Finite Group Databasebeta

← #789625

Subgroups — #789625

Order 512.

2380 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2423C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#16704C212yesmaximal
4#15653C214yesmaximal
5#15646C214yesmaximal, complemented
6#15200C211yesmaximal, characteristic, complemented
7#15195C211yesmaximal, characteristic
8#15145C211yesmaximal, characteristic, complemented
9#15105C212yesmaximal, complemented
10#15021C212yesmaximal, complemented
11#14231C212yesmaximal
12#13969C212yesmaximal
13#13926C212yesmaximal, complemented
14#13884C212yesmaximal, complemented
15#13876C212yesmaximal
16C4×#399C211yesmaximal, characteristic
17Q8×#95C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21#2423—28no—
22C2×#399C2212yes—
23C2×#399C2211yescharacteristic
24C2×#398C2212yes—
25C2×#398C2214yes—
26#2355—28no—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#388C2212yes—
31C2×#388C2214yes—
32C2×#387C2212yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38C2×#382C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47#1445C2211yescharacteristic
48#1445C2212yes—
49#1445C2212yes—
50#1444C2212yes—
51#1444C2211yescharacteristic
52#1444C2212yes—
53#1442C2212yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2211yescharacteristic
68#1439C2212yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2211yescharacteristic
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—28no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—44no—
93C2×C4×Q8D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127#399—28no—
128#399Q818yescomplemented
129#398—28no—
130#398—48no—
131#391—28no—
132#391—24no—
133#389—24no—
134#388—28no—
135#388—48no—
136#387—216no—
137#387—28no—
138#387—48no—
139#384—24no—
140#384—24no—
141#384—44no—
142#382—28no—
143#380—24no—
144#380—44no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—28no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—28no—
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C25C2411yescharacteristic
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8C2×D414yes—
186C22×Q8#4118yes—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C22⋊Q8—416no—
209C4×Q8—232no—
210C4×Q8—432no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D418yes—
257C2×C4⋊C4C2×Q814yes—
258C2×C4⋊C4C2×Q818yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4C2×Q814yes—
277C2×C22⋊C4C2×Q818yes—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42C2×D414yes—
294C2×C42C2×D418yes—
295C2×C42#4118yes—
296#95—22no—
297#95—22no—
298#95—28no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—22no—
314#95—24no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95C2×Q814yes—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—21no—
336C24C22×Q812yes—
337C24C2511yescharacteristic
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8#123116yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q814yes—
428C22×C4C22×Q812yes—
429C22×C4C22×Q814yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4118yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C42−1+412yes—
438C22×C42−1+412yes—
439C22×C42−1+412yes—
440C4⋊C4—216no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—216no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—88no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—88no—
459C22⋊C4—28no—
460C22⋊C4—216no—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—84no—
465C22⋊C4—88no—
466C42—216no—
467C42—28no—
468C42—216no—
469C42—416no—
470C42—44no—
471C42—48no—
472C42—88no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23C23×D411yescharacteristic
497C23C23×Q811yescharacteristic
498C23C22×#4112yes—
499C23C22×#4112yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×2−1+411yescharacteristic
503C23C2×2−1+411yescharacteristic
504C23C2×2−1+411yescharacteristic
505C23#58512yes—
506C23#58511yescharacteristic
507Q8—232no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—88no—
593C2×C4C2×#12318yes—
594C2×C4C2×#12314yes—
595C2×C4#53214yes—
596C2×C4#53714yes—
597C2×C4D4×Q814yes—
598C2×C4#55118yes—
599C2×C4#55218yes—
600C2×C4#55418yes—
601C22—44no—
602C22—44no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—82no—
607C22—81no—
608C22—81no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22C22×#12311yescharacteristic
613C22C2×#53211yescharacteristic
614C22C2×#53711yescharacteristic
615C22C2×D4×Q811yescharacteristic
616C22C2×#55112yes—
617C22C2×#55212yes—
618C22C2×#55412yes—
619C22Q8×#4112yes—
620C22#348711yescharacteristic
621C22#348911yescharacteristic
622C22#349312yes—
623C22#349512yes—
624C22#349811yescharacteristic
625C22#350211yescharacteristic
626C22#350311yescharacteristic
627C22#351112yes—
628C22#351612yes—
629C22#351712yes—
630C22#352212yes—
631C22#355412yes—
632C22#355411yescharacteristic
633C22#355611yescharacteristic
634C22#355812yes—
635C4—28no—
636C4—24no—
637C4—28no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—44no—
643C4—42no—
644C4—84no—
645C4—82no—
646C4—82no—
647C4—84no—
648C4—82no—
649C4—81no—
650C4#249818yes—
651C2—82no—
652C2C2×#249811yescharacteristic
653C2#3600511yescharacteristic
654C2#3606511yescharacteristic
655C2Q8×#12311yescharacteristic
656C2#3655911yescharacteristic
657C2#3959911yescharacteristic
658C2#4288111yescharacteristic
659C2#4525912yes—
660C2#4534612yes—
661C2#5925912yes—
662C2#5927412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2355C2semidirect
5. #15646C2semidirect
6. #15200C2semidirect
8. #15145C2semidirect
9. #15105C2semidirect
10. #15021C2semidirect
13. #13926C2semidirect
14. #13884C2semidirect
17. Q8×#95C2semidirect
128. #399Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24232
C2×#23552
#167042
#156532
#156462
#152002
#151952
#151452
#151052
#150212
#142312
#139692
#139262
#138842
#138762
C4×#3992
Q8×#952