Finite Group Databasebeta

← #449607

Subgroups — #449607

Order 512.

1848 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3C2×#1853C212yesmaximal, complemented
4C2×#1853C212yesmaximal, complemented
5#11327C214yesmaximal, complemented
6#11200C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10964C211yesmaximal, characteristic
9C22×#123—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#106C2214yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16#2487—28no—
17#2486—28no—
18C2×#400—21no—
19C2×#399—21no—
20C2×#398—21no—
21C2×#397—21no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#386—21no—
25C2×#385—21no—
26C2×#381—21no—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31#1853—28no—
32#1853—28no—
33C2×#340C2214yes—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C23×D4D411yescharacteristic
38C24×C4—21no—
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2314yes—
43C2×#123—28no—
44C2×#123—44no—
45C2×#123—44no—
46C2×C4×D4—24no—
47C2×C4×D4—24no—
48C2×C4×D4—42no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—42no—
58C22×C4⋊C4—42no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#106—28no—
72C2×#106—28no—
73C2×#106—44no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—28no—
77C2×C22⋊C8—28no—
78#400—28no—
79#399—28no—
80#398—28no—
81#397—28no—
82#389—24no—
83#389—24no—
84#389—24no—
85#389—24no—
86#386—28no—
87#385—28no—
88#381—28no—
89#380—24no—
90#380—24no—
91#380—24no—
92#380—24no—
93#380D418yes—
94#380D418yes—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101#340—28no—
102#340—28no—
103#340—28no—
104#336—42no—
105C25C2×D411yescharacteristic
106C25C2×D411yescharacteristic
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—41no—
112C22×D4—41no—
113C22×D4SD1614yes—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—84no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D414yes—
141C23×C4C2×D414yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—24no—
145C22×C8—24no—
146C22×C8—24no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—24no—
150C22×C8#4118yes—
151#123—432no—
152C4×D4—416no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
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—28no—
171C2×C4⋊C4—416no—
172C2×C4⋊C4—416no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—84no—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2×D414yes—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—84no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205#106—416no—
206C22⋊C8—216no—
207C22⋊C8—216no—
208#95—24no—
209#95—24no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—28no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—24no—
219C24—24no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—24no—
225C24—24no—
226C24#12012yes—
227C24#12012yes—
228C24C2×SD1612yes—
229C24C2×SD1612yes—
230C24AGL(1,ℤ/8)12yes—
231C24#13712yes—
232C24C22×D411yescharacteristic
233C2×D4—28no—
234C2×D4—28no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
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—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—88no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4—84no—
309C22×C4#12012yes—
310C22×C4#12012yes—
311C22×C4C2×SD1612yes—
312C22×C4C22×D412yes—
313C22×C4C22×D412yes—
314C22×C4C2×#4114yes—
315C22×C4H(5,𝔽2)12yes—
316C22×C42−1+414yes—
317C2×C8—28no—
318C2×C8—28no—
319C2×C8—28no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—44no—
324C4⋊C4—28no—
325C4⋊C4—28no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—416no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—88no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—416no—
344C22⋊C4—88no—
345C42—44no—
346C23—28no—
347C23—28no—
348C23—28no—
349C23—28no—
350C23—22no—
351C23—22no—
352C23—24no—
353C23—24no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23#44814yes—
379C23#45014yes—
380C23C2×#12011yescharacteristic
381C23C2×#12011yescharacteristic
382C23#53411yescharacteristic
383C23#54714yes—
384C23C22×SD1611yescharacteristic
385C23C2×AGL(1,ℤ/8)11yescharacteristic
386C23C2×#13711yescharacteristic
387C23#57511yescharacteristic
388C23#57714yes—
389D4—416no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—816no—
426C2×C4—816no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—84no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—84no—
437C2×C4#52512yes—
438C2×C4#52512yes—
439C2×C4#53612yes—
440C2×C4#53612yes—
441C2×C4#53914yes—
442C2×C4D4214yes—
443C2×C4#54714yes—
444C2×C4#54714yes—
445C2×C4#55912yes—
446C2×C4#56312yes—
447C8—48no—
448C22—28no—
449C22—28no—
450C22—24no—
451C22—24no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22#240311yescharacteristic
467C22#240612yes—
468C22#240612yes—
469C22#272412yes—
470C22C2×#45011yescharacteristic
471C22C2×#44811yescharacteristic
472C22#300611yescharacteristic
473C22#300711yescharacteristic
474C22#300912yes—
475C22#301012yes—
476C22#318711yescharacteristic
477C22#319311yescharacteristic
478C22#320012yes—
479C22#333514yes—
480C22#334414yes—
481C22#334514yes—
482C22#335114yes—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—84no—
487C4—84no—
488C4—84no—
489C4—81no—
490C4—84no—
491C4#245514yes—
492C4#268014yes—
493C2—44no—
494C2—44no—
495C2#1594911yescharacteristic
496C2#2391911yescharacteristic
497C2#2392611yescharacteristic
498C2#2408912yes—
499C2#2409712yes—
500C2#2410112yes—
501C2#2411112yes—
502C2#3245912yes—
503C2#3246812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1853C2semidirect
4. C2×#1853C2semidirect
5. #11327C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#24862
C2×#18532
#113272
#112002
#111992
#109642