Finite Group Databasebeta

← #325450

Subgroups — #325450

Order 512.

533 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33291C211yesmaximal, characteristic, complemented
2#29131C212yesmaximal
3#21631C211yesmaximal, characteristic
4#10626C212yesmaximal, complemented
5#10625C212yesmaximal
6#9907C212yesmaximal, complemented
7#8657C211yesmaximal, characteristic, complemented
8#8657C211yesmaximal, characteristic, complemented
9#8656C211yesmaximal, characteristic
10#8656C211yesmaximal, characteristic
11#8273C211yesmaximal, characteristic, complemented
12C2×#446C2211yescharacteristic
13C2×#436C2211yescharacteristic
14#2859C2212yes—
15C4×#124C2211yescharacteristic
16#2101—22no—
17#2042—22no—
18#1991—22no—
19#1991—22no—
20#1984C2211yescharacteristic
21#1954C2211yescharacteristic
22#1936—22no—
23#1935—22no—
24#1923—22no—
25#1834C2212yes—
26#1834C2211yescharacteristic
27#1834C2211yescharacteristic
28#1831C2211yescharacteristic
29#1803C2212yes—
30#1803C2212yes—
31#1802C2212yes—
32#1802C2212yes—
33#1767C2212yes—
34#1753—22no—
35C42×C8C2211yescharacteristic
36C2×#334C2211yescharacteristic
37C2×#334C2211yescharacteristic
38#1296C2212yes—
39#1296C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1277C2212yes—
43C22×M16—21no—
44C2×#125—22no—
45C2×#124D411yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47C2×C4×D4C2311yescharacteristic
48#446—22no—
49#446—22no—
50#436—22no—
51#436—22no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2311yescharacteristic
56C2×C4⋊C8C2311yescharacteristic
57C2×#103—22no—
58C2×#103—22no—
59C2×#102—22no—
60C2×#102—22no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—22no—
63C2×#97—21no—
64C2×#97—22no—
65C2×C4×C8D412yes—
66C2×C4×C8D412yes—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#391D411yescharacteristic
70#390D411yescharacteristic
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#388—22no—
76#384C2312yes—
77#384C2312yes—
78#382—22no—
79#382D411yescharacteristic
80C4×C22⋊C4—22no—
81C43C2311yescharacteristic
82#340—24no—
83#340—21no—
84#340—21no—
85#340—21no—
86#340—21no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—22no—
94#334—22no—
95#334—22no—
96C22×Q8C2×D411yescharacteristic
97C22×D4C2×D411yescharacteristic
98C23×C4—21no—
99C2×M16—44no—
100C2×M16—42no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8C2×D412yes—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108#125—44no—
109#125—44no—
110#124—42no—
111C4×Q8—21no—
112C4×Q8—21no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×D4—21no—
116C4×D4—21no—
117C4×D4—21no—
118C4×D4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—21no—
133C2×C42—21no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42#4112yes—
138C2×C42C2411yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—42no—
148#103—44no—
149#103—82no—
150#102—44no—
151#102—82no—
152C22⋊C8—44no—
153C22⋊C8—42no—
154#97—42no—
155#97—82no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—42no—
159C4×C8—42no—
160C4×C8#4112yes—
161C4×C8#4112yes—
162#95—22no—
163#95—22no—
164#95—42no—
165#95—42no—
166C24—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—22no—
174C2×D4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4#12112yes—
185C22×C4#12112yes—
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4C2×#4111yescharacteristic
189M16—84no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8#12412yes—
206C2×C8#12412yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—84no—
224C4⋊C4—84no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—42no—
234C42—41no—
235C42—41no—
236C42C2×#4111yescharacteristic
237C42C2×#4111yescharacteristic
238C422−1+411yescharacteristic
239C422−1+411yescharacteristic
240C23—42no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23C2×#12111yescharacteristic
246Q8—42no—
247Q8—42no—
248D4—42no—
249D4—42no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4C2×#12411yescharacteristic
274C2×C4#53611yescharacteristic
275C2×C4#54711yescharacteristic
276C2×C4#54711yescharacteristic
277C2×C4#55611yescharacteristic
278C2×C4#55611yescharacteristic
279C2×C4#57512yes—
280C2×C4#57512yes—
281C2×C4#57512yes—
282C2×C4#57712yes—
283C8—22no—
284C8—22no—
285C8—42no—
286C8—82no—
287C8—82no—
288C8—81no—
289C8—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—82no—
294C22—82no—
295C22#254711yescharacteristic
296C22#304012yes—
297C22#305411yescharacteristic
298C22#306011yescharacteristic
299C22#306712yes—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—162no—
308C4—162no—
309C4#313511yescharacteristic
310C4#314011yescharacteristic
311C4#324012yes—
312C4#335112yes—
313C4#335112yes—
314C2—161no—
315C2#1930312yes—
316C2#1936612yes—
317C2#2871011yescharacteristic
318C2#2872711yescharacteristic
319C2#2873311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33291C2semidirect
4. #10626C2semidirect
6. #9907C2semidirect
7. #8657C2semidirect
8. #8657C2semidirect
11. #8273C2semidirect

Maximal subgroups

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

subgroupindex
#332912
#291312
#216312
#106262
#106252
#99072
#86572
#86562
#82732