Finite Group Databasebeta

← #364758

Subgroups — #364758

Order 512.

1972 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic, complemented
5C2×#1736C211yesmaximal, characteristic, complemented
6#11207C212yesmaximal, complemented
7#11200C212yesmaximal, complemented
8#10884C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal, complemented
11#10722C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×#123—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20#2486—24no—
21#2486C418yescomplemented
22C2×#398—21no—
23C2×#397—21no—
24#2354—24no—
25#2354—24no—
26C2×#390—21no—
27C2×#389C2212yes—
28C2×#388—21no—
29C2×#388—21no—
30C2×#384C2211yescharacteristic
31C2×#381—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C4⋊C4—21no—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1870—24no—
39#1824—24no—
40#1736—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#336C2212yes—
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—24no—
58C22×M16—42no—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×#123—28no—
62C2×#123—44no—
63C2×#123—44no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4D414yes—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—24no—
74C2×#117—24no—
75C2×#117—42no—
76C2×#117—42no—
77C2×#117D414yes—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2×C414yes—
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4—22no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4D411yescharacteristic
102C22×C22⋊C4D411yescharacteristic
103C22×C42—21no—
104C22×C42—21no—
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107C2×C22⋊C8—22no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—42no—
113#398—28no—
114#397—28no—
115#390—28no—
116#389—24no—
117#389C2×C418yes—
118#388—28no—
119#388—28no—
120#384—22no—
121#384—22no—
122#381—28no—
123#380—22no—
124#380—24no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380C2×C418yes—
129#380C2×C414yes—
130#380D414yes—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—24no—
146#340—42no—
147#340—42no—
148#336—24no—
149C25C2×D411yescharacteristic
150C25C2×D411yescharacteristic
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4C22⋊C414yes—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4C22×C412yes—
184C23×C4C22×C412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4#4112yes—
194C23×C4#4112yes—
195C23×C4C2411yescharacteristic
196C2×M16—28no—
197C2×M16—48no—
198C2×M16—48no—
199C2×M16—44no—
200C2×M16—44no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8#4114yes—
215#123—432no—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—28no—
220#117—28no—
221#117—44no—
222#117—48no—
223#117—48no—
224#117—44no—
225#117—44no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
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—22no—
239C2×C4⋊C4—22no—
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—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C22⋊C414yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4#4114yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—82no—
289C2×C22⋊C4—82no—
290C2×C22⋊C4C22⋊C414yes—
291C2×C22⋊C4C22⋊C414yes—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42C22×C412yes—
304C2×C42C2×D412yes—
305C22⋊C8—48no—
306C22⋊C8—44no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—24no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24C2×C22⋊C412yes—
332C24C2×C22⋊C412yes—
333C24#12012yes—
334C24#12312yes—
335C24C22×D411yescharacteristic
336C2×D4—28no—
337C2×D4—28no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—48no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C412yes—
432C22×C4C2×C22⋊C412yes—
433C22×C4C2×C22⋊C412yes—
434C22×C4C4×D414yes—
435C22×C4C4×D414yes—
436C22×C4#12012yes—
437C22×C4#12114yes—
438C22×C4#12114yes—
439C22×C4#12312yes—
440C22×C4AGL(1,ℤ/8)12yes—
441C22×C4AGL(1,ℤ/8)12yes—
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+412yes—
452M16—416no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—416no—
490C4⋊C4—84no—
491C4⋊C4#12314yescomplemented
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—48no—
505C42—48no—
506C42—48no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—42no—
552C23#38614yes—
553C23C22×C22⋊C411yescharacteristic
554C23C2×#12011yescharacteristic
555C23C2×#12311yescharacteristic
556C23#54612yes—
557C23#54712yes—
558C23C2×AGL(1,ℤ/8)11yescharacteristic
559C23C2×AGL(1,ℤ/8)11yescharacteristic
560C23#57511yescharacteristic
561C23#57511yescharacteristic
562C23#57712yes—
563C23#57812yes—
564D4—416no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—816no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4#38614yes—
633C2×C4#41814yes—
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C4×#4112yes—
637C2×C4#52012yes—
638C2×C4C2×#12111yescharacteristic
639C2×C4C2×#12111yescharacteristic
640C2×C4C2×#12311yescharacteristic
641C2×C4#52511yescharacteristic
642C2×C4#52912yes—
643C2×C4#53411yescharacteristic
644C2×C4#53611yescharacteristic
645C2×C4#53912yes—
646C2×C4#54612yes—
647C2×C4#54612yes—
648C2×C4#54712yes—
649C2×C4#54712yes—
650C2×C4#55112yes—
651C2×C4#55212yes—
652C8—44no—
653C8—82no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—81no—
671C22—81no—
672C22—82no—
673C22—82no—
674C22—81no—
675C22—81no—
676C22C2×#38611yescharacteristic
677C22#233912yes—
678C22#240511yescharacteristic
679C22#240611yescharacteristic
680C22#241912yes—
681C22C2×#41811yescharacteristic
682C22#289611yescharacteristic
683C22#289812yes—
684C22#300611yescharacteristic
685C22#300611yescharacteristic
686C22#300911yescharacteristic
687C22#301111yescharacteristic
688C22#309411yescharacteristic
689C22#309411yescharacteristic
690C22#309712yes—
691C22#331512yes—
692C22#331712yes—
693C22#333012yes—
694C22#333212yes—
695C22#334812yes—
696C22#334912yes—
697C22#335312yes—
698C22#335412yes—
699C4—24no—
700C4—24no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—82no—
705C4—84no—
706C4—84no—
707C4—82no—
708C4—82no—
709C4C4×#12311yescharacteristic
710C4#232211yescharacteristic
711C4#235912yes—
712C4#242512yes—
713C4#248611yescharacteristic
714C4#253411yescharacteristic
715C2—44no—
716C2—82no—
717C2#1396811yescharacteristic
718C2#2091211yescharacteristic
719C2#2091311yescharacteristic
720C2#2249612yes—
721C2#2250012yes—
722C2#2404511yescharacteristic
723C2#2405411yescharacteristic
724C2#2410411yescharacteristic
725C2#2410711yescharacteristic
726C2#2495812yes—
727C2#2496212yes—
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
4. C2×#1824C2semidirect
5. C2×#1736C2semidirect
6. #11207C2semidirect
7. #11200C2semidirect
8. #10884C2semidirect
9. #10731C2semidirect
10. #10731C2semidirect
11. #10722C2semidirect
21. #2486C4semidirect
491. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23542
C2×#18702
C2×#18242
C2×#17362
#112072
#112002
#108842
#107312
#107222