Finite Group Databasebeta

← #136116

Subgroups — #136116

Order 512.

271 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 84 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#17819C211yesmaximal, characteristic
2#12035C212yesmaximal, complemented
3#11459C211yesmaximal, characteristic
4#8600C211yesmaximal, characteristic
5#7290C212yesmaximal, complemented
6#2920—21no—
7C2×#431C2211yescharacteristic
8#2162C2212yes—
9#1797—21no—
10#1574—22no—
11#1574—22no—
12#1568—21no—
13#1568C2212yes—
14C2×#335C2211yescharacteristic
15C2×#334C2211yescharacteristic
16#1372—22no—
17#1371—22no—
18#1322C412yes—
19#1314C412yes—
20#1277—21no—
21C22×C42D411yescharacteristic
22#432—22no—
23#432—41no—
24#431—22no—
25#431—22no—
26#431—22no—
27#431—41no—
28#423—21no—
29#423—21no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8Q811yescharacteristic
34#407—41no—
35C2×C22⋊C8—41no—
36C2×C22⋊C8C2×C411yescharacteristic
37C2×C22⋊C8Q811yescharacteristic
38C4×M16—41no—
39C2×C4×C8D411yescharacteristic
40C2×C4×C8C2311yescharacteristic
41C4⋊C16—22no—
42C22⋊C16C2×C414yes—
43C4×C16—22no—
44#336—41no—
45#335—21no—
46#335—21no—
47#335—22no—
48#335D412yes—
49#335D412yes—
50#334—22no—
51#334—22no—
52#334—22no—
53C23×C4C4⋊C411yescharacteristic
54C23×C4D811yescharacteristic
55C23×C4Q1611yescharacteristic
56C2×M16—41no—
57C2×M16—41no—
58C22×C8—41no—
59C22×C8—41no—
60C22×C8C4⋊C411yescharacteristic
61C22×C8C4⋊C411yescharacteristic
62C22×C8C22×C411yescharacteristic
63C22×C8C2×Q811yescharacteristic
64C2×C42—21no—
65C2×C42—21no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42C4⋊C411yescharacteristic
69C2×C42D811yescharacteristic
70C2×C42Q1611yescharacteristic
71C2×C42C2×D411yescharacteristic
72C2×C16—24no—
73C4⋊C8—22no—
74C4⋊C8—22no—
75C4⋊C8—22no—
76C4⋊C8—22no—
77C4⋊C8—41no—
78C4⋊C8—41no—
79C4⋊C8—41no—
80C4⋊C8—41no—
81C4⋊C8—41no—
82C4⋊C8—41no—
83C22⋊C8—22no—
84C22⋊C8—22no—
85C22⋊C8—42no—
86C22⋊C8—42no—
87C22⋊C8—82no—
88#97—42no—
89C4×C8—21no—
90C4×C8—21no—
91C4×C8—42no—
92C4×C8C2×D412yes—
93C4×C8#4112yes—
94C24#10711yescharacteristic
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4#10711yescharacteristic
108C22×C4#10711yescharacteristic
109C22×C4#10711yescharacteristic
110C22×C4C2×C4⋊C411yescharacteristic
111C22×C4C2×D811yescharacteristic
112C22×C4C2×Q1611yescharacteristic
113M16—82no—
114M16—82no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—42no—
122C2×C8—42no—
123C2×C8—81no—
124C2×C8—81no—
125C2×C8—81no—
126C2×C8—81no—
127C2×C8—81no—
128C2×C8—81no—
129C2×C8C4×D412yes—
130C2×C8C22⋊Q812yes—
131C42—21no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42#12011yescharacteristic
138C42#12311yescharacteristic
139C42AGL(1,ℤ/8)11yescharacteristic
140C42#13711yescharacteristic
141C16—44no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—41no—
146C23—41no—
147C23C2×#10711yescharacteristic
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4#36812yes—
169C2×C4#38511yescharacteristic
170C2×C4#42811yescharacteristic
171C2×C4#44711yescharacteristic
172C2×C4#45111yescharacteristic
173C2×C4#48011yescharacteristic
174C2×C4#48411yescharacteristic
175C8—22no—
176C8—22no—
177C8—41no—
178C8—41no—
179C8—82no—
180C8—82no—
181C8—161no—
182C8—161no—
183C22—21no—
184C22—21no—
185C22—42no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—81no—
190C22#185211yescharacteristic
191C22C2×#36811yescharacteristic
192C22#216111yescharacteristic
193C4—22no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—41no—
199C4—41no—
200C4#185811yescharacteristic
201C4#186811yescharacteristic
202C4#219011yescharacteristic
203C4#219111yescharacteristic
204C2—21no—
205C2—21no—
206C2—81no—
207C2#886111yescharacteristic
208C2#1292211yescharacteristic
209C2#1292311yescharacteristic
subgroupG/H
centerC2×C4#447
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1852
prime radicalC24#107
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #12035C2semidirect
5. #7290C2semidirect

Maximal subgroups

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

subgroupindex
#178192
#120352
#114592
#86002
#72902