Finite Group Databasebeta

← #726077

Subgroups — #726077

Order 512.

2708 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic, complemented
2C2×#2324C211yesmaximal, characteristic
3#16986C212yesmaximal, complemented
4#16807C212yesmaximal, complemented
5#15352C214yesmaximal, complemented
6#15348C214yesmaximal, complemented
7#15136C212yesmaximal, complemented
8#15126C212yesmaximal
9#15113C212yesmaximal, complemented
10#14892C212yesmaximal, complemented
11#14340C211yesmaximal, characteristic, complemented
12#14150C211yesmaximal, characteristic, complemented
13#14132C212yesmaximal, complemented
14#13941C211yesmaximal, characteristic, complemented
15#13910C211yesmaximal, characteristic, complemented
16#13876C212yesmaximal, complemented
17#13852C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2212yes—
22C2×#397C2214yes—
23C2×#397C2214yes—
24C2×#394C2214yes—
25#2354—28no—
26#2324—28no—
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#388C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#382C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50#1445C2214yes—
51#1444C2212yes—
52#1444C2212yes—
53#1444C2212yes—
54#1442C2212yes—
55#1442C2214yes—
56#1441C2212yes—
57#1441C2212yes—
58#1441C2211yescharacteristic
59#1441C2211yescharacteristic
60#1441C2212yes—
61#1441C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2211yescharacteristic
66#1439C2214yes—
67#1439C2214yes—
68#1439C2211yescharacteristic
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4D418yes—
95C2×#117—28no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—48no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#397—48no—
137#397—48no—
138#394—48no—
139#390—24no—
140#390—28no—
141#390—28no—
142#390—44no—
143#389—24no—
144#389—28no—
145#389—48no—
146#388—28no—
147#388—216no—
148#388—48no—
149#386—28no—
150#385—28no—
151#384—24no—
152#384—44no—
153#382—24no—
154#380—28no—
155#380—24no—
156#380—44no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—24no—
161C4×C22⋊C4—28no—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4C2×D414yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D418yes—
212C23×C4#4114yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228#121—232no—
229C4×D4—232no—
230#117—432no—
231#117—416no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
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—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4#4114yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
314C2×C42—28no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42C2×D414yes—
329C2×C42#4114yes—
330C2×C42#4114yes—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—44no—
349#95—42no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—22no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—21no—
373C24—22no—
374C24—24no—
375C24—22no—
376C24C22×D414yes—
377C24C2×#4112yes—
378C24C2511yescharacteristic
379C2×D4—216no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—88no—
388C2×D4—84no—
389C2×D4—88no—
390C2×D4—84no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+414yes—
493C22×C42−1+412yes—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—88no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—216no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—84no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C42—416no—
525C42—44no—
526C42—48no—
527C42—48no—
528C42—84no—
529C42#123116yes—
530C23—24no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—41no—
543C23—41no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—41no—
554C23—44no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—41no—
559C23—41no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—42no—
576C23C23×D411yescharacteristic
577C23C22×#4111yescharacteristic
578C23C22×#4111yescharacteristic
579C23C22×#4111yescharacteristic
580C23C22×#4111yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×H(5,𝔽2)12yes—
583C23C2×H(5,𝔽2)12yes—
584C23C2×2−1+411yescharacteristic
585C23C2×2−1+412yes—
586C23#58512yes—
587D4—416no—
588D4—816no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—432no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—416no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—44no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4C2×#12314yes—
674C2×C4C2×#12314yes—
675C2×C4C2×#12314yes—
676C2×C4#52514yes—
677C2×C4#53714yes—
678C2×C4#55214yes—
679C2×C4#55314yes—
680C2×C4#55514yes—
681C2×C4#55614yes—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—42no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—42no—
693C22—44no—
694C22—81no—
695C22—84no—
696C22—82no—
697C22—84no—
698C22—84no—
699C22—84no—
700C22—82no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22C22×#12311yescharacteristic
705C22C2×#52511yescharacteristic
706C22C2×#53711yescharacteristic
707C22#346012yes—
708C22C2×#55211yescharacteristic
709C22C2×#55311yescharacteristic
710C22C2×#55511yescharacteristic
711C22C2×#55611yescharacteristic
712C22#348411yescharacteristic
713C22#348511yescharacteristic
714C22#349212yes—
715C22#349212yes—
716C22#349412yes—
717C22#349811yescharacteristic
718C22#349911yescharacteristic
719C22#350612yes—
720C22#351112yes—
721C22#351412yes—
722C22#352212yes—
723C22#354312yes—
724C22#354812yes—
725C22#355512yes—
726C22#356112yes—
727C4—24no—
728C4—24no—
729C4—44no—
730C4—44no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—42no—
735C4—81no—
736C4—84no—
737C4—82no—
738C4—84no—
739C4—82no—
740C4—81no—
741C4#248518yes—
742C2—84no—
743C2—82no—
744C2C2×#248511yescharacteristic
745C2#3515211yescharacteristic
746C2#3553011yescharacteristic
747C2#3569311yescharacteristic
748C2#3602211yescharacteristic
749C2#3798312yes—
750C2#3828512yes—
751C2#4242511yescharacteristic
752C2#4264911yescharacteristic
753C2#4491112yes—
754C2#4492212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2354C2semidirect
3. #16986C2semidirect
4. #16807C2semidirect
5. #15352C2semidirect
6. #15348C2semidirect
7. #15136C2semidirect
9. #15113C2semidirect
10. #14892C2semidirect
11. #14340C2semidirect
12. #14150C2semidirect
13. #14132C2semidirect
14. #13941C2semidirect
15. #13910C2semidirect
16. #13876C2semidirect
17. #13852C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#23242
#169862
#168072
#153522
#153482
#151362
#151262
#151132
#148922
#143402
#141502
#141322
#139412
#139102
#138762
#138522