Finite Group Databasebeta

← #4457684

Subgroups — #4457684

Order 512.

3928 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#542C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3C2×#2542C211yesmaximal, characteristic, complemented
4C2×#2628C211yesmaximal, characteristic, complemented
5#16674C214yesmaximal, complemented
6#16510C214yesmaximal, complemented
7#16254C214yesmaximal, complemented
8#16251C214yesmaximal, complemented
9#15226C212yesmaximal, complemented
10#15182C212yesmaximal, complemented
11#15143C212yesmaximal, complemented
12#15130C212yesmaximal
13#13942C211yesmaximal, characteristic, complemented
14#13941C211yesmaximal, characteristic, complemented
15#13852C211yesmaximal, characteristic, complemented
16C2×#542—28no—
17C2×#542—28no—
18C2×#542—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2212yes—
22C22×#124C2211yescharacteristic
23C22×#124C2212yes—
24C22×#123C2212yes—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2212yes—
27C22×C22⋊Q8C2214yes—
28C22×#121C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×#117C2211yescharacteristic
32#2628—28no—
33#2567—28no—
34#2542—28no—
35C2×#400C2212yes—
36C2×#399C2214yes—
37C2×#399C2212yes—
38C2×#395C2212yes—
39C2×#395C2214yes—
40C2×#395C2218yes—
41C2×#394C2218yes—
42C2×#393C2212yes—
43C2×#391C2212yes—
44C2×#391C2212yes—
45C2×#390C2212yes—
46C2×#390C2212yes—
47C2×#389C2214yes—
48C2×#389C2212yes—
49C2×#384C2212yes—
50C2×#384C2212yes—
51C2×#382C2211yescharacteristic
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56#1444C2218yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2218yes—
60#1441C2214yes—
61#1440C2218yes—
62#1439C2218yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2218yes—
67#1439C2218yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C23×Q8C2312yes—
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80#542—2128no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×#126—28no—
88C2×#126—28no—
89C2×#126—28no—
90C2×#124—24no—
91C2×#124—24no—
92C2×#124—28no—
93C2×#124—28no—
94C2×#124—24no—
95C2×#124—28no—
96C2×#123—28no—
97C2×#123—28no—
98C2×#123—28no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—216no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—416no—
107C2×#121—24no—
108C2×#121—24no—
109C2×#121—24no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8D418yes—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4D418yes—
116C2×#117—24no—
117C2×#117—24no—
118C2×#117D418yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2318yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2318yes—
133C22×C4⋊C4C2314yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2318yes—
140C22×C22⋊C4C2311yescharacteristic
141C22×C22⋊C4C2311yescharacteristic
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2318yes—
145C22×C42C2312yes—
146C22×C42C2312yes—
147C22×C42C2311yescharacteristic
148C22×C42C2312yes—
149C22×C42C2311yescharacteristic
150#400—28no—
151#399—28no—
152#399—216no—
153#395—216no—
154#395—28no—
155#395—416no—
156#394—416no—
157#393—28no—
158#391—28no—
159#391—44no—
160#390—28no—
161#390—44no—
162#389—216no—
163#389—28no—
164#384—28no—
165#384—28no—
166#382—24no—
167#380—28no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—24no—
170C4×C22⋊C4—44no—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2318yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2318yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2318yes—
182C2×#95C2318yes—
183C2×#95C2314yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—48no—
199C22×Q8—48no—
200C22×Q8C2×D414yes—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—48no—
211C22×D4C2×D414yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—48no—
221C23×C4C2×D414yes—
222C23×C4C2×D418yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C4⋊Q8—232no—
237C4⋊Q8—416no—
238#126—264no—
239#124—232no—
240#124—264no—
241#123—264no—
242C22⋊Q8—264no—
243C22⋊Q8—232no—
244C22⋊Q8—464no—
245#121—232no—
246C4×Q8—232no—
247C4×D4—232no—
248#117—232no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—216no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D418yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
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—216no—
310C2×C22⋊C4—216no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—216no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4C2×D414yes—
321C2×C22⋊C4C2×D414yes—
322C2×C22⋊C4C2×D418yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42C2×D414yes—
334C2×C42C2×D418yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337C2×C42C2×D414yes—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—216no—
344#95—28no—
345#95—216no—
346#95—28no—
347#95—28no—
348#95—44no—
349#95—48no—
350#95—48no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—28no—
356C24—24no—
357C24—21no—
358C24—22no—
359C24—24no—
360C24—28no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C22×D414yes—
368C24C22×D412yes—
369C24C2511yescharacteristic
370C2×Q8—216no—
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—416no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—88no—
381C2×Q8—88no—
382C2×Q8—88no—
383C2×Q8—88no—
384C2×Q8—88no—
385C2×D4—216no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—88no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—216no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—416no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4H(5,𝔽2)18yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+418yes—
473C22×C42−1+412yes—
474C22×C42−1+412yes—
475C22×C42−1+412yes—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—216no—
485C4⋊C4—432no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—816no—
490C4⋊C4—88no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—216no—
498C22⋊C4—416no—
499C22⋊C4—432no—
500C22⋊C4—816no—
501C22⋊C4—88no—
502C42—28no—
503C42—216no—
504C42—48no—
505C42—48no—
506C42#120116yes—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—24no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—44no—
534C23—48no—
535C23—44no—
536C23—48no—
537C23—44no—
538C23—48no—
539C23—44no—
540C23—48no—
541C23C23×D411yescharacteristic
542C23C23×D411yescharacteristic
543C23C23×D411yescharacteristic
544C23C2×H(5,𝔽2)11yescharacteristic
545C23C2×H(5,𝔽2)14yes—
546C23C2×2−1+411yescharacteristic
547C23C2×2−1+411yescharacteristic
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+414yes—
550Q8—416no—
551Q8—816no—
552Q8—816no—
553D4—416no—
554D4—816no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—416no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—816no—
617C2×C4C2×#12014yes—
618C2×C4C2×#12014yes—
619C2×C4C2×#12014yes—
620C2×C4#53614yes—
621C2×C4#53614yes—
622C2×C4#53614yes—
623C2×C4#53714yes—
624C2×C4#53714yes—
625C2×C4#53714yes—
626C22—42no—
627C22—44no—
628C22—42no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—48no—
633C22—48no—
634C22—44no—
635C22—42no—
636C22—82no—
637C22—84no—
638C22—82no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—82no—
643C22—82no—
644C22C22×#12011yescharacteristic
645C22C2×#53611yescharacteristic
646C22C2×#53611yescharacteristic
647C22C2×#53611yescharacteristic
648C22C2×#53711yescharacteristic
649C22C2×#53711yescharacteristic
650C22C2×#53711yescharacteristic
651C22#348412yes—
652C22#348612yes—
653C22#349812yes—
654C22#349812yes—
655C22#349912yes—
656C22#349912yes—
657C22#353614yes—
658C22#354414yes—
659C22#355014yes—
660C22#355214yes—
661C4—24no—
662C4—24no—
663C4—48no—
664C4—42no—
665C4—44no—
666C4—42no—
667C4—48no—
668C4—44no—
669C4—44no—
670C4—42no—
671C4—84no—
672C4—84no—
673C4#261618yes—
674C2—82no—
675C2—84no—
676C2C2×#261611yescharacteristic
677C2#3552711yescharacteristic
678C2#3566611yescharacteristic
679C2#3729912yes—
680C2#3730312yes—
681C2#4242712yes—
682C2#4242812yes—
683C2#4278912yes—
684C2#4279412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2567C2semidirect
3. C2×#2542C2semidirect
4. C2×#2628C2semidirect
5. #16674C2semidirect
6. #16510C2semidirect
7. #16254C2semidirect
8. #16251C2semidirect
9. #15226C2semidirect
10. #15182C2semidirect
11. #15143C2semidirect
13. #13942C2semidirect
14. #13941C2semidirect
15. #13852C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5422
C2×#25672
C2×#25422
C2×#26282
#166742
#165102
#162542
#162512
#152262
#151822
#151432
#151302
#139422
#139412
#138522