Finite Group Databasebeta

← #394828

Subgroups — #394828

Order 512.

2012 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3C2×#2059C212yesmaximal, complemented
4C2×#1853C211yesmaximal, characteristic
5C2×#1852C211yesmaximal, characteristic
6#11328C212yesmaximal, complemented
7#11327C212yesmaximal, complemented
8#11208C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C23×C4⋊C4—21no—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2851—24no—
22#2851—24no—
23C2×#389—21no—
24C2×#389—21no—
25C2×#389—28no—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#387—21no—
31C2×#387—21no—
32C2×#385—21no—
33C2×#385—21no—
34C2×#382C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C22×#95—21no—
38#2059—28no—
39#1853—24no—
40#1852—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#336C2212yes—
48C2×#336C2212yes—
49C24×C4—21no—
50C24×C4—21no—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—24no—
59C2×#125—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—28no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4Q814yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42—24no—
90C22×C42—42no—
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C2×#107—24no—
96C2×#107—24no—
97C2×#107—42no—
98C2×#106—24no—
99C2×#106—24no—
100C2×#106—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8Q814yes—
106C2×C22⋊C8Q814yes—
107#389—28no—
108#389—28no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—24no—
114#389—24no—
115#389—24no—
116#389D418yes—
117#387—28no—
118#387—28no—
119#385—28no—
120#385—28no—
121#382—22no—
122#382—22no—
123#380—22no—
124#380—22no—
125#380D414yes—
126#380D414yes—
127C2×#95—28no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—24no—
142#336—24no—
143#336—42no—
144C25C2×D411yescharacteristic
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×Q812yes—
191C23×C4C2×Q812yes—
192C23×C4C2×Q812yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8C2×Q814yes—
208C22×C8C2×Q814yes—
209#125—48no—
210#125—48no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—216no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—216no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—432no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×Q814yes—
267C2×C4⋊C4#4114yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—48no—
286C2×C42—48no—
287C2×C42—41no—
288C2×C42—41no—
289C2×C42C2×D414yes—
290#107—48no—
291#106—48no—
292C22⋊C8—28no—
293C22⋊C8—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—24no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24#12012yes—
311C24C22⋊Q812yes—
312C24C4⋊Q812yes—
313C24#13512yes—
314C24#13512yes—
315C24AGL(1,ℤ/8)12yes—
316C24#13712yes—
317C24C22×D411yescharacteristic
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—216no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—48no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4#12012yes—
429C22×C4#12012yes—
430C22×C4#12012yes—
431C22×C4C22⋊Q814yes—
432C22×C4C22⋊Q814yes—
433C22×C4C22⋊Q812yes—
434C22×C4C4⋊Q814yes—
435C22×C4C4⋊Q814yes—
436C22×C4C4⋊Q812yes—
437C22×C4AGL(1,ℤ/8)12yes—
438C22×C4#13712yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q811yescharacteristic
443C22×C4C22×Q811yescharacteristic
444C22×C4C22×Q811yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4112yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+412yes—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C4⋊C4—216no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4C22⋊Q818yes—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C42—24no—
490C42—24no—
491C42—44no—
492C42—44no—
493C42—42no—
494C42—42no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23#39514yes—
524C23#44914yes—
525C23#45214yes—
526C23#47814yes—
527C23#47914yes—
528C23#49914yes—
529C23#50114yes—
530C23C2×#12011yescharacteristic
531C23C2×#12011yescharacteristic
532C23C2×C22⋊Q811yescharacteristic
533C23#52511yescharacteristic
534C23C2×C4⋊Q811yescharacteristic
535C23#53411yescharacteristic
536C23#53611yescharacteristic
537C23C2×#13511yescharacteristic
538C23C2×#13511yescharacteristic
539C23C2×AGL(1,ℤ/8)11yescharacteristic
540C23C2×AGL(1,ℤ/8)11yescharacteristic
541C23C2×#13711yescharacteristic
542C23C2×#13711yescharacteristic
543C23#57511yescharacteristic
544C23#57511yescharacteristic
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—432no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—816no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—82no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—82no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—82no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—82no—
629C2×C4#39514yes—
630C2×C4C2×C22⋊Q812yes—
631C2×C4C2×C22⋊Q811yescharacteristic
632C2×C4C2×C22⋊Q811yescharacteristic
633C2×C4#52512yes—
634C2×C4C2×C4⋊Q811yescharacteristic
635C2×C4C2×C4⋊Q811yescharacteristic
636C2×C4#53912yes—
637C2×C4#54412yes—
638C2×C4D4212yes—
639C2×C4D4212yes—
640C2×C4#54712yes—
641C2×C4#54712yes—
642C2×C4D4×Q812yes—
643C2×C4D4×Q812yes—
644C2×C4#55412yes—
645C2×C4#55412yes—
646C8—44no—
647C8—44no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22#205918yescomplemented
663C22C2×#39511yescharacteristic
664C22#240311yescharacteristic
665C22#240611yescharacteristic
666C22#243411yescharacteristic
667C22#258311yescharacteristic
668C22#262711yescharacteristic
669C22#284811yescharacteristic
670C22C2×#44911yescharacteristic
671C22C2×#45211yescharacteristic
672C22#300211yescharacteristic
673C22#300211yescharacteristic
674C22#300311yescharacteristic
675C22#300411yescharacteristic
676C22#300611yescharacteristic
677C22#300711yescharacteristic
678C22C2×#47811yescharacteristic
679C22C2×#47911yescharacteristic
680C22#307711yescharacteristic
681C22#307811yescharacteristic
682C22#310511yescharacteristic
683C22#310811yescharacteristic
684C22#311411yescharacteristic
685C22#311711yescharacteristic
686C22C2×#49911yescharacteristic
687C22C2×#50111yescharacteristic
688C22#316511yescharacteristic
689C22#316611yescharacteristic
690C22#318811yescharacteristic
691C22#319011yescharacteristic
692C22#319411yescharacteristic
693C22#319611yescharacteristic
694C22#323211yescharacteristic
695C22#323211yescharacteristic
696C22#323811yescharacteristic
697C22#323911yescharacteristic
698C4—24no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—48no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—84no—
708C4—82no—
709C4#244012yes—
710C4#244012yes—
711C4#244912yes—
712C4#244912yes—
713C2—28no—
714C2#1472011yescharacteristic
715C2C2×#205911yescharacteristic
716C2#2355411yescharacteristic
717C2#2392011yescharacteristic
718C2#2392211yescharacteristic
719C2#2406611yescharacteristic
720C2#2406711yescharacteristic
721C2#2581511yescharacteristic
722C2#2581611yescharacteristic
723C2#2929811yescharacteristic
724C2#2930311yescharacteristic
725C2#2997211yescharacteristic
726C2#2997311yescharacteristic
727C2#3295911yescharacteristic
728C2#3296011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2059C2semidirect
6. #11328C2semidirect
7. #11327C2semidirect
8. #11208C2semidirect
662. C22#2059semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C22×#3892
C2×#20592
C2×#18532
C2×#18522
#113282
#113272
#112082
#107302
#107122