Finite Group Databasebeta

← #416717

Subgroups — #416717

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2570C211yesmaximal, characteristic, complemented
2C2×#1982C211yesmaximal, characteristic
3C2×#1961C211yesmaximal, characteristic
4#14154C211yesmaximal, characteristic
5#11066C212yesmaximal, complemented
6#11066C212yesmaximal, complemented
7#11051C212yesmaximal, complemented
8#11049C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#97C2211yescharacteristic
19#2570—24no—
20#2570—24no—
21C2×#400—21no—
22C2×#398—21no—
23C2×#390—21no—
24C2×#390—21no—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#387—21no—
30C2×#387—21no—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1982—24no—
38#1961—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1438—21no—
46#1438—21no—
47C23×D4—21no—
48C23×D4—21no—
49C23×D4C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127Q814yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97—22no—
96#400—28no—
97#398—28no—
98#390—24no—
99#390—24no—
100#390—42no—
101#390—42no—
102#390—44no—
103#390—44no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#387—28no—
109#387—28no—
110#384—22no—
111#384—22no—
112#382—22no—
113#382—22no—
114#382—22no—
115#382D414yes—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4D414yes—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132#340—24no—
133#340—24no—
134#340—42no—
135#340—42no—
136#336—42no—
137#336—42no—
138C25—22no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—82no—
163C22×D4—82no—
164C22×D4C2×Q814yes—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8#4114yes—
191#127—28no—
192#127—28no—
193#127—44no—
194#127—44no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
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—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—82no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C4⋊C8—44no—
265#102—48no—
266#102—48no—
267#102—48no—
268#102—48no—
269#97—44no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—22no—
278#95—22no—
279#95—42no—
280C24—24no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24C22×D411yescharacteristic
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—88no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—88no—
334C2×D4—84no—
335C2×D4—84no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
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—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4#12012yes—
388C22×C4#12012yes—
389C22×C4C22⋊Q812yes—
390C22×C4C22⋊Q812yes—
391C22×C4#12412yes—
392C22×C4#12412yes—
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×Q811yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)12yes—
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—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—42no—
450C42—42no—
451C42—84no—
452C42—84no—
453C42C22⋊Q814yes—
454C42C22⋊Q814yes—
455C42#12414yes—
456C42#12414yes—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
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—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23C2×#12011yescharacteristic
493C23C2×C22⋊Q811yescharacteristic
494C23C2×#12411yescharacteristic
495C23#54612yes—
496C23#55012yes—
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23#57511yescharacteristic
500C23#57511yescharacteristic
501C23#57612yes—
502C23#57712yes—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—816no—
508D4—816no—
509D4—82no—
510D4—82no—
511D4—82no—
512D4—82no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4#39314yes—
571C2×C4#39314yes—
572C2×C4C2×C22⋊Q811yescharacteristic
573C2×C4C2×C22⋊Q811yescharacteristic
574C2×C4C2×#12411yescharacteristic
575C2×C4C2×#12411yescharacteristic
576C2×C4#52912yes—
577C2×C4#52912yes—
578C2×C4#53211yescharacteristic
579C2×C4#53211yescharacteristic
580C2×C4#53511yescharacteristic
581C2×C4#53511yescharacteristic
582C2×C4#54012yes—
583C2×C4#54012yes—
584C2×C4#54612yes—
585C2×C4#55012yes—
586C2×C4#55312yes—
587C2×C4#55312yes—
588C2×C4#55412yes—
589C2×C4#55412yes—
590C8—44no—
591C8—82no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—162no—
600C22—162no—
601C22—162no—
602C22—162no—
603C22—162no—
604C22—162no—
605C22—162no—
606C22—162no—
607C22C2×#39311yescharacteristic
608C22#243111yescharacteristic
609C22#243111yescharacteristic
610C22#245412yes—
611C22#249012yes—
612C22#300611yescharacteristic
613C22#300611yescharacteristic
614C22#300811yescharacteristic
615C22#301011yescharacteristic
616C22#307711yescharacteristic
617C22#307711yescharacteristic
618C22#308112yes—
619C22#313612yes—
620C22#313811yescharacteristic
621C22#314011yescharacteristic
622C22#330912yes—
623C22#331012yes—
624C22#331712yes—
625C22#331812yes—
626C22#339712yes—
627C22#339912yes—
628C22#340112yes—
629C22#340312yes—
630C4—24no—
631C4—24no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—84no—
638C4—81no—
639C4—82no—
640C4—81no—
641C4#245412yes—
642C4#249012yes—
643C4#256611yescharacteristic
644C4#256611yescharacteristic
645C4#264111yescharacteristic
646C4#286111yescharacteristic
647C2—164no—
648C2#1522911yescharacteristic
649C2#2345411yescharacteristic
650C2#2345611yescharacteristic
651C2#2558111yescharacteristic
652C2#2558311yescharacteristic
653C2#2564211yescharacteristic
654C2#2564411yescharacteristic
655C2#2653812yes—
656C2#2654112yes—
657C2#2776012yes—
658C2#2776612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2570C2semidirect
5. #11066C2semidirect
6. #11066C2semidirect
7. #11051C2semidirect
8. #11049C2semidirect
9. #10850C2semidirect
10. #10835C2semidirect
11. #10703C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25702
C2×#19822
C2×#19612
#141542
#110662
#110512
#110492
#108502
#108352
#107032