Finite Group Databasebeta

← #401668

Subgroups — #401668

Order 512.

1818 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1862C212yesmaximal
4C2×#1802C211yesmaximal, characteristic
5#15151C211yesmaximal, characteristic, complemented
6#11227C211yesmaximal, characteristic, complemented
7#11219C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#11202C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4—21no—
16C22×#107C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C2×#398—21no—
21C2×#398—21no—
22D4×C4⋊C4—24no—
23D4×C4⋊C4—24no—
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#385—21no—
28C2×#385—21no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#2083—24no—
34#1862—28no—
35#1802—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2211yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—42no—
65C2×C4×D4—42no—
66C2×C4×D4D414yes—
67C2×C4×D4D414yes—
68C22×C4⋊C4—28no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—24no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—42no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102Q818yes—
102#398—44no—
103#398—44no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—24no—
108#389—24no—
109#389D414yes—
110#385—28no—
111#385—28no—
112#384—22no—
113#384—22no—
114#384—22no—
115#384—22no—
116#380—28no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—24no—
131#340—42no—
132#340—42no—
133#336—22no—
134#336—24no—
135#336—42no—
136C25—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4C2×D412yes—
143C22×D4C2×D412yes—
144C22×D4C2×Q814yes—
145C22×D4#4112yes—
146C22×D4#4114yes—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—82no—
168C23×C4—82no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—22no—
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×C8C2×Q814yes—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—44no—
196C4×D4—48no—
197C4×D4—48no—
198C4×D4—44no—
199C4×D4—44no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
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—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—432no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4D814yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×Q814yes—
252C2×C4⋊C4#4112yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42#4112yes—
274#107—48no—
275C4⋊C8—44no—
276#102—216no—
277#102—48no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—42no—
283#95—44no—
284#95—44no—
285#95—42no—
286#95—44no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—42no—
297C24—42no—
298C24C22×D411yescharacteristic
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4#12114yes—
313C2×D4C22⋊Q818yes—
314C2×D4#12314yes—
315C2×D4#12518yes—
316C2×D4C4⋊Q814yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—216no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—48no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—84no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—84no—
387C22×C4—88no—
388C22×C4#12012yes—
389C22×C4#12012yes—
390C22×C4#12112yes—
391C22×C4#12112yes—
392C22×C4C2×D812yes—
393C22×C4C2×D812yes—
394C22×C4C2×D812yes—
395C22×C4AGL(1,ℤ/8)12yes—
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×Q812yes—
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4112yes—
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C42−1+412yes—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
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—
419C2×C8—42no—
420C2×C8—42no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—28no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—44no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—48no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—88no—
447C4⋊C4#12114yes—
448C22⋊C4—216no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—24no—
458C42—41no—
459C42—41no—
460C42—42no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—24no—
465C23—28no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23C2×#12011yescharacteristic
485C23C2×#12111yescharacteristic
486C23#52511yescharacteristic
487C23#54612yes—
488C23D4×Q812yes—
489C23C22×D811yescharacteristic
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23C2×AGL(1,ℤ/8)11yescharacteristic
492C23#57511yescharacteristic
493C23#57712yes—
494C23#57812yes—
495D4—24no—
496D4—24no—
497D4—28no—
498D4—28no—
499D4—24no—
500D4#39818yescomplemented
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—432no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—816no—
549C2×C4—816no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4#44714yes—
566C2×C4#45914yes—
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×C22⋊Q812yes—
570C2×C4C2×#12311yescharacteristic
571C2×C4#52511yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4C2×#12512yes—
574C2×C4C2×C4⋊Q811yescharacteristic
575C2×C4#53411yescharacteristic
576C2×C4#53611yescharacteristic
577C2×C4#53712yes—
578C2×C4#53912yes—
579C2×C4#54612yes—
580C2×C4D4×Q812yes—
581C2×C4#55212yes—
582C2×C4#55212yes—
583C2×C4#55412yes—
584C2×C4#55412yes—
585C8—44no—
586C8—82no—
587C22—28no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—48no—
592C22—44no—
593C22—48no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—44no—
598C22#240511yescharacteristic
599C22#241712yes—
600C22#243411yescharacteristic
601C22#247312yes—
602C22#284711yescharacteristic
603C22C2×#44711yescharacteristic
604C22#300611yescharacteristic
605C22#300911yescharacteristic
606C22#301111yescharacteristic
607C22C2×#45911yescharacteristic
608C22#303611yescharacteristic
609C22#304612yes—
610C22#310011yescharacteristic
611C22#310911yescharacteristic
612C22#312012yes—
613C22#329412yes—
614C22#331012yes—
615C22#331512yes—
616C22#333012yes—
617C22Q8×D812yes—
618C22#338412yes—
619C22#338612yes—
620C22#338812yes—
621C4—24no—
622C4—24no—
623C4—44no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—82no—
628C4—82no—
629C4—84no—
630C4—81no—
631C4—162no—
632C4C2×#39811yescharacteristic
633C4#241712yes—
634C4#247312yes—
635C4#248411yescharacteristic
636C4#258311yescharacteristic
637C4#261811yescharacteristic
638C2—48no—
639C2#1482111yescharacteristic
640C2#2403211yescharacteristic
641C2#2405411yescharacteristic
642C2#2481112yes—
643C2#2482512yes—
644C2#2583811yescharacteristic
645C2#2585111yescharacteristic
646C2#2729412yes—
647C2#2731512yes—
648C2#3290211yescharacteristic
649C2#3291611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×#2083C2semidirect
5. #15151C2semidirect
6. #11227C2semidirect
7. #11219C2semidirect
8. #11207C2semidirect
9. #11202C2semidirect
500. D4#398semidirect

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×#20832
C2×#18622
C2×#18022
#151512
#112272
#112192
#112072
#112022
#107882