Finite Group Databasebeta

← #98984

Subgroups — #98984

Order 512.

398 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 110 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#8579C212yesmaximal
3#8273C212yesmaximal, complemented
4#7049C212yesmaximal, 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—
16C2×#329—22no—
17#1295—22no—
18#1277—22no—
19C23×C8—21no—
20C2×#124D411yescharacteristic
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#391Q811yescharacteristic
38#390Q811yescharacteristic
39#388—22no—
40#382D411yescharacteristic
41C4×C22⋊C4—22no—
42C43C2311yescharacteristic
43#336—22no—
44#336—22no—
45#336—24no—
46#334—28no—
47#329—44no—
48C22×Q8C4⋊C411yescharacteristic
49C22×D4C4⋊C411yescharacteristic
50C23×C4—21no—
51C22×C8—21no—
52C22×C8—21no—
53C22×C8—44no—
54C22×C8—42no—
55C22×C8—44no—
56C22×C8C22⋊C412yes—
57C22×C8C22⋊C411yescharacteristic
58C22×C8C22⋊C411yescharacteristic
59#124—22no—
60#124SD1614yes—
61C4×Q8—41no—
62C4×Q8—41no—
63C4×D4—41no—
64C4×D4—41no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4C4⋊C411yescharacteristic
67C2×C4⋊C4C4⋊C411yescharacteristic
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42C22×C411yescharacteristic
73C2×C42C2×D412yes—
74C2×C42C2×D411yescharacteristic
75C2×C42C2×Q811yescharacteristic
76C2×C42#4112yes—
77C4⋊C8—28no—
78C4⋊C8—41no—
79C4⋊C8—41no—
80C4⋊C8—44no—
81C4⋊C8—42no—
82C22⋊C8—44no—
83C22⋊C8—42no—
84#97—44no—
85#97—44no—
86C4×C8—22no—
87C4×C8—24no—
88C4×C8—41no—
89C4×C8—42no—
90C4×C8—42no—
91#95—22no—
92#95—22no—
93C24—21no—
94C2×Q8—21no—
95C2×Q8—41no—
96C2×Q8—41no—
97C2×Q8#10612yes—
98C2×D4—21no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4#10612yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4C2×C22⋊C411yescharacteristic
110C22×C4C2×C4⋊C411yescharacteristic
111C22×C4#11711yescharacteristic
112C22×C4#12112yes—
113C22×C4C22⋊Q812yes—
114C2×C8—21no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—48no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—84no—
125C2×C8—84no—
126C2×C8—84no—
127C2×C8#10012yes—
128C2×C8#10414yes—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—42no—
132C22⋊C4—28no—
133C22⋊C4—44no—
134C22⋊C4—42no—
135C42—21no—
136C42—21no—
137C42—22no—
138C42—22no—
139C42—41no—
140C42—42no—
141C42—41no—
142C42—41no—
143C42C2×SD1612yes—
144C42#13512yes—
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—21no—
158C2×C4—22no—
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×#10611yescharacteristic
179C2×C4#42711yescharacteristic
180C2×C4#46514yes—
181C8—41no—
182C8—42no—
183C8—88no—
184C8—81no—
185C8—81no—
186C8#32914yescomplemented
187C22—42no—
188C22—44no—
189C22—81no—
190C22—81no—
191C22—82no—
192C22#180011yescharacteristic
193C22#180612yes—
194C22#180711yescharacteristic
195C22#181311yescharacteristic
196C22#181412yes—
197C4—21no—
198C4—21no—
199C4—24no—
200C4—42no—
201C4—41no—
202C4—42no—
203C4—81no—
204C4—81no—
205C4C2×#32911yescharacteristic
206C4#152912yes—
207C4#153611yescharacteristic
208C2—82no—
209C2#857911yescharacteristic
210C2#861411yescharacteristic
211C2#863011yescharacteristic
212C2#866612yes—
213C2#869012yes—
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
3. #8273C2semidirect
4. #7049C2semidirect
186. C8#329semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
#85792
#82732
#70492