Finite Group Databasebeta

← #6484424

Subgroups — #6484424

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#32968C214yesmaximal, complemented
4#29961C214yesmaximal, complemented
5#29267C214yesmaximal, complemented
6#29193C211yesmaximal, characteristic, complemented
7#29192C211yesmaximal, characteristic, complemented
8#29191C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic, complemented
10#29141C211yesmaximal, characteristic, complemented
11#29139C211yesmaximal, characteristic, complemented
12#28515C211yesmaximal, characteristic, complemented
13#28514C211yesmaximal, characteristic, complemented
14#28293C214yesmaximal, complemented
15#21982C211yesmaximal, characteristic, complemented
16#21981C211yesmaximal, characteristic, complemented
17#21805C211yesmaximal, characteristic
18#21805C211yesmaximal, characteristic
19#20809C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×#520C2212yes—
22C2×#479C2214yes—
23C2×#478C2214yes—
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2629C2212yes—
31#2628C2212yes—
32#2570—21no—
33#2570—21no—
34#2550—21no—
35#2550—21no—
36#2441—21no—
37#2441—21no—
38#2439—21no—
39#2439—21no—
40#2361—21no—
41#2361—21no—
42#2361—21no—
43#2361—21no—
44#2354—21no—
45#2354—21no—
46D4×C4⋊C4—21no—
47D4×C4⋊C4—21no—
48#2331C2212yes—
49#2330C2212yes—
50#2326C2212yes—
51#2326C2212yes—
52#2311C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C4×#127C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59C2×#384C2211yescharacteristic
60C4×#117C2211yescharacteristic
61#2097C2214yes—
62#2092C2214yes—
63#2068C2214yes—
64#2067C2214yes—
65#2066C2214yes—
66#2065C2214yes—
67#2024C2214yes—
68#2023C2214yes—
69#1983C2214yes—
70#1983C2214yes—
71#1957C2211yescharacteristic
72#1957C2211yescharacteristic
73#1957C2211yescharacteristic
74#1957C2211yescharacteristic
75#1956C2211yescharacteristic
76#1956C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1948C2214yes—
80#1947C2214yes—
81#1865C2214yes—
82#1864C2214yes—
83#1863C2214yes—
84#1862C2214yes—
85#1853C2214yes—
86#1852C2214yes—
87#1845C2211yescharacteristic
88#1845C2211yescharacteristic
89#1844C2211yescharacteristic
90#1844C2211yescharacteristic
91#1843C2212yes—
92#1834C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1832C2212yes—
97#1803C2214yes—
98#1802C2214yes—
99#1798C2212yes—
100#1798C2212yes—
101#1788C2214yes—
102#1788C2214yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106C4×C4⋊C8C2211yescharacteristic
107C22×#41C2312yes—
108#532—24no—
109#532Q818yescomplemented
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×#127C2311yescharacteristic
115C2×#125—21no—
116C2×#125—21no—
117C2×#125C2312yes—
118C2×#124C2312yes—
119C2×#121—21no—
120C2×#121—21no—
121#520—48no—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2312yes—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117—22no—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—22no—
142C22×C4⋊C4C2312yes—
143C22×C42C2311yescharacteristic
144#479—48no—
145#478—48no—
146#431—24no—
147#431—24no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2312yes—
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172#400—22no—
173#399—22no—
174#398—22no—
175#398—22no—
176#396—22no—
177#396—22no—
178#395—22no—
179#393—22no—
180#391—22no—
181#391—22no—
182#391C2312yes—
183#391C2312yes—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390C2312yes—
189#390C2312yes—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—22no—
203#389—22no—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#388—22no—
209#388—22no—
210#388—22no—
211#388—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#386—22no—
217#386—22no—
218#385—22no—
219#385—22no—
220#385—22no—
221#385—22no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2312yes—
229#384C2312yes—
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#382—21no—
235#382—21no—
236#382—21no—
237#382—21no—
238#382C2312yes—
239#380—21no—
240#380—21no—
241#380—22no—
242#380—22no—
243#380C2312yes—
244#380C2312yes—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261#376C2311yescharacteristic
262#376C2311yescharacteristic
263C43C2311yescharacteristic
264C43C2311yescharacteristic
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C2×#41—24no—
274C2×#41—44no—
275C2×#41—44no—
276C2×#41C2×Q818yes—
277C22×Q8—22no—
278C22×Q8C2412yes—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4—22no—
285C23×C4—22no—
286C23×C4—22no—
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—22no—
295C4⋊Q8—44no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298C4⋊Q8C2×Q814yes—
299#127—22no—
300#127C2×Q814yes—
301#125—28no—
302#125—82no—
303#125—82no—
304#124—24no—
305#124C2×Q818yes—
306#121—28no—
307#121—28no—
308C4×Q8—28no—
309C4×Q8—44no—
310C4×D4—28no—
311C4×D4—28no—
312C4×D4—28no—
313C4×D4—44no—
314C4×D4—48no—
315C4×D4—48no—
316C4×D4—82no—
317C4×D4—82no—
318C4×D4—82no—
319C4×D4—82no—
320#117—24no—
321#117—42no—
322#117—48no—
323#117—82no—
324#117—82no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—44no—
354C2×C4⋊C4—48no—
355C2×C4⋊C4—82no—
356C2×C4⋊C4—82no—
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C22⋊C4C2×D411yescharacteristic
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×Q814yes—
396C2×C42#4114yes—
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#107—44no—
404#107—44no—
405#106—44no—
406#106—44no—
407C4⋊C8—24no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410C4⋊C8—24no—
411C4⋊C8—42no—
412C4⋊C8—42no—
413#103—44no—
414#103—44no—
415#103—44no—
416#103—44no—
417#102—44no—
418#102—44no—
419#102—44no—
420#102—44no—
421C22⋊C8—24no—
422C22⋊C8—24no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449C24—22no—
450C24C22×D411yescharacteristic
451#41—416no—
452#41—416no—
453C2×Q8—22no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—44no—
457C2×Q8—82no—
458C2×Q8—82no—
459C2×Q8C22×Q814yes—
460C2×D4—28no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4C22×Q814yes—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—28no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4C22⋊Q814yes—
499C22×C4C22⋊Q814yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×Q812yes—
514C22×C4C2×#4112yes—
515C22×C4C2511yescharacteristic
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C22⋊C4—28no—
567C22⋊C4—28no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—41no—
571C22⋊C4—41no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—44no—
592C42—44no—
593C42—82no—
594C42—82no—
595C42C22⋊Q814yes—
596C42C22⋊Q814yes—
597C42C22×Q812yes—
598C42C22×Q812yes—
599C42C2×#4112yes—
600C42C2×#4112yes—
601C42H(5,𝔽2)14yes—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—24no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23C2×C22⋊Q812yes—
616C23C23×D411yescharacteristic
617Q8—28no—
618Q8—44no—
619Q8—164no—
620D4—28no—
621D4—416no—
622D4—48no—
623D4—48no—
624D4—44no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—416no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—164no—
680C2×C4—164no—
681C2×C4—164no—
682C2×C4C2×C22⋊Q812yes—
683C2×C4C2×C22⋊Q812yes—
684C2×C4C2×C22⋊Q812yes—
685C2×C4C2×C22⋊Q812yes—
686C2×C4C2×C22⋊Q812yes—
687C2×C4#53512yes—
688C2×C4#53512yes—
689C2×C4#57612yes—
690C2×C4#57712yes—
691C2×C4#57712yes—
692C2×C4#57812yes—
693C2×C4C23×Q811yescharacteristic
694C2×C4C22×#4111yescharacteristic
695C2×C4C2×H(5,𝔽2)12yes—
696C2×C4#58512yes—
697C8—42no—
698C8—42no—
699C8—42no—
700C8—42no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22C22×C22⋊Q811yescharacteristic
716C22C2×#53511yescharacteristic
717C22#344911yescharacteristic
718C22C2×#57611yescharacteristic
719C22C2×#57711yescharacteristic
720C22C2×#57711yescharacteristic
721C22C2×#57811yescharacteristic
722C4—22no—
723C4—22no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—44no—
732C4—41no—
733C4—41no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4#308314yes—
741C4#308414yes—
742C4#350112yes—
743C4#350612yes—
744C2—42no—
745C2—84no—
746C2#3517211yescharacteristic
747C2C2×#308311yescharacteristic
748C2C2×#308411yescharacteristic
749C2#6082211yescharacteristic
750C2#6082411yescharacteristic
751C2#6100611yescharacteristic
752C2#6100811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35294C2semidirect
3. #32968C2semidirect
4. #29961C2semidirect
5. #29267C2semidirect
6. #29193C2semidirect
7. #29192C2semidirect
8. #29191C2semidirect
9. #29190C2semidirect
10. #29141C2semidirect
11. #29139C2semidirect
12. #28515C2semidirect
13. #28514C2semidirect
14. #28293C2semidirect
15. #21982C2semidirect
16. #21981C2semidirect
19. #20809C2semidirect
109. #532Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#342072
#329682
#299612
#292672
#291932
#291922
#291912
#291902
#291412
#291392
#285152
#285142
#282932
#219822
#219812
#218052
#208092