Finite Group Databasebeta

← #99004

Subgroups — #99004

Order 512.

388 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C8×#124C211yesmaximal, characteristic
2#8273C212yesmaximal, complemented
3#7635C212yesmaximal
4#7042C212yesmaximal, complemented
5C2×C8×Q8C411yescharacteristic
6C2×C8×D4C411yescharacteristic
7C4×#124C2211yescharacteristic
8#1984C411yescharacteristic
9#1954C411yescharacteristic
10#1923—22no—
11#1831C2211yescharacteristic
12#1767C2214yes—
13C8×C22⋊C4—22no—
14C42×C8C2211yescharacteristic
15C2×#334—22no—
16#1277—22no—
17#1272—22no—
18#1270—22no—
19C23×C8—21no—
20C2×#124Q811yescharacteristic
21C2×C4×Q8C2×C411yescharacteristic
22C2×C4×D4C2×C411yescharacteristic
23C8×Q8—42no—
24C8×D4—42no—
25C2×C4⋊C8—24no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8C2×C411yescharacteristic
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C22⋊C8—21no—
30C2×C22⋊C8—22no—
31C2×#97—24no—
32C2×C4×C8—21no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D412yes—
36C2×C4×C8D412yes—
37#391D411yescharacteristic
38#390D411yescharacteristic
39#388—22no—
40#382Q811yescharacteristic
41C4×C22⋊C4—22no—
42C43C2311yescharacteristic
43#336—22no—
44#336—22no—
45#336—24no—
46#334—28no—
47C22×Q8C4⋊C411yescharacteristic
48C22×D4C4⋊C411yescharacteristic
49C23×C4—21no—
50C22×C8—21no—
51C22×C8—21no—
52C22×C8—44no—
53C22×C8—42no—
54C22×C8—44no—
55C22×C8C22⋊C412yes—
56C22×C8C22⋊C411yescharacteristic
57C22×C8C22⋊C411yescharacteristic
58#124—22no—
59#124—22no—
60C4×Q8—41no—
61C4×Q8—41no—
62C4×D4—41no—
63C4×D4—41no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C4⋊C411yescharacteristic
66C2×C4⋊C4C4⋊C411yescharacteristic
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C42—21no—
71C2×C42C22×C411yescharacteristic
72C2×C42C2×D412yes—
73C2×C42C2×D411yescharacteristic
74C2×C42C2×Q811yescharacteristic
75C2×C42#4112yes—
76C4⋊C8—28no—
77C4⋊C8—41no—
78C4⋊C8—41no—
79C4⋊C8—44no—
80C4⋊C8—42no—
81C22⋊C8—44no—
82C22⋊C8—42no—
83#97—84no—
84C4×C8—22no—
85C4×C8—24no—
86C4×C8—41no—
87C4×C8—42no—
88C4×C8—42no—
89#95—22no—
90#95—22no—
91C24—21no—
92C2×Q8—21no—
93C2×Q8—41no—
94C2×Q8—41no—
95C2×Q8#10812yes—
96C2×D4—21no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4#10812yes—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4C2×C22⋊C411yescharacteristic
108C22×C4C2×C4⋊C411yescharacteristic
109C22×C4#11711yescharacteristic
110C22×C4#12112yes—
111C22×C4C22⋊Q812yes—
112C2×C8—21no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—48no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—84no—
123C2×C8—84no—
124C2×C8—84no—
125C2×C8#10012yes—
126C2×C8#10414yes—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—42no—
130C22⋊C4—28no—
131C22⋊C4—42no—
132C22⋊C4—44no—
133C42—21no—
134C42—21no—
135C42—22no—
136C42—22no—
137C42—21no—
138C42—21no—
139C42—24no—
140C42—41no—
141C42—41no—
142C42—41no—
143C23—22no—
144C23—41no—
145C23—42no—
146C23—41no—
147C23—42no—
148C23#38011yescharacteristic
149Q8—42no—
150Q8—42no—
151D4—42no—
152D4—42no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—82no—
172C2×C4C2×#10011yescharacteristic
173C2×C4#41311yescharacteristic
174C2×C4C2×#10412yes—
175C2×C4#42112yes—
176C2×C4C2×#10811yescharacteristic
177C2×C4C2×#10811yescharacteristic
178C2×C4#47114yes—
179C8—41no—
180C8—42no—
181C8—88no—
182C8—81no—
183C8—81no—
184C8#32914yes—
185C22—42no—
186C22—44no—
187C22—81no—
188C22—81no—
189C22—82no—
190C22#180011yescharacteristic
191C22#180612yes—
192C22#181011yescharacteristic
193C22#181211yescharacteristic
194C22#181512yes—
195C4—21no—
196C4—21no—
197C4—24no—
198C4—42no—
199C4—41no—
200C4—42no—
201C4—81no—
202C4—81no—
203C4C2×#32911yescharacteristic
204C4#152912yes—
205C4#153611yescharacteristic
206C2—82no—
207C2#857911yescharacteristic
208C2#862011yescharacteristic
209C2#863611yescharacteristic
210C2#867012yes—
211C2#869412yes—
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8273C2semidirect
4. #7042C2semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
#82732
#76352
#70422