Finite Group Databasebeta

← #401858

Subgroups — #401858

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#2087C211yesmaximal, characteristic, complemented
3C2×#2076C211yesmaximal, characteristic, complemented
4C2×#1808C211yesmaximal, characteristic, complemented
5#15181C212yesmaximal
6#11232C211yesmaximal, characteristic, complemented
7#11200C212yesmaximal, complemented
8#11096C211yesmaximal, characteristic, complemented
9#10880C212yesmaximal, complemented
10#10721C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C8×D4—24no—
16C2×C8×D4—24no—
17C2×C8×D4—24no—
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#389C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29#2087—24no—
30#2076—24no—
31#1808—24no—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40#1442—22no—
41#1439—22no—
42#1439—22no—
43#1438—22no—
44C4⋊#95C2212yes—
45C23×D4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—42no—
56C23×C8—42no—
57C23×C8D414yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42C2311yescharacteristic
81C8×D4—232no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—42no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8D414yes—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8D414yes—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99#389—24no—
100#389—24no—
101#380—24no—
102#380—24no—
103#380—24no—
104#380—24no—
105#380—24no—
106#380—24no—
107#380—42no—
108#380—42no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116#340—24no—
117#340—22no—
118#340—24no—
119#340—22no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124C25C2×D411yescharacteristic
125C25C2×D411yescharacteristic
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4Q1614yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—28no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—28no—
177C22×C8—28no—
178C22×C8—44no—
179C22×C8—48no—
180C22×C8—48no—
181C22×C8—48no—
182C22×C8—48no—
183C22×C8—44no—
184C22×C8—44no—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
191C22×C8#4114yes—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—28no—
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—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4#4114yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C2×D412yes—
249C2×C42#4112yes—
250#107—44no—
251#107—44no—
252C4⋊C8—28no—
253C22⋊C8—28no—
254C22⋊C8—28no—
255C4×C8—28no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—82no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24#12112yes—
280C24#12112yes—
281C24C2×Q1612yes—
282C24C2×Q1612yes—
283C24#13512yes—
284C24#13512yes—
285C24C22×D411yescharacteristic
286C2×D4—28no—
287C2×D4—28no—
288C2×D4—28no—
289C2×D4—28no—
290C2×D4—28no—
291C2×D4—28no—
292C2×D4—28no—
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—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
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—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
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#12112yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4#12112yes—
377C22×C4C2×Q1612yes—
378C22×C4#13512yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C2×#4112yes—
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—416no—
402C2×C8—416no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8#12014yes—
410C2×C8#12314yes—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—88no—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
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—24no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23#46714yes—
484C23#46714yes—
485C23C2×#12111yescharacteristic
486C23C2×#12111yescharacteristic
487C23C2×#12111yescharacteristic
488C23#54612yes—
489C23#54712yes—
490C23C22×Q1611yescharacteristic
491C23C2×#13511yescharacteristic
492C23C2×#13511yescharacteristic
493C23C2×#13511yescharacteristic
494C23#57612yes—
495C23#57712yes—
496D4—232no—
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—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
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—82no—
549C2×C4—82no—
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#46714yes—
559C2×C4C2×#12011yescharacteristic
560C2×C4C2×#12311yescharacteristic
561C2×C4#52511yescharacteristic
562C2×C4#52511yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52912yes—
565C2×C4#52912yes—
566C2×C4#53211yescharacteristic
567C2×C4#53411yescharacteristic
568C2×C4#53611yescharacteristic
569C2×C4#53912yes—
570C2×C4#54212yes—
571C2×C4#54612yes—
572C2×C4#54612yes—
573C2×C4#54712yes—
574C2×C4#54812yes—
575C2×C4#55212yes—
576C2×C4#55312yes—
577C8—24no—
578C8—24no—
579C8—42no—
580C8—42no—
581C22—24no—
582C22—24no—
583C22—28no—
584C22—24no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22#242512yes—
604C22#242712yes—
605C22#248611yescharacteristic
606C22#254311yescharacteristic
607C22#284711yescharacteristic
608C22C2×#46711yescharacteristic
609C22C2×#46711yescharacteristic
610C22C2×#46711yescharacteristic
611C22#305011yescharacteristic
612C22#305011yescharacteristic
613C22#305011yescharacteristic
614C22#305811yescharacteristic
615C22#306011yescharacteristic
616C22#306212yes—
617C22#306212yes—
618C22#329612yes—
619C22#329712yes—
620C22#329912yes—
621C22#330512yes—
622C22#333712yes—
623C22#333812yes—
624C22#333912yes—
625C22#334012yes—
626C4—24no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—84no—
631C4—162no—
632C4—162no—
633C4—162no—
634C4#239511yescharacteristic
635C4#240511yescharacteristic
636C4#243211yescharacteristic
637C4#246012yes—
638C4#246812yes—
639C4#263811yescharacteristic
640C2—44no—
641C2—44no—
642C2#1482411yescharacteristic
643C2#2523412yes—
644C2#2523512yes—
645C2#2534012yes—
646C2#2534212yes—
647C2#2765911yescharacteristic
648C2#2766211yescharacteristic
649C2#2855211yescharacteristic
650C2#2855311yescharacteristic
651C2#3293111yescharacteristic
652C2#3293911yescharacteristic
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×#2087C2semidirect
3. C2×#2076C2semidirect
4. C2×#1808C2semidirect
6. #11232C2semidirect
7. #11200C2semidirect
8. #11096C2semidirect
9. #10880C2semidirect
10. #10721C2semidirect
11. #10721C2semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#20872
C2×#20762
C2×#18082
#151812
#112322
#112002
#110962
#108802
#107212