Finite Group Databasebeta

← #325414

Subgroups — #325414

Order 512.

549 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33256C211yesmaximal, characteristic, complemented
2#29122C212yesmaximal
3#21631C211yesmaximal, characteristic
4#10624C211yesmaximal, characteristic, complemented
5#10623C211yesmaximal, characteristic, complemented
6#10622C211yesmaximal, characteristic
7#10621C211yesmaximal, characteristic
8#9901C212yesmaximal, complemented
9#8651C212yesmaximal, complemented
10#8650C212yesmaximal
11C4×#335C211yesmaximal, characteristic, complemented
12C2×#446C2211yescharacteristic
13C2×#436C2211yescharacteristic
14#2859C2212yes—
15C4×#124C2211yescharacteristic
16#2101—22no—
17#2042—22no—
18#1984C2211yescharacteristic
19#1954C2211yescharacteristic
20#1941—22no—
21#1941—22no—
22#1934—22no—
23#1933—22no—
24#1923—22no—
25#1844C2212yes—
26#1834C2211yescharacteristic
27#1834C2211yescharacteristic
28#1831C2211yescharacteristic
29#1803C2212yes—
30#1803C2212yes—
31#1802C2212yes—
32#1802C2212yes—
33C4×C4⋊C8C2212yes—
34#1753—22no—
35C42×C8C2211yescharacteristic
36C2×#335C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2212yes—
40#1296C2212yes—
41#1277C2211yescharacteristic
42#1277C2211yescharacteristic
43C22×M16—21no—
44C2×#125—22no—
45C2×#124D411yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47C2×C4×D4C2311yescharacteristic
48#446—24no—
49#436—24no—
50C2×#106—22no—
51C2×#106—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×C4×C8—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—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#336—22no—
89#336—22no—
90#336—22no—
91#335—24no—
92C22×Q8C2×D411yescharacteristic
93C22×D4C2×D411yescharacteristic
94C23×C4—21no—
95C2×M16—42no—
96C2×M16—44no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8C2×D412yes—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104#125—84no—
105#124—42no—
106C4×Q8—22no—
107C4×Q8—22no—
108C4×D4—22no—
109C4×D4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C2×D411yescharacteristic
120C2×C4⋊C4C2×D411yescharacteristic
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D412yes—
127C2×C42C2×D411yescharacteristic
128C2×C42#4112yes—
129C2×C42C2411yescharacteristic
130#106—44no—
131#106—44no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—42no—
139#103—44no—
140#103—82no—
141#102—44no—
142#102—82no—
143C22⋊C8—44no—
144C22⋊C8—42no—
145#97—42no—
146C4×C8—24no—
147C4×C8—22no—
148C4×C8—42no—
149C4×C8—82no—
150C4×C8#4114yes—
151#95—22no—
152#95—22no—
153#95—42no—
154#95—42no—
155C24—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4#12112yes—
174C22×C4#12112yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C2×#4111yescharacteristic
178M16—84no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8#12412yes—
194C2×C8#12412yes—
195C2×C8#13514yes—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—84no—
210C4⋊C4—84no—
211C22⋊C4—44no—
212C22⋊C4—42no—
213C22⋊C4—44no—
214C42—22no—
215C42—22no—
216C42—22no—
217C42—21no—
218C42—21no—
219C42—24no—
220C42—41no—
221C42—41no—
222C42C2×#4112yes—
223C422−1+412yes—
224C23—42no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—42no—
229C23C2×#12111yescharacteristic
230Q8—42no—
231Q8—42no—
232D4—42no—
233D4—42no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4#47114yes—
257C2×C4C2×#12411yescharacteristic
258C2×C4#53611yescharacteristic
259C2×C4#54712yes—
260C2×C4#55612yes—
261C2×C4C2×#13512yes—
262C2×C4#57512yes—
263C2×C4#57512yes—
264C2×C4#57611yescharacteristic
265C2×C4#57811yescharacteristic
266C8—22no—
267C8—42no—
268C8—82no—
269C8—82no—
270C8—81no—
271C8—81no—
272C8#48814yes—
273C22—82no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—82no—
278C22#254711yescharacteristic
279C22#303512yes—
280C22#305411yescharacteristic
281C22#306111yescharacteristic
282C22C2×#47112yes—
283C4—24no—
284C4—22no—
285C4—42no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4—162no—
290C4—162no—
291C4C2×#48811yescharacteristic
292C4#314211yescharacteristic
293C4#323312yes—
294C4#335212yes—
295C4#335512yes—
296C2—161no—
297C2#1915912yes—
298C2#1922912yes—
299C2#2870911yescharacteristic
300C2#2872111yescharacteristic
301C2#2873211yescharacteristic
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. #33256C2semidirect
4. #10624C2semidirect
5. #10623C2semidirect
8. #9901C2semidirect
9. #8651C2semidirect
11. C4×#335C2semidirect

Maximal subgroups

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

subgroupindex
#332562
#291222
#216312
#106242
#106232
#106222
#106212
#99012
#86512
#86502
C4×#3352