Finite Group Databasebeta

← #361439

Subgroups — #361439

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×SD16C211yesmaximal, characteristic
2C2×#1877C212yesmaximal
3C2×#1864C212yesmaximal
4#13935C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11122C212yesmaximal, complemented
8#10861C211yesmaximal, characteristic, complemented
9#10797C211yesmaximal, characteristic, complemented
10#10789C211yesmaximal, characteristic, complemented
11#10788C211yesmaximal, characteristic
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2212yes—
15C22×C4×D4C2211yescharacteristic
16C2×C4×SD16—28no—
17C2×C4×SD16C418yescomplemented
18C22×#106C2212yes—
19C22×#106C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#389C2212yes—
27C22×#95—21no—
28#1877—28no—
29#1864—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1441—22no—
37C4⋊#95—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41#1434—21no—
42C22⋊#95—21no—
43C22⋊#95—22no—
44C4×#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4—22no—
48C24×C4—21no—
49C22×SD16—24no—
50C22×SD16—24no—
51C22×SD16C2×C414yes—
52C22×SD16D414yes—
53C22×SD16Q814yes—
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C4×Q8—22no—
58C2×C4×Q8—24no—
59C2×C4×Q8—42no—
60C2×C4×Q8C2×C414yescomplemented
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—22no—
67C2×C4×D4—42no—
68C2×C4×D4—42no—
69C2×C4×D4—42no—
70C2×C4×D4C2×C414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—22no—
78C22×C22⋊C4—21no—
79C22×C42—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C4×SD16—232no—
83C2×#106—24no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—42no—
87C2×#106C2×C414yescomplemented
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103C2×C414yescomplemented
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102C2×C414yes—
97C2×#102D418yes—
98C2×C4×C8—24no—
99C2×C4×C8C2×C414yescomplemented
100#391—24no—
101#391—42no—
102#389—24no—
103#389—42no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—44no—
114C2×#95D411yescharacteristic
115C2×#95D412yes—
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—42no—
120#336—24no—
121#336—24no—
122#336—22no—
123C25—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8C22×C412yes—
132C22×Q8C2×D412yes—
133C22×Q8C2×Q812yes—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C22×C412yes—
140C22×D4C2×D414yes—
141C22×D4C2×D412yes—
142C22×D4C2×Q812yes—
143C22×D4#4114yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—28no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—82no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C2×SD16—28no—
167C2×SD16—28no—
168C2×SD16—216no—
169C2×SD16—216no—
170C2×SD16C4⋊C4116yescomplemented
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C22×C412yes—
184C22×C8C22×C412yes—
185C22×C8C2×D414yes—
186C22×C8C2×D412yes—
187C22×C8C2×Q812yes—
188C4×Q8—28no—
189C4×Q8—44no—
190C4×Q8—48no—
191C4×D4—28no—
192C4×D4—44no—
193C4×D4—48no—
194C4×D4—48no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4—48no—
198C2×C4⋊C4—21no—
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—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
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—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—44no—
238C2×C42C22×C412yes—
239C2×C42#4112yes—
240#106—28no—
241#106—48no—
242#103—28no—
243#103—48no—
244#102—48no—
245#102C22⋊C4116yes—
246C4×C8—28no—
247#95—22no—
248#95—42no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—42no—
254#95—42no—
255#95—81no—
256#95—81no—
257#95—82no—
258#95SD1614yescomplemented
259C24—22no—
260C24—24no—
261C24—24no—
262C24—22no—
263C24—22no—
264C24—41no—
265C24—41no—
266C24—42no—
267C24—41no—
268C24—42no—
269C24—41no—
270C24C22×D411yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8C2×C4⋊C414yes—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—22no—
302C2×D4C2×C22⋊C414yes—
303C2×D4C2×C4⋊C414yes—
304C2×D4#11714yes—
305C2×D4#12118yes—
306C2×D4C22⋊Q818yes—
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—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—48no—
334C22×C4—416no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—416no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—84no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4C2×SD1614yes—
370C22×C4C2×SD1612yes—
371C22×C4#13512yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4111yescharacteristic
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C42−1+411yescharacteristic
382SD16—232no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8C2×C22⋊C414yes—
399C2×C8C2×C4⋊C414yes—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—48no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—88no—
415C4⋊C4C2×C22⋊C414yes—
416C22⋊C4—48no—
417C22⋊C4—416no—
418C42—22no—
419C42—22no—
420C42—24no—
421C42—48no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—42no—
426C23—24no—
427C23—24no—
428C23—28no—
429C23—28no—
430C23—24no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—44no—
436C23—42no—
437C23—44no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—42no—
447C23—44no—
448C23—42no—
449C23C2×C4×D411yescharacteristic
450C23#52512yes—
451C23D4212yes—
452C23D4211yescharacteristic
453C23D4×Q811yescharacteristic
454C23C22×SD1611yescharacteristic
455C23C2×#13511yescharacteristic
456C23#57512yes—
457C23#57711yescharacteristic
458C23#57712yes—
459C23#57711yescharacteristic
460Q8—24no—
461Q8—24no—
462Q8—42no—
463Q8—42no—
464Q8—44no—
465Q8—44no—
466D4—24no—
467D4—24no—
468D4—24no—
469D4—28no—
470D4—28no—
471D4#38018yes—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—816no—
512C2×C4—832no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4C4×SD1614yes—
527C2×C4C22×C22⋊C411yescharacteristic
528C2×C4C22×C4⋊C411yescharacteristic
529C2×C4C2×#11711yescharacteristic
530C2×C4C4×#4111yescharacteristic
531C2×C4#51811yescharacteristic
532C2×C4#52011yescharacteristic
533C2×C4C2×#12112yes—
534C2×C4C2×C22⋊Q812yes—
535C2×C4#52912yes—
536C2×C4#53812yes—
537C2×C4#53912yes—
538C2×C4#54612yes—
539C2×C4#54612yes—
540C2×C4#54612yes—
541C2×C4#54712yes—
542C2×C4#54711yescharacteristic
543C2×C4#55412yes—
544C2×C4#55411yescharacteristic
545C8—24no—
546C8—24no—
547C8—82no—
548C22—28no—
549C22—48no—
550C22—48no—
551C22—48no—
552C22—48no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22#229011yescharacteristic
560C22D4×C22⋊C411yescharacteristic
561C22D4×C4⋊C411yescharacteristic
562C22#242412yes—
563C22#244912yes—
564C22C2×C4×SD1611yescharacteristic
565C22#294311yescharacteristic
566C22#294911yescharacteristic
567C22#295211yescharacteristic
568C22#310112yes—
569C22#310212yes—
570C22#312312yes—
571C22#312512yes—
572C22#326712yes—
573C22#326712yes—
574C22#327512yes—
575C22D4×SD1612yes—
576C22D4×SD1611yescharacteristic
577C22#328212yes—
578C22#328411yescharacteristic
579C22Q8×SD1611yescharacteristic
580C22#338111yescharacteristic
581C22#338812yes—
582C4—24no—
583C4—44no—
584C4—44no—
585C4—42no—
586C4—44no—
587C4—84no—
588C4—82no—
589C4—82no—
590C4—162no—
591C4C2×#38011yescharacteristic
592C4#231811yescharacteristic
593C4#233911yescharacteristic
594C4#234911yescharacteristic
595C4#242512yes—
596C4#245012yes—
597C2—48no—
598C2#1389911yescharacteristic
599C2#2111711yescharacteristic
600C2#2113311yescharacteristic
601C2#2243811yescharacteristic
602C2C22⋊C4×SD1611yescharacteristic
603C2#2267311yescharacteristic
604C2C4⋊C4×SD1611yescharacteristic
605C2#2515612yes—
606C2#2515812yes—
607C2#2637612yes—
608C2#2637812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13935C2semidirect
6. #11220C2semidirect
7. #11122C2semidirect
8. #10861C2semidirect
9. #10797C2semidirect
10. #10789C2semidirect
17. C2×C4×SD16C4semidirect
60. C2×C4×Q8C2×C4semidirect
87. C2×#106C2×C4semidirect
92. C2×#103C2×C4semidirect
99. C2×C4×C8C2×C4semidirect
170. C2×SD16C4⋊C4semidirect
258. #95SD16semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SD162
C2×#18772
C2×#18642
#139352
D4×#952
#112202
#111222
#108612
#107972
#107892
#107882