Finite Group Databasebeta

← #433460

Subgroups — #433460

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2077C212yesmaximal
2C2×#1787C212yesmaximal
3#15150C211yesmaximal, characteristic
4#13922C211yesmaximal, characteristic
5#11122C212yesmaximal, complemented
6#11122C212yesmaximal, complemented
7#11050C212yesmaximal, complemented
8#10926C212yesmaximal, complemented
9#10921C211yesmaximal, characteristic
10C24×Q8C2211yescharacteristic
11C22×#106C2212yes—
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2211yescharacteristic
18C2×#397—22no—
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#384C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#2077—28no—
26#1787—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32#1446—22no—
33#1445—22no—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C22⋊#95—22no—
37C23×Q8—24no—
38C23×Q8—21no—
39C23×Q8—24no—
40C23×Q8—24no—
41C23×Q8—24no—
42C23×Q8—41no—
43C23×Q8—41no—
44C23×Q8—42no—
45C23×Q8—42no—
46C23×Q8C2312yes—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4D412yes—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#106—24no—
61C2×#106—42no—
62C2×#106—42no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×C22⋊C8—42no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8D418yes—
78C2×C4×C8—41no—
79C2×C4×C8—42no—
80#397—48no—
81#391—24no—
82#391—24no—
83#391—42no—
84#391—42no—
85#384—41no—
86#384—41no—
87#380—24no—
88#380—24no—
89#380—24no—
90C4×C22⋊C4—41no—
91C4×C22⋊C4—41no—
92C2×#95—24no—
93C2×#95—22no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—24no—
102#336—41no—
103#336—42no—
104C25C2×D411yescharacteristic
105C22×Q8—24no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—216no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—48no—
121C22×Q8—48no—
122C22×Q8—48no—
123C22×Q8—48no—
124C22×Q8—42no—
125C22×Q8—48no—
126C22×Q8—48no—
127C22×Q8—48no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8—48no—
131C22×Q8—48no—
132C22×Q8—48no—
133C22×Q8—44no—
134C22×Q8—44no—
135C22×Q8—48no—
136C22×Q8—42no—
137C22×Q8#4114yes—
138C22×Q8#4114yes—
139C22×Q8#4114yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D414yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2411yescharacteristic
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×C8—22no—
172C22×C8—22no—
173C22×C8C2×D414yes—
174C22×C8C2×D414yes—
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—22no—
182C2×C4⋊C4—22no—
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—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4SD1618yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211#106—48no—
212#103—48no—
213#103—48no—
214#103—48no—
215#103—48no—
216C22⋊C8—48no—
217C4×C8—44no—
218#95—42no—
219#95—48no—
220#95—48no—
221#95—44no—
222#95—42no—
223#95—44no—
224#95—48no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—24no—
231C24#12712yes—
232C24C2×SD1614yes—
233C24C22×D411yescharacteristic
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—416no—
250C2×Q8—48no—
251C2×Q8—416no—
252C2×Q8—432no—
253C2×Q8—432no—
254C2×Q8—432no—
255C2×Q8—432no—
256C2×Q8—48no—
257C2×Q8—48no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8#12618yes—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—416no—
293C22×C4—48no—
294C22×C4—416no—
295C22×C4—48no—
296C22×C4—48no—
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—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4#12712yes—
327C22×C4C2×SD1614yes—
328C22×C4C2×SD1614yes—
329C22×C4C22×D412yes—
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C4C2×#4112yes—
333C22×C4H(5,𝔽2)12yes—
334C22×C4H(5,𝔽2)12yes—
335C22×C4H(5,𝔽2)11yescharacteristic
336C22×C4H(5,𝔽2)11yescharacteristic
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—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—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—88no—
364C22⋊C4—48no—
365C22⋊C4—84no—
366C22⋊C4—88no—
367C42—42no—
368C42—42no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C23—24no—
374C23—22no—
375C23—24no—
376C23—24no—
377C23—28no—
378C23—28no—
379C23—22no—
380C23—24no—
381C23—42no—
382C23—41no—
383C23—42no—
384C23—41no—
385C23—42no—
386C23—44no—
387C23—44no—
388C23#49214yes—
389C23#52511yescharacteristic
390C23C2×#12711yescharacteristic
391C23#53411yescharacteristic
392C23#54612yes—
393C23#54612yes—
394C23C22×SD1612yes—
395C23#57512yes—
396C23#57814yes—
397Q8—28no—
398Q8—28no—
399Q8—432no—
400Q8—44no—
401Q8—44no—
402Q8—44no—
403Q8—44no—
404Q8—44no—
405Q8—44no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—432no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—84no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4#52512yes—
449C2×C4C2×#12612yes—
450C2×C4#53412yes—
451C2×C4#53812yes—
452C2×C4#53812yes—
453C2×C4#53812yes—
454C2×C4#54612yes—
455C2×C4#54612yes—
456C2×C4#54612yes—
457C2×C4#54612yes—
458C2×C4#55112yes—
459C2×C4#55112yes—
460C2×C4#55111yescharacteristic
461C2×C4#55111yescharacteristic
462C2×C4#56011yescharacteristic
463C2×C4H(3,𝔽4)11yescharacteristic
464C8—44no—
465C8—44no—
466C22—24no—
467C22—28no—
468C22—44no—
469C22—42no—
470C22—48no—
471C22—42no—
472C22—44no—
473C22—44no—
474C22#248012yes—
475C22#253911yescharacteristic
476C22#253912yes—
477C22#268411yescharacteristic
478C22#269711yescharacteristic
479C22#310312yes—
480C22#312712yes—
481C22C2×#49211yescharacteristic
482C22#315311yescharacteristic
483C22#315612yes—
484C22#318712yes—
485C22#320212yes—
486C22#329814yes—
487C22#329814yes—
488C22#331314yes—
489C22#332714yes—
490C4—44no—
491C4—42no—
492C4—48no—
493C4—44no—
494C4—162no—
495C4—162no—
496C4—162no—
497C4#248012yes—
498C4#248012yes—
499C4#268412yes—
500C4#269712yes—
501C2—44no—
502C2#1548011yescharacteristic
503C2#2752214yes—
504C2#2830511yescharacteristic
505C2#2832211yescharacteristic
506C2#2833612yes—
507C2#2835912yes—
508C2#3126212yes—
509C2#3166812yes—
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
5. #11122C2semidirect
6. #11122C2semidirect
7. #11050C2semidirect
8. #10926C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20772
C2×#17872
#151502
#139222
#111222
#110502
#109262
#109212