Finite Group Databasebeta

← #409640

Subgroups — #409640

Order 512.

1320 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14712C211yesmaximal, characteristic, complemented
2C4×#390C211yesmaximal, characteristic
3#11292C212yesmaximal, complemented
4#11267C211yesmaximal, characteristic, complemented
5#11267C211yesmaximal, characteristic, complemented
6#11066C212yesmaximal
7#11049C211yesmaximal, characteristic
8#11049C211yesmaximal, characteristic
9#10806C212yesmaximal, complemented
10#10795C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×#102C2212yes—
14C22×#102C2212yes—
15C2×#390C2211yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#389—21no—
18C2×#389—21no—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4—21no—
24C2×C4×C22⋊C4—21no—
25C2×C43C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1437—21no—
37#1437—21no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—24no—
50C2×C4×D4—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C42—21no—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80#390—22no—
81#390—22no—
82#390—22no—
83#390Q814yes—
84#389—42no—
85#389—42no—
86#389—44no—
87#389—44no—
88#384—24no—
89#384—42no—
90C4×C4⋊C4—24no—
91C4×C22⋊C4—44no—
92C4×C22⋊C4—44no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C43—42no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336—22no—
109#336—22no—
110#336—24no—
111#336—42no—
112C25—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4C2×Q812yes—
121C22×D4#4112yes—
122C22×D4#4112yes—
123C23×C4—24no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C4×D4—48no—
157C4×D4—48no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4C2×Q812yes—
194C2×C4⋊C4#4112yes—
195C2×C4⋊C4#4112yes—
196C2×C4⋊C4#4114yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42—41no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42D814yes—
235C2×C42SD1614yes—
236C2×C42C2×Q812yes—
237C2×C42#4112yes—
238C2×C42#4114yes—
239#102—48no—
240#102—48no—
241#95—44no—
242#95—44no—
243#95—42no—
244#95—42no—
245#95—42no—
246#95—44no—
247#95—42no—
248#95—42no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24C22×D411yescharacteristic
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4#12514yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
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—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4C22⋊Q812yes—
354C22×C4C22⋊Q812yes—
355C22×C4#12412yes—
356C22×C4#12412yes—
357C22×C4C2×D812yes—
358C22×C4C2×D812yes—
359C22×C4C2×D812yes—
360C22×C4C2×SD1612yes—
361C22×C4C2×SD1612yes—
362C22×C4C2×SD1612yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4#13712yes—
365C22×C4C22×Q811yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)12yes—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C4⋊C4—88no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C42—24no—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—44no—
418C42—44no—
419C42—44no—
420C42—44no—
421C42—42no—
422C42—42no—
423C42—84no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23C2×C22⋊Q811yescharacteristic
460C23#52511yescharacteristic
461C23#52511yescharacteristic
462C23#52511yescharacteristic
463C23C2×#12411yescharacteristic
464C23#53411yescharacteristic
465C23#53511yescharacteristic
466C23C22×D811yescharacteristic
467C23C22×SD1611yescharacteristic
468C23C2×AGL(1,ℤ/8)11yescharacteristic
469C23C2×#13711yescharacteristic
470C23#57511yescharacteristic
471C23#57511yescharacteristic
472C23#57511yescharacteristic
473C23#57511yescharacteristic
474D4—28no—
475D4—44no—
476D4—44no—
477D4—44no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—816no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4#47414yes—
554C2×C4#47614yes—
555C2×C4#48614yes—
556C2×C4C2×#12511yescharacteristic
557C2×C4#52912yes—
558C2×C4#52911yescharacteristic
559C2×C4#52912yes—
560C2×C4#52911yescharacteristic
561C2×C4#53311yescharacteristic
562C2×C4#53812yes—
563C2×C4#54012yes—
564C2×C4#55112yes—
565C2×C4#55112yes—
566C2×C4#55112yes—
567C2×C4#55312yes—
568C2×C4#55312yes—
569C2×C4#55412yes—
570C2×C4#55412yes—
571C2×C4#55512yes—
572C8—82no—
573C8—82no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22#239811yescharacteristic
590C22#256111yescharacteristic
591C22#256311yescharacteristic
592C22#257911yescharacteristic
593C22#259811yescharacteristic
594C22#263511yescharacteristic
595C22#285111yescharacteristic
596C22C2×#47411yescharacteristic
597C22C2×#47611yescharacteristic
598C22#308011yescharacteristic
599C22#308011yescharacteristic
600C22#310011yescharacteristic
601C22#310011yescharacteristic
602C22#310011yescharacteristic
603C22#310111yescharacteristic
604C22#310111yescharacteristic
605C22#310311yescharacteristic
606C22#311411yescharacteristic
607C22#311511yescharacteristic
608C22#311511yescharacteristic
609C22#311611yescharacteristic
610C22#311611yescharacteristic
611C22#311711yescharacteristic
612C22C2×#48611yescharacteristic
613C22#313511yescharacteristic
614C22#314011yescharacteristic
615C22#314111yescharacteristic
616C22#318611yescharacteristic
617C22#318711yescharacteristic
618C22#319411yescharacteristic
619C22#319611yescharacteristic
620C22#320511yescharacteristic
621C22#320611yescharacteristic
622C22#321911yescharacteristic
623C22#321911yescharacteristic
624C4—24no—
625C4—24no—
626C4—24no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—82no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4—162no—
635C4#250512yes—
636C4#251812yes—
637C4#252212yes—
638C4#252612yes—
639C2—84no—
640C2#1504511yescharacteristic
641C2#2381511yescharacteristic
642C2#2382111yescharacteristic
643C2#2910311yescharacteristic
644C2#2911311yescharacteristic
645C2#2913911yescharacteristic
646C2#2914611yescharacteristic
647C2#2923411yescharacteristic
648C2#2923511yescharacteristic
649C2#2936411yescharacteristic
650C2#2936811yescharacteristic
651C2#3010511yescharacteristic
652C2#3010611yescharacteristic
653C2#3306611yescharacteristic
654C2#3306811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
1. #14712C2semidirect
3. #11292C2semidirect
4. #11267C2semidirect
5. #11267C2semidirect
9. #10806C2semidirect
10. #10795C2semidirect

Maximal subgroups

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

subgroupindex
#147122
C4×#3902
#112922
#112672
#110662
#110492
#108062
#107952
#107882