Finite Group Databasebeta

← #402832

Subgroups — #402832

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1918C211yesmaximal, characteristic, complemented
4#15129C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11199C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10964C211yesmaximal, characteristic
9#10806C211yesmaximal, characteristic, complemented
10#10730C212yesmaximal, complemented
11#10730C212yesmaximal, complemented
12C22×C4×D4—21no—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C22×#107C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C2×#394—21no—
19D4×C4⋊C4—24no—
20D4×C4⋊C4—24no—
21C2×#389—21no—
22C2×#389C2211yescharacteristic
23C2×#385—22no—
24C2×#384C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2099—24no—
30#1918—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1441—21no—
43#1440—22no—
44C4⋊#95C2211yescharacteristic
45C23×D4D411yescharacteristic
46C24×C4—22no—
47C24×C4C2312yes—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—28no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—44no—
57C2×C4×D4—44no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90#394—44no—
91#389—28no—
92#389—22no—
93#389—22no—
94#385—216no—
95#384—22no—
96#384D414yes—
97#380—24no—
98#380—24no—
99#380—24no—
100#380—24no—
101#380—24no—
102#380—42no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—42no—
119#340—42no—
120#340—42no—
121#336—22no—
122#336—22no—
123C25C2×D412yes—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4D814yes—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—28no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4#4114yes—
158C23×C4#4114yes—
159C23×C4C2411yescharacteristic
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
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—21no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8#4112yes—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—216no—
180C4×D4—432no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4#4112yes—
229C2×C4⋊C4#4114yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
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×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249#107—28no—
250C22⋊C8—48no—
251C22⋊C8—48no—
252#95—28no—
253#95—24no—
254#95—44no—
255#95—42no—
256#95—44no—
257#95—42no—
258#95—81no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24#12314yes—
272C24C2×D814yes—
273C24#13714yes—
274C24C22×D411yescharacteristic
275C2×D4—28no—
276C2×D4—216no—
277C2×D4—28no—
278C2×D4—216no—
279C2×D4—28no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—416no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—416no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4#12314yes—
368C22×C4C2×D812yes—
369C22×C4AGL(1,ℤ/8)12yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C42−1+411yescharacteristic
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—28no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—416no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—48no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4#12414yes—
424C22⋊C4—216no—
425C22⋊C4—416no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C42—22no—
429C42—22no—
430C42—24no—
431C42—48no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23#48018yes—
467C23C2×#12011yescharacteristic
468C23C2×#12312yes—
469C23#54611yescharacteristic
470C23#54612yes—
471C23#54711yescharacteristic
472C23C22×D811yescharacteristic
473C23C2×AGL(1,ℤ/8)11yescharacteristic
474C23C2×#13712yes—
475C23#57712yes—
476C23#57711yescharacteristic
477C23#57811yescharacteristic
478D4—232no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—832no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#44714yes—
543C2×C4#52511yescharacteristic
544C2×C4#52511yescharacteristic
545C2×C4#52511yescharacteristic
546C2×C4#52512yes—
547C2×C4C2×#12411yescharacteristic
548C2×C4#52912yes—
549C2×C4#52912yes—
550C2×C4#53511yescharacteristic
551C2×C4#53711yescharacteristic
552C2×C4#53812yes—
553C2×C4#53912yes—
554C2×C4#54612yes—
555C2×C4#54811yescharacteristic
556C2×C4#55011yescharacteristic
557C2×C4#55112yes—
558C2×C4#55212yes—
559C2×C4#55312yes—
560C2×C4#55312yes—
561C8—42no—
562C8—81no—
563C8—82no—
564C22—28no—
565C22—28no—
566C22—28no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22#240711yescharacteristic
580C22#241912yes—
581C22#243511yescharacteristic
582C22#244412yes—
583C22#284911yescharacteristic
584C22C2×#44711yescharacteristic
585C22#300411yescharacteristic
586C22#300911yescharacteristic
587C22#301311yescharacteristic
588C22C2×#48012yes—
589C22#309212yes—
590C22#309712yes—
591C22#309912yes—
592C22#329412yes—
593C22#329411yescharacteristic
594C22#331212yes—
595C22#331612yes—
596C22#332012yes—
597C22#332412yes—
598C22#333011yescharacteristic
599C22#333411yescharacteristic
600C22#334412yes—
601C22#335411yescharacteristic
602C4—24no—
603C4—44no—
604C4—44no—
605C4—42no—
606C4—84no—
607C4—82no—
608C4—82no—
609C4—162no—
610C4—161no—
611C4#239811yescharacteristic
612C4#246012yes—
613C4#246712yes—
614C4#256211yescharacteristic
615C4#259811yescharacteristic
616C4#263611yescharacteristic
617C2—48no—
618C2#1486011yescharacteristic
619C2#2411611yescharacteristic
620C2#2413411yescharacteristic
621C2#2493312yes—
622C2#2495512yes—
623C2#2588511yescharacteristic
624C2#2589411yescharacteristic
625C2#2619812yes—
626C2#2622012yes—
627C2#3297811yescharacteristic
628C2#3298811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×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
2. C2×#2099C2semidirect
3. C2×#1918C2semidirect
5. #11357C2semidirect
6. #11199C2semidirect
7. #11096C2semidirect
9. #10806C2semidirect
10. #10730C2semidirect
11. #10730C2semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#20992
C2×#19182
#151292
#113572
#111992
#110962
#109642
#108062
#107302