Finite Group Databasebeta

← #128479

Subgroups — #128479

Order 512.

343 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 104 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
2C4×#356C211yesmaximal, characteristic
3#11478C211yesmaximal, characteristic
4#8165C212yesmaximal
5#7206C212yesmaximal
6C2×C8×Q8C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8C4×C4⋊Q8C2211yescharacteristic
9C2×#356C412yes—
10#1984C411yescharacteristic
11#1984C2211yescharacteristic
12#1916—22no—
13#1916—22no—
14#1831C411yescharacteristic
15C4×#101C2212yes—
16#1744C411yescharacteristic
17#1461C2212yes—
18#1451—22no—
19#1394C412yes—
20#1325C412yes—
21#1319C412yes—
22#1283—22no—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×C4×Q8C2×C411yescharacteristic
25C2×C4×Q8C2311yescharacteristic
26C8×Q8—42no—
27C8×Q8—42no—
28#431—22no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×C4⋊C8D412yes—
33C2×#101C2×C414yes—
34C4×M16—22no—
35C2×C4×C8—21no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D412yes—
39#391C2×C411yescharacteristic
40#391C2×C411yescharacteristic
41#389—22no—
42#384C2×C411yescharacteristic
43C4×C4⋊C4—22no—
44C43C2×C411yescharacteristic
45#356—24no—
46#343—22no—
47#336—22no—
48#336—22no—
49#336—21no—
50#336—21no—
51#336—22no—
52#324—22no—
53#324C2×C414yes—
54#322—44no—
55C22×Q8C4211yescharacteristic
56C22×Q8C22×C411yescharacteristic
57C2×M16—24no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8C22⋊C412yes—
61C22×C8C22⋊C411yescharacteristic
62C22×C8C22⋊C411yescharacteristic
63C4⋊Q8—22no—
64C4⋊Q8M1614yes—
65C4×Q8—41no—
66C4×Q8—41no—
67C4×Q8—41no—
68C4×Q8—41no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4C4211yescharacteristic
73C2×C4⋊C4C22×C411yescharacteristic
74C2×C4⋊C4#4112yes—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C4211yescharacteristic
79C2×C42C4211yescharacteristic
80C2×C42C22×C411yescharacteristic
81C2×C42C2×D412yes—
82C4⋊C8—41no—
83C4⋊C8—42no—
84C4⋊C8—41no—
85C4⋊C8—42no—
86C4⋊C8—42no—
87#101—24no—
88#101—24no—
89C22⋊C8—24no—
90#97—42no—
91C4×C8—24no—
92C4×C8—22no—
93C4×C8—41no—
94C4×C8—42no—
95C4×C8—42no—
96C4×C8—42no—
97#95—22no—
98#95—22no—
99C2×Q8—21no—
100C2×Q8—41no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×Q8—41no—
104C2×Q8#9712yes—
105C2×Q8C2×M1612yes—
106C2×Q8#13112yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4C2×C4211yescharacteristic
112C22×C4C2×C22⋊C411yescharacteristic
113C22×C4#11711yescharacteristic
114C22×C4C4×D412yes—
115C22×C4C4×D412yes—
116M16—44no—
117M16—44no—
118C2×C8—21no—
119C2×C8—22no—
120C2×C8—41no—
121C2×C8—42no—
122C2×C8—41no—
123C2×C8—44no—
124C2×C8—44no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—42no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8#9912yes—
131C4⋊C4—28no—
132C4⋊C4—41no—
133C4⋊C4—42no—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—82no—
137C42—21no—
138C42—22no—
139C42—21no—
140C42—21no—
141C42—22no—
142C42—41no—
143C42—41no—
144C42—41no—
145C42—41no—
146C42—41no—
147C42—81no—
148C42#9712yes—
149C23C4×C22⋊C411yescharacteristic
150Q8—42no—
151Q8—42no—
152Q8—42no—
153Q8—42no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4C2×#9711yescharacteristic
176C2×C4#40511yescharacteristic
177C2×C4C2×#9911yescharacteristic
178C2×C4#41011yescharacteristic
179C2×C4#43514yes—
180C8—22no—
181C8—41no—
182C8—42no—
183C8—81no—
184C8—84no—
185C8—84no—
186C8—81no—
187C22—21no—
188C22—21no—
189C22#175211yescharacteristic
190C22C4×#9911yescharacteristic
191C22#175711yescharacteristic
192C4—21no—
193C4—24no—
194C4—21no—
195C4—21no—
196C4—42no—
197C4—42no—
198C4—41no—
199C4—42no—
200C4—81no—
201C4—81no—
202C4#213311yescharacteristic
203C4#213511yescharacteristic
204C2—21no—
205C2—21no—
206C2#816011yescharacteristic
207C2#1263111yescharacteristic
208C2#1263411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C4×#99
prime radicalC23C4×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ 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
C4×#3562
#114782
#81652
#72062