Finite Group Databasebeta

← #674764

Subgroups — #674764

Order 512.

3614 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2445C212yesmaximal
2C2×#2349C211yesmaximal, characteristic, complemented
3C2×C4×#127C211yesmaximal, characteristic
4#15300C214yesmaximal, complemented
5#15271C214yesmaximal, complemented
6#15138C212yesmaximal, complemented
7#15133C212yesmaximal, complemented
8#14824C214yesmaximal, complemented
9#14707C211yesmaximal, characteristic, complemented
10#13968C214yesmaximal, complemented
11#13941C211yesmaximal, characteristic, complemented
12#13941C211yesmaximal, characteristic, complemented
13#13941C211yesmaximal, characteristic, complemented
14#13916C212yesmaximal, complemented
15C4×#389C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#123C2214yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2214yes—
22#2445—216no—
23C2×#398C2214yes—
24C2×#397C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27#2349—28no—
28C4×#127—28no—
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#381C2214yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C22×#95C2212yes—
49C2×C43C2211yescharacteristic
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2212yes—
66#1436C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C4×#95C2212yes—
78C23×D4C2311yescharacteristic
79C23×D4C2314yes—
80C23×D4C2311yescharacteristic
81C24×C4C2314yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×#127—24no—
85C2×#127—24no—
86C2×#127D418yes—
87C2×#123—216no—
88C2×#123—216no—
89C2×#123—48no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—216no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—48no—
100C2×C4×D4—48no—
101C2×C4×D4D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#398—48no—
133#397—48no—
134#394—216no—
135#394—216no—
136#390—24no—
137#390—28no—
138#390—28no—
139#390—28no—
140#390—24no—
141#390—216no—
142#389—28no—
143#389—28no—
144#389—28no—
145#386—216no—
146#385—216no—
147#384—24no—
148#382—24no—
149#382—24no—
150#381—48no—
151#380—28no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C2×#95—28no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C43—24no—
180C25C2412yes—
181C25C2412yes—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—28no—
190C22×D4—22no—
191C22×D4—28no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4C2×D414yes—
199C22×D4C2×D4116yes—
200C22×D4C2×D414yes—
201C22×D4C2×D414yes—
202C22×D4#4114yes—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2×D418yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233#127—232no—
234#123—464no—
235C4×D4—232no—
236C4×D4—464no—
237C4×D4—432no—
238C4×D4—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4#4118yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D418yes—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—42no—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42#4114yes—
326C2×C42#4114yes—
327#95—22no—
328#95—24no—
329#95—28no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24C22×D414yes—
369C24C22×D414yes—
370C24C22×D414yes—
371C24C2×#4114yes—
372C24C2511yescharacteristic
373C2×D4—232no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—232no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—416no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4#121116yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—216no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—84no—
466C22×C4—88no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—88no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C42−1+412yes—
493C22×C42−1+412yes—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—48no—
498C4⋊C4—416no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C22⋊C4—216no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C42—216no—
518C42—28no—
519C42—48no—
520C42—48no—
521C42—44no—
522C42—48no—
523C42#123116yescomplemented
524C23—24no—
525C23—24no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—24no—
537C23—28no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23C2×#12118yes—
570C23C23×D411yescharacteristic
571C23C23×D411yescharacteristic
572C23C22×#4111yescharacteristic
573C23C22×#4111yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)12yes—
578C23C2×2−1+411yescharacteristic
579C23C2×2−1+411yescharacteristic
580C23#58512yes—
581D4—232no—
582D4—464no—
583D4—416no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—816no—
658C2×C4—816no—
659C2×C4—832no—
660C2×C4C2×#12114yes—
661C2×C4C2×#12314yes—
662C2×C4C2×#12314yes—
663C2×C4C2×#12314yes—
664C2×C4#53614yes—
665C2×C4#54618yes—
666C2×C4#54714yes—
667C2×C4#54714yes—
668C2×C4#55214yes—
669C2×C4#55214yes—
670C22—216no—
671C22—28no—
672C22—44no—
673C22—48no—
674C22—44no—
675C22—48no—
676C22—44no—
677C22—44no—
678C22—48no—
679C22—44no—
680C22—44no—
681C22—48no—
682C22—44no—
683C22—48no—
684C22—44no—
685C22—44no—
686C22—82no—
687C22—84no—
688C22—82no—
689C22—82no—
690C22—84no—
691C22—82no—
692C22C22×#12111yescharacteristic
693C22C22×#12311yescharacteristic
694C22#344411yescharacteristic
695C22C2×#53611yescharacteristic
696C22#345511yescharacteristic
697C22C2×#54612yes—
698C22C2×#54711yescharacteristic
699C22C2×#54711yescharacteristic
700C22D4×#4112yes—
701C22C2×#55211yescharacteristic
702C22C2×#55211yescharacteristic
703C22#348012yes—
704C22#348112yes—
705C22#348412yes—
706C22#349612yes—
707C22#350811yescharacteristic
708C22#350812yes—
709C22#350811yescharacteristic
710C22#351112yes—
711C22#351312yes—
712C22#352212yes—
713C22#355412yes—
714C22#355912yes—
715C4—28no—
716C4—24no—
717C4—24no—
718C4—44no—
719C4—44no—
720C4—44no—
721C4—44no—
722C4—84no—
723C4—82no—
724C4—84no—
725C4—82no—
726C4—82no—
727C4#248618yes—
728C2—48no—
729C2—84no—
730C2#3474411yescharacteristic
731C2C2×#248611yescharacteristic
732C2#3555411yescharacteristic
733C2#3614911yescharacteristic
734C2#3621412yes—
735C2#3637011yescharacteristic
736C2#3896911yescharacteristic
737C2#3937211yescharacteristic
738C2#3987712yes—
739C2#5941112yes—
740C2#5944612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2349C2semidirect
4. #15300C2semidirect
5. #15271C2semidirect
6. #15138C2semidirect
7. #15133C2semidirect
8. #14824C2semidirect
9. #14707C2semidirect
10. #13968C2semidirect
11. #13941C2semidirect
12. #13941C2semidirect
13. #13941C2semidirect
14. #13916C2semidirect
15. C4×#389C2semidirect
523. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24452
C2×#23492
C2×C4×#1272
#153002
#152712
#151382
#151332
#148242
#147072
#139682
#139412
#139162
C4×#3892