Finite Group Databasebeta

← #128459

Subgroups — #128459

Order 512.

359 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 120 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
4C8×#101C212yesmaximal
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—
18C8×M16—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#356C818yescomplemented
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—42no—
55C82—42no—
56C22×Q8C4211yescharacteristic
57C22×Q8C22×C411yescharacteristic
58C2×M16—24no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8C22⋊C412yes—
62C22×C8C22⋊C411yescharacteristic
63C22×C8C22⋊C411yescharacteristic
64C4⋊Q8—22no—
65C4⋊Q8C2×C814yes—
66C4×Q8—41no—
67C4×Q8—41no—
68C4×Q8—41no—
69C4×Q8—41no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4C4211yescharacteristic
74C2×C4⋊C4C22×C411yescharacteristic
75C2×C4⋊C4#4112yes—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42C4211yescharacteristic
80C2×C42C4211yescharacteristic
81C2×C42C22×C411yescharacteristic
82C2×C42C2×D412yes—
83C4⋊C8—41no—
84C4⋊C8—42no—
85C4⋊C8—41no—
86C4⋊C8—42no—
87C4⋊C8—42no—
88#101—24no—
89#101C2×C818yes—
90C22⋊C8—24no—
91#97—42no—
92C4×C8—22no—
93C4×C8—22no—
94C4×C8—22no—
95C4×C8—41no—
96C4×C8—42no—
97C4×C8—42no—
98C4×C8—42no—
99#95—22no—
100#95—22no—
101C2×Q8—21no—
102C2×Q8—41no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8C4×C812yes—
107C2×Q8C22×C812yes—
108C2×Q8#13112yes—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4C2×C4211yescharacteristic
114C22×C4C2×C22⋊C411yescharacteristic
115C22×C4#11711yescharacteristic
116C22×C4C4×D412yes—
117C22×C4C4×D412yes—
118M16—28no—
119M16—44no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—41no—
123C2×C8—44no—
124C2×C8—42no—
125C2×C8—41no—
126C2×C8—44no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8#9912yes—
133C4⋊C4—28no—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—41no—
137C4⋊C4—42no—
138C4⋊C4—82no—
139C42—21no—
140C42—22no—
141C42—21no—
142C42—21no—
143C42—22no—
144C42—41no—
145C42—41no—
146C42—41no—
147C42—41no—
148C42—41no—
149C42—81no—
150C42C4×C812yes—
151C23C4×C22⋊C411yescharacteristic
152Q8—42no—
153Q8—42no—
154Q8—42no—
155Q8—42no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4C2×C4×C811yescharacteristic
178C2×C4#40511yescharacteristic
179C2×C4C2×#9911yescharacteristic
180C2×C4#41011yescharacteristic
181C2×C4C8×D414yes—
182C8—22no—
183C8—48no—
184C8—41no—
185C8—42no—
186C8—81no—
187C8—84no—
188C8—81no—
189C22—21no—
190C22—21no—
191C22C8×C22⋊C411yescharacteristic
192C22C4×#9911yescharacteristic
193C22#175711yescharacteristic
194C4—21no—
195C4—24no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—42no—
200C4—41no—
201C4—42no—
202C4—81no—
203C4—81no—
204C4#213311yescharacteristic
205C4#213511yescharacteristic
206C2—21no—
207C2—21no—
208C2C8×#9911yescharacteristic
209C2#1263111yescharacteristic
210C2#1263411yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC2×C4×Q8C23
socleC22C4×#99
prime radicalC23C4×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
45. #356C8semidirect

Maximal subgroups

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

subgroupindex
#222382
C4×#3562
#114782
C8×#1012
#72062