Finite Group Databasebeta

← #6531995

Subgroups — #6531995

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic
3#27614C214yesmaximal
4#27586C214yesmaximal
5#25062C218yesmaximal, complemented
6#24996C218yesmaximal, complemented
7#21982C212yesmaximal
8#21818C211yesmaximal, characteristic
9#20734C212yesmaximal
10C2×#519—21no—
11C22×C4×Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2212yes—
14#2582—24no—
15#2582C2218yes—
16C2×#398—22no—
17#2351—24no—
18Q8×C4⋊C4C2214yes—
19Q8×C4⋊C4C2214yes—
20#2316—22no—
21#2316—21no—
22C2×#391—21no—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389C2212yes—
26#2285C2214yes—
27#2285C2212yes—
28C2×#384C2211yescharacteristic
29#2100C2218yes—
30#2095C2218yes—
31#2083C2218yes—
32#2072C2218yes—
33#1984C2218yes—
34#1947C2218yes—
35#1928C22116yes—
36#1921C22116yes—
37#1852C2218yes—
38#1845C2214yes—
39#1834C2214yes—
40#1824C2214yes—
41#1817C2214yes—
42#1803C2214yes—
43#1803C2214yes—
44#1787C2214yes—
45C4×#103C2218yes—
46C23×Q8D411yescharacteristic
47C23×Q8C2311yescharacteristic
48#519—48no—
49C2×C4×Q8—24no—
50C2×C4×Q8—24no—
51C2×C4×Q8—22no—
52C2×C4×Q8—28no—
53C2×C4×Q8—28no—
54C2×C4×Q8D414yes—
55C2×C4×Q8C2314yes—
56C2×C4×Q8C2314yes—
57C2×#117—24no—
58C2×#117—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#431—28no—
71C2×#107C23116yes—
72C2×C4⋊C8C2314yes—
73C2×#103C2318yes—
74C2×#103C2318yes—
75C2×C22⋊C8C2314yes—
76C2×C4×C8C2314yes—
77#398—216no—
78#391—28no—
79#391—24no—
80#391C2318yes—
81#389—24no—
82#389—24no—
83#389—28no—
84#389—28no—
85#389—28no—
86#389—24no—
87#389C2318yes—
88#384—22no—
89#384C2312yes—
90#384C2312yes—
91#384C2314yes—
92#384C2314yes—
93#384C2314yes—
94#382—24no—
95#382—22no—
96#380—24no—
97#380—22no—
98#380C2314yes—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4C2318yes—
102C4×C4⋊C4C2314yes—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106#340C2318yes—
107#340C2318yes—
108#340C23116yes—
109#336C2318yes—
110#336C2318yes—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—21no—
115C22×Q8—24no—
116C22×Q8—42no—
117C22×Q8Q1618yes—
118C22×Q8C2×D414yes—
119C22×Q8C2412yes—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8C2418yes—
129C4×Q8—216no—
130C4×Q8—28no—
131C4×Q8—28no—
132C4×Q8—28no—
133C4×Q8—28no—
134C4×Q8—416no—
135C4×Q8—416no—
136C4×Q8—48no—
137#117—416no—
138#117—48no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4C2×D414yes—
162C2×C4⋊C4C2×D418yes—
163C2×C4⋊C4C2412yes—
164C2×C4⋊C4C2414yes—
165C2×C4⋊C4C2414yes—
166C2×C4⋊C4C2414yes—
167C2×C22⋊C4—24no—
168C2×C42—28no—
169C2×C42—28no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—44no—
173C2×C42C2×D414yes—
174C2×C42C2×D412yes—
175C2×C42#4118yes—
176C2×C42C2414yes—
177C2×C42C2412yes—
178#107—416no—
179C4⋊C8—28no—
180#103—216no—
181#103—48no—
182C22⋊C8—28no—
183C4×C8—28no—
184#95—28no—
185#95—28no—
186#95—28no—
187#95—24no—
188#95—28no—
189#95—24no—
190C24C22×D411yescharacteristic
191C2×Q8—24no—
192C2×Q8—28no—
193C2×Q8—28no—
194C2×Q8—22no—
195C2×Q8—232no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—416no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8C2×Q16116yes—
202C2×Q82−1+414yes—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4#12314yes—
227C22×C4#12314yes—
228C22×C4C2×Q1618yes—
229C22×C4C2×Q1614yes—
230C22×C4AGL(1,ℤ/8)14yes—
231C22×C4C22×D414yes—
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D412yes—
234C22×C4C22×D414yes—
235C22×C4C22×D412yes—
236C22×C4C2×#4114yes—
237C22×C4C2511yescharacteristic
238C2×C8—28no—
239C2×C8—28no—
240C2×C8—28no—
241C4⋊C4—216no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—28no—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—44no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—42no—
251C4⋊C4—48no—
252C4⋊C4—48no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C4⋊C4—84no—
260C4⋊C4—88no—
261C22⋊C4—48no—
262C42—28no—
263C42—28no—
264C42—416no—
265C42—48no—
266C42—48no—
267C42#12318yes—
268C42C2×#4114yes—
269C42C2×#4114yes—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23C2×#12312yes—
275C23C22×Q1612yes—
276C23C2×AGL(1,ℤ/8)12yes—
277C23C23×D411yescharacteristic
278Q8—232no—
279Q8—28no—
280Q8—48no—
281Q8—44no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—216no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—28no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—44no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—88no—
313C2×C4#484116yes—
314C2×C4C2×#12312yes—
315C2×C4C2×#12314yes—
316C2×C4C2×#12314yes—
317C2×C4C22×Q1618yes—
318C2×C4C22×Q1614yes—
319C2×C4C2×AGL(1,ℤ/8)14yes—
320C2×C4C22×#4112yes—
321C2×C4C2×2−1+412yes—
322C2×C4#58512yes—
323C8—48no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22C2×#48414yes—
331C22C22×#12311yescharacteristic
332C22#348611yescharacteristic
333C22#348711yescharacteristic
334C22C23×Q1611yescharacteristic
335C22C22×AGL(1,ℤ/8)11yescharacteristic
336C22#358511yescharacteristic
337C22#358611yescharacteristic
338C4—22no—
339C4—28no—
340C4—24no—
341C4—44no—
342C4—44no—
343C4—82no—
344C4—84no—
345C4—84no—
346C4—84no—
347C4C2×#48418yes—
348C4#349514yes—
349C2—24no—
350C2#3567211yescharacteristic
351C2C22×#48411yescharacteristic
352C2#6052011yescharacteristic
353C2#6182111yescharacteristic
354C2#6182811yescharacteristic
355C2#6188411yescharacteristic
356C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #25062C2semidirect
6. #24996C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
#341822
#276142
#275862
#250622
#249962
#219822
#218182
#207342