Finite Group Databasebeta

← #369377

Subgroups — #369377

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#486C211yesmaximal, characteristic
2C2×C4×#127C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic
5#14714C211yesmaximal, characteristic, complemented
6#11189C214yesmaximal, complemented
7#10841C211yesmaximal, characteristic, complemented
8#10788C214yesmaximal, complemented
9C4×#336C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12C22×C4×D4—21no—
13C22×C4×D4—21no—
14C22×C4×D4C2212yes—
15C2×#486—24no—
16C2×#486—24no—
17C2×#486C418yescomplemented
18C22×#102C2214yes—
19C22×#102C2214yes—
20C22×C4×C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C4×#127—24no—
23C4×#127—24no—
24C2×#391—21no—
25C2×#391—21no—
26C2×#390—21no—
27C2×#390—21no—
28C2×#390C2212yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1990—24no—
33#1957—24no—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×Q8—21no—
43C23×D4—21no—
44C23×D4—21no—
45C23×D4C2312yes—
46C24×C4—22no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yescomplemented
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127C2×C414yes—
59C2×#127D414yes—
60C2×#127D414yes—
61C2×C4×D4—28no—
62C2×C4×D4—28no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—28no—
66C2×C4×D4—44no—
67C2×C4×D4—44no—
68C2×C4×D4—44no—
69C2×C4×D4—44no—
70C2×C4×D4—44no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#486—232no—
90C2×#102—28no—
91C2×#102—28no—
92C2×#102—28no—
93C2×#102—28no—
94C2×#102—44no—
95C2×#102C2×C4116yescomplemented
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8C2×C414yescomplemented
102#391—44no—
103#391—44no—
104#390—28no—
105#390—28no—
106#390—24no—
107#390—24no—
108#384—22no—
109#384—22no—
110#384—22no—
111#384D414yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C43—22no—
117C43—22no—
118#340—28no—
119#340—44no—
120#336—24no—
121#336—24no—
122C25—22no—
123C22×Q8—24no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—24no—
135C22×D4—24no—
136C22×D4—22no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4C22×C414yes—
152C22×D4C2×D414yes—
153C22×D4C2×D414yes—
154C23×C4—28no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C22×C414yes—
184C22×C8#4114yes—
185C4⋊Q8—28no—
186C4⋊Q8—28no—
187C4⋊Q8—44no—
188C4⋊Q8—44no—
189#127—28no—
190#127—28no—
191#127—28no—
192#127C22⋊C4116yes—
193C4×D4—416no—
194C4×D4—416no—
195C4×D4—432no—
196C4×D4—432no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42D814yes—
253C2×C42SD1614yes—
254C2×C42C22×C412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42#4114yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261#102—232no—
262#102—416no—
263C4×C8—28no—
264C4×C8—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24C22×D411yescharacteristic
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—48no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×D4—24no—
293C2×D4—28no—
294C2×D4—28no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—28no—
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—216no—
302C2×D4—216no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4C2×C22⋊C418yes—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—416no—
347C22×C4—416no—
348C22×C4—416no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4C4×D412yes—
373C22×C4C4×D412yes—
374C22×C4#12012yes—
375C22×C4#12012yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12412yes—
379C22×C4#12412yes—
380C22×C4C2×D812yes—
381C22×C4C2×D812yes—
382C22×C4C2×D812yes—
383C22×C4C2×SD1612yes—
384C22×C4C2×SD1612yes—
385C22×C4C2×SD1612yes—
386C22×C4#13512yes—
387C22×C4#13512yes—
388C22×C4AGL(1,ℤ/8)12yes—
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4AGL(1,ℤ/8)12yes—
391C22×C4#13712yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4111yescharacteristic
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+412yes—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—28no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C42—24no—
433C42—24no—
434C42—28no—
435C42—28no—
436C42—42no—
437C42—42no—
438C42—44no—
439C42—44no—
440C42C2×C22⋊C414yes—
441C42C4×D414yes—
442C42#12114yes—
443C42#12314yes—
444C23—28no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23C2×C4×D411yescharacteristic
480C23C2×#12011yescharacteristic
481C23C2×#12111yescharacteristic
482C23C2×#12411yescharacteristic
483C23#53211yescharacteristic
484C23#53511yescharacteristic
485C23#53611yescharacteristic
486C23C22×D811yescharacteristic
487C23C22×SD1611yescharacteristic
488C23C2×#13511yescharacteristic
489C23C2×#13511yescharacteristic
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23C2×AGL(1,ℤ/8)11yescharacteristic
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×#13711yescharacteristic
494Q8—816no—
495D4—216no—
496D4—48no—
497D4—432no—
498D4—432no—
499D4—48no—
500D4—48no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—832no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#38614yes—
551C2×C4#38614yes—
552C2×C4C4×D814yes—
553C2×C4C4×SD1614yes—
554C2×C4#44014yes—
555C2×C4#44214yes—
556C2×C4#44714yes—
557C2×C4#44914yes—
558C2×C4#45014yes—
559C2×C4#45214yes—
560C2×C4#45914yes—
561C2×C4#46114yes—
562C2×C4#46314yes—
563C2×C4#46314yes—
564C2×C4#48614yescomplemented
565C2×C4C22×C22⋊C411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C4×#4112yes—
568C2×C4#52012yes—
569C2×C4C2×#12111yescharacteristic
570C2×C4C2×#12311yescharacteristic
571C2×C4#52912yes—
572C2×C4#54212yes—
573C2×C4#54612yes—
574C2×C4#54612yes—
575C2×C4#54612yes—
576C2×C4#54712yes—
577C2×C4#54712yes—
578C2×C4#54812yes—
579C2×C4#55612yes—
580C2×C4#55912yes—
581C8—44no—
582C8—44no—
583C22—48no—
584C22—48no—
585C22—48no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22C2×#38611yescharacteristic
599C22C4×#12411yescharacteristic
600C22#232611yescharacteristic
601C22#243211yescharacteristic
602C22#243511yescharacteristic
603C22#254411yescharacteristic
604C22#254711yescharacteristic
605C22C2×C4×D811yescharacteristic
606C22C2×C4×SD1611yescharacteristic
607C22C2×#44011yescharacteristic
608C22C2×#44211yescharacteristic
609C22C2×#44711yescharacteristic
610C22C2×#45011yescharacteristic
611C22C2×#44911yescharacteristic
612C22C2×#45211yescharacteristic
613C22C2×#45911yescharacteristic
614C22C2×#46111yescharacteristic
615C22C2×#46311yescharacteristic
616C22C2×#46311yescharacteristic
617C22C2×#48611yescharacteristic
618C22#313111yescharacteristic
619C22#313811yescharacteristic
620C22#314111yescharacteristic
621C22#316711yescharacteristic
622C22#316911yescharacteristic
623C22#318211yescharacteristic
624C22#318411yescharacteristic
625C22#320511yescharacteristic
626C22#320711yescharacteristic
627C22#320811yescharacteristic
628C22#321011yescharacteristic
629C22#322611yescharacteristic
630C22#322711yescharacteristic
631C22#322811yescharacteristic
632C22#322811yescharacteristic
633C4—24no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—84no—
638C4—84no—
639C4—162no—
640C4#187718yes—
641C4#187918yes—
642C4#233912yes—
643C4#236712yes—
644C4#242712yes—
645C4#244712yes—
646C2—88no—
647C2#1402711yescharacteristic
648C2C2×#187711yescharacteristic
649C2C2×#187911yescharacteristic
650C2C4×#48611yescharacteristic
651C2#2164611yescharacteristic
652C2#2221911yescharacteristic
653C2#2222011yescharacteristic
654C2#2567111yescharacteristic
655C2#2567311yescharacteristic
656C2#2585511yescharacteristic
657C2#2585711yescharacteristic
658C2#2861311yescharacteristic
659C2#2861511yescharacteristic
660C2#2871611yescharacteristic
661C2#2871811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1990C2semidirect
5. #14714C2semidirect
6. #11189C2semidirect
7. #10841C2semidirect
8. #10788C2semidirect
9. C4×#336C2semidirect
17. C2×#486C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
95. C2×#102C2×C4semidirect
101. C2×C4×C8C2×C4semidirect
564. C2×C4#486semidirect

Maximal subgroups

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

subgroupindex
C22×#4862
C2×C4×#1272
C2×#19902
C2×#19572
#147142
#111892
#108412
#107882
C4×#3362