Finite Group Databasebeta

← #364673

Subgroups — #364673

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic, complemented
4C2×#1736C211yesmaximal, characteristic, complemented
5#13871C211yesmaximal, characteristic
6#11228C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10881C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic, complemented
10#10722C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#123—21no—
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#435—24no—
19C2×#435—24no—
20C2×#435—24no—
21C22×C4⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×#97C2211yescharacteristic
25#2486—24no—
26#2486C418yescomplemented
27C2×#398—21no—
28C2×#397—21no—
29C2×#389C2212yes—
30C2×#381—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35#1824—24no—
36#1736—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1440—21no—
46#1438—21no—
47#1437—21no—
48#1436—21no—
49C4⋊#95C2212yes—
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—42no—
58C22×M16D414yes—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—24no—
63C23×C8—42no—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×#123—28no—
69C2×#123—48no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4C2×C414yes—
75C2×C4×D4D414yes—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—24no—
79C2×#117—42no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4C2×C414yes—
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4D411yescharacteristic
99C22×C22⋊C4D411yescharacteristic
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102#435—232no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8D414yes—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115#398—28no—
116#397—28no—
117#389—24no—
118#389C2×C418yes—
119#381—28no—
120#380—24no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—24no—
125#380—42no—
126#380C2×C418yes—
127#380C2×C414yes—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#340—42no—
142#340—42no—
143C25C2×D411yescharacteristic
144C25C2×D411yescharacteristic
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—24no—
148C22×D4—42no—
149C22×D4C22⋊C414yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4#4112yes—
182C23×C4#4112yes—
183C23×C4C2411yescharacteristic
184C2×M16—28no—
185C2×M16—28no—
186C2×M16—44no—
187C2×M16—48no—
188C2×M16—48no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—28no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—48no—
206C22×C8—48no—
207C22×C8—44no—
208C22×C8—44no—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8#4114yes—
214#123—432no—
215C4×D4—28no—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219#117—28no—
220#117—48no—
221#117—44no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C22⋊C414yes—
256C2×C4⋊C4C22×C414yes—
257C2×C4⋊C4C22×C414yes—
258C2×C4⋊C4#4114yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4C22⋊C414yes—
276C2×C22⋊C4C22⋊C414yes—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—44no—
282C2×C42—42no—
283C2×C42C22×C412yes—
284C2×C42C2×D412yes—
285C4⋊C8—28no—
286C22⋊C8—28no—
287C22⋊C8—28no—
288#97—28no—
289#95—28no—
290#95—24no—
291#95—28no—
292#95—44no—
293#95—44no—
294#95—42no—
295#95—82no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24C2×C22⋊C412yes—
310C24C2×C22⋊C412yes—
311C24#12112yes—
312C24#12112yes—
313C24AGL(1,ℤ/8)14yes—
314C24C22×D411yescharacteristic
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—44no—
318C2×D4—48no—
319C2×D4—48no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—48no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C4×D414yes—
404C22×C4C4×D414yes—
405C22×C4#12014yes—
406C22×C4#12112yes—
407C22×C4#12112yes—
408C22×C4#12314yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+412yes—
419M16—416no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—416no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8#12314yescomplemented
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—416no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C22⋊C4—28no—
461C22⋊C4—28no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—416no—
465C22⋊C4—88no—
466C22⋊C4—84no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—48no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23#38614yes—
513C23#41814yes—
514C23C22×C22⋊C411yescharacteristic
515C23C2×#12111yescharacteristic
516C23C2×#12111yescharacteristic
517C23#54612yes—
518C23#54712yes—
519C23C2×AGL(1,ℤ/8)12yes—
520C23#57512yes—
521C23#57712yes—
522C23#57812yes—
523D4—416no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—816no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4#38614yes—
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C4×#4112yes—
588C2×C4#52012yes—
589C2×C4C2×#12011yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4C2×#12311yescharacteristic
592C2×C4#52511yescharacteristic
593C2×C4#52912yes—
594C2×C4#53411yescharacteristic
595C2×C4#53611yescharacteristic
596C2×C4#53912yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4#54712yes—
600C2×C4#54712yes—
601C2×C4#55112yes—
602C2×C4#55212yes—
603C8—24no—
604C8—42no—
605C8—42no—
606C8—42no—
607C22—24no—
608C22—28no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22C2×#38611yescharacteristic
626C22#233912yes—
627C22#242512yes—
628C22#248611yescharacteristic
629C22#253411yescharacteristic
630C22C2×#41811yescharacteristic
631C22#289611yescharacteristic
632C22#289812yes—
633C22#305512yes—
634C22#305512yes—
635C22#305711yescharacteristic
636C22#305911yescharacteristic
637C22#306312yes—
638C22#330612yes—
639C22#330812yes—
640C22#331512yes—
641C22#331712yes—
642C22#334312yes—
643C22#334512yes—
644C22#334812yes—
645C22#334912yes—
646C4—24no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—84no—
651C4—84no—
652C4—84no—
653C4—162no—
654C4C4×#12311yescharacteristic
655C4#232211yescharacteristic
656C4#235912yes—
657C4#240511yescharacteristic
658C4#240611yescharacteristic
659C4#241912yes—
660C2—44no—
661C2—44no—
662C2#1396811yescharacteristic
663C2#2097311yescharacteristic
664C2#2097411yescharacteristic
665C2#2249512yes—
666C2#2249612yes—
667C2#2524212yes—
668C2#2524612yes—
669C2#2766912yes—
670C2#2810112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1824C2semidirect
4. C2×#1736C2semidirect
6. #11228C2semidirect
7. #11199C2semidirect
8. #10881C2semidirect
9. #10731C2semidirect
10. #10722C2semidirect
11. #10721C2semidirect
26. #2486C4semidirect
440. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C2×#24862
C2×#18242
C2×#17362
#138712
#112282
#111992
#108812
#107312
#107222
#107212