Finite Group Databasebeta

← #4457062

Subgroups — #4457062

Order 512.

2266 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#17074C212yesmaximal, complemented
4#17042C212yesmaximal, complemented
5#16990C212yesmaximal, complemented
6#16820C212yesmaximal, complemented
7#16301C214yesmaximal
8#16300C214yesmaximal
9#15218C212yesmaximal, complemented
10#15194C212yesmaximal, complemented
11#15139C212yesmaximal, complemented
12#15137C212yesmaximal, complemented
13#14264C211yesmaximal, characteristic
14#13951C211yesmaximal, characteristic
15#13924C211yesmaximal, characteristic
16#13904C211yesmaximal, characteristic
17#13844C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2441—28no—
23C2×#400C2214yes—
24C2×#399C2212yes—
25C2×#398C2214yes—
26C2×#396C2212yes—
27#2316—28no—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#387C2212yes—
38C2×#384C2211yescharacteristic
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2214yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#1445C2212yes—
48#1445C2212yes—
49#1445C2214yes—
50#1445C2212yes—
51#1445C2212yes—
52#1444C2211yescharacteristic
53#1444C2211yescharacteristic
54#1444C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2214yes—
59#1443C2214yes—
60#1442C2211yescharacteristic
61#1442C2211yescharacteristic
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1439C2212yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#125—24no—
88C2×#125—44no—
89C2×#117—24no—
90C2×#117—44no—
91C2×#117—44no—
92C2×#117D418yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4Q818yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129#400—48no—
130#399—28no—
131#398—48no—
132#396—28no—
133#389—28no—
134#389—28no—
135#389—24no—
136#389—24no—
137#389—28no—
138#389—28no—
139#389—28no—
140#389—44no—
141#389—44no—
142#387—28no—
143#384—24no—
144#381—44no—
145#381—44no—
146#380—24no—
147#380—24no—
148#380—216no—
149C4×C4⋊C4—44no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C25C2411yescharacteristic
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4C2×D414yes—
192C23×C4C2×Q814yes—
193C23×C4C2×Q818yes—
194C23×C4#4114yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211#125—416no—
212#117—416no—
213#117—416no—
214C2×C4⋊C4—232no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
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—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×Q818yes—
267C2×C4⋊C4C2×Q818yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4C2×D418yes—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—44no—
298C2×C42C2×D418yes—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—44no—
319#95—42no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24C22×D412yes—
338C24C22×Q812yes—
339C24C2×#4112yes—
340C24C2511yescharacteristic
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—216no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4C22⋊Q8116yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×Q818yes—
436C22×C4C22×Q812yes—
437C22×C4C22×Q812yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C42−1+414yes—
444C22×C42−1+414yes—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C42—48no—
474C42—48no—
475C42—44no—
476C42—44no—
477C42—48no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23C2×C22⋊Q814yes—
501C23C23×D411yescharacteristic
502C23C23×Q811yescharacteristic
503C23C22×#4111yescharacteristic
504C23C2×H(5,𝔽2)12yes—
505C23C2×H(5,𝔽2)11yescharacteristic
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×2−1+412yes—
508C23C2×2−1+412yes—
509C23#58511yescharacteristic
510C23#58511yescharacteristic
511C23#58512yes—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—232no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—816no—
600C2×C4C2×C22⋊Q814yes—
601C2×C4C2×C22⋊Q814yes—
602C2×C4#53414yes—
603C2×C4#53614yes—
604C2×C4#53918yes—
605C2×C4#54418yes—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22C22×C22⋊Q811yescharacteristic
618C22C2×#53411yescharacteristic
619C22C2×#53611yescharacteristic
620C22C2×#53912yes—
621C22C2×#54412yes—
622C22#348811yescharacteristic
623C22#348811yescharacteristic
624C22#349512yes—
625C22#349911yescharacteristic
626C22#350011yescharacteristic
627C22#350111yescharacteristic
628C22#350111yescharacteristic
629C22#350212yes—
630C22#350811yescharacteristic
631C22#351011yescharacteristic
632C22#353912yes—
633C22#354112yes—
634C22#354412yes—
635C22#354612yes—
636C22#354812yes—
637C22#355912yes—
638C22#356312yes—
639C22#356512yes—
640C4—24no—
641C4—44no—
642C4—44no—
643C4—48no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4—82no—
651C4—84no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4#262718yes—
656C2—44no—
657C2C2×#262711yescharacteristic
658C2#3551511yescharacteristic
659C2#3566811yescharacteristic
660C2#3733512yes—
661C2#3741512yes—
662C2#4248912yes—
663C2#4249112yes—
664C2#4291611yescharacteristic
665C2#4294411yescharacteristic
666C2#4296411yescharacteristic
667C2#4298511yescharacteristic
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
3. #17074C2semidirect
4. #17042C2semidirect
5. #16990C2semidirect
6. #16820C2semidirect
9. #15218C2semidirect
10. #15194C2semidirect
11. #15139C2semidirect
12. #15137C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#23162
#170742
#170422
#169902
#168202
#163012
#163002
#152182
#151942
#151392
#151372
#142642
#139512
#139242
#139042
#138442