Finite Group Databasebeta

← #6477620

Subgroups — #6477620

Order 512.

1942 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C212yesmaximal
2#35011C211yesmaximal, characteristic
3C2×#2346C211yesmaximal, characteristic
4#33108C214yesmaximal, complemented
5#29757C214yesmaximal, complemented
6#24598C218yesmaximal, complemented
7#24498C218yesmaximal, complemented
8#21934C211yesmaximal, characteristic
9#21765C212yesmaximal
10C22×C22⋊Q8C2212yes—
11C22×#117C2211yescharacteristic
12#3095—216no—
13C2×#484C2214yes—
14C2×#485C2218yes—
15C2×#483C2214yes—
16C2×#428C2214yes—
17C2×#419C2214yes—
18C2×#407C2212yes—
19#2857C2214yes—
20#2659—21no—
21#2659—21no—
22#2583—24no—
23#2502C2214yes—
24#2434C2214yes—
25#2346—24no—
26#2346C2218yes—
27#2315—22no—
28C2×#389C2212yes—
29C2×#388—22no—
30C2×#382—21no—
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×#376—21no—
35#2083C2218yescomplemented
36#2068C22116yescomplemented
37#2043C2218yescomplemented
38#1981C2218yescomplemented
39#1948C2218yescomplemented
40#1927C22116yescomplemented
41#1894C2218yes—
42#1872C2218yes—
43#1851C2218yes—
44#1823C2214yes—
45#1797C2214yes—
46#1787C2214yes—
47C23×Q8C2312yes—
48C24×C4C2311yescharacteristic
49C22×M16C2314yes—
50C2×C4⋊Q8C2314yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—28no—
54C2×C22⋊Q8—28no—
55C2×C22⋊Q8C2314yes—
56C2×C22⋊Q8C2314yes—
57C2×#117—22no—
58C2×#117—24no—
59C2×#117D418yes—
60C2×#117C2314yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4D412yes—
71C22×C42—21no—
72C22×C42—22no—
73C22×C42C2312yes—
74#485—232no—
75#484—216no—
76#483—216no—
77#428—216no—
78C2×#107C2318yes—
79C2×#106C2318yes—
80C2×C4⋊C8C2318yes—
81#419—216no—
82C2×#103C2318yes—
83C2×#103C2318yes—
84C2×#103C23116yes—
85#407—28no—
86C2×C22⋊C8C2314yes—
87C2×C22⋊C8C2314yes—
88#398—24no—
89#398—24no—
90#395—24no—
91#395—24no—
92#391C2318yes—
93#389—24no—
94#389—24no—
95#389—24no—
96#389C2318yes—
97#388—216no—
98#385—28no—
99#385—28no—
100#384C2314yes—
101#382—28no—
102#380—24no—
103#380—24no—
104#380—22no—
105#380C2314yes—
106C4×C4⋊C4—24no—
107C4×C4⋊C4C2318yes—
108C4×C22⋊C4D414yes—
109C4×C22⋊C4C2314yes—
110#376—28no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340C23116yes—
117#336C2318yes—
118C25C2×D411yescharacteristic
119C22×Q8—22no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—48no—
123C22×Q8C2414yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—44no—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D414yes—
134C23×C4C2×D412yes—
135C23×C4#4114yes—
136C23×C4C2411yescharacteristic
137C23×C4C2412yes—
138C2×M16—28no—
139C2×M16—48no—
140C22×C8C2418yes—
141C4⋊Q8—48no—
142C22⋊Q8—28no—
143C22⋊Q8—28no—
144C22⋊Q8—28no—
145C22⋊Q8—28no—
146C22⋊Q8—48no—
147C22⋊Q8—48no—
148#117—216no—
149#117—28no—
150#117C2×D4116yes—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2414yes—
172C2×C4⋊C4C2414yes—
173C2×C4⋊C4C2412yes—
174C2×C4⋊C4C2412yes—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—48no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4C2×D418yes—
184C2×C42—28no—
185C2×C42—24no—
186C2×C42—28no—
187C2×C42—44no—
188C2×C42—48no—
189C2×C42C2×D414yes—
190C2×C42C2414yes—
191#107—216no—
192#106—216no—
193C4⋊C8—48no—
194#103—216no—
195#103—216no—
196#103—416no—
197C22⋊C8—28no—
198C22⋊C8—28no—
199#95—24no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—28no—
204#95C2×D414yes—
205#95C2×D414yes—
206C24—24no—
207C24—41no—
208C24—42no—
209C24—41no—
210C24#12114yes—
211C24C22×D412yes—
212C24C22×D411yescharacteristic
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—216no—
216C2×Q8—416no—
217C2×Q8—44no—
218C2×Q8—84no—
219C2×Q8—88no—
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4#12114yes—
250C22×C4#12118yes—
251C22×C4#13718yes—
252C22×C4C22×D414yes—
253C22×C4C22×D412yes—
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D412yes—
256C22×C4C22×D414yes—
257C22×C4C22×D414yes—
258C22×C4C2×#4112yes—
259C22×C4C2×#4114yes—
260C22×C4C2511yescharacteristic
261M16—416no—
262C2×C8—28no—
263C2×C8—216no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—28no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4—88no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4C22×D414yes—
281C4⋊C4C22×D414yes—
282C22⋊C4—216no—
283C22⋊C4—216no—
284C22⋊C4—48no—
285C22⋊C4—48no—
286C22⋊C4—84no—
287C42—216no—
288C42—28no—
289C42—44no—
290C42—416no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—24no—
297C23—24no—
298C23—28no—
299C23—44no—
300C23—44no—
301C23—84no—
302C23—84no—
303C23C2×#12114yes—
304C23C2×#12112yes—
305C23C2×#13714yes—
306C23C23×D411yescharacteristic
307Q8—48no—
308Q8—48no—
309Q8—88no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—28no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—48no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—48no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—84no—
340C2×C4—88no—
341C2×C4—88no—
342C2×C4—88no—
343C2×C4C2×#12114yes—
344C2×C4C2×#12112yes—
345C2×C4C2×#13718yes—
346C2×C4C23×D411yescharacteristic
347C2×C4C22×#4111yescharacteristic
348C2×C4#58514yes—
349C8—48no—
350C22—22no—
351C22—24no—
352C22—22no—
353C22—24no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—44no—
358C22—44no—
359C22—88no—
360C22#303914yes—
361C22C22×#12111yescharacteristic
362C22#348512yes—
363C22C22×#13712yes—
364C22#358511yescharacteristic
365C22#358611yescharacteristic
366C4—28no—
367C4—22no—
368C4—24no—
369C4—44no—
370C4—44no—
371C4—44no—
372C4—84no—
373C4—84no—
374C4—84no—
375C4#344912yes—
376C4#349112yes—
377C2—24no—
378C2—44no—
379C2#3508911yescharacteristic
380C2C2×#303911yescharacteristic
381C2#6042511yescharacteristic
382C2#6170512yes—
383C2#6172112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33108C2semidirect
5. #29757C2semidirect
6. #24598C2semidirect
7. #24498C2semidirect
35. #2083C22semidirect
36. #2068C22semidirect
37. #2043C22semidirect
38. #1981C22semidirect
39. #1948C22semidirect
40. #1927C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
#350112
C2×#23462
#331082
#297572
#245982
#244982
#219342
#217652