Finite Group Databasebeta

← #6393399

Subgroups — #6393399

Order 512.

3730 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
1C2×#3506C211yesmaximal, characteristic
2C2×#2975C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
4#22504C214yesmaximal, complemented
5#22503C214yesmaximal, complemented
6#22432C214yesmaximal, complemented
7#22431C214yesmaximal, complemented
8#21860C211yesmaximal, characteristic, complemented
9#21859C211yesmaximal, characteristic, complemented
10#21769C214yesmaximal, complemented
11#21386C211yesmaximal, characteristic, complemented
12#21385C211yesmaximal, characteristic, complemented
13#20909C214yesmaximal, complemented
14#3506—28no—
15#3506C4116yescomplemented
16C2×#559—22no—
17C2×#556C2212yes—
18C2×#555C2212yes—
19C2×#553—22no—
20C2×#550C2212yes—
21C2×#547C2212yes—
22C2×#538—22no—
23C2×#535—21no—
24C2×#533C2211yescharacteristic
25C2×#532C2211yescharacteristic
26C2×#525C2214yes—
27C2×#520—21no—
28C22×C4×D4—21no—
29C22×C4×D4C2211yescharacteristic
30C23×C4⋊C4C2212yes—
31#2975—28no—
32C2×#436C2214yes—
33C2×#436C2214yes—
34C2×#431C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C2×#417C2218yescomplemented
37C2×#407C2214yes—
38D4×C4⋊C4—22no—
39D4×C4⋊C4—22no—
40D4×C4⋊C4C2214yes—
41D4×C4⋊C4C2214yes—
42C2×#389—21no—
43C2×#385—22no—
44C2×#384C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47#1990C2212yescomplemented
48#1990C2212yescomplemented
49#1988C2212yescomplemented
50#1987C2212yescomplemented
51#1982C2212yescomplemented
52#1981C2212yescomplemented
53#1957C2212yescomplemented
54#1956C2212yescomplemented
55#1895C2218yescomplemented
56#1894C2218yescomplemented
57#1874C2212yes—
58#1874C2214yes—
59#1874C2212yes—
60#1874C2218yescomplemented
61#1872C2212yes—
62#1872C2214yes—
63#1872C2212yes—
64#1872C2218yescomplemented
65#1825C2218yescomplemented
66#1803C2214yescomplemented
67#1802C2214yescomplemented
68#1788C2218yescomplemented
69C4×#103C2214yescomplemented
70C4×#102C2214yescomplemented
71#1736C2218yescomplemented
72C22×#41—21no—
73C22×#41C2314yes—
74C23×D4D411yescharacteristic
75C24×C4—22no—
76C24×C4C2312yes—
77C22×M16C2318yes—
78#559—232no—
79#556—28no—
80#556C2×C4116yescomplemented
81#555—28no—
82#555C2×C4116yescomplemented
83#553—232no—
84#550—28no—
85#550C2×C4116yescomplemented
86#547—28no—
87#547C2×C4116yescomplemented
88#538—232no—
89#535—216no—
90#533—24no—
91#533C2×C418yescomplemented
92#532—24no—
93#532C2×C418yescomplemented
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127—22no—
98C2×#127C2311yescharacteristic
99C2×#126—24no—
100C2×#125—21no—
101C2×#124—22no—
102C2×#124—24no—
103C2×#124—21no—
104#525—216no—
105#525C2×C4132yescomplemented
106C2×#123—24no—
107C2×#123—24no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2314yes—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121—21no—
114C2×#121—24no—
115C2×#121C2312yes—
116C2×#121C2314yes—
117C2×#120—24no—
118#520—216no—
119C2×C4×Q8—22no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×C4×D4—24no—
123C2×C4×D4—24no—
124C2×C4×D4—24no—
125C2×C4×D4—24no—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—48no—
129C2×C4×D4C2×C418yes—
130C2×C4×D4D418yes—
131C2×C4×D4D412yes—
132C2×C4×D4D412yes—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×#117—21no—
137C2×#117—22no—
138C2×#117C2314yes—
139C22×C4⋊C4—22no—
140C22×C4⋊C4—24no—
141C22×C4⋊C4—22no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—48no—
145C22×C4⋊C4D411yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2318yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2311yescharacteristic
150C22×C22⋊C4—22no—
151C22×C22⋊C4D412yes—
152C22×C42—21no—
153C22×C42C2311yescharacteristic
154#436—216no—
155#436—216no—
156#431—24no—
157#431—24no—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160#417—416no—
161C2×#103C2314yes—
162C2×#103C2318yes—
163C2×#103C2314yes—
164C2×#102C2314yes—
165C2×#102C2318yes—
166C2×#102C2314yes—
167#407—216no—
168C2×C22⋊C8C2318yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C4×C8C2312yes—
172C2×C4×C8C2312yes—
173#389—24no—
174#389—24no—
175#385—28no—
176#385—28no—
177#384—21no—
178#384—21no—
179#384C2312yes—
180#384C2312yes—
181#380—24no—
182#380C2318yes—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C2×#95—21no—
188C2×#95—22no—
189#340C2318yes—
190#340C2318yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195C25C2×D412yes—
196C2×#41—28no—
197C2×#41—28no—
198C2×#41—48no—
199C2×#41—48no—
200C2×#41—44no—
201C2×#41—44no—
202C2×#41C22×C4116yescomplemented
203C22×Q8—21no—
204C22×Q8—21no—
205C22×Q8C2412yes—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4C22⋊C418yes—
214C22×D4C2×D412yes—
215C22×D4C2×D412yes—
216C22×D4C2412yes—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—28no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—48no—
226C23×C4—48no—
227C23×C4—84no—
228C23×C4C22×C418yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2×D414yes—
232C23×C4C2×D418yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2414yes—
236C2×M16—216no—
237C2×M16—48no—
238C2×M16—48no—
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244C4⋊Q8C22×C414yescomplemented
245C4⋊Q8C22×C414yescomplemented
246C4⋊Q8C22×C414yescomplemented
247#127—216no—
248#127—22no—
249#127C22×C414yescomplemented
250#126—232no—
251#125—28no—
252#124—216no—
253#124—232no—
254#124—28no—
255#123—232no—
256#123—232no—
257C22⋊Q8—28no—
258C22⋊Q8—24no—
259C22⋊Q8—28no—
260C22⋊Q8C22×C418yes—
261C22⋊Q8C22×C4116yescomplemented
262#121—216no—
263#121—216no—
264#121—28no—
265#121—24no—
266#121—28no—
267#121—232no—
268#121C22×C418yes—
269#121C22×C4116yescomplemented
270#120—232no—
271C4×Q8—216no—
272C4×Q8—28no—
273C4×Q8—24no—
274C4×Q8C22×C418yescomplemented
275C4×D4—232no—
276C4×D4—232no—
277C4×D4—216no—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—416no—
281C4×D4—416no—
282C4×D4C22⋊C4116yes—
283C4×D4C22⋊C4116yes—
284C4×D4C22×C418yes—
285C4×D4C22×C418yes—
286C4×D4C22×C418yescomplemented
287C4×D4C2×D418yes—
288C4×D4C2×D418yes—
289#117—28no—
290#117—28no—
291#117—216no—
292#117C22×C4116yescomplemented
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—216no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—416no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C22⋊C418yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—48no—
334C2×C22⋊C4C22⋊C4116yes—
335C2×C22⋊C4C2×D414yes—
336C2×C22⋊C4C2×D414yes—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42C22×C414yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D414yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C4⋊C8—24no—
349C4⋊C8—24no—
350#103—28no—
351#103—44no—
352#103—48no—
353#102—28no—
354#102—44no—
355#102—48no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C22⋊C8—216no—
359C4×C8—24no—
360C4×C8—24no—
361#95—22no—
362#95—22no—
363#95—28no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24C2×C22⋊C418yes—
372C24C22×D414yes—
373C24C22×D411yescharacteristic
374#41—432no—
375#41—432no—
376#41—816no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—81no—
385C2×Q8—81no—
386C2×Q8C23×C414yes—
387C2×D4—28no—
388C2×D4—216no—
389C2×D4—216no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—232no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4C2×C22⋊C418yes—
411C2×D4C2×C22⋊C418yes—
412C2×D4C23×C414yes—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—216no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—416no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—416no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—88no—
449C22×C4—832no—
450C22×C4C2×C22⋊C414yes—
451C22×C4C2×C22⋊C418yes—
452C22×C4C2×C22⋊C4116yes—
453C22×C4C2×C22⋊C414yes—
454C22×C4AGL(1,ℤ/8)14yes—
455C22×C4#13714yes—
456C22×C4C23×C412yes—
457C22×C4C23×C418yes—
458C22×C4C22×D414yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D418yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2511yescharacteristic
467M16—816no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—216no—
477C4⋊C4—24no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—28no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—416no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4C2×C22⋊C418yes—
496C4⋊C4C2×C22⋊C418yes—
497C4⋊C4C23×C414yes—
498C4⋊C4C23×C414yes—
499C4⋊C4H(5,𝔽2)14yes—
500C22⋊C4—28no—
501C22⋊C4—216no—
502C22⋊C4—216no—
503C22⋊C4—216no—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—416no—
507C22⋊C4C2×C22⋊C4116yes—
508C22⋊C4C2×C22⋊C4116yes—
509C42—216no—
510C42—28no—
511C42—44no—
512C42—44no—
513C42C2×C22⋊C414yes—
514C42C2×C22⋊C414yes—
515C42C23×C412yes—
516C42C23×C412yes—
517C42C22×D412yes—
518C42C22×D412yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—24no—
524C23—24no—
525C23—216no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C22×C22⋊C418yes—
544C23C22×C22⋊C412yes—
545C23C2×AGL(1,ℤ/8)12yes—
546C23C2×#13712yes—
547C23C23×D411yescharacteristic
548Q8—28no—
549Q8—48no—
550Q8—44no—
551Q8—84no—
552D4—216no—
553D4—216no—
554D4—28no—
555D4—416no—
556D4—48no—
557D4—44no—
558D4—84no—
559D4—816no—
560D4—816no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—216no—
571C2×C4—216no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—416no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—416no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—816no—
597C2×C4—832no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4#417116yescomplemented
601C2×C4C22×C22⋊C418yes—
602C2×C4C22×C22⋊C412yes—
603C2×C4C22×C22⋊C412yes—
604C2×C4C22×C22⋊C412yes—
605C2×C4C22×C22⋊C412yes—
606C2×C4C22×C22⋊C412yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4C2×#13712yes—
610C2×C4C2×#13712yes—
611C2×C4C24×C411yescharacteristic
612C2×C4C23×D411yescharacteristic
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4#58512yes—
615C8—44no—
616C8—44no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—432no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22C2×#41714yes—
639C22C23×C22⋊C411yescharacteristic
640C22#348411yescharacteristic
641C22#348511yescharacteristic
642C22C22×AGL(1,ℤ/8)11yescharacteristic
643C22C22×#13711yescharacteristic
644C22#358511yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—28no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—84no—
656C4C2×#41714yes—
657C4C2×#41714yes—
658C4C4×H(5,𝔽2)12yes—
659C4#349612yes—
660C2—24no—
661C2—48no—
662C2—84no—
663C2#3425611yescharacteristic
664C2C22×#41711yescharacteristic
665C2#6008911yescharacteristic
666C2#6158211yescharacteristic
667C2#6158411yescharacteristic
668C2#6171711yescharacteristic
669C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3506C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2975C2semidirect
3. C2×D4×C4⋊C4C2semidirect
4. #22504C2semidirect
5. #22503C2semidirect
6. #22432C2semidirect
7. #22431C2semidirect
8. #21860C2semidirect
9. #21859C2semidirect
10. #21769C2semidirect
11. #21386C2semidirect
12. #21385C2semidirect
13. #20909C2semidirect
15. #3506C4semidirect
36. C2×#417C22semidirect
47. #1990C22semidirect
48. #1990C22semidirect
49. #1988C22semidirect
50. #1987C22semidirect
51. #1982C22semidirect
52. #1981C22semidirect
53. #1957C22semidirect
54. #1956C22semidirect
55. #1895C22semidirect
56. #1894C22semidirect
60. #1874C22semidirect
64. #1872C22semidirect
65. #1825C22semidirect
66. #1803C22semidirect
67. #1802C22semidirect
68. #1788C22semidirect
69. C4×#103C22semidirect
70. C4×#102C22semidirect
71. #1736C22semidirect
80. #556C2×C4semidirect
82. #555C2×C4semidirect
85. #550C2×C4semidirect
87. #547C2×C4semidirect
91. #533C2×C4semidirect
93. #532C2×C4semidirect
105. #525C2×C4semidirect
202. C2×#41C22×C4semidirect
244. C4⋊Q8C22×C4semidirect
245. C4⋊Q8C22×C4semidirect
246. C4⋊Q8C22×C4semidirect
249. #127C22×C4semidirect
261. C22⋊Q8C22×C4semidirect
269. #121C22×C4semidirect
274. C4×Q8C22×C4semidirect
286. C4×D4C22×C4semidirect
292. #117C22×C4semidirect
600. C2×C4#417semidirect

Maximal subgroups

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

subgroupindex
C2×#35062
C2×#29752
C2×D4×C4⋊C42
#225042
#225032
#224322
#224312
#218602
#218592
#217692
#213862
#213852
#209092