Finite Group Databasebeta

← #696713

Subgroups — #696713

Order 512.

3910 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2486C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic, complemented
5#16607C214yesmaximal, complemented
6#16024C214yesmaximal, complemented
7#15949C214yesmaximal
8#15236C212yesmaximal, complemented
9#15228C212yesmaximal, complemented
10#14907C214yesmaximal, complemented
11#14258C211yesmaximal, characteristic, complemented
12#13976C212yesmaximal, complemented
13#13939C211yesmaximal, characteristic, complemented
14#13880C212yesmaximal, complemented
15#13839C211yesmaximal, characteristic
16C2×#552—28no—
17C2×#552—28no—
18C2×#552—28no—
19C22×#126C2212yes—
20C22×#125C2211yescharacteristic
21C22×#125C2212yes—
22C22×#123C2212yes—
23C22×#123C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C22×#117C2212yes—
30C22×#117C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#2502—28no—
34#2486—28no—
35C2×#398C2214yes—
36C2×#398C2212yes—
37C2×#397C2212yes—
38#2317—28no—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#389C2214yes—
42C2×#389C2212yes—
43C2×#385C2214yes—
44C2×#385C2214yes—
45C2×#385C2218yes—
46C2×#384C2212yes—
47C2×#384C2212yes—
48C2×#382C2212yes—
49C2×#381C2212yes—
50C2×#380C2212yes—
51C2×#380C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54#1444C2218yes—
55#1444C2218yes—
56#1444C2218yes—
57#1442C2218yes—
58#1441C2212yes—
59#1441C2218yes—
60#1441C2214yes—
61#1440C2212yes—
62#1440C2218yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2212yes—
66#1438C2212yes—
67#1436C2218yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79#552—2128no—
80C2×#126—28no—
81C2×#126—28no—
82C2×#126—28no—
83C2×#125—28no—
84C2×#125—28no—
85C2×#125—24no—
86C2×#125—24no—
87C2×#125—28no—
88C2×#125—24no—
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123—48no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8D418yes—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4D418yes—
106C2×#117—24no—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117—28no—
110C2×#117—28no—
111C2×#117—28no—
112C2×#117—28no—
113C2×#117D418yes—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2318yes—
136C22×C4⋊C4C2318yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2311yescharacteristic
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2311yescharacteristic
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2312yes—
147C22×C42C2312yes—
148C22×C42C2312yes—
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151C22×C42C2312yes—
152#398—28no—
153#398—216no—
154#397—28no—
155#391—28no—
156#391—28no—
157#389—216no—
158#389—28no—
159#385—216no—
160#385—216no—
161#385—416no—
162#384—28no—
163#384—28no—
164#382—28no—
165#381—28no—
166#380—28no—
167#380—24no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—28no—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2318yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2318yes—
177C2×#95C2318yes—
178C2×#95C2314yes—
179C2×#95C2318yes—
180C2×#95C2314yes—
181C2×#95C2318yes—
182C2×#95C2318yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8C2×D414yes—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—42no—
194C22×D4C2×D414yes—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228#126—264no—
229#125—264no—
230#125—232no—
231#123—264no—
232#123—432no—
233C22⋊Q8—232no—
234C22⋊Q8—232no—
235C4×Q8—232no—
236C4×D4—232no—
237#117—232no—
238#117—264no—
239#117—232no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
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—22no—
257C2×C4⋊C4—22no—
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—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—216no—
279C2×C4⋊C4—216no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—416no—
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×D418yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D414yes—
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—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—216no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4C2×D418yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42C2×D418yes—
330C2×C42C2×D414yes—
331C2×C42#4118yes—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—28no—
336#95—216no—
337#95—216no—
338#95—28no—
339#95—216no—
340#95—28no—
341#95—216no—
342#95—44no—
343#95—48no—
344#95—48no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24C22×D412yes—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24H(5,𝔽2)14yes—
361C24C2511yescharacteristic
362C2×Q8—28no—
363C2×Q8—216no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—48no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—48no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4#120116yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4114yes—
457C22×C42−1+418yes—
458C22×C42−1+414yes—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—28no—
470C4⋊C4—416no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—816no—
475C4⋊C4—816no—
476C22⋊C4—216no—
477C22⋊C4—232no—
478C22⋊C4—216no—
479C22⋊C4—216no—
480C22⋊C4—216no—
481C22⋊C4—28no—
482C22⋊C4—216no—
483C22⋊C4—28no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C42—216no—
487C42—216no—
488C42—28no—
489C42—216no—
490C42—48no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23C2×#12014yes—
522C23#53514yes—
523C23C23×D411yescharacteristic
524C23C23×D411yescharacteristic
525C23C23×D411yescharacteristic
526C23C22×#4112yes—
527C23C2×H(5,𝔽2)12yes—
528C23C2×2−1+412yes—
529C23C2×2−1+414yes—
530C23#58512yes—
531Q8—416no—
532D4—416no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—232no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—432no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—416no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—816no—
598C2×C4C2×#12014yes—
599C2×C4C2×#12014yes—
600C2×C4#52514yes—
601C2×C4#53614yes—
602C2×C4#54718yes—
603C2×C4#54718yes—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—28no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—48no—
615C22—48no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22C22×#12011yescharacteristic
621C22C2×#52511yescharacteristic
622C22C2×#53511yescharacteristic
623C22C2×#53611yescharacteristic
624C22#344911yescharacteristic
625C22C2×#54712yes—
626C22C2×#54712yes—
627C22#348612yes—
628C22#348812yes—
629C22#348812yes—
630C22#349214yes—
631C22#349912yes—
632C22#349912yes—
633C22#350714yes—
634C22#354414yes—
635C22#356314yes—
636C4—24no—
637C4—42no—
638C4—44no—
639C4—48no—
640C4—48no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—84no—
647C4—84no—
648C4#240618yes—
649C2—44no—
650C2—44no—
651C2C2×#240611yescharacteristic
652C2#3525011yescharacteristic
653C2#3566611yescharacteristic
654C2#3568612yes—
655C2#3786312yes—
656C2#3948412yes—
657C2#4296412yes—
658C2#4492114yes—
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×#2502C2semidirect
4. C2×#2317C2semidirect
5. #16607C2semidirect
6. #16024C2semidirect
8. #15236C2semidirect
9. #15228C2semidirect
10. #14907C2semidirect
11. #14258C2semidirect
12. #13976C2semidirect
13. #13939C2semidirect
14. #13880C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#25022
C2×#24862
C2×#23172
#166072
#160242
#159492
#152362
#152282
#149072
#142582
#139762
#139392
#138802
#138392