Finite Group Databasebeta

← #6463595

Subgroups — #6463595

Order 512.

2022 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29262C212yesmaximal, complemented
4#29261C214yesmaximal, complemented
5#29260C212yesmaximal, complemented
6#29259C211yesmaximal, characteristic, complemented
7#29258C211yesmaximal, characteristic, complemented
8#29257C212yesmaximal, complemented
9#29256C212yesmaximal, complemented
10#29255C211yesmaximal, characteristic, complemented
11#29254C211yesmaximal, characteristic, complemented
12#23730C212yesmaximal, complemented
13#23451C212yesmaximal, complemented
14#21257C212yesmaximal
15#21169C212yesmaximal, complemented
16#20872C214yesmaximal, complemented
17C2×#1831C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×C4×#41C2212yes—
21C2×#427C2214yes—
22C22×C4⋊C8C2212yes—
23C2×#416C2214yes—
24C2×#416C2214yes—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2211yescharacteristic
30C4×C4⋊Q8C2211yescharacteristic
31C4×#127C2211yescharacteristic
32C4×#127C2211yescharacteristic
33C4×#125C2212yes—
34C4×#125C2212yes—
35C4×#124C2212yes—
36C4×#124C2212yes—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8—21no—
40C4×C22⋊Q8—21no—
41C4×#121—21no—
42C4×#121—21no—
43C4×#121—21no—
44C4×#121—21no—
45#2297C2212yes—
46#2296C2212yes—
47#2295—21no—
48#2295—21no—
49#2293—21no—
50#2293—21no—
51#2285C2211yescharacteristic
52#2285C2211yescharacteristic
53C42×Q8—21no—
54C42×Q8—21no—
55C42×D4—21no—
56C42×D4—21no—
57C4×#117C2212yes—
58C2×C43C2211yescharacteristic
59#2097C2214yes—
60#2096C2212yes—
61#2095C2212yes—
62#2094C2212yes—
63#2093C2212yes—
64#2092C2214yes—
65#2026C2214yes—
66#2025C2214yes—
67#2024C2212yes—
68#2024C2212yes—
69#2023C2212yes—
70#2023C2212yes—
71#1983C2214yes—
72#1983C2214yes—
73#1982C2212yes—
74#1982C2212yes—
75#1981C2212yes—
76#1981C2212yes—
77#1949C2214yes—
78#1949C2214yes—
79#1948C2212yes—
80#1948C2212yes—
81#1947C2212yes—
82#1947C2212yes—
83#1865C2214yes—
84#1865C2214yes—
85#1864C2214yes—
86#1864C2214yes—
87#1863C2214yes—
88#1863C2214yes—
89#1862C2214yes—
90#1862C2214yes—
91#1831C2211yescharacteristic
92#1831C2211yescharacteristic
93#1831C2212yes—
94#1831C2212yes—
95#1831C2212yes—
96#1735C2214yes—
97C2×#336C2212yes—
98C2×#336C2212yes—
99C22×#41C2312yes—
100C23×C8C2314yes—
101#533—28no—
102#532—24no—
103#532Q818yescomplemented
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#125C2312yes—
109C2×#124C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×#121—21no—
113C2×#121—21no—
114C4×#41—48no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×#117—22no—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C2×#117C2312yes—
137C22×C42—22no—
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42D411yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#427—48no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8—28no—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154#416—216no—
155#416—48no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#386—22no—
197#386—22no—
198#385—22no—
199#385—22no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#382C2312yes—
205#382C2312yes—
206#380—21no—
207#380—21no—
208#380—21no—
209#380—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240#376—22no—
241C43—22no—
242C43D411yescharacteristic
243C43D411yescharacteristic
244C43D411yescharacteristic
245C43D411yescharacteristic
246C43C2311yescharacteristic
247C43C2311yescharacteristic
248C43C2311yescharacteristic
249C43C2311yescharacteristic
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#336C2314yes—
255#336C2312yes—
256#336C2312yes—
257#336C2312yes—
258#336C2314yes—
259#336C2312yes—
260C2×#41—24no—
261C2×#41—28no—
262C2×#41—28no—
263C2×#41C2×Q818yes—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8C2412yes—
267C22×D4—22no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4C2412yes—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C22×C8—28no—
282C22×C8—28no—
283C22×C8—28no—
284C22×C8C2414yes—
285C22×C8C2414yes—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—22no—
289C4⋊Q8C2×Q814yes—
290#127—22no—
291#127C2×Q814yes—
292#125—24no—
293#125—24no—
294#124—24no—
295#124C2×Q818yes—
296C22⋊Q8—28no—
297C22⋊Q8—28no—
298#121—28no—
299#121—28no—
300C4×Q8—28no—
301C4×Q8—28no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4—44no—
315C4×D4—48no—
316C4×D4—48no—
317C4×D4—44no—
318#117—24no—
319#117—28no—
320#117—24no—
321#117—24no—
322#117—48no—
323#117#4118yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
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—22no—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C42—22no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—42no—
374C2×C42—42no—
375C2×C42—42no—
376C2×C42—48no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
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×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×Q814yes—
400C2×C42#4114yes—
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407#107—44no—
408#107—44no—
409#106—44no—
410#106—44no—
411C4⋊C8—28no—
412C4⋊C8—24no—
413C4⋊C8—24no—
414#103—28no—
415#103—28no—
416#103—44no—
417#103—44no—
418#102—28no—
419#102—28no—
420#102—44no—
421#102—44no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—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—
438C24—22no—
439C24C22×D411yescharacteristic
440#41—232no—
441#41—416no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8C22×Q814yes—
452C2×D4—28no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4C22×Q814yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—28no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4C22⋊Q814yes—
498C22×C4C22⋊Q814yes—
499C22×C4C22×D411yescharacteristic
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×Q812yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4114yes—
515C22×C4H(5,𝔽2)14yes—
516C22×C4C2511yescharacteristic
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—416no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4C2×#4114yes—
553C4⋊C4C2×#4114yes—
554C22⋊C4—28no—
555C22⋊C4—28no—
556C22⋊C4—28no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C42—22no—
566C42—22no—
567C42—28no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—44no—
590C42—44no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—44no—
596C42—44no—
597C42C22⋊Q814yes—
598C42C22⋊Q814yes—
599C42#13514yes—
600C42C22×Q812yes—
601C42C22×Q812yes—
602C42C2×#4112yes—
603C42C2×#4112yes—
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—
616C23#52512yes—
617C23#53512yes—
618C23C23×D411yescharacteristic
619Q8—28no—
620Q8—48no—
621Q8—48no—
622Q8—44no—
623D4—28no—
624D4—48no—
625D4—416no—
626D4—48no—
627D4—44no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—28no—
643C2×C4—24no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—24no—
647C2×C4—24no—
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—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—416no—
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—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—88no—
683C2×C4—88no—
684C2×C4C2×C22⋊Q812yes—
685C2×C4C2×C22⋊Q812yes—
686C2×C4C2×C22⋊Q812yes—
687C2×C4C2×C22⋊Q812yes—
688C2×C4C2×C22⋊Q812yes—
689C2×C4#52512yes—
690C2×C4#53512yes—
691C2×C4#55418yes—
692C2×C4#55918yes—
693C2×C4C2×#13512yes—
694C2×C4C2×#13512yes—
695C2×C4C2×#13512yes—
696C2×C4#57512yes—
697C2×C4#57512yes—
698C2×C4#57512yes—
699C2×C4C23×Q811yescharacteristic
700C2×C4C22×#4112yes—
701C2×C4C22×#4111yescharacteristic
702C2×C4C2×H(5,𝔽2)12yes—
703C8—44no—
704C8—44no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22#258014yes—
719C22C22×C22⋊Q811yescharacteristic
720C22C2×#52511yescharacteristic
721C22C2×#53511yescharacteristic
722C22C22×#13511yescharacteristic
723C22C2×#57511yescharacteristic
724C22C2×#57511yescharacteristic
725C22C2×#57511yescharacteristic
726C4—22no—
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—44no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—44no—
738C4—41no—
739C4—44no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4#258014yes—
745C4#307614yes—
746C4C2×#55412yes—
747C4C2×#55912yes—
748C2—24no—
749C2—84no—
750C2C2×#258011yescharacteristic
751C2C2×#307611yescharacteristic
752C2#6048411yescharacteristic
753C2#6053611yescharacteristic
754C2#6054611yescharacteristic
755C2#6078211yescharacteristic
756C2#6079011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#533C2semidirect
3. #29262C2semidirect
4. #29261C2semidirect
5. #29260C2semidirect
6. #29259C2semidirect
7. #29258C2semidirect
8. #29257C2semidirect
9. #29256C2semidirect
10. #29255C2semidirect
11. #29254C2semidirect
12. #23730C2semidirect
13. #23451C2semidirect
15. #21169C2semidirect
16. #20872C2semidirect
17. C2×#1831C2semidirect
103. #532Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C4×#5322
#292622
#292612
#292602
#292592
#292582
#292572
#292562
#292552
#292542
#237302
#234512
#212572
#211692
#208722
C2×#18312