Finite Group Databasebeta

← #369430

Subgroups — #369430

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic, complemented
4C2×#1989C211yesmaximal, characteristic, complemented
5C2×#1955C211yesmaximal, characteristic
6#11200C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10730C212yesmaximal, complemented
10#10730C212yesmaximal, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×#97C2211yescharacteristic
20#2562—24no—
21#2562C418yescomplemented
22C2×#400—21no—
23C2×#394—21no—
24D4×C4⋊C4—24no—
25D4×C4⋊C4—24no—
26C2×#389—21no—
27C2×#388—22no—
28C2×#385—22no—
29C2×#384C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1992—24no—
36#1989—24no—
37#1955—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C23×D4D411yescharacteristic
49C24×C4—22no—
50C24×C4C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125C2×C414yescomplemented
60C2×C4×D4—28no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—48no—
66C2×C4×D4C2×C414yes—
67C2×C4×D4D414yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4D412yes—
90C22×C42—21no—
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—42no—
101C2×C22⋊C8—42no—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—22no—
105#400—28no—
106#394—28no—
107#389—28no—
108#388—216no—
109#385—216no—
110#384—22no—
111#384D414yes—
112#380—24no—
113#380—24no—
114#380—24no—
115#380—24no—
116#380C2×C418yescomplemented
117#380C2×C418yescomplemented
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C414yescomplemented
121C4×C4⋊C4D414yes—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—22no—
127#340—24no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—42no—
136#340—42no—
137C25C2×D412yes—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4C22⋊C414yes—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—28no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—82no—
167C23×C4—82no—
168C23×C4C22×C414yes—
169C23×C4C2×D412yes—
170C23×C4C2×D414yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8#4114yes—
191#125—28no—
192#125—28no—
193#125—44no—
194#125—44no—
195C4×D4—28no—
196C4×D4—28no—
197C4×D4—28no—
198C4×D4—28no—
199C4×D4—432no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—48no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—81no—
244C2×C4⋊C4—81no—
245C2×C4⋊C4C22⋊C414yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C22×C414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4#4114yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—82no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4C22⋊C418yes—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—44no—
268C2×C42—81no—
269C2×C42—81no—
270C2×C42C22×C412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42#4112yes—
275C2×C42#4112yes—
276C4⋊C8—44no—
277C22⋊C8—48no—
278C22⋊C8—48no—
279#97—44no—
280#95—28no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—24no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24C2×C22⋊C414yes—
298C24AGL(1,ℤ/8)14yes—
299C24#13714yes—
300C24C22×D411yescharacteristic
301C2×D4—28no—
302C2×D4—216no—
303C2×D4—28no—
304C2×D4—216no—
305C2×D4—28no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
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—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—416no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
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—416no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4#12012yes—
393C22×C4#12012yes—
394C22×C4#12312yes—
395C22×C4#12312yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4H(5,𝔽2)12yes—
405C22×C42−1+412yes—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—416no—
437C4⋊C4—48no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—84no—
450C4⋊C4#12414yes—
451C22⋊C4—216no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—88no—
455C42—48no—
456C42—84no—
457C42C4×D414yes—
458C42C4×D414yes—
459C42#12114yes—
460C42#12114yes—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23#41718yes—
496C23C22×C22⋊C411yescharacteristic
497C23C2×#12011yescharacteristic
498C23C2×#12311yescharacteristic
499C23#54612yes—
500C23#54712yes—
501C23C2×AGL(1,ℤ/8)12yes—
502C23C2×#13712yes—
503C23#57611yescharacteristic
504C23#57711yescharacteristic
505C23#57711yescharacteristic
506C23#57811yescharacteristic
507D4—232no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—832no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4#38614yes—
571C2×C4#38614yes—
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C4×#4112yes—
575C2×C4#52012yes—
576C2×C4C2×#12111yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C2×C4C2×#12411yescharacteristic
579C2×C4#52912yes—
580C2×C4#53211yescharacteristic
581C2×C4#53511yescharacteristic
582C2×C4#53611yescharacteristic
583C2×C4#54212yes—
584C2×C4#54612yes—
585C2×C4#54612yes—
586C2×C4#54712yes—
587C2×C4#54812yes—
588C2×C4#55612yes—
589C2×C4#55912yes—
590C8—44no—
591C8—81no—
592C8—81no—
593C22—28no—
594C22—28no—
595C22—28no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22C2×#38611yescharacteristic
609C22#233912yes—
610C22#243211yescharacteristic
611C22#243511yescharacteristic
612C22#244712yes—
613C22C2×#41712yes—
614C22#289711yescharacteristic
615C22#289811yescharacteristic
616C22#300311yescharacteristic
617C22#300411yescharacteristic
618C22#301211yescharacteristic
619C22#301311yescharacteristic
620C22#309212yes—
621C22#309811yescharacteristic
622C22#309911yescharacteristic
623C22#331012yes—
624C22#331212yes—
625C22#331812yes—
626C22#332012yes—
627C22#334312yes—
628C22#334412yes—
629C22#334512yes—
630C22#334612yes—
631C4—24no—
632C4—24no—
633C4—44no—
634C4—44no—
635C4—84no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4C4×#12411yescharacteristic
641C4#232611yescharacteristic
642C4#236712yes—
643C4#242712yes—
644C4#254411yescharacteristic
645C4#254711yescharacteristic
646C2—48no—
647C2#1402711yescharacteristic
648C2#2090911yescharacteristic
649C2#2091511yescharacteristic
650C2#2249112yes—
651C2#2249312yes—
652C2#2570911yescharacteristic
653C2#2571211yescharacteristic
654C2#2589111yescharacteristic
655C2#2589411yescharacteristic
656C2#2630812yes—
657C2#2631012yes—
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×#1992C2semidirect
4. C2×#1989C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
8. #10849C2semidirect
9. #10730C2semidirect
10. #10730C2semidirect
11. #10703C2semidirect
21. #2562C4semidirect
59. C2×#125C2×C4semidirect
116. #380C2×C4semidirect
117. #380C2×C4semidirect
120. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×D4×C4⋊C42
C2×#19922
C2×#19892
C2×#19552
#112002
#111992
#108492
#107302
#107032