Finite Group Databasebeta

← #440915

Subgroups — #440915

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#1993C211yesmaximal, characteristic, complemented
4#15125C211yesmaximal, characteristic
5#11384C212yesmaximal, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11219C212yesmaximal, complemented
8#11096C212yesmaximal
9#10922C211yesmaximal, characteristic
10#10881C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18#2439—24no—
19#2439—24no—
20C2×#396—21no—
21C2×#393—21no—
22C2×#393—21no—
23C2×#391—21no—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#387—21no—
27C2×#382C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#2057—24no—
32#1993—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1446—21no—
43#1442—21no—
44#1439—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8—21no—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—42no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—24no—
63C2×#117—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8Q818yescomplemented
88C2×#97—22no—
89C2×#97—42no—
90#396—28no—
91#393—28no—
92#393—44no—
93#391—28no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389D414yes—
98#387—28no—
99#382—22no—
100#382D414yes—
101#380—22no—
102#380—22no—
103#380—24no—
104#380—42no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#340—42no—
126#336—22no—
127#336—42no—
128C25C2×D411yescharacteristic
129C22×Q8—24no—
130C22×Q8—44no—
131C22×Q8—44no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—82no—
138C22×Q8—82no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×Q814yes—
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C2×Q814yes—
172C22×C8C2×Q814yes—
173C4⋊Q8—28no—
174C4⋊Q8—48no—
175C4⋊Q8—44no—
176C4⋊Q8—48no—
177C4⋊Q8—44no—
178C4⋊Q8—44no—
179#117—48no—
180#117—44no—
181#117—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
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—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4#4114yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242#107—48no—
243C22⋊C8—48no—
244#97—44no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—44no—
253#95—42no—
254#95—42no—
255#95—44no—
256#95—82no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24C4⋊Q812yes—
268C24C22×D411yescharacteristic
269C2×Q8—48no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—88no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8—84no—
276C2×Q8—88no—
277C2×Q8—88no—
278C2×Q8—88no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
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—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
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#12012yes—
350C22×C4#12012yes—
351C22×C4C4⋊Q812yes—
352C22×C4AGL(1,ℤ/8)12yes—
353C22×C4#13712yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q812yes—
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C42−1+412yes—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—84no—
399C4⋊C4#12414yes—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—48no—
403C22⋊C4—84no—
404C22⋊C4—88no—
405C42—22no—
406C42—22no—
407C42—24no—
408C42—48no—
409C42—41no—
410C42—41no—
411C42—42no—
412C23—22no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—42no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23C2×#12011yescharacteristic
436C23C2×C4⋊Q811yescharacteristic
437C23#53611yescharacteristic
438C23#54612yes—
439C23#54612yes—
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×#13711yescharacteristic
442C23#57511yescharacteristic
443C23#57511yescharacteristic
444C23#57612yes—
445C23#57812yes—
446Q8—816no—
447Q8—816no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4#52511yescharacteristic
512C2×C4C2×#12411yescharacteristic
513C2×C4#53312yes—
514C2×C4#53511yescharacteristic
515C2×C4#53511yescharacteristic
516C2×C4#54212yes—
517C2×C4#54212yes—
518C2×C4#54312yes—
519C2×C4#54612yes—
520C2×C4#54612yes—
521C2×C4D4×Q812yes—
522C2×C4D4×Q812yes—
523C2×C4#55412yes—
524C2×C4#55412yes—
525C2×C4#55612yes—
526C2×C4#56212yes—
527C8—44no—
528C8—44no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22#240411yescharacteristic
541C22#243411yescharacteristic
542C22#243511yescharacteristic
543C22#254912yes—
544C22#270012yes—
545C22#300611yescharacteristic
546C22#300711yescharacteristic
547C22#300811yescharacteristic
548C22#301111yescharacteristic
549C22#316211yescharacteristic
550C22#316412yes—
551C22#316511yescharacteristic
552C22#323511yescharacteristic
553C22#323711yescharacteristic
554C22#324312yes—
555C22#330812yes—
556C22#331312yes—
557C22#331312yes—
558C22#331412yes—
559C22#331912yes—
560C22#331912yes—
561C22#332512yes—
562C22#332812yes—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—84no—
573C4—162no—
574C4#246512yes—
575C4#249212yes—
576C4#269212yes—
577C4#269212yes—
578C2—82no—
579C2#1567111yescharacteristic
580C2#2397611yescharacteristic
581C2#2398011yescharacteristic
582C2#2581811yescharacteristic
583C2#2582111yescharacteristic
584C2#2589611yescharacteristic
585C2#2589711yescharacteristic
586C2#2877712yes—
587C2#2878412yes—
588C2#3176312yes—
589C2#3176912yes—
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
2. C2×#2057C2semidirect
3. C2×#1993C2semidirect
5. #11384C2semidirect
6. #11358C2semidirect
7. #11219C2semidirect
10. #10881C2semidirect
87. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#20572
C2×#19932
#151252
#113842
#113582
#112192
#110962
#109222
#108812
#107312