Finite Group Databasebeta

← #99006

Subgroups — #99006

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#8274C212yesmaximal, 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×#335—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#335—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—42no—
133C22⋊C4—42no—
134C42—21no—
135C42—21no—
136C42—22no—
137C42—22no—
138C42—21no—
139C42—21no—
140C42—22no—
141C42—22no—
142C42—41no—
143C42—41no—
144C42—41no—
145C23—22no—
146C23—41no—
147C23—42no—
148C23—41no—
149C23—42no—
150C23#38011yescharacteristic
151Q8—42no—
152Q8—42no—
153D4—42no—
154D4—42no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—82no—
174C2×C4C2×#10011yescharacteristic
175C2×C4#41311yescharacteristic
176C2×C4C2×#10412yes—
177C2×C4#42112yes—
178C2×C4C2×#10811yescharacteristic
179C2×C4C2×#10811yescharacteristic
180C2×C4#47212yes—
181C2×C4#47312yes—
182C8—41no—
183C8—42no—
184C8—88no—
185C8—81no—
186C8—81no—
187C8#32914yes—
188C22—42no—
189C22—44no—
190C22—81no—
191C22—81no—
192C22—82no—
193C22#180011yescharacteristic
194C22#180612yes—
195C22#181011yescharacteristic
196C22#181211yescharacteristic
197C22#181612yes—
198C4—21no—
199C4—21no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—41no—
204C4—42no—
205C4—81no—
206C4—81no—
207C4C2×#32911yescharacteristic
208C4#152912yes—
209C4#153611yescharacteristic
210C2—82no—
211C2#857911yescharacteristic
212C2#862211yescharacteristic
213C2#863811yescharacteristic
214C2#867112yes—
215C2#869512yes—
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. #8274C2semidirect
4. #7042C2semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
#82742
#76352
#70422