Finite Group Databasebeta

← #5021440

Subgroups — #5021440

Order 512.

3422 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#563C211yesmaximal, characteristic
2C2×#2822C211yesmaximal, characteristic, complemented
3C2×#2820C211yesmaximal, characteristic
4C2×#2859C211yesmaximal, characteristic, complemented
5#15940C214yesmaximal, complemented
6#15928C212yesmaximal, complemented
7#15927C212yesmaximal, complemented
8#15670C212yesmaximal, complemented
9#15614C214yesmaximal, complemented
10#15430C212yesmaximal, complemented
11#15318C214yesmaximal, complemented
12#15271C214yesmaximal
13#14709C211yesmaximal, characteristic, complemented
14#14708C211yesmaximal, characteristic
15#14664C211yesmaximal, characteristic, complemented
16C2×#563—28no—
17C2×#563—28no—
18C2×#563—28no—
19C22×C4⋊Q8C2212yes—
20C22×#126C2212yes—
21C22×#126C2212yes—
22C22×#125C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×#125C2211yescharacteristic
25C22×#124C2211yescharacteristic
26C22×#123C2212yes—
27C22×C22⋊Q8C2212yes—
28C22×C22⋊Q8C2212yes—
29#2859—28no—
30#2822—28no—
31#2820—28no—
32C2×#400C2212yes—
33C2×#400C2212yes—
34C2×#399C2212yes—
35C2×#398C2212yes—
36C2×#398C2214yes—
37C2×#398C2212yes—
38C2×#397C2212yes—
39C2×#396C2214yes—
40C2×#395C2212yes—
41C2×#395C2214yes—
42C2×#394C2212yes—
43C2×#391C2212yes—
44C2×#390C2212yes—
45C2×#389C2214yes—
46C2×#389C2214yes—
47C2×#385C2214yes—
48C2×#384C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#382C2211yescharacteristic
54C2×C43C2211yescharacteristic
55#1446C2214yes—
56#1445C2212yes—
57#1445C2212yes—
58#1444C2214yes—
59#1444C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1437C2212yes—
71#1437C2212yes—
72#1437C2214yes—
73#1436C2212yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2212yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C22⋊#95C2214yes—
81C23×Q8C2312yes—
82C23×Q8C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4C2312yes—
85#563—2128no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—28no—
89C2×#126—28no—
90C2×#126—28no—
91C2×#126—28no—
92C2×#126—28no—
93C2×#126—28no—
94C2×#126—28no—
95C2×#125—24no—
96C2×#125—24no—
97C2×#125—24no—
98C2×#125—24no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125D418yes—
103C2×#125D418yes—
104C2×#124—24no—
105C2×#124—24no—
106C2×#124D418yes—
107C2×#123—28no—
108C2×#123—28no—
109C2×#123—28no—
110C2×C22⋊Q8—28no—
111C2×C22⋊Q8—28no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—28no—
115C2×C22⋊Q8—28no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2314yes—
145C22×C42C2312yes—
146C22×C42C2311yescharacteristic
147C22×C42C2311yescharacteristic
148C22×C42C2312yes—
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151#400—28no—
152#400—28no—
153#399—28no—
154#398—28no—
155#398—28no—
156#398—216no—
157#397—28no—
158#396—216no—
159#395—28no—
160#395—48no—
161#394—28no—
162#391—44no—
163#390—44no—
164#389—216no—
165#389—216no—
166#385—48no—
167#384—24no—
168#384—24no—
169#382—24no—
170#382—24no—
171#382—24no—
172#382—24no—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C43—24no—
194C25C2412yes—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8C2×D414yes—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4C2×D414yes—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C4⋊Q8—264no—
230#126—264no—
231#126—264no—
232#125—232no—
233#125—232no—
234#125—416no—
235#124—232no—
236#123—264no—
237C22⋊Q8—264no—
238C22⋊Q8—264no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D418yes—
295C2×C4⋊C4C2×D418yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D414yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4C2×D418yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342#95—28no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24C22×D414yes—
371C24C2511yescharacteristic
372C2×Q8—216no—
373C2×Q8—28no—
374C2×Q8—48no—
375C2×Q8—84no—
376C2×Q8—82no—
377C2×Q8—82no—
378C2×Q8—88no—
379C2×Q8—88no—
380C2×Q8—84no—
381C2×Q8—84no—
382C2×D4—28no—
383C2×D4—48no—
384C2×D4—84no—
385C2×D4—82no—
386C2×D4—82no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
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—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—216no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—88no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—88no—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C42−1+414yes—
468C22×C42−1+414yes—
469C22×C42−1+414yes—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—416no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C42—28no—
500C42—28no—
501C42—216no—
502C42—48no—
503C42—48no—
504C42#120116yes—
505C23—24no—
506C23—28no—
507C23—24no—
508C23—28no—
509C23—24no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C23×D411yescharacteristic
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+412yes—
531C23C2×2−1+412yes—
532C23C2×2−1+412yes—
533C23#58512yes—
534C23#58512yes—
535Q8—88no—
536Q8—816no—
537D4—88no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—816no—
613C2×C4C2×#12014yes—
614C2×C4C2×#12014yes—
615C2×C4C2×#12014yes—
616C2×C4#56318yescomplemented
617C22—48no—
618C22—48no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22C22×#12011yescharacteristic
627C22#348611yescharacteristic
628C22#348611yescharacteristic
629C22#348812yes—
630C22#348912yes—
631C22#349612yes—
632C22#349712yes—
633C22#349811yescharacteristic
634C22#349911yescharacteristic
635C22#349912yes—
636C22#349912yes—
637C22#350011yescharacteristic
638C22#350011yescharacteristic
639C22C2×#56312yes—
640C22#353112yes—
641C22#355012yes—
642C22#355312yes—
643C22#355912yes—
644C22#356112yes—
645C22#356312yes—
646C22#356312yes—
647C4—24no—
648C4—24no—
649C4—28no—
650C4—24no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—84no—
658C4—82no—
659C4—84no—
660C4—82no—
661C2—84no—
662C2#3551311yescharacteristic
663C2#3566011yescharacteristic
664C2#3566611yescharacteristic
665C2#4597912yes—
666C2#4599812yes—
667C2#4614712yes—
668C2#4614912yes—
669C2#4615012yes—
670C2#4622712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2822C2semidirect
4. C2×#2859C2semidirect
5. #15940C2semidirect
6. #15928C2semidirect
7. #15927C2semidirect
8. #15670C2semidirect
9. #15614C2semidirect
10. #15430C2semidirect
11. #15318C2semidirect
13. #14709C2semidirect
15. #14664C2semidirect
616. C2×C4#563semidirect

Maximal subgroups

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

subgroupindex
C22×#5632
C2×#28222
C2×#28202
C2×#28592
#159402
#159282
#159272
#156702
#156142
#154302
#153182
#152712
#147092
#147082
#146642