Finite Group Databasebeta

← #124739

Subgroups — #124739

Order 512.

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

normal subgroups only

This group has 277 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#21346C211yesmaximal, characteristic
2#12882C211yesmaximal, characteristic
3#12659C211yesmaximal, characteristic
4#7133C212yesmaximal
5#7124C212yesmaximal
6C42×Q8C411yescharacteristic
7C2×#364—21no—
8C2×#364—21no—
9C2×#347—21no—
10#1985C411yescharacteristic
11#1985C2211yescharacteristic
12#1940C411yescharacteristic
13#1940C2211yescharacteristic
14#1916—21no—
15#1916—21no—
16#1767—21no—
17#1767C411yescharacteristic
18#1745—21no—
19C4×#97C2211yescharacteristic
20#1377C2212yes—
21#1377C2212yes—
22#1375—21no—
23C2×C4×Q8—21no—
24C2×C4×Q8C2×C411yescharacteristic
25C2×M32—22no—
26C2×M32—22no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31C2×#97—21no—
32C2×#97C2×C411yescharacteristic
33C2×#97D411yescharacteristic
34C2×#97D411yescharacteristic
35C2×#97C2311yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C4×C4⋊C4—21no—
39C4×C4⋊C4—21no—
40C4×C4⋊C4—21no—
41C4×C4⋊C4—21no—
42C4×C4⋊C4C2×C411yescharacteristic
43C43—21no—
44C43C2×C411yescharacteristic
45#364—24no—
46#364—42no—
47#347—42no—
48#336—21no—
49#336—21no—
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55C22×Q8M1611yescharacteristic
56C22×C8—21no—
57C22×C8—21no—
58C22×C8C22⋊C411yescharacteristic
59C22×C8C22⋊C411yescharacteristic
60C22×C8C2×D411yescharacteristic
61C22×C8C2×D411yescharacteristic
62C4×Q8—41no—
63C4×Q8—41no—
64C4×Q8—44no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4M1611yescharacteristic
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C22⋊C411yescharacteristic
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22×C411yescharacteristic
81M32—24no—
82M32—42no—
83M32—42no—
84M32—42no—
85C2×C16—42no—
86C2×C16—42no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—42no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92#97—41no—
93#97—41no—
94#97—41no—
95#97—42no—
96#97SD1612yes—
97#97Q1612yes—
98C4×C8—22no—
99C4×C8—21no—
100C4×C8—21no—
101#95—21no—
102#95—21no—
103#95M1611yescharacteristic
104#95M1611yescharacteristic
105C2×Q8—42no—
106C2×Q8—41no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4C2×C22⋊C411yescharacteristic
114C22×C4C2×M1611yescharacteristic
115C22×C4C2×M1611yescharacteristic
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8#10312yes—
131C2×C8#10412yes—
132C2×C8C2×SD1611yescharacteristic
133C2×C8C2×Q1611yescharacteristic
134C2×C8AGL(1,ℤ/8)11yescharacteristic
135C2×C8#13711yescharacteristic
136C4⋊C4—24no—
137C4⋊C4—41no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—42no—
147C42—42no—
148C42—42no—
149C42—41no—
150C42—41no—
151C42—41no—
152C42—41no—
153C42—81no—
154C42—81no—
155C42—81no—
156C42#10312yes—
157C16—44no—
158C16—82no—
159C23#40711yescharacteristic
160Q8—42no—
161Q8—42no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4#41011yescharacteristic
185C2×C4#41311yescharacteristic
186C2×C4C2×#10311yescharacteristic
187C2×C4#41711yescharacteristic
188C2×C4C2×#10411yescharacteristic
189C2×C4#42111yescharacteristic
190C8—22no—
191C8—21no—
192C8—21no—
193C8—21no—
194C8—21no—
195C8—22no—
196C8—81no—
197C8—81no—
198C22—21no—
199C22—21no—
200C22#146411yescharacteristic
201C22#148711yescharacteristic
202C22#148911yescharacteristic
203C4—42no—
204C4—42no—
205C4—42no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—81no—
210C4—81no—
211C4#150311yescharacteristic
212C4#151211yescharacteristic
213C4#153211yescharacteristic
214C4#155011yescharacteristic
215C2—21no—
216C2—21no—
217C2#761511yescharacteristic
218C2#1152111yescharacteristic
219C2#1152311yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
socleC22#1464
prime radicalC23#407
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#213462
#128822
#126592
#71332
#71242