Finite Group Databasebeta

← #1395654

Subgroups — #1395654

Order 512.

2752 conjugacy classes of subgroups.

The subgroup lattice has 775 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#16655C212yesmaximal, complemented
4#16592C214yesmaximal, complemented
5#16540C214yesmaximal, complemented
6#16516C212yesmaximal, complemented
7#15244C211yesmaximal, characteristic, complemented
8#15220C212yesmaximal, complemented
9#15152C211yesmaximal, characteristic, complemented
10#15126C212yesmaximal, complemented
11#15123C211yesmaximal, characteristic
12#15110C212yesmaximal, complemented
13#14291C212yesmaximal, complemented
14#14136C212yesmaximal, complemented
15#13918C211yesmaximal, characteristic, complemented
16#13879C212yesmaximal, complemented
17C4×#394C211yesmaximal, characteristic, complemented
18C22×#126C2212yes—
19C22×#121C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22#2512—28no—
23C2×#400C2212yes—
24C2×#397C2212yes—
25C2×#396C2212yes—
26C2×#394C2211yescharacteristic
27C2×#394C2214yes—
28C2×#392C2212yes—
29#2354—28no—
30C2×#390C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#384C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#382C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50#1445C2212yes—
51#1443C2214yes—
52#1442C2211yescharacteristic
53#1442C2212yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2211yescharacteristic
57#1441C2214yes—
58#1440C2214yes—
59#1440C2212yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1438C2214yes—
71#1438C2214yes—
72#1438C2214yes—
73#1438C2211yescharacteristic
74#1438C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C23×D4C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C23×D4C2314yes—
88C24×C4C2312yes—
89C2×#126—28no—
90C2×#126—28no—
91C2×#126—44no—
92C2×#121—28no—
93C2×#121—44no—
94C2×#121—44no—
95C2×C4×D4—28no—
96C2×C4×D4—24no—
97C2×#117—28no—
98C2×#117—44no—
99C2×#117—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2314yes—
137#400—44no—
138#397—28no—
139#396—28no—
140#394—24no—
141#394—48no—
142#392—44no—
143#390—24no—
144#390—24no—
145#390—28no—
146#390—24no—
147#390—44no—
148#389—24no—
149#389—44no—
150#388—28no—
151#388—216no—
152#384—24no—
153#384—24no—
154#382—44no—
155#382—44no—
156#380—28no—
157#380—44no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4D418yescomplemented
161C4×C22⋊C4—28no—
162C4×C22⋊C4—44no—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C25C2412yes—
188C25C2412yes—
189C22×D4—24no—
190C22×D4—28no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4#4118yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228#126—432no—
229#121—432no—
230C4×D4—232no—
231#117—432no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4#4114yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D414yes—
333C2×C42C2×D418yes—
334#95—28no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—28no—
344#95—44no—
345#95—42no—
346#95—44no—
347#95—44no—
348#95—42no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—42no—
357#95C2×D418yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—22no—
376C24—22no—
377C24—24no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24C2×#4114yes—
384C24C2511yescharacteristic
385C2×D4—28no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—416no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—88no—
397C2×D4—88no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—216no—
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4114yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C4H(5,𝔽2)14yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—88no—
512C22⋊C4—216no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—88no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C42—28no—
530C42—216no—
531C42—416no—
532C42—44no—
533C42—44no—
534C42—48no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—42no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—42no—
590C23—42no—
591C23—44no—
592C23—44no—
593C23C23×D411yescharacteristic
594C23C22×#4112yes—
595C23C22×#4111yescharacteristic
596C23C2×H(5,𝔽2)12yes—
597C23C2×H(5,𝔽2)11yescharacteristic
598C23C2×H(5,𝔽2)11yescharacteristic
599C23C2×H(5,𝔽2)11yescharacteristic
600C23C2×2−1+411yescharacteristic
601C23#58512yes—
602C23#58511yescharacteristic
603C23#58512yes—
604D4—416no—
605D4—832no—
606D4—88no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—432no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4#52514yes—
692C2×C4#53514yes—
693C2×C4#53714yes—
694C2×C4#53818yes—
695C2×C4#53918yes—
696C2×C4#54614yes—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—84no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—84no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—84no—
721C22—84no—
722C22—82no—
723C22C2×#52511yescharacteristic
724C22C2×#53511yescharacteristic
725C22C2×#53711yescharacteristic
726C22C2×#53812yes—
727C22C2×#53912yes—
728C22C2×#54611yescharacteristic
729C22#347512yes—
730C22#348711yescharacteristic
731C22#348811yescharacteristic
732C22#349412yes—
733C22#349412yes—
734C22#350011yescharacteristic
735C22#350611yescharacteristic
736C22#351111yescharacteristic
737C22#351212yes—
738C22#351712yes—
739C22#354012yes—
740C22#354612yes—
741C22#354612yes—
742C22#354711yescharacteristic
743C22#354911yescharacteristic
744C22#356212yes—
745C22#356512yes—
746C4—24no—
747C4—44no—
748C4—44no—
749C4—44no—
750C4—42no—
751C4—42no—
752C4—42no—
753C4—42no—
754C4—82no—
755C4—82no—
756C4—81no—
757C4—84no—
758C4—82no—
759C4—84no—
760C4#263618yes—
761C2—84no—
762C2—162no—
763C2C2×#263611yescharacteristic
764C2#3608311yescharacteristic
765C2#3642611yescharacteristic
766C2#3895611yescharacteristic
767C2#3946911yescharacteristic
768C2#3993712yes—
769C2#4000912yes—
770C2#4288211yescharacteristic
771C2#4288911yescharacteristic
772C2#4294712yes—
773C2#4297512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15123C2
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×#2512C2semidirect
2. C2×#2354C2semidirect
3. #16655C2semidirect
4. #16592C2semidirect
5. #16540C2semidirect
6. #16516C2semidirect
7. #15244C2semidirect
8. #15220C2semidirect
9. #15152C2semidirect
10. #15126C2semidirect
12. #15110C2semidirect
13. #14291C2semidirect
14. #14136C2semidirect
15. #13918C2semidirect
16. #13879C2semidirect
17. C4×#394C2semidirect
160. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#23542
#166552
#165922
#165402
#165162
#152442
#152202
#151522
#151262
#151232
#151102
#142912
#141362
#139182
#138792
C4×#3942