Finite Group Databasebeta

← #1006646

Subgroups — #1006646

Order 512.

3174 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C212yesmaximal, complemented
2C2×#2434C212yesmaximal
3#17112C212yesmaximal, complemented
4#16524C214yesmaximal, complemented
5#16339C212yesmaximal
6#15510C212yesmaximal, complemented
7#15440C214yesmaximal
8#15190C212yesmaximal, complemented
9#15188C212yesmaximal, complemented
10#15178C211yesmaximal, characteristic
11#14575C212yesmaximal
12#14142C212yesmaximal, complemented
13#13997C212yesmaximal
14#13922C211yesmaximal, characteristic
15C4×#393C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22#2568—216no—
23#2434—216no—
24C2×#400C2214yes—
25C2×#400C2214yes—
26C2×#395C2214yes—
27C2×#395C2214yes—
28C2×#393C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2214yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#389C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#382C2214yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1445C2214yes—
49#1444C2212yes—
50#1444C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2211yescharacteristic
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1439C2212yes—
64#1439C2211yescharacteristic
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1437C2214yes—
69#1436C2214yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2214yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2211yescharacteristic
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—44no—
80C23×Q8—42no—
81C23×Q8—42no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×Q8C2311yescharacteristic
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—44no—
92C2×C4⋊Q8—42no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8—44no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—44no—
99C2×C4×Q8—28no—
100C2×C4×Q8—42no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124#400—216no—
125#400—216no—
126#395—216no—
127#395—216no—
128#393Q818yescomplemented
129#391—28no—
130#391—28no—
131#391—28no—
132#391—28no—
133#391—216no—
134#391—44no—
135#391—44no—
136#391D4116yes—
137#389—44no—
138#385—216no—
139#385—216no—
140#382—216no—
141#380—42no—
142C4×C4⋊C4—28no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—42no—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C25C2411yescharacteristic
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—41no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8—44no—
197C22×Q8—41no—
198C22×Q8—432no—
199C22×Q8—416no—
200C22×Q8—416no—
201C22×Q8—432no—
202C22×Q8—416no—
203C22×Q8C2×D418yes—
204C22×Q8C2×Q814yescomplemented
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4#4114yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C4⋊Q8—432no—
230C4⋊Q8—432no—
231C4⋊Q8—416no—
232C22⋊Q8—432no—
233C4×Q8—416no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D418yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4C2×Q818yes—
284C2×C22⋊C4C2×Q814yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—44no—
297C2×C42C2×D418yes—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—42no—
315#95—41no—
316#95—41no—
317#95—41no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×D4116yes—
325#95C2×D4116yes—
326#95C2×Q814yescomplemented
327#95C2×Q818yescomplemented
328C24—22no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—21no—
336C24C22×Q812yes—
337C24C2×#4112yes—
338C242−1+412yes—
339C24C2511yescharacteristic
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—216no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—48no—
347C2×Q8—44no—
348C2×Q8—48no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—48no—
352C2×Q8—4128no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—216no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4C22×D418yes—
445C22×C4C22×D414yes—
446C22×C4C22×D418yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q812yes—
452C22×C4C22×Q814yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C42−1+414yes—
460C22×C42−1+414yes—
461C4⋊C4—216no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—416no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—88no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—416no—
478C22⋊C4—84no—
479C42—216no—
480C42—416no—
481C42—48no—
482C42—44no—
483C42—44no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23#55714yes—
507C23C23×D412yes—
508C23C23×Q811yescharacteristic
509C23C22×#4111yescharacteristic
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×H(5,𝔽2)12yes—
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×2−1+412yes—
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+412yes—
516C23C2×2−1+411yescharacteristic
517C23#58511yescharacteristic
518Q8—416no—
519Q8—432no—
520Q8—432no—
521Q8—816no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—432no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—416no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4#53518yes—
605C2×C4#53814yes—
606C2×C4D4214yes—
607C2×C4#54618yes—
608C2×C4D4×Q818yes—
609C2×C4H(3,𝔽4)14yes—
610C22—28no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22C2×#53512yes—
622C22C2×#53811yescharacteristic
623C22C2×D4211yescharacteristic
624C22C2×#54612yes—
625C22C2×D4×Q812yes—
626C22C2×#55711yescharacteristic
627C22#348612yes—
628C22#348912yes—
629C22#349912yes—
630C22#350111yescharacteristic
631C22#350312yes—
632C22#350311yescharacteristic
633C22#350711yescharacteristic
634C22#350912yes—
635C22C2×H(3,𝔽4)11yescharacteristic
636C22#353811yescharacteristic
637C22#353811yescharacteristic
638C22#355011yescharacteristic
639C22#355212yes—
640C22#355212yes—
641C22#355212yes—
642C22#356111yescharacteristic
643C22#356112yes—
644C4—28no—
645C4—44no—
646C4—42no—
647C4—44no—
648C4—48no—
649C4—42no—
650C4—48no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—81no—
655C4—84no—
656C4—84no—
657C4#267018yes—
658C2—44no—
659C2#3961612yes—
660C2#3964011yescharacteristic
661C2#4281212yes—
662C2#4294111yescharacteristic
663C2#4514412yes—
664C2#4532111yescharacteristic
665C2#4600411yescharacteristic
666C2C2×#267011yescharacteristic
667C2#5303211yescharacteristic
668C2#5899811yescharacteristic
669C2#5914412yes—
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
1. C2×#2568C2semidirect
3. #17112C2semidirect
4. #16524C2semidirect
6. #15510C2semidirect
8. #15190C2semidirect
9. #15188C2semidirect
12. #14142C2semidirect
128. #393Q8semidirect
204. C22×Q8C2×Q8semidirect
326. #95C2×Q8semidirect
327. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#24342
#171122
#165242
#163392
#155102
#154402
#151902
#151882
#151782
#145752
#141422
#139972
#139222
C4×#3932