Finite Group Databasebeta

← #126456

Subgroups — #126456

Order 512.

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

normal subgroups only

This group has 226 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#32518C211yesmaximal, characteristic
2#13510C211yesmaximal, characteristic
3#13510C211yesmaximal, characteristic
4#10687C211yesmaximal, characteristic
5#7127C211yesmaximal, characteristic
6#7126C211yesmaximal, characteristic
7#7124C211yesmaximal, characteristic
8#2508—21no—
9C2×#365—21no—
10C2×#365—21no—
11#2088—21no—
12#2088—21no—
13#2085—21no—
14#1949—21no—
15#1944C2211yescharacteristic
16#1944C2211yescharacteristic
17#1940C2211yescharacteristic
18#1919—21no—
19#1849C2211yescharacteristic
20#1387—21no—
21#1387—21no—
22#1377C2211yescharacteristic
23#1377C2211yescharacteristic
24#1303—21no—
25#1301—21no—
26#1275C2211yescharacteristic
27C2×M32—21no—
28C2×M32—21no—
29#431—21no—
30C2×#108—21no—
31C2×#108—21no—
32C2×#107D411yescharacteristic
33C2×#107D411yescharacteristic
34C2×#106—21no—
35C2×#106D411yescharacteristic
36C2×#106D411yescharacteristic
37#423—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#97C2311yescharacteristic
41#400—41no—
42#398—41no—
43#389—21no—
44#389—21no—
45#389—21no—
46#389—21no—
47#387—41no—
48#382—21no—
49#382—41no—
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
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C42—21no—
74C2×C42—41no—
75C2×C42C2×D411yescharacteristic
76M32—22no—
77M32—22no—
78M32—41no—
79M32—41no—
80C2×C16—41no—
81C2×C16—41no—
82#108—41no—
83#108—41no—
84#108—41no—
85#108—41no—
86#107—41no—
87#107—41no—
88#106—21no—
89#106—21no—
90#106—21no—
91#106—81no—
92#106SD1612yes—
93C4⋊C8—41no—
94C4⋊C8—42no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—41no—
98C22⋊C8—41no—
99#97—41no—
100C4×C8—41no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95SD1611yescharacteristic
109#95SD1611yescharacteristic
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4#12011yescharacteristic
117C22×C4C2×SD1611yescharacteristic
118C22×C4AGL(1,ℤ/8)11yescharacteristic
119M16—81no—
120M16—81no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—81no—
129C2×C8—81no—
130C2×C8C2×SD1611yescharacteristic
131C2×C8#13511yescharacteristic
132C2×C8AGL(1,ℤ/8)11yescharacteristic
133C2×C8#13711yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—81no—
143C4⋊C4—82no—
144C4⋊C4—161no—
145C4⋊C4—161no—
146C4⋊C4—161no—
147C42—21no—
148C42—21no—
149C42—41no—
150C42—41no—
151C42—82no—
152C16—42no—
153C16—42no—
154C23#45011yescharacteristic
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
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—81no—
177C2×C4#44811yescharacteristic
178C2×C4#44911yescharacteristic
179C2×C4#45411yescharacteristic
180C2×C4#45611yescharacteristic
181C2×C4#45611yescharacteristic
182C2×C4#45811yescharacteristic
183C8—21no—
184C8—21no—
185C8—21no—
186C8—21no—
187C8—81no—
188C8—161no—
189C8—161no—
190C22—21no—
191C22—21no—
192C22#161611yescharacteristic
193C22#165111yescharacteristic
194C22#165411yescharacteristic
195C4—41no—
196C4—41no—
197C4—81no—
198C4—81no—
199C4—82no—
200C4—161no—
201C4—161no—
202C4—161no—
203C4#160911yescharacteristic
204C4#161111yescharacteristic
205C4#162611yescharacteristic
206C4#166311yescharacteristic
207C2—21no—
208C2—21no—
209C2#781511yescharacteristic
210C2#1230611yescharacteristic
211C2#1230911yescharacteristic
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
#325182
#135102
#106872
#71272
#71262
#71242