Finite Group Databasebeta

← #368065

Subgroups — #368065

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4C2×#1854C212yesmaximal, complemented
5C2×#1853C211yesmaximal, characteristic, complemented
6#13872C211yesmaximal, characteristic
7#11323C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10722C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#428—28no—
18C2×#428—24no—
19C22×#107C2212yes—
20C22×#106C2212yes—
21C22×#106C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2212yes—
24#2354—24no—
25#2354—24no—
26C2×#390—21no—
27C2×#388—21no—
28C2×#388—21no—
29C2×#386—21no—
30C2×#384C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1870C418yescomplemented
38#1854—28no—
39#1853—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46#1441—21no—
47#1439—21no—
48C4⋊#95C2212yes—
49C22⋊#95—21no—
50C23×D4D411yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—24no—
57C22×M16—24no—
58C22×M16C2×C414yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C414yes—
66C2×C4×D4Q814yes—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—24no—
72C2×#117—42no—
73C2×#117—42no—
74C2×#117D414yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4D411yescharacteristic
96C22×C22⋊C4D411yescharacteristic
97C22×C42—21no—
98C22×C42—21no—
99C22×C42—21no—
100C22×C42C2311yescharacteristic
101#428—232no—
102C2×#107—24no—
103C2×#107—24no—
104C2×#107D418yescomplemented
105C2×#106—24no—
106C2×#106—24no—
107C2×#106—24no—
108C2×#106—24no—
109C2×#106—24no—
110C2×#106—42no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—24no—
115C2×C22⋊C8C2×C414yes—
116C2×C22⋊C8C2×C418yescomplemented
117#390—28no—
118#388—28no—
119#388—28no—
120#386—44no—
121#384—22no—
122#384—22no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380—24no—
129#380—42no—
130#380D414yes—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95—22no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—42no—
146#336C2×C418yescomplemented
147C25C2×D411yescharacteristic
148C25C2×D411yescharacteristic
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—41no—
154C22×D4—41no—
155C22×D4C4⋊C414yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—82no—
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×Q812yes—
188C23×C4C2×Q812yes—
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C2×M16—216no—
193C2×M16—28no—
194C2×M16—28no—
195C2×M16—28no—
196C2×M16—216no—
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×C8C22×C414yes—
208C22×C8C22×C414yes—
209C22×C8C2×D414yes—
210C22×C8#4114yes—
211C4×D4—28no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—28no—
215#117—28no—
216#117—44no—
217#117—48no—
218#117—48no—
219#117—44no—
220#117—44no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
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—24no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
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—24no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4C4⋊C414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D412yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—82no—
278C2×C22⋊C4—82no—
279C2×C22⋊C4—82no—
280C2×C22⋊C4C4⋊C414yes—
281C2×C22⋊C4C4⋊C414yes—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22×C412yes—
294C2×C42C2×Q812yes—
295#107—216no—
296#106—216no—
297#106—48no—
298C22⋊C8—28no—
299C22⋊C8—216no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—82no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24C2×C4⋊C412yes—
326C24C2×C4⋊C412yes—
327C24#12012yes—
328C24#12312yes—
329C24C22×D411yescharacteristic
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—48no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4C2×C4⋊C412yes—
423C22×C4C2×C4⋊C412yes—
424C22×C4C2×C4⋊C412yes—
425C22×C4C2×C4⋊C412yes—
426C22×C4C4×D414yes—
427C22×C4C4×D414yes—
428C22×C4#12012yes—
429C22×C4C22⋊Q814yes—
430C22×C4C22⋊Q814yes—
431C22×C4#12312yes—
432C22×C4AGL(1,ℤ/8)12yes—
433C22×C4#13712yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×Q811yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C42−1+412yes—
444M16—232no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8C4×D418yes—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—84no—
482C4⋊C4#12014yes—
483C22⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—88no—
490C22⋊C4—84no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—48no—
496C42—48no—
497C42—48no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23#38514yes—
544C23C22×C4⋊C411yescharacteristic
545C23C2×#12011yescharacteristic
546C23C2×#12311yescharacteristic
547C23D4212yes—
548C23#54712yes—
549C23C2×AGL(1,ℤ/8)11yescharacteristic
550C23C2×#13711yescharacteristic
551C23#57511yescharacteristic
552C23#57511yescharacteristic
553C23#57712yes—
554C23#57712yes—
555D4—416no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
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—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—816no—
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—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4#38514yes—
624C2×C4#42814yescomplemented
625C2×C4C2×C4×D411yescharacteristic
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C2×C4×D412yes—
628C2×C4#52012yes—
629C2×C4C2×#12011yescharacteristic
630C2×C4C2×C22⋊Q811yescharacteristic
631C2×C4C2×C22⋊Q811yescharacteristic
632C2×C4#52511yescharacteristic
633C2×C4#52512yes—
634C2×C4#53411yescharacteristic
635C2×C4#53611yescharacteristic
636C2×C4#53912yes—
637C2×C4D4212yes—
638C2×C4#54712yes—
639C2×C4#54812yes—
640C2×C4D4×Q812yes—
641C2×C4#55312yes—
642C2×C4#55412yes—
643C8—28no—
644C8—44no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—81no—
662C22—82no—
663C22—82no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22C2×#38511yescharacteristic
668C22D4×C4⋊C412yes—
669C22#240311yescharacteristic
670C22#240611yescharacteristic
671C22#244312yes—
672C22C2×#42811yescharacteristic
673C22#291111yescharacteristic
674C22#291212yes—
675C22#300611yescharacteristic
676C22#300711yescharacteristic
677C22#300911yescharacteristic
678C22#301011yescharacteristic
679C22#309411yescharacteristic
680C22#309511yescharacteristic
681C22#309912yes—
682C22#326712yes—
683C22#327112yes—
684C22#327212yes—
685C22#327812yes—
686C22#334812yes—
687C22#335012yes—
688C22#335412yes—
689C22#335612yes—
690C4—24no—
691C4—24no—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—82no—
696C4—84no—
697C4—82no—
698C4—82no—
699C4—162no—
700C4C4×#12011yescharacteristic
701C4#232211yescharacteristic
702C4#236112yes—
703C4#240911yescharacteristic
704C4#244912yes—
705C4#257911yescharacteristic
706C2—44no—
707C2—82no—
708C2#1401511yescharacteristic
709C2#2084611yescharacteristic
710C2#2084811yescharacteristic
711C2#2266212yes—
712C2#2266312yes—
713C2#2386411yescharacteristic
714C2#2387011yescharacteristic
715C2#2410411yescharacteristic
716C2#2410811yescharacteristic
717C2#2615312yes—
718C2#2615512yes—
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
1. C22×#428C2semidirect
4. C2×#1854C2semidirect
5. C2×#1853C2semidirect
7. #11323C2semidirect
8. #11095C2semidirect
9. #10907C2semidirect
10. #10731C2semidirect
11. #10722C2semidirect
37. #1870C4semidirect
104. C2×#107D4semidirect
116. C2×C22⋊C8C2×C4semidirect
146. #336C2×C4semidirect
624. C2×C4#428semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#23542
C2×#18702
C2×#18542
C2×#18532
#138722
#113232
#110952
#109072
#107312
#107222