Finite Group Databasebeta

← #400703

Subgroups — #400703

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic, complemented
2C2×#2056C211yesmaximal, characteristic, complemented
3C2×#2036C212yesmaximal
4C2×#2001C211yesmaximal, characteristic
5C2×#1923C211yesmaximal, characteristic
6#13969C212yesmaximal
7#11232C211yesmaximal, characteristic, complemented
8#11124C212yesmaximal, complemented
9#11096C211yesmaximal, characteristic
10#10862C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C4×Q8C2212yes—
15C2×C4×Q16—28no—
16C2×C4×Q16—24no—
17C22×#107C2211yescharacteristic
18C22×#107C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#391C2212yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#2056—24no—
30#2036—28no—
31#2001—24no—
32#1923—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1441—22no—
40#1439—22no—
41#1437—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×Q8C2312yes—
46C24×C4C2311yescharacteristic
47C22×Q16—24no—
48C22×Q16—24no—
49C22×Q16—22no—
50C22×Q16—22no—
51C22×Q16D414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—42no—
61C2×C22⋊Q8—42no—
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—24no—
66C2×C4×Q8—42no—
67C2×C4×Q8—42no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—22no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C4×Q16—232no—
81C2×#107—22no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103D418yes—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—22no—
103C2×C4×C8—24no—
104C2×C4×C8D414yes—
105#391—24no—
106#391—42no—
107#380—24no—
108#380—42no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120#340—22no—
121#340—22no—
122#340—24no—
123#340—42no—
124#336—22no—
125#336—22no—
126C25C2×D411yescharacteristic
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C2×D414yes—
139C22×Q8C2×D414yes—
140C22×Q8#4114yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C2×Q16—28no—
162C2×Q16—216no—
163C2×Q16—28no—
164C2×Q16—216no—
165C2×Q16—28no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8C2×D412yes—
182C22×C8C2×D414yes—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C22×C8#4112yes—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C4×Q8—216no—
191C4×Q8—48no—
192C4×Q8—48no—
193C4×Q8—48no—
194C2×C4⋊C4—22no—
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—28no—
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—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4#4114yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4Q1614yescomplemented
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42C2×D412yes—
241C2×C42#4112yes—
242#107—28no—
243#107—48no—
244C4⋊C8—44no—
245#103—216no—
246#103—48no—
247#103—48no—
248C22⋊C8—44no—
249C22⋊C8—44no—
250C4×C8—28no—
251#95—22no—
252#95—22no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—82no—
259#95—82no—
260C24—21no—
261C24—22no—
262C24—22no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24C2×Q1612yes—
271C24C22×D411yescharacteristic
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8#12118yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
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—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
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#12112yes—
369C22×C4#12112yes—
370C22×C4C2×Q1612yes—
371C22×C4C2×Q1612yes—
372C22×C4#13512yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+411yescharacteristic
383Q16—232no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—42no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8#12014yes—
403C2×C8#12114yes—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—416no—
409C4⋊C4—416no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C22⋊C4—28no—
419C22⋊C4—44no—
420C22⋊C4—416no—
421C22⋊C4—88no—
422C42—22no—
423C42—22no—
424C42—24no—
425C42—416no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C23—22no—
431C23—24no—
432C23—22no—
433C23—24no—
434C23—22no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23C2×#12111yescharacteristic
454C23#52512yes—
455C23D4212yes—
456C23D4211yescharacteristic
457C23#54711yescharacteristic
458C23C22×Q1611yescharacteristic
459C23C2×#13511yescharacteristic
460C23#57512yes—
461C23#57611yescharacteristic
462C23#57811yescharacteristic
463C23#57812yes—
464Q8—28no—
465Q8—28no—
466Q8—44no—
467Q8—44no—
468Q8—44no—
469Q8—44no—
470Q8—44no—
471Q8—44no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—28no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4#46714yes—
533C2×C4C2×#12011yescharacteristic
534C2×C4C2×#12112yes—
535C2×C4C2×#12111yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4#52512yes—
538C2×C4#52912yes—
539C2×C4#53211yescharacteristic
540C2×C4#53511yescharacteristic
541C2×C4#53711yescharacteristic
542C2×C4#53812yes—
543C2×C4#53912yes—
544C2×C4#54612yes—
545C2×C4#54612yes—
546C2×C4#54612yes—
547C2×C4#54711yescharacteristic
548C2×C4#54712yes—
549C2×C4#55511yescharacteristic
550C2×C4#55912yes—
551C8—24no—
552C8—24no—
553C8—81no—
554C8—81no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—44no—
559C22—42no—
560C22—81no—
561C22—82no—
562C22—81no—
563C22—82no—
564C22—81no—
565C22—81no—
566C22#239211yescharacteristic
567C22#240811yescharacteristic
568C22#242412yes—
569C22#245512yes—
570C22#253711yescharacteristic
571C22C2×#46711yescharacteristic
572C22#305411yescharacteristic
573C22#305911yescharacteristic
574C22#306111yescharacteristic
575C22#310212yes—
576C22#310412yes—
577C22#312412yes—
578C22#312712yes—
579C22#326812yes—
580C22#326912yes—
581C22#327712yes—
582C22#328512yes—
583C22D4×Q1611yescharacteristic
584C22D4×Q1612yes—
585C22#328711yescharacteristic
586C22#333811yescharacteristic
587C22#334211yescharacteristic
588C22#335212yes—
589C4—24no—
590C4—42no—
591C4—42no—
592C4—48no—
593C4—81no—
594C4—84no—
595C4—84no—
596C4—81no—
597C4—162no—
598C4#242512yes—
599C4#243011yescharacteristic
600C4#245912yes—
601C4#254411yescharacteristic
602C4#263611yescharacteristic
603C4#284711yescharacteristic
604C2—82no—
605C2#1480611yescharacteristic
606C2#2369811yescharacteristic
607C2#2370511yescharacteristic
608C2#2426811yescharacteristic
609C2#2427011yescharacteristic
610C2#2516112yes—
611C2#2516312yes—
612C2#2655912yes—
613C2#2656112yes—
614C2#2822811yescharacteristic
615C2#2823411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×Q16C2semidirect
2. C2×#2056C2semidirect
7. #11232C2semidirect
8. #11124C2semidirect
10. #10862C2semidirect
11. #10789C2semidirect
228. C2×C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×#20562
C2×#20362
C2×#20012
C2×#19232
#139692
#112322
#111242
#110962
#108622
#107892