Finite Group Databasebeta

← #948614

Subgroups — #948614

Order 512.

2676 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C212yesmaximal, complemented
2C2×#2629C212yesmaximal, complemented
3#17133C214yesmaximal, complemented
4#17070C212yesmaximal, complemented
5#16877C212yesmaximal, complemented
6#16358C212yesmaximal, complemented
7#16305C212yesmaximal, complemented
8#15309C214yesmaximal
9#15228C212yesmaximal, complemented
10#15220C212yesmaximal, complemented
11#15131C212yesmaximal, complemented
12#14005C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic
14#13872C211yesmaximal, characteristic
15#13839C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2629—216no—
21#2442—216no—
22C2×#400C2214yes—
23C2×#398C2214yes—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#396C2214yes—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#386C2212yes—
36C2×#385C2214yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2214yes—
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1441C2211yescharacteristic
53#1441C2211yescharacteristic
54#1441C2214yes—
55#1441C2214yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2212yes—
62#1439C2211yescharacteristic
63#1439C2211yescharacteristic
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2212yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#125—28no—
82C2×#125—28no—
83C2×#125—44no—
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C2×C4×D4Q818yes—
87C2×#117—28no—
88C2×#117—42no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#400—216no—
124#398—216no—
125#398—216no—
126#398—216no—
127#396—216no—
128#396—216no—
129#396—48no—
130#390—44no—
131#389—28no—
132#389—28no—
133#389—48no—
134#389—44no—
135#389D4116yescomplemented
136#386—44no—
137#385—216no—
138#384—28no—
139#384—42no—
140#382—28no—
141#380—28no—
142#380—28no—
143#380—28no—
144#380—216no—
145#380—42no—
146C4×C4⋊C4—42no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—42no—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4C2×Q814yes—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4C2×Q814yes—
195C23×C4C2×Q814yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209#125—432no—
210C4×D4—416no—
211#117—416no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D4116yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C2×Q814yes—
292C2×C22⋊C4C2×Q814yes—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42C2×D418yes—
304C2×C42C2×Q814yes—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—42no—
318#95—44no—
319#95—42no—
320#95—44no—
321#95—41no—
322#95—41no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95C2×D4116yes—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24C22×Q812yes—
347C24C22×Q812yes—
348C24H(5,𝔽2)12yes—
349C242−1+412yes—
350C24C2511yescharacteristic
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—88no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4C22×D418yes—
442C22×C4C22×D418yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×Q812yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q812yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C42−1+414yes—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—84no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C4⋊C4—84no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—84no—
484C42—28no—
485C42—216no—
486C42—48no—
487C42—84no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23#54414yes—
534C23C23×D412yes—
535C23C23×Q811yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×2−1+412yes—
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23#58512yes—
544C23#58512yes—
545D4—416no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—816no—
626C2×C4#53718yes—
627C2×C4#54314yes—
628C2×C4#54414yes—
629C2×C4D4214yes—
630C2×C4D4×Q818yes—
631C2×C4#56314yes—
632C22—28no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—81no—
649C22—81no—
650C22—82no—
651C22—81no—
652C22—82no—
653C22—81no—
654C22C2×#53712yes—
655C22C2×#54311yescharacteristic
656C22C2×#54411yescharacteristic
657C22C2×#54411yescharacteristic
658C22C2×D4211yescharacteristic
659C22C2×D4×Q812yes—
660C22#348812yes—
661C22#348912yes—
662C22#349612yes—
663C22#349712yes—
664C22#350412yes—
665C22C2×#56311yescharacteristic
666C22#353611yescharacteristic
667C22#353611yescharacteristic
668C22#353711yescharacteristic
669C22#353811yescharacteristic
670C22#353912yes—
671C22#354712yes—
672C22#355111yescharacteristic
673C22#355111yescharacteristic
674C22#355512yes—
675C22#355912yes—
676C22#356312yes—
677C4—24no—
678C4—44no—
679C4—48no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—48no—
685C4—82no—
686C4—84no—
687C4—82no—
688C4—84no—
689C4#268318yes—
690C2—44no—
691C2—82no—
692C2#3742511yescharacteristic
693C2#3747711yescharacteristic
694C2#3768712yes—
695C2#3937012yes—
696C2#4244512yes—
697C2#4280312yes—
698C2#4510311yescharacteristic
699C2#4524411yescharacteristic
700C2C2×#268311yescharacteristic
701C2#5296111yescharacteristic
702C2#5301711yescharacteristic
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
1. C2×#2442C2semidirect
2. C2×#2629C2semidirect
3. #17133C2semidirect
4. #17070C2semidirect
5. #16877C2semidirect
6. #16358C2semidirect
7. #16305C2semidirect
9. #15228C2semidirect
10. #15220C2semidirect
11. #15131C2semidirect
12. #14005C2semidirect
135. #389D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C2×#26292
#171332
#170702
#168772
#163582
#163052
#153092
#152282
#152202
#151312
#140052
#139062
#138722
#138392