Finite Group Databasebeta

← #134818

Subgroups — #134818

Order 512.

295 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 76 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#22238C211yesmaximal, characteristic
2#11532C212yesmaximal
3#8623C212yesmaximal
4#7248C212yesmaximal
5C2×C8×Q8C411yescharacteristic
6C2×C8×Q8C2211yescharacteristic
7C4×C4⋊Q8C2211yescharacteristic
8#1984C411yescharacteristic
9#1984C2211yescharacteristic
10#1916—22no—
11#1916—22no—
12#1831C411yescharacteristic
13#1744C411yescharacteristic
14#1568—22no—
15#1467C2214yes—
16#1325—22no—
17#1319—22no—
18#1290—22no—
19C2×C4⋊Q8Q811yescharacteristic
20C2×C4×Q8C2×C411yescharacteristic
21C2×C4×Q8C2311yescharacteristic
22C8×Q8—42no—
23C8×Q8—42no—
24#432—24no—
25C2×C4⋊C8—21no—
26C2×C4⋊C8C2×C411yescharacteristic
27C2×C4⋊C8C2×C411yescharacteristic
28C2×C4⋊C8D412yes—
29C2×C4×C8—21no—
30C2×C4×C8C2×C411yescharacteristic
31C2×C4×C8C2×C411yescharacteristic
32C2×C4×C8D412yes—
33#391D411yescharacteristic
34#391Q811yescharacteristic
35#389—22no—
36#384D411yescharacteristic
37C4×C4⋊C4—22no—
38C43C2×C411yescharacteristic
39#336—22no—
40#336—22no—
41#336—21no—
42#336—21no—
43#336—22no—
44#335—42no—
45#334—42no—
46#324—24no—
47#324—24no—
48C22×Q8C4⋊C411yescharacteristic
49C22×Q8C2×Q811yescharacteristic
50C22×C8—21no—
51C22×C8—21no—
52C22×C8C22⋊C412yes—
53C22×C8C22⋊C411yescharacteristic
54C22×C8C22⋊C411yescharacteristic
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C4×Q8—41no—
58C4×Q8—41no—
59C4×Q8—41no—
60C4×Q8—41no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4C4⋊C411yescharacteristic
65C2×C4⋊C4C2×D411yescharacteristic
66C2×C4⋊C4#4112yes—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42C4⋊C411yescharacteristic
71C2×C42C4⋊C411yescharacteristic
72C2×C42C22×C411yescharacteristic
73C2×C42C2×D412yes—
74C4⋊C8—41no—
75C4⋊C8—42no—
76C4⋊C8—41no—
77C4⋊C8—44no—
78C4⋊C8—42no—
79C22⋊C8—44no—
80#97—44no—
81C4×C8—22no—
82C4×C8—22no—
83C4×C8—22no—
84C4×C8—41no—
85C4×C8—42no—
86#95—22no—
87#95—22no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—41no—
92C2×Q8—41no—
93C2×Q8—41no—
94C2×Q8—41no—
95C2×Q8#10812yes—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4C2×C22⋊C411yescharacteristic
101C22×C4C2×C4⋊C411yescharacteristic
102C22×C4#11711yescharacteristic
103C22×C4#12112yes—
104C22×C4C22⋊Q812yes—
105C2×C8—21no—
106C2×C8—22no—
107C2×C8—41no—
108C2×C8—42no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—42no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—84no—
116C2×C8#9912yes—
117C4⋊C4—28no—
118C4⋊C4—41no—
119C4⋊C4—42no—
120C4⋊C4—41no—
121C4⋊C4—42no—
122C4⋊C4—82no—
123C42—22no—
124C42—21no—
125C42—21no—
126C42—22no—
127C42—21no—
128C42—41no—
129C42—41no—
130C42—41no—
131C42—41no—
132C42—41no—
133C42—81no—
134C42#10812yes—
135C23#38011yescharacteristic
136Q8—42no—
137Q8—42no—
138Q8—42no—
139Q8—42no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4C2×#9911yescharacteristic
163C2×C4#41011yescharacteristic
164C2×C4C2×#10811yescharacteristic
165C2×C4C2×#10811yescharacteristic
166C8—22no—
167C8—41no—
168C8—42no—
169C8—81no—
170C8—81no—
171C8—164no—
172C22—21no—
173C22—21no—
174C22#179911yescharacteristic
175C22#181011yescharacteristic
176C22#181311yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—42no—
181C4—42no—
182C4—41no—
183C4—42no—
184C4—42no—
185C4—81no—
186C4—81no—
187C4#213311yescharacteristic
188C4#213511yescharacteristic
189C2—21no—
190C2—21no—
191C2#861211yescharacteristic
192C2#1280411yescharacteristic
193C2#1280811yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC23#380
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#222382
#115322
#86232
#72482