Finite Group Databasebeta

← #325456

Subgroups — #325456

Order 512.

577 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 577 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic, complemented
2#29148C212yesmaximal
3#21631C211yesmaximal, characteristic
4#10623C212yesmaximal, complemented
5#10621C212yesmaximal
6#9903C212yesmaximal, complemented
7#8657C212yesmaximal, complemented
8#8656C212yesmaximal
9#8273C211yesmaximal, characteristic, complemented
10C2×#446C2211yescharacteristic
11C2×#436C2211yescharacteristic
12#2868C2212yes—
13C4×#124C2211yescharacteristic
14#2024—22no—
15#2023—22no—
16#1990—22no—
17#1990—22no—
18#1984C2211yescharacteristic
19#1959—22no—
20#1954C2211yescharacteristic
21#1923—22no—
22#1836—22no—
23#1834C2212yes—
24#1834C2212yes—
25#1831C2211yescharacteristic
26#1803C2214yes—
27#1802C2214yes—
28#1767C2212yes—
29#1753—22no—
30C42×C8C2211yescharacteristic
31C2×#334C2212yes—
32#1296C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C22×M16—21no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—22no—
39C2×#124D411yescharacteristic
40C2×C4×Q8C2311yescharacteristic
41C2×C4×D4C2311yescharacteristic
42#446—24no—
43#436—24no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8C2314yes—
46C2×C4⋊C8C2311yescharacteristic
47C2×C4⋊C8C2311yescharacteristic
48C2×#103—24no—
49C2×#102—24no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—22no—
52C2×#97—21no—
53C2×#97—22no—
54C2×C4×C8D412yes—
55C2×C4×C8D412yes—
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#391D411yescharacteristic
59#390D411yescharacteristic
60#388—22no—
61#384—22no—
62#384—22no—
63#384—22no—
64#384C2314yes—
65#382D411yescharacteristic
66C4×C22⋊C4—22no—
67C43C2311yescharacteristic
68#340—24no—
69#340—22no—
70#340—22no—
71#340—24no—
72#340—24no—
73#336—22no—
74#336—22no—
75#336—22no—
76#334—24no—
77C22×Q8—42no—
78C22×Q8C2×D411yescharacteristic
79C22×D4C2×D411yescharacteristic
80C23×C4—21no—
81C2×M16—44no—
82C2×M16—42no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—24no—
86C22×C8C2×D412yes—
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C4⋊Q8—44no—
90C4⋊Q8—44no—
91C4⋊Q8—84no—
92C4⋊Q8—84no—
93#124—24no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×D4—22no—
97C4×D4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4C2×D411yescharacteristic
106C2×C4⋊C4C2×D411yescharacteristic
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—21no—
110C2×C42—21no—
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42#4112yes—
115C2×C42C2411yescharacteristic
116C4⋊C8—24no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—24no—
120C4⋊C8—42no—
121#103—84no—
122#102—84no—
123C22⋊C8—44no—
124C22⋊C8—42no—
125#97—42no—
126#97—44no—
127C4×C8—22no—
128C4×C8—42no—
129C4×C8—42no—
130C4×C8#4114yes—
131#95—22no—
132#95—22no—
133C24—21no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—84no—
138C2×Q8—84no—
139C2×Q8—84no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4#12112yes—
152C22×C4#12112yes—
153C22×C4C22×D411yescharacteristic
154C22×C4C22×D411yescharacteristic
155C22×C4C2×#4111yescharacteristic
156M16—84no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—44no—
165C2×C8—44no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—44no—
169C2×C8#12412yes—
170C2×C8#12412yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C4⋊C4—84no—
181C4⋊C4—84no—
182C22⋊C4—42no—
183C22⋊C4—44no—
184C22⋊C4—44no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—24no—
189C42—41no—
190C42—41no—
191C42#13714yes—
192C42C2×#4112yes—
193C422−1+412yes—
194C23—42no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—42no—
199C23C2×#12111yescharacteristic
200Q8—44no—
201Q8—168no—
202D4—44no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—82no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4#47014yes—
222C2×C4#47114yes—
223C2×C4C2×#12411yescharacteristic
224C2×C4#53611yescharacteristic
225C2×C4#54712yes—
226C2×C4#55612yes—
227C2×C4C2×#13712yes—
228C2×C4#57512yes—
229C2×C4#57512yes—
230C2×C4#57612yes—
231C8—22no—
232C8—22no—
233C8—42no—
234C8—84no—
235C8—81no—
236C8—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—82no—
241C22—82no—
242C22#254711yescharacteristic
243C22#304012yes—
244C22C2×#47011yescharacteristic
245C22#305811yescharacteristic
246C22C2×#47112yes—
247C4—22no—
248C4—24no—
249C4—24no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4—164no—
254C4#313511yescharacteristic
255C4#314011yescharacteristic
256C4#324012yes—
257C4#335714yes—
258C2—161no—
259C2#1930212yes—
260C2#1936412yes—
261C2#2871111yescharacteristic
262C2#2872711yescharacteristic
263C2#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. #33289C2semidirect
4. #10623C2semidirect
6. #9903C2semidirect
7. #8657C2semidirect
9. #8273C2semidirect

Maximal subgroups

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

subgroupindex
#332892
#291482
#216312
#106232
#106212
#99032
#86572
#86562
#82732