Finite Group Databasebeta

← #102902

Subgroups — #102902

Order 512.

494 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 112 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#29287C212yesmaximal
2#28255C211yesmaximal, characteristic
3#9870C212yesmaximal
4#9637C211yesmaximal, characteristic
5#7050C211yesmaximal, characteristic
6C2×#462—22no—
7C4×C4⋊Q8C2212yes—
8C4×C4⋊Q8C2211yescharacteristic
9#2088—24no—
10#2056—22no—
11#1988—22no—
12#1987—22no—
13#1948—22no—
14#1948—22no—
15#1842C2211yescharacteristic
16#1842C2212yes—
17#1803—22no—
18C42×C8C2211yescharacteristic
19C2×Q8⋊C8—22no—
20#1279—24no—
21C22×Q16—22no—
22C2×C4⋊Q8D412yes—
23C2×C4⋊Q8D411yescharacteristic
24C2×C4×Q8—22no—
25C2×C4×Q8—21no—
26#462—44no—
27#462—44no—
28C2×#106—24no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8D412yes—
33C2×C4⋊C8D414yes—
34C2×#103—24no—
35C2×C4×C8—21no—
36C2×C4×C8—22no—
37#391—22no—
38#391—21no—
39#389—22no—
40#389—22no—
41#389—22no—
42#384D412yes—
43#384D411yescharacteristic
44C4×C4⋊C4—22no—
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C43C2311yescharacteristic
48#340—22no—
49#340—24no—
50#340—24no—
51#340—24no—
52Q8⋊C8—44no—
53C22×Q8—22no—
54C22×Q8—21no—
55C2×Q16—48no—
56C2×Q16—82no—
57C22×C8—22no—
58C22×C8—24no—
59C22×C8—41no—
60C4⋊Q8—22no—
61C4⋊Q8—21no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65C4⋊Q8SD1614yes—
66C4⋊Q8Q1612yescomplemented
67C4×Q8—42no—
68C4×Q8—42no—
69C4×Q8—42no—
70C4×Q8—42no—
71C4×Q8—42no—
72C4×Q8—42no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C42—42no—
84C2×C42—41no—
85C2×C42C2×D411yescharacteristic
86C2×C42C2×D412yes—
87C2×C42C2×D412yes—
88C2×C42C2×D411yescharacteristic
89C2×C42#4111yescharacteristic
90#106—84no—
91C4⋊C8—42no—
92C4⋊C8—44no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99#103—48no—
100C4×C8—22no—
101C4×C8—24no—
102C4×C8—41no—
103C4×C8—42no—
104#95—42no—
105#95—42no—
106#95—42no—
107C2×Q8—24no—
108C2×Q8—22no—
109C2×Q8—42no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8—41no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C22×C4—42no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—41no—
121C22×C4#12012yes—
122C22×C4#12011yescharacteristic
123C22×C4#12112yes—
124C22×C4#12111yescharacteristic
125C22×C4#12711yescharacteristic
126Q16—88no—
127C2×C8—42no—
128C2×C8—44no—
129C2×C8—44no—
130C2×C8—44no—
131C2×C8—44no—
132C2×C8—42no—
133C2×C8—44no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—44no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C42—21no—
152C42—21no—
153C42—22no—
154C42—22no—
155C42—41no—
156C42—82no—
157C42—82no—
158C42—81no—
159C42—81no—
160C42C2×SD1612yes—
161C42C2×Q1611yescharacteristic
162C42#13512yes—
163C42#13511yescharacteristic
164C42AGL(1,ℤ/8)12yes—
165C42#13712yes—
166C42#13712yes—
167C23#39211yescharacteristic
168Q8—44no—
169Q8—44no—
170Q8—44no—
171Q8—82no—
172Q8—82no—
173Q8—82no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—82no—
183C2×C4—81no—
184C2×C4—82no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4—81no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4#44812yes—
192C2×C4#44912yes—
193C2×C4#45012yes—
194C2×C4#45112yes—
195C2×C4#45212yes—
196C2×C4#45212yes—
197C2×C4#45311yescharacteristic
198C2×C4#45411yescharacteristic
199C2×C4#45512yes—
200C2×C4#45612yes—
201C2×C4#46512yes—
202C2×C4#46711yescharacteristic
203C2×C4#46812yes—
204C2×C4#47011yescharacteristic
205C8—44no—
206C8—82no—
207C8—84no—
208C22#200012yes—
209C22#200111yescharacteristic
210C22#200611yescharacteristic
211C22#200712yes—
212C22#201711yescharacteristic
213C4—22no—
214C4—22no—
215C4—21no—
216C4—21no—
217C4—41no—
218C4—82no—
219C4—84no—
220C4—162no—
221C4—161no—
222C4#162911yescharacteristic
223C4#163411yescharacteristic
224C4#163512yes—
225C4#164012yes—
226C4#164512yes—
227C4#164812yes—
228C4#165312yes—
229C2#1023412yes—
230C2#1023511yescharacteristic
231C2#1025712yes—
232C2#1025811yescharacteristic
233C2#1030211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
66. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#292872
#282552
#98702
#96372
#70502