Finite Group Databasebeta

← #6439309

Subgroups — #6439309

Order 512.

2142 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3097C211yesmaximal, characteristic
2C22×C4×Q16C211yesmaximal, characteristic, complemented
3#35182C211yesmaximal, characteristic
4#34266C211yesmaximal, characteristic
5#26561C214yesmaximal, complemented
6#26560C214yesmaximal, complemented
7#26559C214yesmaximal, complemented
8#26558C214yesmaximal, complemented
9#23736C211yesmaximal, characteristic, complemented
10#23542C211yesmaximal, characteristic, complemented
11#23487C211yesmaximal, characteristic
12#23389C211yesmaximal, characteristic
13#21173C211yesmaximal, characteristic, complemented
14#21151C211yesmaximal, characteristic, complemented
15#21066C211yesmaximal, characteristic, complemented
16#21046C211yesmaximal, characteristic
17#20874C211yesmaximal, characteristic, complemented
18#20874C211yesmaximal, characteristic, complemented
19#20847C211yesmaximal, characteristic
20C23×Q16C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26#3097—28no—
27C2×#484C2212yes—
28C2×#485C2212yes—
29C2×#485C2212yes—
30C2×#483C2212yes—
31C2×C4×Q16—28no—
32C2×C4×Q16C2214yes—
33C2×C4×Q16C2214yes—
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#427C2211yescharacteristic
37C2×#427C2211yescharacteristic
38C22×#107C2211yescharacteristic
39C2×#424C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C2×#419C2211yescharacteristic
42C2×#419C2211yescharacteristic
43C2×#419C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#103C2211yescharacteristic
46C2×#408C2211yescharacteristic
47C2×#408C2211yescharacteristic
48C22×C4×C8C2211yescharacteristic
49#2853—22no—
50#2623—22no—
51#2448—22no—
52#2446—22no—
53#2423C2214yes—
54#2421C2214yes—
55#2367—22no—
56#2365—22no—
57#2362C2214yes—
58#2360C2214yes—
59#2322—22no—
60#2317C2211yescharacteristic
61#2311C2211yescharacteristic
62#2310C2211yescharacteristic
63#2310C2211yescharacteristic
64C4×#123—22no—
65#2297C2211yescharacteristic
66#2296C2211yescharacteristic
67#2296C2211yescharacteristic
68#2274C2211yescharacteristic
69#2097C2212yes—
70#2097C2212yes—
71#2096C2212yes—
72#2095C2212yes—
73#2056C2214yes—
74#2056C2214yes—
75#2036C2214yes—
76#2035C2214yes—
77#2001C2214yes—
78#2001C2214yes—
79#1949C2214yes—
80#1947C2214yes—
81#1942C2212yes—
82#1942C2212yes—
83#1941C2212yes—
84#1941C2212yes—
85#1928C2214yes—
86#1927C2214yes—
87#1923C2214yes—
88#1923C2214yes—
89#1865C2214yes—
90#1865C2214yes—
91#1863C2214yes—
92#1863C2214yes—
93#1854C2214yes—
94#1852C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C23×Q8C2311yescharacteristic
104C22×Q16—24no—
105C22×Q16—24no—
106C22×Q16—24no—
107C22×Q16C2314yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112#536—24no—
113#536—24no—
114#536—24no—
115#536—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×#124—21no—
119C2×#124—21no—
120C2×#123—22no—
121C2×#123—22no—
122C2×C22⋊Q8C2314yes—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2312yes—
125#520—24no—
126#520—42no—
127C2×C4×Q8—21no—
128C2×C4×Q8—22no—
129C2×C4×Q8—22no—
130C2×C4×Q8—24no—
131C2×C4×Q8—24no—
132C2×C4×Q8—42no—
133C2×C4×Q8—42no—
134C2×C4×Q8C2314yes—
135C2×C4×Q8C2314yes—
136C2×C4×D4—22no—
137C2×C4×D4—21no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2312yes—
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C22×C4⋊C4—21no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2311yescharacteristic
149C22×C42—21no—
150C22×C42—21no—
151C22×C42C2311yescharacteristic
152#485—28no—
153#485—28no—
154#484—28no—
155#483—28no—
156C4×Q16—216no—
157C4×Q16—216no—
158#428—24no—
159#428—42no—
160#427—24no—
161#427—24no—
162C2×#107—24no—
163C2×#107C2311yescharacteristic
164C2×#107C2312yes—
165C2×#107C2311yescharacteristic
166C2×#107C2314yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#106C2311yescharacteristic
170C2×#106C2312yes—
171C2×#106C2311yescharacteristic
172C2×#106C2312yes—
173C2×#106C2312yes—
174#424—24no—
175C2×C4⋊C8C2314yes—
176#419—24no—
177#419—24no—
178#419—24no—
179#419—42no—
180C2×#103—24no—
181C2×#103D418yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2314yes—
188C2×#103C2312yes—
189C2×#103C2312yes—
190C2×#103C2312yes—
191C2×#103C2312yes—
192#408—24no—
193#408—24no—
194C2×C22⋊C8C2312yes—
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2314yes—
197C2×C4×C8—24no—
198C2×C4×C8C2312yes—
199C2×C4×C8C2312yes—
200#400—24no—
201#399—24no—
202#398—24no—
203#397—24no—
204#391—21no—
205#391—22no—
206#391—22no—
207#391D412yes—
208#391C2312yes—
209#391C2312yes—
210#391C2314yes—
211#390—21no—
212#390D412yes—
213#389—22no—
214#389—24no—
215#389C2312yes—
216#389C2312yes—
217#389C2312yes—
218#389C2312yes—
219#388—24no—
220#388—24no—
221#388—24no—
222#388—24no—
223#387—24no—
224#387—24no—
225#387—24no—
226#387—24no—
227#386—22no—
228#386D412yes—
229#386D412yes—
230#385—21no—
231#385—21no—
232#385—21no—
233#385—21no—
234#385—22no—
235#385—24no—
236#384C2312yes—
237#382C2312yes—
238#381—22no—
239#381—22no—
240#380C2314yes—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—22no—
244C4×C4⋊C4—24no—
245C4×C4⋊C4C2314yes—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—24no—
248C4×C22⋊C4C2314yes—
249#340C2312yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#336C2312yes—
256#336C2312yes—
257#336C2314yes—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8—22no—
272C22×Q8—22no—
273C22×Q8—22no—
274C22×Q8—22no—
275C22×Q8—22no—
276C22×Q8C2×D414yes—
277C22×Q8C2×D411yescharacteristic
278C22×Q8#4114yes—
279C22×Q8C2412yes—
280C22×Q8C2412yes—
281C22×D4C2×D412yes—
282C22×D4C2×D411yescharacteristic
283C23×C4—21no—
284C23×C4—21no—
285C23×C4—21no—
286C23×C4—21no—
287C23×C4—22no—
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2412yes—
291C23×C4C2411yescharacteristic
292C2×Q16—232no—
293C2×Q16—28no—
294C2×Q16—28no—
295C2×Q16—28no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—22no—
300C2×M16—22no—
301C2×M16—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C2×D414yes—
307C22×C8#4114yes—
308C22×C8C2412yes—
309C22×C8C2412yes—
310C22×C8C2411yescharacteristic
311C22×C8C2411yescharacteristic
312C22×C8C2412yes—
313C4⋊Q8—28no—
314C4⋊Q8—44no—
315#124—28no—
316#124—44no—
317#123—216no—
318#123—48no—
319C22⋊Q8—28no—
320C22⋊Q8—28no—
321C22⋊Q8—24no—
322C22⋊Q8—24no—
323C22⋊Q8—42no—
324C22⋊Q8—42no—
325C4×Q8—28no—
326C4×Q8—28no—
327C4×Q8—44no—
328C4×Q8—44no—
329C4×Q8—48no—
330C4×Q8—48no—
331C4×Q8—44no—
332C4×D4—44no—
333C4×D4—48no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—22no—
343#117—24no—
344#117—42no—
345#117#4114yes—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—44no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2×D414yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2411yescharacteristic
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D414yes—
387C2×C22⋊C4C2×D414yes—
388C2×C42—21no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—24no—
396C2×C42—24no—
397C2×C42—24no—
398C2×C42—44no—
399C2×C42—44no—
400C2×C42C2×D412yes—
401C2×C42C2412yes—
402C2×C42C2412yes—
403#107—22no—
404#107—22no—
405#107—28no—
406#107—24no—
407#107—24no—
408#107—42no—
409#106—22no—
410#106—22no—
411#106—24no—
412#106—24no—
413#106—42no—
414C4⋊C8—28no—
415#103—24no—
416#103—24no—
417#103—28no—
418#103—24no—
419#103—24no—
420#103—28no—
421#103—42no—
422#103—42no—
423#103C2×D418yes—
424#103C2×D418yes—
425C22⋊C8—24no—
426C22⋊C8—24no—
427C22⋊C8—28no—
428C4×C8—24no—
429C4×C8—24no—
430#95—22no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—24no—
438#95—24no—
439#95—24no—
440#95—24no—
441#95C2×D414yes—
442#95C2×D414yes—
443C24C22×D412yes—
444C24C22×D411yescharacteristic
445#41—44no—
446#41—44no—
447#41—44no—
448#41—44no—
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—28no—
457C2×Q8—28no—
458C2×Q8—28no—
459C2×Q8—28no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—42no—
469C2×Q8—44no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8—44no—
476C2×Q8C22×D414yes—
477C2×Q8C2×#4114yes—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—42no—
507C22×C4—44no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—82no—
514C22×C4—82no—
515C22×C4—84no—
516C22×C4C22×D414yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D412yes—
522C22×C4C2×#4112yes—
523C22×C4C2×#4112yes—
524C22×C4C2×#4112yes—
525C22×C4H(5,𝔽2)12yes—
526C22×C42−1+412yes—
527C22×C4C2511yescharacteristic
528Q16—232no—
529M16—44no—
530M16—44no—
531C2×C8—28no—
532C2×C8—28no—
533C2×C8—22no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—22no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8C22×D412yes—
546C2×C8C22×D412yes—
547C2×C8C2×#4114yes—
548C4⋊C4—216no—
549C4⋊C4—24no—
550C4⋊C4—24no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—44no—
566C4⋊C4—48no—
567C4⋊C4—48no—
568C4⋊C4—44no—
569C4⋊C4—44no—
570C4⋊C4—42no—
571C4⋊C4—44no—
572C4⋊C4—44no—
573C4⋊C4—84no—
574C4⋊C4—82no—
575C4⋊C4—82no—
576C4⋊C4C22×D414yes—
577C4⋊C4C2×#4114yes—
578C22⋊C4—216no—
579C22⋊C4—24no—
580C22⋊C4—24no—
581C22⋊C4—28no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C22⋊C4—42no—
587C22⋊C4—42no—
588C22⋊C4—42no—
589C22⋊C4—84no—
590C42—24no—
591C42—24no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—48no—
597C42—48no—
598C42—44no—
599C42—42no—
600C42—42no—
601C42—42no—
602C42—84no—
603C23—22no—
604C23—22no—
605C23—24no—
606C23—22no—
607C23—22no—
608C23—22no—
609C23—24no—
610C23#52512yes—
611C23D4212yes—
612C23#54712yes—
613C23C23×D411yescharacteristic
614Q8—24no—
615Q8—24no—
616Q8—24no—
617Q8—24no—
618Q8—42no—
619Q8—42no—
620Q8—48no—
621Q8—48no—
622Q8—44no—
623Q8—42no—
624Q8—42no—
625D4—48no—
626D4—44no—
627C2×C4—28no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—48no—
651C2×C4—48no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—164no—
673C2×C4#52514yes—
674C2×C4#52512yes—
675C2×C4#53914yes—
676C2×C4D4212yes—
677C2×C4#54714yes—
678C2×C4#54712yes—
679C2×C4#55914yes—
680C2×C4#57814yes—
681C2×C4C23×D411yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C22×#4111yescharacteristic
684C2×C4C22×#4111yescharacteristic
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4C2×2−1+411yescharacteristic
687C8—24no—
688C8—22no—
689C8—22no—
690C8—42no—
691C8—42no—
692C22—24no—
693C22—22no—
694C22—22no—
695C22—24no—
696C22—42no—
697C22—44no—
698C22—42no—
699C22—42no—
700C22—44no—
701C22#245514yes—
702C22C2×#52511yescharacteristic
703C22C2×D4211yescharacteristic
704C22C2×#54711yescharacteristic
705C22C2×#57812yes—
706C22#358412yes—
707C4—24no—
708C4—42no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—42no—
713C4—44no—
714C4—44no—
715C4—81no—
716C4—82no—
717C4—81no—
718C4—81no—
719C4—81no—
720C4—82no—
721C4#245514yes—
722C4C2×#52511yescharacteristic
723C4C2×#53911yescharacteristic
724C4C2×#54711yescharacteristic
725C4C2×#55911yescharacteristic
726C2—24no—
727C2—82no—
728C2C2×#245511yescharacteristic
729C2#6057312yes—
730C2#6105211yescharacteristic
731C2#6105311yescharacteristic
732C2#6128612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×Q16C2semidirect
5. #26561C2semidirect
6. #26560C2semidirect
7. #26559C2semidirect
8. #26558C2semidirect
9. #23736C2semidirect
10. #23542C2semidirect
13. #21173C2semidirect
14. #21151C2semidirect
15. #21066C2semidirect
17. #20874C2semidirect
18. #20874C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30972
C22×C4×Q162
#351822
#342662
#265612
#265602
#265592
#265582
#237362
#235422
#234872
#233892
#211732
#211512
#210662
#210462
#208742
#208472