Finite Group Databasebeta

← #126458

Subgroups — #126458

Order 512.

240 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 54 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#32516C211yesmaximal, characteristic
2#13510C211yesmaximal, characteristic
3#13510C211yesmaximal, characteristic
4#10687C211yesmaximal, characteristic
5#7127C211yesmaximal, characteristic
6#7127C211yesmaximal, characteristic
7#7124C211yesmaximal, characteristic
8#2506—21no—
9C2×#365—21no—
10C2×#365—21no—
11#2088—21no—
12#2087—21no—
13#2083—21no—
14#1947—21no—
15#1944C2211yescharacteristic
16#1944C2211yescharacteristic
17#1940C2211yescharacteristic
18#1920—21no—
19#1849C2211yescharacteristic
20#1387—21no—
21#1387—21no—
22#1377C2211yescharacteristic
23#1377C2211yescharacteristic
24#1303—21no—
25#1301—21no—
26#1275C2211yescharacteristic
27C2×#125—41no—
28C2×M32—21no—
29C2×M32—21no—
30#431—21no—
31C2×#108—21no—
32C2×#108—21no—
33C2×#107—21no—
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#106D411yescharacteristic
37C2×#106D411yescharacteristic
38#423—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×#97C2311yescharacteristic
42#400—41no—
43#398—41no—
44#389—21no—
45#389—21no—
46#389—21no—
47#389—21no—
48#387—41no—
49#384—21no—
50C4×C4⋊C4—21no—
51C4×C4⋊C4D411yescharacteristic
52#365—42no—
53#365—42no—
54#340—21no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—21no—
59#336—21no—
60C2×M16—41no—
61C22×C8—21no—
62C22×C8C2×D411yescharacteristic
63C22×C8C2×D411yescharacteristic
64#125—84no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C42—21no—
77C2×C42—41no—
78C2×C42C2×D411yescharacteristic
79M32—22no—
80M32—22no—
81M32—41no—
82M32—41no—
83C2×C16—41no—
84C2×C16—41no—
85#108—41no—
86#108—41no—
87#108—41no—
88#108—41no—
89#107—41no—
90#107—41no—
91#107—81no—
92#106—21no—
93#106—21no—
94#106—21no—
95#106SD1612yes—
96C4⋊C8—41no—
97C4⋊C8—42no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C22⋊C8—41no—
102#97—41no—
103C4×C8—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95SD1611yescharacteristic
110#95SD1611yescharacteristic
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4#12011yescharacteristic
118C22×C4C2×SD1611yescharacteristic
119C22×C4AGL(1,ℤ/8)11yescharacteristic
120M16—81no—
121M16—81no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—81no—
130C2×C8—81no—
131C2×C8C2×SD1611yescharacteristic
132C2×C8#13511yescharacteristic
133C2×C8AGL(1,ℤ/8)11yescharacteristic
134C2×C8#13711yescharacteristic
135C4⋊C4—22no—
136C4⋊C4—42no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—82no—
145C4⋊C4—82no—
146C4⋊C4—82no—
147C4⋊C4—82no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C42—21no—
151C42—21no—
152C42—41no—
153C42—41no—
154C42—82no—
155C16—42no—
156C16—42no—
157C23#45011yescharacteristic
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4#44811yescharacteristic
181C2×C4#44911yescharacteristic
182C2×C4#45411yescharacteristic
183C2×C4#45611yescharacteristic
184C2×C4#45611yescharacteristic
185C2×C4#45811yescharacteristic
186C8—21no—
187C8—21no—
188C8—21no—
189C8—21no—
190C8—81no—
191C8—161no—
192C8—161no—
193C22—21no—
194C22—21no—
195C22#161611yescharacteristic
196C22#165111yescharacteristic
197C22#165411yescharacteristic
198C4—42no—
199C4—41no—
200C4—41no—
201C4—81no—
202C4—82no—
203C4—161no—
204C4—161no—
205C4—161no—
206C4#160911yescharacteristic
207C4#161111yescharacteristic
208C4#162611yescharacteristic
209C4#166311yescharacteristic
210C2—21no—
211C2—21no—
212C2#781511yescharacteristic
213C2#1230511yescharacteristic
214C2#1230711yescharacteristic
subgroupG/H
center, socleC22#1616
commutator, Frattini, cosocleC2×#97C23
prime radicalC23#450
LCSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#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
#325162
#135102
#106872
#71272
#71242