Finite Group Databasebeta

← #325425

Subgroups — #325425

Order 512.

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

normal subgroups only

This group has 549 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#33258C211yesmaximal, characteristic, complemented
2#29140C212yesmaximal
3#21631C211yesmaximal, characteristic
4#10617C212yesmaximal, complemented
5#10615C212yesmaximal
6#9897C212yesmaximal, complemented
7#8657C212yesmaximal, complemented
8#8656C212yesmaximal
9C4×#334C211yesmaximal, characteristic, complemented
10C2×#446C2211yescharacteristic
11C2×#436C2211yescharacteristic
12#2859C2212yes—
13C4×#124C2211yescharacteristic
14#2026—22no—
15#2025—22no—
16#1984C2211yescharacteristic
17#1958—22no—
18#1954C2211yescharacteristic
19#1942—22no—
20#1942—22no—
21#1923—22no—
22#1845C2212yes—
23#1835—22no—
24#1834C2212yes—
25#1831C2211yescharacteristic
26#1803C2214yes—
27#1802C2214yes—
28C4×C4⋊C8C2212yes—
29#1753—22no—
30C42×C8C2211yescharacteristic
31C2×#334C2212yes—
32#1296C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C22×M16—21no—
36C2×#125—22no—
37C2×#124D411yescharacteristic
38C2×C4×Q8C2311yescharacteristic
39C2×C4×D4C2311yescharacteristic
40#446—24no—
41#436—24no—
42C2×#107—22no—
43C2×#107—22no—
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×C4×C8—22no—
54C2×C4×C8D412yes—
55C2×C4×C8D412yes—
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#391D411yescharacteristic
59#390D411yescharacteristic
60#389—24no—
61#389—24no—
62#388—22no—
63#384C2314yes—
64#382—22no—
65#382D411yescharacteristic
66C4×C22⋊C4—22no—
67C43C2311yescharacteristic
68#340—22no—
69#340—22no—
70#340—24no—
71#340—24no—
72#336—22no—
73#336—22no—
74#336—22no—
75#334—24no—
76C22×Q8C2×D411yescharacteristic
77C22×D4C2×D411yescharacteristic
78C23×C4—21no—
79C2×M16—44no—
80C2×M16—42no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—24no—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D412yes—
87#125—84no—
88#124—24no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×D4—22no—
92C4×D4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4C2×D411yescharacteristic
99C2×C4⋊C4C2×D411yescharacteristic
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C42—21no—
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D412yes—
106C2×C42C2×D411yescharacteristic
107C2×C42#4112yes—
108C2×C42C2411yescharacteristic
109#107—44no—
110#107—44no—
111C4⋊C8—24no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—24no—
115C4⋊C8—42no—
116#103—84no—
117#102—84no—
118C22⋊C8—42no—
119C22⋊C8—44no—
120#97—42no—
121C4×C8—24no—
122C4×C8—22no—
123C4×C8—42no—
124C4×C8—44no—
125C4×C8#4114yes—
126#95—22no—
127#95—22no—
128#95—44no—
129C24—21no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4#12112yes—
145C22×C4#12112yes—
146C22×C4C22×D411yescharacteristic
147C22×C4C22×D411yescharacteristic
148C22×C4C2×#4111yescharacteristic
149M16—84no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—44no—
157C2×C8—42no—
158C2×C8—44no—
159C2×C8—42no—
160C2×C8—44no—
161C2×C8#12412yes—
162C2×C8#12412yes—
163C2×C8#13514yes—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—84no—
172C4⋊C4—88no—
173C22⋊C4—44no—
174C22⋊C4—42no—
175C22⋊C4—44no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—42no—
180C42—41no—
181C42—41no—
182C42#13714yes—
183C42C2×#4112yes—
184C422−1+412yes—
185C23—42no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—42no—
190C23C2×#12111yescharacteristic
191Q8—44no—
192D4—44no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—84no—
212C2×C4#47014yes—
213C2×C4C2×#12411yescharacteristic
214C2×C4#53611yescharacteristic
215C2×C4#54712yes—
216C2×C4#55612yes—
217C2×C4C2×#13512yes—
218C2×C4C2×#13712yes—
219C2×C4#57512yes—
220C2×C4#57612yes—
221C8—22no—
222C8—22no—
223C8—24no—
224C8—84no—
225C8—81no—
226C8—81no—
227C22—82no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—82no—
232C22#254711yescharacteristic
233C22#303512yes—
234C22C2×#47011yescharacteristic
235C22#305811yescharacteristic
236C22#306712yes—
237C4—22no—
238C4—24no—
239C4—42no—
240C4—81no—
241C4—81no—
242C4—81no—
243C4—164no—
244C4#313111yescharacteristic
245C4#314011yescharacteristic
246C4#323212yes—
247C4#335714yes—
248C2—161no—
249C2#1915812yes—
250C2#1921912yes—
251C2#2871111yescharacteristic
252C2#2872011yescharacteristic
253C2#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. #33258C2semidirect
4. #10617C2semidirect
6. #9897C2semidirect
7. #8657C2semidirect
9. C4×#334C2semidirect

Maximal subgroups

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

subgroupindex
#332582
#291402
#216312
#106172
#106152
#98972
#86572
#86562
C4×#3342