Finite Group Databasebeta

← #125270

Subgroups — #125270

Order 512.

259 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 68 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#33285C211yesmaximal, characteristic
2#12656C211yesmaximal, characteristic
3#12596C211yesmaximal, characteristic
4#7135C212yesmaximal
5#7108C212yesmaximal
6#2647C411yescharacteristic
7#2642C411yescharacteristic
8C2×#364—21no—
9C2×#347—21no—
10C4×M32—21no—
11#1992C411yescharacteristic
12#1991C2211yescharacteristic
13#1960C411yescharacteristic
14#1956C2211yescharacteristic
15#1935—21no—
16#1935—21no—
17#1933—21no—
18#1933—21no—
19#1726C2211yescharacteristic
20#1377—21no—
21#1376C2212yes—
22#1376C2212yes—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×#125C2×C411yescharacteristic
25C2×M32—22no—
26C2×M32—22no—
27C2×#103—21no—
28C2×#103—21no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#97C2×C411yescharacteristic
32C2×#97D411yescharacteristic
33C2×#97D411yescharacteristic
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8C2311yescharacteristic
37#400—41no—
38#400—41no—
39#400—41no—
40#398—41no—
41#391—21no—
42#391—21no—
43#389—21no—
44#389—21no—
45#384C2×C411yescharacteristic
46#382C2×C411yescharacteristic
47#376C2×C411yescharacteristic
48#364—24no—
49#347—24no—
50#346—41no—
51C4×C16—41no—
52#340—21no—
53#340—21no—
54#340—21no—
55#340—21no—
56C22×Q8—21no—
57C22×C8C22⋊C411yescharacteristic
58C22×C8C22⋊C411yescharacteristic
59C22×C8C2×D411yescharacteristic
60C22×C8C2×D411yescharacteristic
61C4⋊Q8—22no—
62C4⋊Q8—41no—
63#125—22no—
64#125—41no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C22⋊C411yescharacteristic
75C2×C42C22×C411yescharacteristic
76M32—24no—
77M32—24no—
78M32—42no—
79M32—42no—
80C2×C16—42no—
81C2×C16—42no—
82#103—42no—
83#103—42no—
84#103—42no—
85#103—41no—
86#103—41no—
87#97—22no—
88#97—22no—
89#97—22no—
90C4×C8—22no—
91C4×C8—21no—
92C4×C8—21no—
93C4×C8—41no—
94#95—21no—
95#95—21no—
96#95—41no—
97#95—41no—
98#95—42no—
99#95—41no—
100#95—41no—
101#95—41no—
102C2×Q8—41no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×Q8—41no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4#10011yescharacteristic
115C22×C4#10011yescharacteristic
116C22×C4C2×C22⋊C411yescharacteristic
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8#10412yes—
126C2×C8#10412yes—
127C2×C8#13512yes—
128C2×C8AGL(1,ℤ/8)11yescharacteristic
129C2×C8#13711yescharacteristic
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—82no—
137C4⋊C4—82no—
138C4⋊C4—161no—
139C4⋊C4—161no—
140C4⋊C4—161no—
141C42—21no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42#13112yes—
147C16—44no—
148C16—44no—
149C23C2×#10011yescharacteristic
150Q8—81no—
151Q8—81no—
152Q8—81no—
153Q8—81no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4#40811yescharacteristic
177C2×C4#41011yescharacteristic
178C2×C4#41611yescharacteristic
179C2×C4#41711yescharacteristic
180C2×C4C2×#10411yescharacteristic
181C2×C4C2×#10411yescharacteristic
182C8—22no—
183C8—22no—
184C8—22no—
185C8—21no—
186C8—21no—
187C22—21no—
188C22—21no—
189C22#146911yescharacteristic
190C22#152511yescharacteristic
191C22#155111yescharacteristic
192C4—42no—
193C4—41no—
194C4—41no—
195C4—81no—
196C4—161no—
197C4—161no—
198C4—161no—
199C4—161no—
200C4#148311yescharacteristic
201C4#149411yescharacteristic
202C4#150011yescharacteristic
203C4#150911yescharacteristic
204C2—21no—
205C2—21no—
206C2#767611yescharacteristic
207C2#1199011yescharacteristic
208C2#1199111yescharacteristic
subgroupG/H
center, socleC22#1551
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23C2×#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332852
#126562
#125962
#71352
#71082