Finite Group Databasebeta

← #137892

Subgroups — #137892

Order 512.

345 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 112 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#33496C211yesmaximal, characteristic
2#22238C211yesmaximal, characteristic
3#11637C212yesmaximal
4#11493C211yesmaximal, characteristic
5#7142C212yesmaximal
6C2×C8×Q8C2211yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8C4×C4⋊Q8C411yescharacteristic
9Q8×C16—24no—
10#2162—22no—
11#2162C2212yes—
12#2162C2212yes—
13#1984C411yescharacteristic
14#1984C411yescharacteristic
15#1916—22no—
16#1916—22no—
17#1831C411yescharacteristic
18#1744C2211yescharacteristic
19Q8⋊C16—24no—
20#1322—22no—
21C2×C4⋊Q8C811yescharacteristic
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×Q8C2×C411yescharacteristic
24C8×Q8—22no—
25C8×Q8—22no—
26C8×Q8—22no—
27C8×Q8—22no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8D412yes—
32C2×C4×C8—21no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8D412yes—
35C2×C4×C8C2311yescharacteristic
36#391C811yescharacteristic
37#391C811yescharacteristic
38#389—22no—
39#384C811yescharacteristic
40C4×C4⋊C4—22no—
41C43C2×C411yescharacteristic
42C4⋊C16—24no—
43C4⋊C16—22no—
44C4⋊C16—22no—
45C22⋊C16—22no—
46C22⋊C16—22no—
47C4×C16—24no—
48C4×C16—22no—
49C4×C16D414yes—
50#336—22no—
51#336—22no—
52#336—21no—
53#336—21no—
54#336—22no—
55C22×Q8C2×C811yescharacteristic
56C22×Q8C2×C811yescharacteristic
57C22×C8—21no—
58C22×C8—21no—
59C22×C8C22⋊C412yes—
60C22×C8C2×D411yescharacteristic
61C22×C8C2×D411yescharacteristic
62C4⋊Q8—22no—
63C4⋊Q8C1614yescomplemented
64C4×Q8—22no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×Q8—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4C2×C811yescharacteristic
72C2×C4⋊C4C2×C811yescharacteristic
73C2×C4⋊C4M1612yes—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42C22⋊C412yes—
78C2×C42C2×C811yescharacteristic
79C2×C42C2×C811yescharacteristic
80C2×C42C22×C411yescharacteristic
81C2×C16—24no—
82C2×C16—42no—
83C4⋊C8—24no—
84C4⋊C8—22no—
85C4⋊C8—22no—
86C4⋊C8—41no—
87C4⋊C8—41no—
88C4×C8—22no—
89C4×C8—22no—
90C4×C8—22no—
91C4×C8—42no—
92C4×C8C2×D412yes—
93C4×C8#4112yes—
94#95—22no—
95#95—22no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8C2×C1612yes—
102C2×Q8C2×C1612yes—
103C2×Q8M3212yes—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4C22⋊C812yes—
109C22×C4C22⋊C812yes—
110C22×C4C2×C22⋊C411yescharacteristic
111C22×C4C22×C811yescharacteristic
112C22×C4C2×M1611yescharacteristic
113C2×C8—22no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—42no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—42no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8#12011yescharacteristic
123C2×C8#12112yes—
124C2×C8#12311yescharacteristic
125C4⋊C4—28no—
126C4⋊C4—22no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—42no—
130C4⋊C4—82no—
131C42—22no—
132C42—22no—
133C42—21no—
134C42—22no—
135C42—41no—
136C42—41no—
137C42—41no—
138C42—41no—
139C42—81no—
140C42C2×C1612yes—
141C42C4×D412yes—
142C42#13112yes—
143C16—44no—
144C16—82no—
145C23C2×C22⋊C811yescharacteristic
146Q8—24no—
147Q8—24no—
148Q8—24no—
149Q8—42no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4C22⋊C1614yes—
169C2×C4#38611yescharacteristic
170C2×C4#40811yescharacteristic
171C2×C4C8×D412yes—
172C2×C4#43512yes—
173C2×C4C22×C1611yescharacteristic
174C2×C4C2×M3211yescharacteristic
175C2×C4#50412yes—
176C8—22no—
177C8—21no—
178C8—21no—
179C8—42no—
180C8—81no—
181C8—81no—
182C22—21no—
183C22—21no—
184C22#186911yescharacteristic
185C22C2×C22⋊C1611yescharacteristic
186C22#211311yescharacteristic
187C4—21no—
188C4—24no—
189C4—22no—
190C4—42no—
191C4—42no—
192C4—42no—
193C4—81no—
194C4—81no—
195C4#190911yescharacteristic
196C4#191211yescharacteristic
197C4D4×C1612yes—
198C4#216812yes—
199C2—21no—
200C2—21no—
201C2#910011yescharacteristic
202C2#1297611yescharacteristic
203C2#1298811yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC23C2×C22⋊C8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
63. C4⋊Q8C16semidirect

Maximal subgroups

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

subgroupindex
#334962
#222382
#116372
#114932
#71422