Finite Group Databasebeta

← #96793

Subgroups — #96793

Order 512.

385 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 126 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#21266C211yesmaximal, characteristic
2#8260C211yesmaximal, characteristic, complemented
3#8133C211yesmaximal, characteristic, complemented
4#7560C212yesmaximal
5#7027C212yesmaximal, complemented
6C2×#433—21no—
7#2293C2211yescharacteristic
8#1954C412yes—
9#1939C412yes—
10#1923—22no—
11#1767C2211yescharacteristic
12C4×C4⋊C8C2212yes—
13#1750—21no—
14#1744C2212yes—
15C42×C8C2211yescharacteristic
16C2×#322—21no—
17#1273—21no—
18#1273—22no—
19#1271C414yes—
20#1270C414yes—
21C2×C4×D4C2×C411yescharacteristic
22C2×#117—21no—
23#433—42no—
24#433—42no—
25C2×C4⋊C8—22no—
26C2×C4⋊C8—22no—
27C2×C4⋊C8C2×C412yes—
28C2×C22⋊C8—22no—
29C2×C22⋊C8—22no—
30C2×#97—21no—
31C2×C4×C8—22no—
32C2×C4×C8—22no—
33C2×C4×C8C2×C412yes—
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D412yes—
36C2×C4×C8D411yescharacteristic
37#390C2×C412yes—
38#389C2×C412yes—
39#388—22no—
40C4×C4⋊C4C2×C411yescharacteristic
41C4×C22⋊C4—21no—
42C43C2311yescharacteristic
43#336—22no—
44#336—21no—
45#336—22no—
46#336C2×C414yes—
47#336C2×C414yes—
48#322—42no—
49C22×D4C4211yescharacteristic
50C23×C4—21no—
51C22×C8—22no—
52C22×C8—24no—
53C22×C8—24no—
54C22×C8C22⋊C412yes—
55C22×C8C22⋊C412yes—
56C4×D4—24no—
57#117—41no—
58#117—41no—
59#117—42no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4C4211yescharacteristic
62C2×C4⋊C4C4212yes—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C42—21no—
67C2×C42C22×C412yes—
68C2×C42C22×C411yescharacteristic
69C2×C42C2×D411yescharacteristic
70C2×C42C2×D411yescharacteristic
71C2×C42#4112yes—
72C4⋊C8—24no—
73C4⋊C8—44no—
74C4⋊C8—44no—
75C22⋊C8—44no—
76C22⋊C8—44no—
77#97—41no—
78#97—41no—
79C4×C8—24no—
80C4×C8—22no—
81C4×C8—42no—
82C4×C8—44no—
83C4×C8—42no—
84C4×C8—42no—
85C4×C8D812yes—
86C4×C8SD1612yes—
87#95—21no—
88#95—22no—
89C24—21no—
90C2×D4—22no—
91C2×D4—42no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—42no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4C2×C4211yescharacteristic
99C22×C4C2×C22⋊C411yescharacteristic
100C22×C4#11711yescharacteristic
101C22×C4C4×D412yes—
102C22×C4C4×D412yes—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—44no—
109C2×C8—44no—
110C2×C8—44no—
111C2×C8—44no—
112C2×C8—42no—
113C2×C8—42no—
114C2×C8—42no—
115C2×C8—44no—
116C2×C8—44no—
117C2×C8#10214yes—
118C4⋊C4—22no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C22⋊C4—42no—
123C22⋊C4—42no—
124C22⋊C4—42no—
125C22⋊C4—44no—
126C42—22no—
127C42—24no—
128C42—42no—
129C42—41no—
130C42—41no—
131C42#13114yes—
132C42C2×D811yescharacteristic
133C42C2×SD1611yescharacteristic
134C42#13512yes—
135C23—41no—
136C23—42no—
137C23—41no—
138C23—41no—
139C23—42no—
140C23C4×C22⋊C411yescharacteristic
141D4—82no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4#40512yes—
157C2×C4#40812yes—
158C2×C4C2×#10211yescharacteristic
159C2×C4#41611yescharacteristic
160C2×C4#42112yes—
161C2×C4#43312yes—
162C2×C4C4×D812yes—
163C2×C4C4×SD1612yes—
164C2×C4#44414yes—
165C8—24no—
166C8—42no—
167C8—84no—
168C8—84no—
169C8—82no—
170C22—82no—
171C22—82no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22#175011yescharacteristic
176C22#175712yes—
177C22#175911yescharacteristic
178C22C4×#10211yescharacteristic
179C22#176512yes—
180C4—24no—
181C4—22no—
182C4—22no—
183C4—22no—
184C4—22no—
185C4—82no—
186C4—81no—
187C4#148012yes—
188C4#148212yes—
189C2—161no—
190C2#815811yescharacteristic
191C2#818311yescharacteristic
192C2#818811yescharacteristic
193C2#824812yes—
194C2#825512yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC43C23
socleC23C4×C22⋊C4
prime radicalC22×D4C42
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8260C2semidirect
3. #8133C2semidirect
5. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#212662
#82602
#81332
#75602
#70272