Finite Group Databasebeta

← #136122

Subgroups — #136122

Order 512.

268 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 82 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#17815C211yesmaximal, characteristic
2#12030C212yesmaximal, complemented
3#11460C211yesmaximal, characteristic
4#8600C211yesmaximal, characteristic
5#7292C212yesmaximal, complemented
6#2920—21no—
7C2×#431C2211yescharacteristic
8#2163C2212yes—
9#1797—21no—
10#1572—24no—
11#1567—21no—
12#1566C2212yes—
13C2×#335C2211yescharacteristic
14C2×#334C2211yescharacteristic
15#1368—24no—
16#1322C412yes—
17#1314C412yes—
18#1277—21no—
19C22×C42D411yescharacteristic
20#432—22no—
21#432—41no—
22#431—22no—
23#431—22no—
24#431—22no—
25#431—41no—
26#423—22no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8Q811yescharacteristic
31#407—41no—
32C2×C22⋊C8—41no—
33C2×C22⋊C8C2×C411yescharacteristic
34C2×C22⋊C8Q811yescharacteristic
35C4×M16—41no—
36C2×C4×C8D411yescharacteristic
37C2×C4×C8C2311yescharacteristic
38C4⋊C16—22no—
39C22⋊C16C2×C414yes—
40#346—22no—
41#336—41no—
42#335—24no—
43#335—22no—
44#334—22no—
45#334—22no—
46#334D414yes—
47C23×C4C4⋊C411yescharacteristic
48C23×C4SD1611yescharacteristic
49C23×C4SD1611yescharacteristic
50C2×M16—42no—
51C22×C8—41no—
52C22×C8—41no—
53C22×C8C4⋊C411yescharacteristic
54C22×C8C4⋊C411yescharacteristic
55C22×C8C22×C411yescharacteristic
56C22×C8C2×Q811yescharacteristic
57C2×C42—21no—
58C2×C42—21no—
59C2×C42—21no—
60C2×C42—21no—
61C2×C42C4⋊C411yescharacteristic
62C2×C42SD1611yescharacteristic
63C2×C42SD1611yescharacteristic
64C2×C42C2×D411yescharacteristic
65C2×C16—24no—
66C4⋊C8—22no—
67C4⋊C8—22no—
68C4⋊C8—24no—
69C4⋊C8—41no—
70C4⋊C8—41no—
71C4⋊C8—42no—
72C4⋊C8—41no—
73C4⋊C8—41no—
74C22⋊C8—22no—
75C22⋊C8—22no—
76C22⋊C8—42no—
77C22⋊C8—42no—
78C22⋊C8—82no—
79#97—42no—
80C4×C8—21no—
81C4×C8—21no—
82C4×C8—42no—
83C4×C8C2×D412yes—
84C4×C8#4112yes—
85C24#10611yescharacteristic
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4#10611yescharacteristic
99C22×C4#10611yescharacteristic
100C22×C4#10611yescharacteristic
101C22×C4C2×C4⋊C411yescharacteristic
102C22×C4C2×SD1611yescharacteristic
103C22×C4C2×SD1611yescharacteristic
104M16—82no—
105M16—82no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—44no—
113C2×C8—81no—
114C2×C8—81no—
115C2×C8—81no—
116C2×C8—81no—
117C2×C8—81no—
118C2×C8—81no—
119C2×C8C4×D412yes—
120C2×C8C22⋊Q812yes—
121C42—21no—
122C42—21no—
123C42—21no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42#12011yescharacteristic
128C42#12311yescharacteristic
129C42AGL(1,ℤ/8)11yescharacteristic
130C42#13711yescharacteristic
131C16—44no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—41no—
136C23—41no—
137C23C2×#10611yescharacteristic
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4#38511yescharacteristic
160C2×C4#42811yescharacteristic
161C2×C4#44811yescharacteristic
162C2×C4#45011yescharacteristic
163C2×C4#48111yescharacteristic
164C2×C4#48311yescharacteristic
165C8—22no—
166C8—22no—
167C8—41no—
168C8—41no—
169C8—84no—
170C8—161no—
171C8—161no—
172C22—21no—
173C22—21no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—81no—
179C22#185311yescharacteristic
180C22#215511yescharacteristic
181C22#215511yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—41no—
188C4—41no—
189C4#185811yescharacteristic
190C4#186811yescharacteristic
191C4#223811yescharacteristic
192C4#223911yescharacteristic
193C2—21no—
194C2—21no—
195C2—81no—
196C2#886211yescharacteristic
197C2#1293011yescharacteristic
198C2#1293111yescharacteristic
subgroupG/H
center, socleC22#1853
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#106
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12030C2semidirect
5. #7292C2semidirect

Maximal subgroups

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

subgroupindex
#178152
#120302
#114602
#86002
#72922