Finite Group Databasebeta

← #126503

Subgroups — #126503

Order 512.

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

normal subgroups only

This group has 222 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#32514C211yesmaximal, characteristic
2#13541C211yesmaximal, characteristic
3#13541C211yesmaximal, characteristic
4#10688C211yesmaximal, characteristic
5#7126C211yesmaximal, characteristic
6#7126C211yesmaximal, characteristic
7#7124C211yesmaximal, characteristic
8#2507—21no—
9#2088—21no—
10#2086—21no—
11#2083—21no—
12#1947—21no—
13#1944C2211yescharacteristic
14#1944C2211yescharacteristic
15#1940C2211yescharacteristic
16#1919—21no—
17#1849C2211yescharacteristic
18#1387—21no—
19#1387—21no—
20#1383—21no—
21#1383—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×#107—21no—
33C2×#107D411yescharacteristic
34C2×#107D411yescharacteristic
35C2×#106D411yescharacteristic
36C2×#106D411yescharacteristic
37#423—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#97C2311yescharacteristic
41#398—41no—
42#398—41no—
43#389—21no—
44#389—21no—
45#389—21no—
46#389—21no—
47#389—41no—
48#387—41no—
49#384—21no—
50C4×C4⋊C4—21no—
51C4×C4⋊C4D411yescharacteristic
52#340—21no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340—21no—
57#336—21no—
58C2×M16—41no—
59C22×C8—21no—
60C22×C8C2×D411yescharacteristic
61C22×C8C2×D411yescharacteristic
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
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—41no—
77M32—41no—
78M32—41no—
79M32—41no—
80C2×C16—41no—
81C2×C16—41no—
82#108—42no—
83#108—42no—
84#107—41no—
85#107—41no—
86#107—81no—
87#106—21no—
88#106—21no—
89#106—21no—
90#106Q1612yes—
91C4⋊C8—41no—
92C4⋊C8—42no—
93C4⋊C8—41no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C22⋊C8—41no—
97#97—41no—
98C4×C8—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95SD1611yescharacteristic
105#95SD1611yescharacteristic
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4#12011yescharacteristic
113C22×C4C2×SD1611yescharacteristic
114C22×C4AGL(1,ℤ/8)11yescharacteristic
115M16—81no—
116M16—81no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—81no—
125C2×C8—81no—
126C2×C8C2×Q1611yescharacteristic
127C2×C8#13511yescharacteristic
128C2×C8#13711yescharacteristic
129C2×C8#13711yescharacteristic
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—82no—
140C4⋊C4—161no—
141C4⋊C4—161no—
142C4⋊C4—161no—
143C4⋊C4—161no—
144C4⋊C4—161no—
145C42—21no—
146C42—21no—
147C42—41no—
148C42—41no—
149C42—161no—
150C16—81no—
151C16—81no—
152C23#45011yescharacteristic
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
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#45111yescharacteristic
176C2×C4#45211yescharacteristic
177C2×C4#45411yescharacteristic
178C2×C4#45411yescharacteristic
179C2×C4#45411yescharacteristic
180C2×C4#45811yescharacteristic
181C8—21no—
182C8—21no—
183C8—21no—
184C8—21no—
185C8—81no—
186C8—161no—
187C8—161no—
188C22—21no—
189C22—21no—
190C22#161611yescharacteristic
191C22#163511yescharacteristic
192C22#163711yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—81no—
197C4—82no—
198C4—161no—
199C4—161no—
200C4—161no—
201C4#161511yescharacteristic
202C4#161711yescharacteristic
203C4#163611yescharacteristic
204C4#165811yescharacteristic
205C2—21no—
206C2—21no—
207C2#782711yescharacteristic
208C2#1230411yescharacteristic
209C2#1230811yescharacteristic
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
#325142
#135412
#106882
#71262
#71242