Finite Group Databasebeta

← #125525

Subgroups — #125525

Order 512.

237 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 64 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#12897C211yesmaximal, characteristic
3#12884C211yesmaximal, characteristic
4#7135C212yesmaximal
5#7108C212yesmaximal
6#2647C411yescharacteristic
7#2642C411yescharacteristic
8#2150—21no—
9#1992C411yescharacteristic
10#1991C2211yescharacteristic
11#1960C411yescharacteristic
12#1956C2211yescharacteristic
13#1935—21no—
14#1935—21no—
15#1933—21no—
16#1933—21no—
17#1726C2211yescharacteristic
18#1377—21no—
19#1376—21no—
20#1376C2212yes—
21#1376C2212yes—
22#1375—21no—
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
48C4⋊C16—42no—
49#340—21no—
50#340—21no—
51#340—21no—
52#340—21no—
53C22×Q8—21no—
54C22×C8C22⋊C411yescharacteristic
55C22×C8C22⋊C411yescharacteristic
56C22×C8C2×D411yescharacteristic
57C22×C8C2×D411yescharacteristic
58C4⋊Q8—22no—
59C4⋊Q8—41no—
60#125—22no—
61#125—41no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C42C22⋊C411yescharacteristic
71C2×C42C22⋊C411yescharacteristic
72C2×C42C22×C411yescharacteristic
73M32—42no—
74M32—42no—
75M32—42no—
76M32—42no—
77C2×C16—42no—
78C2×C16—42no—
79#103—42no—
80#103—42no—
81#103—42no—
82#103—41no—
83#103—41no—
84#97—22no—
85#97—22no—
86#97—41no—
87C4×C8—21no—
88C4×C8—21no—
89C4×C8—41no—
90C4×C8—41no—
91#95—21no—
92#95—21no—
93#95—41no—
94#95—41no—
95#95—42no—
96#95—41no—
97#95—41no—
98#95—41no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×Q8—41no—
104C2×Q8—41no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4#10011yescharacteristic
112C22×C4#10011yescharacteristic
113C22×C4C2×C22⋊C411yescharacteristic
114C2×C8—21no—
115C2×C8—21no—
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#13512yes—
125C2×C8AGL(1,ℤ/8)11yescharacteristic
126C2×C8#13711yescharacteristic
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—82no—
134C4⋊C4—82no—
135C4⋊C4—161no—
136C4⋊C4—161no—
137C4⋊C4—161no—
138C42—21no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42#13112yes—
144C16—82no—
145C16—82no—
146C23C2×#10011yescharacteristic
147Q8—81no—
148Q8—81no—
149Q8—81no—
150Q8—81no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4—81no—
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#40811yescharacteristic
174C2×C4#41011yescharacteristic
175C2×C4#41611yescharacteristic
176C2×C4#41711yescharacteristic
177C2×C4#42111yescharacteristic
178C2×C4#42111yescharacteristic
179C8—21no—
180C8—21no—
181C8—22no—
182C8—41no—
183C8—41no—
184C22—21no—
185C22—21no—
186C22#146911yescharacteristic
187C22#153611yescharacteristic
188C22#155111yescharacteristic
189C4—42no—
190C4—41no—
191C4—41no—
192C4—81no—
193C4—161no—
194C4—161no—
195C4—161no—
196C4—161no—
197C4#148211yescharacteristic
198C4#149511yescharacteristic
199C4#151211yescharacteristic
200C4#152111yescharacteristic
201C2—21no—
202C2—21no—
203C2#771611yescharacteristic
204C2#1199011yescharacteristic
205C2#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
#128972
#128842
#71352
#71082