Finite Group Databasebeta

← #6468424

Subgroups — #6468424

Order 512.

2110 conjugacy classes of subgroups.

The subgroup lattice has 671 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#29290C212yesmaximal, complemented
4#29273C212yesmaximal, complemented
5#29254C212yesmaximal, complemented
6#29237C214yesmaximal, complemented
7#28280C212yesmaximal, complemented
8#27677C212yesmaximal, complemented
9#24299C212yesmaximal, complemented
10#23781C214yesmaximal, complemented
11#21614C214yesmaximal, complemented
12#21395C212yesmaximal
13#21365C212yesmaximal, complemented
14#20810C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×C4×#41—21no—
18C22×C4×Q8—21no—
19C2×#476C2214yes—
20C2×#432C2212yes—
21C4×C4⋊Q8—21no—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8C2212yes—
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2212yes—
29C4×C4⋊Q8C2212yes—
30C4×C4⋊Q8C2212yes—
31C4×#127C2211yescharacteristic
32C4×#127C2211yescharacteristic
33C4×#124—21no—
34C4×#124—21no—
35C4×#121—21no—
36C4×#121—21no—
37C4×#121C2212yes—
38C4×#121C2212yes—
39#2297—21no—
40C2×#391—21no—
41#2293—21no—
42#2293C2212yes—
43C2×#389—21no—
44C2×#389C2212yes—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C42×D4—21no—
48C42×D4C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C43C2211yescharacteristic
52#2099C2214yes—
53#2093C2214yes—
54#2087C2218yes—
55#2065C2218yes—
56#2058C2214yes—
57#2023C2214yes—
58#1987C2214yes—
59#1987C2214yes—
60#1982C2214yes—
61#1948C2214yes—
62#1947C2214yes—
63#1947C2214yes—
64#1919C2218yes—
65#1874C2214yes—
66#1874C2214yes—
67#1862C2218yes—
68#1853C2214yes—
69#1852C2214yes—
70#1842C2212yes—
71#1831C2212yes—
72#1802C2214yes—
73#1797C2212yes—
74#1797C2212yes—
75#1776C2218yes—
76#1767C2212yes—
77#1767C2212yes—
78#1762C2218yes—
79C22×#41—21no—
80C23×Q8—21no—
81#532—24no—
82#532Q818yescomplemented
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×#127C2311yescharacteristic
90C2×#124—21no—
91C2×#121—21no—
92C2×#121C2312yes—
93C4×#41—48no—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8—48no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117—21no—
112C2×#117—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C42—22no—
120C22×C42—22no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#476—216no—
129#432—28no—
130C2×#107C2318yes—
131C2×#106C2318yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#102C2318yes—
136C2×#102C2318yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2314yes—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—21no—
144#391—21no—
145#390—21no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390C2312yes—
150#390C2312yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—21no—
159#389C2314yes—
160#389C2314yes—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#386—22no—
168#386—22no—
169#384—21no—
170#384—21no—
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#384C2312yes—
174#384C2312yes—
175#382—21no—
176#380—21no—
177#380C2312yes—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2314yes—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C2×#95—22no—
199C2×#95—22no—
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214C2×#41—28no—
215C2×#41—44no—
216C2×#41—44no—
217C22×Q8—28no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2412yes—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C22×C8C2418yes—
240C4⋊Q8—28no—
241C4⋊Q8—28no—
242C4⋊Q8—28no—
243C4⋊Q8—24no—
244C4⋊Q8—22no—
245C4⋊Q8C2×Q814yescomplemented
246C4⋊Q8#4118yescomplemented
247#127—22no—
248#127C2×Q814yescomplemented
249#124—28no—
250#121—28no—
251#121—24no—
252#121C2×Q818yes—
253C4×Q8—48no—
254C4×Q8—48no—
255C4×Q8—48no—
256C4×Q8—44no—
257C4×Q8—44no—
258C4×Q8—48no—
259C4×D4—28no—
260C4×D4—24no—
261C4×D4—28no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—44no—
265C4×D4—48no—
266C4×D4—48no—
267C4×D4—44no—
268C4×D4—44no—
269C4×D4C2×Q818yescomplemented
270#117—48no—
271#117—44no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2414yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—42no—
319C2×C42—48no—
320C2×C42—42no—
321C2×C42—48no—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
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×Q814yes—
345C2×C42#4114yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—48no—
353#106—216no—
354C4⋊C8—28no—
355C4⋊C8—28no—
356C4⋊C8—44no—
357#102—216no—
358#102—48no—
359C22⋊C8—28no—
360#97—28no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371C24—22no—
372C24C22×D411yescharacteristic
373#41—432no—
374C2×Q8—28no—
375C2×Q8—24no—
376C2×Q8—28no—
377C2×Q8—44no—
378C2×Q8—42no—
379C2×Q8—432no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×D4—28no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4C22×Q814yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4C22⋊Q814yes—
435C22×C4C22⋊Q814yes—
436C22×C4AGL(1,ℤ/8)14yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
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×Q812yes—
451C22×C4C2×#4112yes—
452C22×C4C2511yescharacteristic
453C2×C8—28no—
454C2×C8—28no—
455C2×C8—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4C22×Q814yes—
480C4⋊C4C2×#4118yes—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—44no—
512C42—44no—
513C42—48no—
514C42—44no—
515C42—48no—
516C42C22⋊Q814yes—
517C42C22⋊Q814yes—
518C42AGL(1,ℤ/8)14yes—
519C42#13714yes—
520C42C22×Q812yes—
521C42C22×Q812yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C42C2×#4114yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C2×C22⋊Q812yes—
537C23C2×AGL(1,ℤ/8)12yes—
538C23#57512yes—
539C23C23×D411yescharacteristic
540Q8—416no—
541Q8—416no—
542Q8—48no—
543D4—28no—
544D4—48no—
545D4—416no—
546D4—48no—
547D4—44no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
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—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—88no—
598C2×C4—88no—
599C2×C4—88no—
600C2×C4C2×C22⋊Q812yes—
601C2×C4C2×C22⋊Q812yes—
602C2×C4C2×C22⋊Q812yes—
603C2×C4C2×C22⋊Q812yes—
604C2×C4C2×C22⋊Q812yes—
605C2×C4#52512yes—
606C2×C4#52512yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×#13712yes—
613C2×C4C2×#13712yes—
614C2×C4C2×#13712yes—
615C2×C4C23×Q811yescharacteristic
616C2×C4C22×#4112yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4#58512yes—
619C8—48no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22#307714yes—
634C22C22×C22⋊Q811yescharacteristic
635C22C2×#52511yescharacteristic
636C22#344911yescharacteristic
637C22C22×AGL(1,ℤ/8)11yescharacteristic
638C22C22×AGL(1,ℤ/8)11yescharacteristic
639C22C22×#13711yescharacteristic
640C22C2×#57511yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—44no—
653C4—48no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4#307914yes—
658C4#311214yes—
659C4Q8×#4112yes—
660C4#348012yes—
661C2—24no—
662C2—84no—
663C2#3495211yescharacteristic
664C2C2×#307711yescharacteristic
665C2C2×#307911yescharacteristic
666C2C2×#311211yescharacteristic
667C2#6055811yescharacteristic
668C2#6096311yescharacteristic
669C2#6096811yescharacteristic
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. #29290C2semidirect
4. #29273C2semidirect
5. #29254C2semidirect
6. #29237C2semidirect
7. #28280C2semidirect
8. #27677C2semidirect
9. #24299C2semidirect
10. #23781C2semidirect
11. #21614C2semidirect
13. #21365C2semidirect
14. #20810C2semidirect
82. #532Q8semidirect
245. C4⋊Q8C2×Q8semidirect
246. C4⋊Q8#41semidirect
248. #127C2×Q8semidirect
269. C4×D4C2×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
#292902
#292732
#292542
#292372
#282802
#276772
#242992
#237812
#216142
#213952
#213652
#208102