Finite Group Databasebeta

← #471716

Subgroups — #471716

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic, complemented
2C2×#1923C211yesmaximal, characteristic
3#15195C211yesmaximal, characteristic
4#15125C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11371C211yesmaximal, characteristic, complemented
7#11315C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#11199C212yesmaximal
10#11095C212yesmaximal
11#10861C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#393—21no—
19C2×#384C2211yescharacteristic
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1990—24no—
24#1923—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1446—21no—
35#1442—21no—
36#1442—21no—
37#1442—21no—
38#1439—21no—
39#1439—21no—
40#1439—21no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C24×C4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#106—24no—
69C2×#106—42no—
70C2×#106—42no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—24no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—42no—
77C2×C4×C8—42no—
78C2×C4×C8D414yes—
79#393—44no—
80#384—22no—
81#384—22no—
82#380—24no—
83#380—42no—
84#380—42no—
85#380—42no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D411yescharacteristic
99C2×#95D411yescharacteristic
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—22no—
108#336—22no—
109C25C2×D411yescharacteristic
110C22×Q8—44no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8—82no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8#4114yes—
155C4⋊Q8—28no—
156C4⋊Q8—48no—
157C4⋊Q8—44no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4#4114yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4SD1614yes—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42C2×D412yes—
207C2×C42#4112yes—
208#106—48no—
209C4⋊C8—44no—
210C22⋊C8—44no—
211C22⋊C8—44no—
212C4×C8—44no—
213#95—22no—
214#95—22no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—42no—
219#95—42no—
220#95—42no—
221#95—42no—
222#95—82no—
223#95—82no—
224#95—82no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—22no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24C2×SD1612yes—
236C24C22×D411yescharacteristic
237C2×Q8—48no—
238C2×Q8—88no—
239C2×Q8—88no—
240C2×Q8—84no—
241C2×Q8—84no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4C2×SD1612yes—
309C22×C4C2×SD1612yes—
310C22×C4#13512yes—
311C22×C4#13712yes—
312C22×C4#13712yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4C2×#4112yes—
316C22×C4C2×#4111yescharacteristic
317C22×C4H(5,𝔽2)12yes—
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C42−1+412yes—
321C22×C42−1+411yescharacteristic
322C22×C42−1+411yescharacteristic
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8#12414yes—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—48no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C22⋊C4—28no—
363C22⋊C4—44no—
364C22⋊C4—88no—
365C22⋊C4—88no—
366C42—22no—
367C42—22no—
368C42—24no—
369C42—41no—
370C42—41no—
371C42—42no—
372C23—24no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—22no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—42no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23C2×#12111yescharacteristic
396C23#53511yescharacteristic
397C23#53611yescharacteristic
398C23#54612yes—
399C23#54712yes—
400C23C22×SD1611yescharacteristic
401C23C2×#13511yescharacteristic
402C23C2×#13711yescharacteristic
403C23C2×#13711yescharacteristic
404C23#57612yes—
405C23#57712yes—
406Q8—816no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4#46514yes—
465C2×C4#47014yes—
466C2×C4#52511yescharacteristic
467C2×C4C2×#12411yescharacteristic
468C2×C4#53411yescharacteristic
469C2×C4#53611yescharacteristic
470C2×C4#53711yescharacteristic
471C2×C4#53711yescharacteristic
472C2×C4#53812yes—
473C2×C4#53912yes—
474C2×C4#54112yes—
475C2×C4#54212yes—
476C2×C4#54211yescharacteristic
477C2×C4#54211yescharacteristic
478C2×C4#55112yes—
479C2×C4#55612yes—
480C2×C4#55612yes—
481C2×C4#55912yes—
482C2×C4#56212yes—
483C2×C4#56312yes—
484C8—24no—
485C8—42no—
486C8—81no—
487C8—81no—
488C22—42no—
489C22—44no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—81no—
494C22—82no—
495C22—81no—
496C22—81no—
497C22—82no—
498C22—81no—
499C22#253411yescharacteristic
500C22#254711yescharacteristic
501C22#262011yescharacteristic
502C22#268912yes—
503C22#271012yes—
504C22C2×#46511yescharacteristic
505C22C2×#47011yescharacteristic
506C22#305711yescharacteristic
507C22#305811yescharacteristic
508C22#320611yescharacteristic
509C22#321011yescharacteristic
510C22#322412yes—
511C22#322911yescharacteristic
512C22#323111yescharacteristic
513C22#324112yes—
514C22#329512yes—
515C22#329712yes—
516C22#331612yes—
517C22#332012yes—
518C22#333512yes—
519C22#334012yes—
520C22#335712yes—
521C22#335712yes—
522C4—24no—
523C4—24no—
524C4—42no—
525C4—81no—
526C4—81no—
527C4—84no—
528C4—162no—
529C4—162no—
530C4—162no—
531C4#259211yescharacteristic
532C4#263711yescharacteristic
533C4#263911yescharacteristic
534C4#275912yes—
535C4#279812yes—
536C4#281411yescharacteristic
537C2—82no—
538C2#1680511yescharacteristic
539C2#2809711yescharacteristic
540C2#2810811yescharacteristic
541C2#2870611yescharacteristic
542C2#2871111yescharacteristic
543C2#2984511yescharacteristic
544C2#2984711yescharacteristic
545C2#3142612yes—
546C2#3142912yes—
547C2#3204712yes—
548C2#3204812yes—
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. C2×#1990C2semidirect
5. #11384C2semidirect
6. #11371C2semidirect
7. #11315C2semidirect
8. #11231C2semidirect
11. #10861C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19902
C2×#19232
#151952
#151252
#113842
#113712
#113152
#112312
#111992
#110952
#108612