Finite Group Databasebeta

← #4453579

Subgroups — #4453579

Order 512.

2902 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C212yesmaximal, complemented
2C2×#2392C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic, complemented
4#16688C212yesmaximal, complemented
5#16684C212yesmaximal, complemented
6#16515C214yesmaximal, complemented
7#16470C212yesmaximal, complemented
8#16288C212yesmaximal
9#16286C214yesmaximal, complemented
10#15220C212yesmaximal, complemented
11#15181C212yesmaximal, complemented
12#15149C212yesmaximal, complemented
13#15131C212yesmaximal, complemented
14#14340C211yesmaximal, characteristic, complemented
15#13872C212yesmaximal, complemented
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2485—216no—
22#2392—28no—
23C2×#398C2211yescharacteristic
24C2×#398C2214yes—
25C2×#397C2211yescharacteristic
26C2×#397C2214yes—
27C2×#396C2212yes—
28C2×#396C2214yes—
29C2×#394C2211yescharacteristic
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#392C2211yescharacteristic
33#2317—28no—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2214yes—
39C2×#387C2214yes—
40C2×#386C2214yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#384C2212yes—
44C2×#384C2211yescharacteristic
45C2×#382C2212yes—
46C2×#382C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C22⋊#95C2214yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4×D4—28no—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C2×#117—24no—
90C2×#117—24no—
91C2×#117—42no—
92C2×#117—42no—
93C2×#117D418yes—
94C2×#117D418yescomplemented
95C22×C4⋊C4—24no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#398—24no—
132#398—48no—
133#397—24no—
134#397—216no—
135#396—28no—
136#396—48no—
137#394—24no—
138#394—28no—
139#394—216no—
140#392—24no—
141#389—28no—
142#389—28no—
143#389—28no—
144#389—28no—
145#388—216no—
146#387—216no—
147#386—48no—
148#385—28no—
149#385—28no—
150#384—28no—
151#384—42no—
152#382—28no—
153#382—42no—
154#380—28no—
155#380—28no—
156#380—28no—
157#380—44no—
158#380—44no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—28no—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—42no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4#4114yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C4×D4—416no—
229C4×D4—432no—
230#117—232no—
231#117—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
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⋊C4C2×D418yes—
316C2×C22⋊C4C2×D414yes—
317C2×C22⋊C4C2×D414yes—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×D418yes—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—42no—
348#95—42no—
349#95—44no—
350#95—44no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
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—
373C24C22×D412yes—
374C24C22×D412yes—
375C24C2×#4112yes—
376C242−1+414yes—
377C24C2511yescharacteristic
378C2×D4—216no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—48no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4#121116yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D418yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C42−1+412yes—
492C4⋊C4—216no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—88no—
504C4⋊C4—88no—
505C4⋊C4—88no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C22⋊C4—216no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C42—216no—
522C42—28no—
523C42—28no—
524C42—48no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
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—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23C2×#12114yes—
583C23C23×D411yescharacteristic
584C23C23×D411yescharacteristic
585C23C22×#4111yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)11yescharacteristic
588C23C2×H(5,𝔽2)12yes—
589C23C2×2−1+411yescharacteristic
590C23C2×2−1+411yescharacteristic
591C23C2×2−1+412yes—
592C23#58512yes—
593C23#58512yes—
594D4—432no—
595D4—416no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—232no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
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—816no—
673C2×C4—816no—
674C2×C4C2×#12114yes—
675C2×C4C2×#12114yes—
676C2×C4#53414yes—
677C2×C4#53614yes—
678C2×C4#53714yes—
679C2×C4#53714yes—
680C2×C4#53914yes—
681C2×C4#53914yes—
682C22—28no—
683C22—24no—
684C22—42no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22—44no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22C22×#12111yescharacteristic
709C22C2×#53411yescharacteristic
710C22C2×#53611yescharacteristic
711C22C2×#53711yescharacteristic
712C22C2×#53711yescharacteristic
713C22C2×#53911yescharacteristic
714C22C2×#53911yescharacteristic
715C22#348812yes—
716C22#348812yes—
717C22#348912yes—
718C22#348912yes—
719C22#349212yes—
720C22#349312yes—
721C22#353811yescharacteristic
722C22#354011yescharacteristic
723C22#354112yes—
724C22#354612yes—
725C22#354712yes—
726C22#354911yescharacteristic
727C22#355111yescharacteristic
728C22#355212yes—
729C22#356312yes—
730C22#356512yes—
731C4—24no—
732C4—42no—
733C4—42no—
734C4—48no—
735C4—44no—
736C4—42no—
737C4—42no—
738C4—84no—
739C4—84no—
740C4—82no—
741C4—84no—
742C4—82no—
743C4#261818yes—
744C2—44no—
745C2—84no—
746C2C2×#261811yescharacteristic
747C2#3551812yes—
748C2#3727311yescharacteristic
749C2#3727912yes—
750C2#3737411yescharacteristic
751C2#4250211yescharacteristic
752C2#4250712yes—
753C2#4268411yescharacteristic
754C2#4292812yes—
755C2#4297111yescharacteristic
756C2#4298011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2485C2semidirect
3. C2×#2317C2semidirect
4. #16688C2semidirect
5. #16684C2semidirect
6. #16515C2semidirect
7. #16470C2semidirect
9. #16286C2semidirect
10. #15220C2semidirect
11. #15181C2semidirect
12. #15149C2semidirect
13. #15131C2semidirect
14. #14340C2semidirect
15. #13872C2semidirect
94. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
C2×#23922
C2×#23172
#166882
#166842
#165152
#164702
#162882
#162862
#152202
#151812
#151492
#151312
#143402
#138722