Finite Group Databasebeta

← #409284

Subgroups — #409284

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14682C211yesmaximal, characteristic
2C2×C4⋊#95C211yesmaximal, characteristic
3#11372C212yesmaximal, complemented
4#11316C212yesmaximal, complemented
5#11315C212yesmaximal, complemented
6#11268C212yesmaximal, complemented
7#10730C211yesmaximal, characteristic
8#10730C211yesmaximal, characteristic
9#10730C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10712C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C22×#95—21no—
22C22×#95—21no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C4⋊#95—21no—
34C4⋊#95—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39#1434—21no—
40#1434—21no—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C24×C4—21no—
44C24×C4—21no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—42no—
57C22×C4⋊C4—42no—
58C22×C4⋊C4—42no—
59C22×C4⋊C4Q814yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—24no—
71C22×C42—42no—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82#382—24no—
83#382—42no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—44no—
103C2×#95—44no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—42no—
114C25C2×D411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×Q812yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C2×Q814yes—
216C2×C4⋊C4#4114yes—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—48no—
235C2×C42—48no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42#4114yes—
239C22⋊C8—44no—
240C22⋊C8—44no—
241#95—42no—
242#95—42no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—44no—
254#95—42no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24C22⋊Q812yes—
264C24#12312yes—
265C24#12412yes—
266C24#13512yes—
267C24#13512yes—
268C24AGL(1,ℤ/8)12yes—
269C24#13712yes—
270C24C22×D411yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—48no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4C22⋊Q812yes—
382C22×C4#12312yes—
383C22×C4#12312yes—
384C22×C4#12312yes—
385C22×C4#12412yes—
386C22×C4#12514yes—
387C22×C4#12514yes—
388C22×C4#12614yes—
389C22×C4#12614yes—
390C22×C4AGL(1,ℤ/8)12yes—
391C22×C4#13712yes—
392C22×C4C22×Q811yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
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—24no—
419C4⋊C4—24no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C42—24no—
437C42—24no—
438C42—44no—
439C42—44no—
440C42—42no—
441C42—42no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23#40014yes—
471C23#47814yes—
472C23#47914yes—
473C23#48214yes—
474C23#48514yes—
475C23#48814yes—
476C23#48914yes—
477C23C2×C22⋊Q811yescharacteristic
478C23C2×#12311yescharacteristic
479C23C2×#12311yescharacteristic
480C23#52511yescharacteristic
481C23C2×#12411yescharacteristic
482C23#53411yescharacteristic
483C23#53511yescharacteristic
484C23C2×#13511yescharacteristic
485C23C2×#13511yescharacteristic
486C23C2×AGL(1,ℤ/8)11yescharacteristic
487C23C2×AGL(1,ℤ/8)11yescharacteristic
488C23C2×#13711yescharacteristic
489C23C2×#13711yescharacteristic
490C23#57511yescharacteristic
491C23#57511yescharacteristic
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
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—416no—
515C2×C4—416no—
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—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—816no—
539C2×C4—816no—
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—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4#40014yes—
577C2×C4C2×#12511yescharacteristic
578C2×C4C2×#12511yescharacteristic
579C2×C4C2×#12611yescharacteristic
580C2×C4C2×#12611yescharacteristic
581C2×C4#52912yes—
582C2×C4#53312yes—
583C2×C4#53812yes—
584C2×C4#54312yes—
585C2×C4#55112yes—
586C2×C4#55112yes—
587C2×C4#55112yes—
588C2×C4#55112yes—
589C2×C4#55312yes—
590C2×C4#55312yes—
591C2×C4#55312yes—
592C2×C4#55312yes—
593C8—82no—
594C8—82no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22#210118yes—
610C22C2×#40011yescharacteristic
611C22#248211yescharacteristic
612C22#248311yescharacteristic
613C22#250411yescharacteristic
614C22#259111yescharacteristic
615C22#262611yescharacteristic
616C22#285511yescharacteristic
617C22C2×#47811yescharacteristic
618C22C2×#47911yescharacteristic
619C22#307711yescharacteristic
620C22#307811yescharacteristic
621C22C2×#48211yescharacteristic
622C22C2×#48511yescharacteristic
623C22#309111yescharacteristic
624C22#309111yescharacteristic
625C22#309211yescharacteristic
626C22#309211yescharacteristic
627C22#309411yescharacteristic
628C22#309511yescharacteristic
629C22#310511yescharacteristic
630C22#310811yescharacteristic
631C22#311411yescharacteristic
632C22#311711yescharacteristic
633C22C2×#48811yescharacteristic
634C22C2×#48911yescharacteristic
635C22#314111yescharacteristic
636C22#314211yescharacteristic
637C22#318811yescharacteristic
638C22#319011yescharacteristic
639C22#319411yescharacteristic
640C22#319611yescharacteristic
641C22#321111yescharacteristic
642C22#321111yescharacteristic
643C22#321711yescharacteristic
644C22#321811yescharacteristic
645C4—24no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—84no—
651C4—84no—
652C4—161no—
653C4—161no—
654C4—161no—
655C4—161no—
656C4#251012yes—
657C4#251012yes—
658C4#256912yes—
659C4#256912yes—
660C2—28no—
661C2#1501511yescharacteristic
662C2C2×#210111yescharacteristic
663C2#2374711yescharacteristic
664C2#2758811yescharacteristic
665C2#2758911yescharacteristic
666C2#2760211yescharacteristic
667C2#2760411yescharacteristic
668C2#2797711yescharacteristic
669C2#2797811yescharacteristic
670C2#2933011yescharacteristic
671C2#2933611yescharacteristic
672C2#2995611yescharacteristic
673C2#2995711yescharacteristic
674C2#3315011yescharacteristic
675C2#3315111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11372C2semidirect
4. #11316C2semidirect
5. #11315C2semidirect
6. #11268C2semidirect

Maximal subgroups

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

subgroupindex
#146822
C2×C4⋊#952
#113722
#113162
#113152
#112682
#107302
#107122