Finite Group Databasebeta

← #360146

Subgroups — #360146

Order 512.

1940 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#440C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic, complemented
3C2×#1880C212yesmaximal
4C2×#1803C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6#13906C211yesmaximal, characteristic
7#11219C212yesmaximal, complemented
8#11066C212yesmaximal
9#10843C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#440—24no—
17C2×#440—24no—
18C2×#440C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#102C2212yes—
24C22×#102C2211yescharacteristic
25C22×#97C2211yescharacteristic
26#2351—24no—
27#2351—24no—
28C2×#390C2212yes—
29C2×#389—22no—
30C2×#389C2212yes—
31C2×#384C2211yescharacteristic
32C2×#382—21no—
33C2×#382C2211yescharacteristic
34C2×#380—22no—
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1880—28no—
38#1804—24no—
39#1803—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1439—21no—
47#1439—21no—
48C4⋊#95C2212yes—
49C23×Q8C2311yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16C2×C414yes—
57C22×SD16D414yes—
58C22×SD16Q814yes—
59C23×C8C2312yes—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8—42no—
70C2×C4×Q8—42no—
71C2×C4×Q8C2×C414yes—
72C2×C4×Q8D414yes—
73C2×C4×Q8D414yes—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4C2×C414yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D412yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C42—21no—
94C22×C42—22no—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97#440—232no—
98C2×#107—22no—
99C2×#107—22no—
100C2×#107C2×C414yes—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103C2×C414yes—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—24no—
113C2×#102—24no—
114C2×#102—42no—
115C2×#102C2×C414yes—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97C2×C414yes—
119#390—24no—
120#390—42no—
121#389—216no—
122#389—24no—
123#389—24no—
124#384—22no—
125#384—22no—
126#382—28no—
127#382—22no—
128#382D414yes—
129#380—48no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95D411yescharacteristic
139C2×#95D411yescharacteristic
140#340—24no—
141#340—24no—
142#340—24no—
143#340—42no—
144#336—24no—
145#336—42no—
146C25—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8C22×C412yes—
155C22×Q8C2×D412yes—
156C22×Q8C2×D412yes—
157C22×Q8C2×D412yes—
158C22×Q8C2×Q812yes—
159C22×Q8#4112yes—
160C22×Q8#4112yes—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C22×C412yes—
173C22×D4C2×D412yes—
174C22×D4C2×Q812yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2411yescharacteristic
190C2×SD16—28no—
191C2×SD16—28no—
192C2×SD16—28no—
193C2×SD16—28no—
194C2×SD16—28no—
195C2×SD16—28no—
196C2×SD16C4⋊C4116yescomplemented
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8C22×C412yes—
215C22×C8C22×C412yes—
216C22×C8C2×D412yes—
217C22×C8C2×Q812yes—
218C4×Q8—28no—
219C4×Q8—28no—
220C4×Q8—28no—
221C4×Q8—48no—
222C4×Q8—48no—
223C4×Q8—48no—
224C4×Q8—48no—
225C4×Q8C22⋊C4116yescomplemented
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—44no—
229C4×D4—44no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—82no—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—82no—
278C2×C22⋊C4—82no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C22×C412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42#4112yes—
296C2×C42#4112yes—
297#107—28no—
298C4⋊C8—44no—
299#103—28no—
300#103—48no—
301#102—28no—
302#102—48no—
303#97—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—44no—
310#95—44no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—41no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24C22×D411yescharacteristic
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8C2×C22⋊C414yes—
345C2×Q8C2×C4⋊C414yes—
346C2×Q8#11714yes—
347C2×Q8#12114yes—
348C2×Q8#12114yes—
349C2×Q8C22⋊Q814yes—
350C2×Q8C22⋊Q814yes—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4C2×C4⋊C414yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—416no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4C4×D412yes—
433C22×C4C4×D412yes—
434C22×C4#12012yes—
435C22×C4#12012yes—
436C22×C4#12112yes—
437C22×C4#12112yes—
438C22×C4AGL(1,ℤ/8)14yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×Q811yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C4H(5,𝔽2)11yescharacteristic
451C22×C42−1+411yescharacteristic
452C22×C42−1+411yescharacteristic
453SD16—232no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8C2×C4⋊C414yes—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—216no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—416no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4C2×C22⋊C414yes—
497C22⋊C4—48no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C42—216no—
501C42—416no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42C2×C22⋊C414yes—
507C42C4×D414yes—
508C42#12114yes—
509C42#12314yes—
510C23—24no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—81no—
528C23—81no—
529C23—81no—
530C23—81no—
531C23—82no—
532C23—82no—
533C23C2×C4×D411yescharacteristic
534C23C2×#12011yescharacteristic
535C23C2×#12111yescharacteristic
536C23D4211yescharacteristic
537C23#54611yescharacteristic
538C23#54811yescharacteristic
539C23D4×Q811yescharacteristic
540C23C2×AGL(1,ℤ/8)12yes—
541C23#57512yes—
542C23#57712yes—
543C23#57812yes—
544Q8—24no—
545Q8—24no—
546Q8—24no—
547Q8—24no—
548Q8—24no—
549Q8—24no—
550Q8—24no—
551Q8#38018yescomplemented
552D4—24no—
553D4—24no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—816no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4#38614yes—
615C2×C4#38614yes—
616C2×C4C22×C22⋊C411yescharacteristic
617C2×C4C22×C4⋊C411yescharacteristic
618C2×C4C2×#11711yescharacteristic
619C2×C4C2×C4×D411yescharacteristic
620C2×C4#52011yescharacteristic
621C2×C4#52011yescharacteristic
622C2×C4C2×#12111yescharacteristic
623C2×C4C2×#12111yescharacteristic
624C2×C4C2×#12111yescharacteristic
625C2×C4C2×C22⋊Q811yescharacteristic
626C2×C4C2×C22⋊Q811yescharacteristic
627C2×C4C2×#12311yescharacteristic
628C2×C4#53511yescharacteristic
629C2×C4#53611yescharacteristic
630C2×C4#53711yescharacteristic
631C2×C4#53711yescharacteristic
632C2×C4D4211yescharacteristic
633C2×C4#54611yescharacteristic
634C2×C4#54711yescharacteristic
635C2×C4#54711yescharacteristic
636C2×C4#54811yescharacteristic
637C2×C4D4×Q811yescharacteristic
638C2×C4#55211yescharacteristic
639C2×C4#55211yescharacteristic
640C2×C4#55311yescharacteristic
641C2×C4#55311yescharacteristic
642C2×C4#55411yescharacteristic
643C2×C4#55411yescharacteristic
644C8—24no—
645C8—42no—
646C8—82no—
647C22—44no—
648C22—48no—
649C22—84no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—84no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—84no—
658C22C2×#38611yescharacteristic
659C22D4×C4⋊C411yescharacteristic
660C22#235211yescharacteristic
661C22#240811yescharacteristic
662C22#240911yescharacteristic
663C22#241611yescharacteristic
664C22#241711yescharacteristic
665C22C4×AGL(1,ℤ/8)12yes—
666C22#294712yes—
667C22#300612yes—
668C22#301312yes—
669C22#303811yescharacteristic
670C22#304011yescharacteristic
671C22#304311yescharacteristic
672C22#304411yescharacteristic
673C22#326512yes—
674C22#326912yes—
675C22#330812yes—
676C22#332112yes—
677C22#336812yes—
678C22#337712yes—
679C22#338612yes—
680C22#338812yes—
681C4—28no—
682C4—24no—
683C4—24no—
684C4—44no—
685C4—44no—
686C4—82no—
687C4—82no—
688C4—84no—
689C4—162no—
690C4C2×#38011yescharacteristic
691C4#231111yescharacteristic
692C4D4×C4⋊C411yescharacteristic
693C4#235211yescharacteristic
694C4#241611yescharacteristic
695C4#241711yescharacteristic
696C4#248611yescharacteristic
697C4#248711yescharacteristic
698C2—84no—
699C2#1387211yescharacteristic
700C2#2092811yescharacteristic
701C2#2093011yescharacteristic
702C2#2267712yes—
703C2#2278112yes—
704C2#2425612yes—
705C2#2434612yes—
706C2#2474211yescharacteristic
707C2#2474311yescharacteristic
708C2#2482711yescharacteristic
709C2#2482811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2351C2semidirect
4. C2×#1803C2semidirect
7. #11219C2semidirect
9. #10843C2semidirect
10. #10835C2semidirect
11. #10789C2semidirect
18. C2×#440C4semidirect
196. C2×SD16C4⋊C4semidirect
225. C4×Q8C22⋊C4semidirect
551. Q8#380semidirect

Maximal subgroups

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

subgroupindex
C22×#4402
C2×#23512
C2×#18802
C2×#18032
C2×#18042
#139062
#112192
#110662
#108432
#108352
#107892