Finite Group Databasebeta

← #6463607

Subgroups — #6463607

Order 512.

2202 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#532C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#29262C214yesmaximal, complemented
4#29260C214yesmaximal, complemented
5#29259C212yesmaximal, complemented
6#29258C212yesmaximal, complemented
7#29255C212yesmaximal, complemented
8#29254C212yesmaximal, complemented
9#23734C212yesmaximal, complemented
10#23452C212yesmaximal, complemented
11#21258C212yesmaximal
12#21170C212yesmaximal, complemented
13#20875C214yesmaximal, complemented
14#20752C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×C4×#41C2212yes—
18C22×C4×Q8—21no—
19C2×#428C2214yes—
20C2×#423C2212yes—
21C2×#417C2214yes—
22C2×#417C2214yes—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8—21no—
27C4×C4⋊Q8C2212yes—
28C4×C4⋊Q8C2212yes—
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2212yes—
31C4×C4⋊Q8C2211yescharacteristic
32C4×C4⋊Q8C2211yescharacteristic
33C4×#127C2211yescharacteristic
34C4×#127C2211yescharacteristic
35C4×#124C2212yes—
36C4×#124C2212yes—
37C4×#121—21no—
38C4×#121—21no—
39C4×#121—21no—
40C4×#121—21no—
41#2297C2212yes—
42C2×#391—21no—
43#2293—21no—
44#2293—21no—
45C2×#389—21no—
46C2×#389C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C42×D4—21no—
50C42×D4—21no—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2212yes—
53C2×C43C2211yescharacteristic
54#2096C2214yes—
55#2095C2214yes—
56#2094C2214yes—
57#2093C2214yes—
58#2024C2214yes—
59#2024C2214yes—
60#2023C2214yes—
61#2023C2214yes—
62#1983C2218yes—
63#1982C2214yes—
64#1981C2214yes—
65#1949C2218yes—
66#1948C2214yes—
67#1947C2214yes—
68#1865C2218yes—
69#1864C2218yes—
70#1863C2218yes—
71#1862C2218yes—
72#1831C2212yes—
73#1831C2212yes—
74#1831C2212yes—
75#1831C2212yes—
76#1736C2214yes—
77#1728C2212yes—
78#1728C2212yes—
79C22×#41C2312yes—
80C23×Q8—21no—
81C22×M16C2314yes—
82#532—24no—
83#532Q818yescomplemented
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#127C2311yescharacteristic
91C2×#124C2312yes—
92C2×#121—21no—
93C2×#121—21no—
94C4×#41—48no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—48no—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C2×#117C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C42—22no—
121C22×C42—22no—
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#428—48no—
130C2×#107C2318yes—
131C2×#106C2318yes—
132#423—28no—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135#417—216no—
136#417—48no—
137C2×#103C2318yes—
138C2×#103C2318yes—
139C2×#102C2318yes—
140C2×#102C2318yes—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391C2312yes—
146#391C2312yes—
147#390—21no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389C2314yes—
164#389C2314yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#386—22no—
170#386—22no—
171#384—21no—
172#384—21no—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2312yes—
176#384C2312yes—
177#382C2312yes—
178#380—21no—
179#380—21no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2314yes—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C2×#95—22no—
201C2×#95—22no—
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43D411yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209C43C2311yescharacteristic
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216C2×#41—24no—
217C2×#41—28no—
218C2×#41—28no—
219C2×#41C2×Q818yes—
220C22×Q8—28no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8C2412yes—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×M16—28no—
243C2×M16—28no—
244C2×M16—28no—
245C22×C8C2418yes—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—22no—
251C4⋊Q8C2×Q814yescomplemented
252#127—22no—
253#127C2×Q814yescomplemented
254#124—24no—
255#124C2×Q818yescomplemented
256#121—28no—
257#121—28no—
258C4×Q8—44no—
259C4×Q8—44no—
260C4×Q8—48no—
261C4×Q8—48no—
262C4×Q8—48no—
263C4×Q8—48no—
264C4×D4—28no—
265C4×D4—28no—
266C4×D4—44no—
267C4×D4—44no—
268C4×D4—44no—
269C4×D4—44no—
270C4×D4—48no—
271C4×D4—48no—
272C4×D4—44no—
273C4×D4—44no—
274#117—28no—
275#117—48no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4#4118yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—48no—
327C2×C42—48no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×Q814yes—
351C2×C42#4114yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—48no—
359#106—48no—
360C4⋊C8—28no—
361C4⋊C8—28no—
362#103—216no—
363#103—48no—
364#102—216no—
365#102—48no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376C24—22no—
377C24C22×D411yescharacteristic
378#41—232no—
379#41—416no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—44no—
386C2×Q8—432no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8C22×Q814yes—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4C22×Q814yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4C22⋊Q814yes—
441C22×C4C22⋊Q814yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×Q812yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4114yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4C2511yescharacteristic
460M16—416no—
461C2×C8—28no—
462C2×C8—28no—
463C2×C8—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4C2×#4118yes—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—44no—
520C42—44no—
521C42—48no—
522C42—44no—
523C42—48no—
524C42C22⋊Q814yes—
525C42C22⋊Q814yes—
526C42AGL(1,ℤ/8)14yes—
527C42#13714yes—
528C42C22×Q812yes—
529C42C22×Q812yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—24no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C2×C22⋊Q812yes—
544C23#52512yes—
545C23#53512yes—
546C23C23×D411yescharacteristic
547Q8—28no—
548Q8—416no—
549Q8—416no—
550Q8—44no—
551D4—28no—
552D4—48no—
553D4—416no—
554D4—48no—
555D4—44no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—416no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—88no—
606C2×C4C2×C22⋊Q812yes—
607C2×C4C2×C22⋊Q812yes—
608C2×C4C2×C22⋊Q812yes—
609C2×C4C2×C22⋊Q812yes—
610C2×C4C2×C22⋊Q812yes—
611C2×C4#52512yes—
612C2×C4#53512yes—
613C2×C4#55418yes—
614C2×C4#55918yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4C2×AGL(1,ℤ/8)12yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4#57512yes—
622C2×C4#57512yes—
623C2×C4C23×Q811yescharacteristic
624C2×C4C22×#4112yes—
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)12yes—
627C8—48no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22#258014yes—
642C22C22×C22⋊Q811yescharacteristic
643C22C2×#52511yescharacteristic
644C22C2×#53511yescharacteristic
645C22C22×AGL(1,ℤ/8)11yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57511yescharacteristic
648C22C2×#57511yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—44no—
659C4—41no—
660C4—41no—
661C4—48no—
662C4—82no—
663C4—84no—
664C4—82no—
665C4#258014yes—
666C4#307914yes—
667C4C2×#55412yes—
668C4C2×#55912yes—
669C2—24no—
670C2—84no—
671C2C2×#258011yescharacteristic
672C2C2×#307911yescharacteristic
673C2#6048411yescharacteristic
674C2#6055811yescharacteristic
675C2#6056011yescharacteristic
676C2#6080311yescharacteristic
677C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. #29262C2semidirect
4. #29260C2semidirect
5. #29259C2semidirect
6. #29258C2semidirect
7. #29255C2semidirect
8. #29254C2semidirect
9. #23734C2semidirect
10. #23452C2semidirect
12. #21170C2semidirect
13. #20875C2semidirect
14. #20752C2semidirect
83. #532Q8semidirect
251. C4⋊Q8C2×Q8semidirect
253. #127C2×Q8semidirect
255. #124C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5322
C2×C4×C4⋊Q82
#292622
#292602
#292592
#292582
#292552
#292542
#237342
#234522
#212582
#211702
#208752
#207522