8H(5,𝔽2)14yes—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—81no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—81no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4C2×SD1614yes—
479C22×C4C2×SD1618yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C42−1+412yes—
491C22×C4C2511yescharacteristic
492SD16—48no—
493SD16—416no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—82no—
530C4⋊C4—84no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4C2×SD1614yes—
534C4⋊C4C2×SD1614yes—
535C4⋊C4C2×SD1618yes—
536C4⋊C4C2×#4114yes—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—81no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C42—28no—
553C42—28no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—24no—
558C42—24no—
559C42—42no—
560C42—42no—
561C42—48no—
562C42—48no—
563C42—44no—
564C42—42no—
565C42C2×#4114yes—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—41no—
571C23—42no—
572C23—41no—
573C23—42no—
574C23—41no—
575C23—42no—
576C23—42no—
577C23—41no—
578C23—41no—
579C23—42no—
580C23—42no—
581C23—41no—
582C23—41no—
583C23—42no—
584C23#54812yes—
585C23C22×SD1612yes—
586C23#57612yes—
587C23C23×D411yescharacteristic
588Q8—216no—
589Q8—24no—
590Q8—24no—
591Q8—24no—
592Q8#55318yescomplemented
593D4—42no—
594D4—44no—
595D4—42no—
596D4—44no—
597D4—88no—
598D4—84no—
599D4—84no—
600D4—164no—
601D4—162no—
602D4—164no—
603C2×C4—216no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—21no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—28no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—84no—
651C2×C4—88no—
652C2×C4—84no—
653C2×C4—164no—
654C2×C4#52512yes—
655C2×C4#52512yes—
656C2×C4#54812yes—
657C2×C4C22×SD1614yes—
658C2×C4C22×SD1612yes—
659C2×C4C22×SD1614yes—
660C2×C4C22×SD1612yes—
661C2×C4#57512yes—
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4112yes—
664C2×C4C2×H(5,𝔽2)12yes—
665C2×C4C2×2−1+411yescharacteristic
666C8—42no—
667C8—44no—
668C8—42no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—81no—
676C22—82no—
677C22—81no—
678C22—82no—
679C22—82no—
680C22—81no—
681C22—81no—
682C22—82no—
683C22—81no—
684C22—82no—
685C22—82no—
686C22—81no—
687C22—81no—
688C22—82no—
689C22#336214yes—
690C22C2×#52511yescharacteristic
691C22C2×#54811yescharacteristic
692C22#348411yescharacteristic
693C22C23×SD1611yescharacteristic
694C22C2×#57511yescharacteristic
695C22C2×#57611yescharacteristic
696C22#358611yescharacteristic
697C4—22no—
698C4—24no—
699C4—24no—
700C4—22no—
701C4—28no—
702C4—22no—
703C4—42no—
704C4—42no—
705C4—41no—
706C4—44no—
707C4—81no—
708C4—82no—
709C4—81no—
710C4—81no—
711C4—84no—
712C4#310314yes—
713C4C2×#55312yes—
714C4#349212yes—
715C2—24no—
716C2—161no—
717C2—162no—
718C2#3658111yescharacteristic
719C2C2×#310311yescharacteristic
720C2#6058411yescharacteristic
721C2C2×#336211yescharacteristic
722C2#6132011yescharacteristic
723C2#6156211yescharacteristic
724C2#6160511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36596C2
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
3. #32903C2semidirect
4. #32903C2semidirect
5. #29328C2semidirect
6. #29235C2semidirect
7. #27973C2semidirect
8. #27678C2semidirect
9. #27554C2semidirect
10. #26163C2semidirect
11. #26108C2semidirect
12. #24728C2semidirect
13. #22997C2semidirect
14. #22767C2semidirect
15. #22625C2semidirect
16. C4×#448C2semidirect
17. C4×#448C2semidirect
329. C2×C4⋊C4SD16semidirect
592. Q8#553semidirect

Maximal subgroups

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

subgroupindex
#365962
C2×Q8×C4⋊C42
#329032
#293282
#292352
#279732
#276782
#275542
#261632
#261082
#247282
#229972
#227672
#226252
C4×#4482