Finite Group Databasebeta

← #471773

Subgroups — #471773

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C212yesmaximal
2C2×#2035C212yesmaximal
3#14702C211yesmaximal, characteristic
4#14686C211yesmaximal, characteristic
5#11378C212yesmaximal, complemented
6#11324C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11124C212yesmaximal, complemented
9#11050C211yesmaximal, characteristic, complemented
10#11050C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#380—22no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26#2075—28no—
27#2035—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1436—22no—
36C4⋊#95—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C22⋊#95—22no—
41C23×Q8C2311yescharacteristic
42C24×C4D412yes—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—42no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42—42no—
64C22×C42—42no—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—24no—
69C2×#107—42no—
70C2×#107—42no—
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—42no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8D418yes—
83#391—22no—
84#391—22no—
85#391—42no—
86#384—42no—
87#384—42no—
88#380—24no—
89#380—24no—
90#380—24no—
91#380—48no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—24no—
98C2×#95—21no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—22no—
104#336—22no—
105#336—42no—
106C25C2×D411yescharacteristic
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8#4112yes—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D414yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8C2×D414yes—
157C22×C8C2×D414yes—
158C22⋊Q8—48no—
159C22⋊Q8—48no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
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—21no—
171C2×C4⋊C4—21no—
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—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4#4112yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—48no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—48no—
208C2×C42—48no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42#4112yes—
212C2×C42#4112yes—
213#107—48no—
214#106—48no—
215#103—48no—
216#103—48no—
217C22⋊C8—48no—
218#95—48no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—42no—
223#95—44no—
224#95—44no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24#12712yes—
235C24AGL(1,ℤ/8)14yes—
236C24C22×D411yescharacteristic
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—216no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4#12314yes—
325C22×C4#12712yes—
326C22×C4#13514yes—
327C22×C4#13714yes—
328C22×C4#13714yes—
329C22×C4C22×D412yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C4H(5,𝔽2)12yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C42−1+411yescharacteristic
338C22×C42−1+412yes—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—88no—
369C4⋊C4—88no—
370C22⋊C4—48no—
371C22⋊C4—84no—
372C22⋊C4—88no—
373C42—24no—
374C42—24no—
375C42—48no—
376C42—42no—
377C42—42no—
378C23—24no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—22no—
389C23—22no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—42no—
399C23—41no—
400C23#49614yes—
401C23C2×#12311yescharacteristic
402C23#52511yescharacteristic
403C23C2×#12711yescharacteristic
404C23#53411yescharacteristic
405C23#53411yescharacteristic
406C23#53611yescharacteristic
407C23#53711yescharacteristic
408C23C2×#13512yes—
409C23C2×AGL(1,ℤ/8)12yes—
410C23C2×#13712yes—
411C23C2×#13712yes—
412Q8—44no—
413Q8—44no—
414Q8—82no—
415Q8—82no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—432no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—84no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—84no—
466C2×C4—82no—
467C2×C4—84no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—84no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4#48518yes—
474C2×C4#53412yes—
475C2×C4#53712yes—
476C2×C4#53812yes—
477C2×C4#53812yes—
478C2×C4#53912yes—
479C2×C4#54212yes—
480C2×C4#54612yes—
481C2×C4#54612yes—
482C2×C4#54712yes—
483C2×C4#54712yes—
484C2×C4#55211yescharacteristic
485C2×C4#55311yescharacteristic
486C2×C4#55511yescharacteristic
487C2×C4#55611yescharacteristic
488C2×C4#56212yes—
489C2×C4#56312yes—
490C8—44no—
491C8—44no—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22#248511yescharacteristic
504C22#250111yescharacteristic
505C22#253911yescharacteristic
506C22#262211yescharacteristic
507C22#263811yescharacteristic
508C22#265611yescharacteristic
509C22#281411yescharacteristic
510C22C2×#48512yes—
511C22#309212yes—
512C22#310712yes—
513C22#311212yes—
514C22#314711yescharacteristic
515C22C2×#49611yescharacteristic
516C22#315211yescharacteristic
517C22#315211yescharacteristic
518C22#318812yes—
519C22#319012yes—
520C22#319312yes—
521C22#319512yes—
522C22#323112yes—
523C22#323812yes—
524C22#324712yes—
525C22#325512yes—
526C4—28no—
527C4—42no—
528C4—44no—
529C4—82no—
530C4—82no—
531C4—161no—
532C4—162no—
533C4—161no—
534C4—162no—
535C4#268612yes—
536C4#269512yes—
537C4#270812yes—
538C4#271512yes—
539C2—44no—
540C2#1680711yescharacteristic
541C2#2763112yes—
542C2#2790811yescharacteristic
543C2#2791411yescharacteristic
544C2#2830811yescharacteristic
545C2#2831411yescharacteristic
546C2#2989412yes—
547C2#3017612yes—
548C2#3024311yescharacteristic
549C2#3025011yescharacteristic
550C2#3274512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11378C2semidirect
6. #11324C2semidirect
7. #11323C2semidirect
8. #11124C2semidirect
9. #11050C2semidirect
10. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
C2×#20352
#147022
#146862
#113782
#113242
#113232
#111242
#110502
#107122