Finite Group Databasebeta

← #828536

Subgroups — #828536

Order 512.

2698 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2509C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#16985C212yesmaximal, complemented
4#15477C214yesmaximal, complemented
5#15341C214yesmaximal, complemented
6#15256C212yesmaximal, complemented
7#15197C211yesmaximal, characteristic, complemented
8#15152C211yesmaximal, characteristic, complemented
9#15129C212yesmaximal
10#15129C211yesmaximal, characteristic
11#14190C212yesmaximal
12#13994C212yesmaximal, complemented
13#13907C212yesmaximal, complemented
14#13907C212yesmaximal, complemented
15#13875C212yesmaximal, complemented
16C4×#394C211yesmaximal, characteristic
17#13818C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2212yes—
21#2509—28no—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#394C2211yescharacteristic
26C2×#394C2212yes—
27D4×C4⋊C4—28no—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#388C2214yes—
36C2×#387C2212yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2214yes—
44C2×#380C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50#1445C2212yes—
51#1443C2214yes—
52#1442C2211yescharacteristic
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2212yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2211yescharacteristic
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2212yes—
69#1439C2212yes—
70#1438C2211yescharacteristic
71#1437C2214yes—
72#1436C2214yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80#1434C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2314yes—
88C2×C4×D4—28no—
89C2×C4×D4—24no—
90C2×C4×D4—28no—
91C2×C4×D4—48no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2314yes—
125C22×C42C2312yes—
126#400—28no—
127#400—28no—
128#400—44no—
129#394—24no—
130#394—28no—
131#389—24no—
132#389—24no—
133#389—24no—
134#389—28no—
135#389—28no—
136#389—28no—
137#389—24no—
138#388—48no—
139#387—28no—
140#385—216no—
141#384—24no—
142#382—24no—
143#380—28no—
144#380—28no—
145#380—48no—
146#380—48no—
147#380—44no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—44no—
150C4×C4⋊C4D418yes—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—44no—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C25C2412yes—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4#4114yes—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4#4118yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C4×D4—232no—
221C4×D4—432no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
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—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—432no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4#4118yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—28no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42C2×D414yes—
306C2×C42C2×D418yes—
307#95—22no—
308#95—22no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—22no—
321#95—24no—
322#95—28no—
323#95—22no—
324#95—24no—
325#95—44no—
326#95—42no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95C2×D418yes—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24C2×#4114yes—
351C24C2×#4114yes—
352C24H(5,𝔽2)14yes—
353C24C2511yescharacteristic
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—28no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4118yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+414yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—84no—
480C4⋊C4—88no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4#124116yes—
484C22⋊C4—216no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—88no—
492C42—28no—
493C42—48no—
494C42—44no—
495C42—44no—
496C42—48no—
497C42—88no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23#55318yes—
533C23C23×D411yescharacteristic
534C23C22×#4112yes—
535C23C22×#4112yes—
536C23C22×#4111yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×2−1+411yescharacteristic
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+412yes—
543C23#58511yescharacteristic
544D4—232no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—432no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—832no—
634C2×C4#52514yes—
635C2×C4C2×#12418yes—
636C2×C4C2×#12414yes—
637C2×C4#53714yes—
638C2×C4#54814yes—
639C2×C4#55218yes—
640C2×C4#55318yes—
641C22—28no—
642C22—28no—
643C22—28no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22C2×#52511yescharacteristic
657C22C22×#12411yescharacteristic
658C22C2×#53711yescharacteristic
659C22C2×#54811yescharacteristic
660C22C2×#55212yes—
661C22C2×#55312yes—
662C22C2×#55312yes—
663C22#348212yes—
664C22#348411yescharacteristic
665C22#348811yescharacteristic
666C22#349212yes—
667C22#349512yes—
668C22#349911yescharacteristic
669C22#351112yes—
670C22#351312yes—
671C22#351511yescharacteristic
672C22#351911yescharacteristic
673C22#351912yes—
674C22#352312yes—
675C22#353612yes—
676C22#354412yes—
677C22#354411yescharacteristic
678C22#354711yescharacteristic
679C4—28no—
680C4—24no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—84no—
688C4—84no—
689C4—81no—
690C4—82no—
691C4—81no—
692C4—82no—
693C4—81no—
694C4#256218yes—
695C2—48no—
696C2C2×#256211yescharacteristic
697C2#3590311yescharacteristic
698C2#3608911yescharacteristic
699C2#3644611yescharacteristic
700C2#3658111yescharacteristic
701C2#3912611yescharacteristic
702C2#4242811yescharacteristic
703C2#4491712yes—
704C2#4537112yes—
705C2#5920912yes—
706C2#5967712yes—
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
1. C2×#2509C2semidirect
3. #16985C2semidirect
4. #15477C2semidirect
5. #15341C2semidirect
6. #15256C2semidirect
7. #15197C2semidirect
8. #15152C2semidirect
12. #13994C2semidirect
13. #13907C2semidirect
14. #13907C2semidirect
15. #13875C2semidirect
17. #13818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25092
C2×D4×C4⋊C42
#169852
#154772
#153412
#152562
#151972
#151522
#151292
#141902
#139942
#139072
#138752
C4×#3942
#138182