Finite Group Databasebeta

← #6506469

Subgroups — #6506469

Order 512.

2243 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic, complemented
2#35242C211yesmaximal, characteristic
3#29118C214yesmaximal, complemented
4#27285C218yesmaximal, complemented
5#27277C218yesmaximal, complemented
6#25866C214yesmaximal, complemented
7#21766C212yesmaximal
8#20599C212yesmaximal, complemented
9#20529C211yesmaximal, characteristic, complemented
10C24×Q8C2211yescharacteristic
11C2×#536C2212yes—
12C2×#525C2211yescharacteristic
13C2×#489C2214yescomplemented
14C2×#487C2214yescomplemented
15C2×#419C2218yes—
16C2×#407C2212yes—
17#2849—21no—
18#2849—21no—
19#2842—21no—
20#2840—21no—
21#2665—24no—
22#2628C2214yes—
23#2614—22no—
24#2607—22no—
25#2439C2218yes—
26#2436C2214yes—
27#2434C2214yes—
28#2409—24no—
29#2310C2212yes—
30#2280C2212yes—
31#2280C2211yescharacteristic
32#2074C2218yescomplemented
33#2043C2218yescomplemented
34#2042C2218yescomplemented
35#2024C2218yescomplemented
36#1960C2214yescomplemented
37#1958C2214yescomplemented
38#1935C2218yescomplemented
39#1933C2218yescomplemented
40#1895C2214yes—
41#1894C2214yes—
42#1875C2214yes—
43#1873C2214yes—
44#1825C2214yes—
45#1788C2214yes—
46#1787C2218yes—
47#1787C2218yes—
48#1753C2218yes—
49#1750C2214yes—
50C22×#41C2311yescharacteristic
51C23×Q8—21no—
52C23×Q8—22no—
53C23×Q8—28no—
54C23×Q8—48no—
55C23×Q8C2312yes—
56C23×Q8C2314yes—
57C24×C4—21no—
58C24×C4C2311yescharacteristic
59C22×M16C2314yes—
60#536—48no—
61C2×C4⋊Q8C2314yes—
62C2×C4⋊Q8C2314yes—
63C2×#124C2314yes—
64#525—28no—
65C2×#123—21no—
66C2×#123—22no—
67C2×#123—22no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8C2312yes—
70C2×#121C2312yes—
71C2×#120—21no—
72C2×C4×D4—22no—
73C2×#117C2311yescharacteristic
74C2×#117C2314yes—
75C2×#117C2312yes—
76#489—48no—
77#487—48no—
78#419—416no—
79C2×#103C2318yes—
80C2×#103C2318yes—
81C2×#103C2318yes—
82C2×#103C2318yes—
83C2×#103C2318yes—
84C2×#102C2318yes—
85#407—28no—
86C2×C22⋊C8C2314yes—
87C2×C22⋊C8C2314yes—
88C2×#97C2314yes—
89C2×C4×C8C2314yes—
90#400—24no—
91#399—24no—
92#396—28no—
93#395—24no—
94#395—28no—
95#393—24no—
96#393—28no—
97#393—28no—
98#391—24no—
99#391—28no—
100#391C2318yes—
101#389C2318yes—
102#387—28no—
103#386—24no—
104#386—24no—
105#385—24no—
106#385—24no—
107#382C2314yes—
108#381—22no—
109#381—22no—
110#381—22no—
111#381D412yes—
112#380D414yes—
113#380C2314yes—
114#380C2314yes—
115#379—22no—
116#379D411yescharacteristic
117#379D411yescharacteristic
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4C2314yes—
121#340C2318yes—
122#340C2318yes—
123#336C2318yes—
124C25C2×D411yescharacteristic
125C2×#41—24no—
126C2×#41—44no—
127C22×Q8—28no—
128C22×Q8—22no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—216no—
132C22×Q8—28no—
133C22×Q8—28no—
134C22×Q8—28no—
135C22×Q8—48no—
136C22×Q8—44no—
137C22×Q8—48no—
138C22×Q8—464no—
139C22×Q8—416no—
140C22×Q8—48no—
141C22×Q8—44no—
142C22×Q8C2411yescharacteristic
143C22×Q8C2414yes—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4C2411yescharacteristic
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—21no—
150C23×C4—44no—
151C23×C4—48no—
152C23×C4—44no—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4#4118yes—
156C23×C4C2411yescharacteristic
157C23×C4C2412yes—
158C2×M16—28no—
159C2×M16—48no—
160C22×C8C2418yes—
161C4⋊Q8—28no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165#124—28no—
166#124—44no—
167#123—48no—
168#123—48no—
169#123—44no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—416no—
173#121—24no—
174#121—24no—
175#120—44no—
176C4×D4—48no—
177#117—24no—
178#117—44no—
179#117—48no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2412yes—
187C2×C4⋊C4C2418yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4C2×D412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D414yes—
199C2×C22⋊C4C2×D414yes—
200C2×C42—21no—
201C2×C42—24no—
202C2×C42C2414yes—
203#103—216no—
204#103—216no—
205#103—48no—
206#103—48no—
207#103—48no—
208#102—48no—
209C22⋊C8—28no—
210C22⋊C8—28no—
211#97—44no—
212C4×C8—44no—
213#95—24no—
214#95—24no—
215#95—28no—
216#95—28no—
217#95—24no—
218#95—24no—
219#95—28no—
220#95C2×D418yes—
221C24—21no—
222C24—42no—
223C24—44no—
224C24#12414yes—
225C24C22×D412yes—
226C24C22×D411yescharacteristic
227#41—88no—
228C2×Q8—28no—
229C2×Q8—24no—
230C2×Q8—232no—
231C2×Q8—28no—
232C2×Q8—22no—
233C2×Q8—28no—
234C2×Q8—24no—
235C2×Q8—416no—
236C2×Q8—432no—
237C2×Q8—432no—
238C2×Q8—44no—
239C2×Q8—416no—
240C2×Q8—432no—
241C2×Q8—464no—
242C2×Q8—44no—
243C2×Q8—48no—
244C2×Q8—84no—
245C2×Q8—88no—
246C2×Q8H(5,𝔽2)18yes—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—44no—
250C2×D4—48no—
251C2×D4—44no—
252C2×D4—42no—
253C22×C4—28no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—28no—
259C22×C4—21no—
260C22×C4—416no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—416no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—432no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—42no—
270C22×C4#12418yes—
271C22×C4#12414yes—
272C22×C4C22×D414yes—
273C22×C4C22×D414yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D412yes—
276C22×C4C2×#4114yes—
277C22×C4C2×#4114yes—
278C22×C4C2×#4114yes—
279C22×C4C2511yescharacteristic
280M16—88no—
281C2×C8—28no—
282C2×C8—28no—
283C2×C8—28no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—28no—
287C4⋊C4—28no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—44no—
292C4⋊C4—88no—
293C4⋊C4—84no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C22⋊C4—84no—
306C42—24no—
307C42—24no—
308C42—42no—
309C42—84no—
310C23—24no—
311C23—24no—
312C23—24no—
313C23—22no—
314C23—24no—
315C23—44no—
316C23—48no—
317C23—416no—
318C23—44no—
319C23—41no—
320C23—42no—
321C23C2×#12412yes—
322C23C2×#12414yes—
323C23C23×D411yescharacteristic
324Q8—216no—
325Q8—416no—
326Q8—432no—
327Q8—48no—
328Q8—44no—
329Q8—168no—
330D4—44no—
331D4—88no—
332D4—84no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—416no—
346C2×C4—432no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—84no—
357C2×C4C2×#12412yes—
358C2×C4C2×#12414yes—
359C2×C4C22×#4112yes—
360C2×C4C2×H(5,𝔽2)14yes—
361C8—48no—
362C22—22no—
363C22—24no—
364C22—24no—
365C22—24no—
366C22—24no—
367C22—416no—
368C22—48no—
369C22—44no—
370C22—81no—
371C22—82no—
372C22—84no—
373C22C22×#12411yescharacteristic
374C22#348412yes—
375C22#358614yes—
376C4—22no—
377C4—24no—
378C4—48no—
379C4—44no—
380C4—44no—
381C4—81no—
382C4—82no—
383C4—88no—
384C4#349014yes—
385C2—42no—
386C2—44no—
387C2—161no—
388C2#3536711yescharacteristic
389C2#6062012yes—
390C2#6164614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37480C2semidirect
3. #29118C2semidirect
4. #27285C2semidirect
5. #27277C2semidirect
6. #25866C2semidirect
8. #20599C2semidirect
9. #20529C2semidirect
13. C2×#489C22semidirect
14. C2×#487C22semidirect
32. #2074C22semidirect
33. #2043C22semidirect
34. #2042C22semidirect
35. #2024C22semidirect
36. #1960C22semidirect
37. #1958C22semidirect
38. #1935C22semidirect
39. #1933C22semidirect

Maximal subgroups

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

subgroupindex
#374802
#352422
#291182
#272852
#272772
#258662
#217662
#205992
#205292