Finite Group Databasebeta

← #101784

Subgroups — #101784

Order 512.

343 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 84 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#29221C211yesmaximal, characteristic
2#23266C211yesmaximal, characteristic
3#8584C211yesmaximal, characteristic
4#7683C212yesmaximal
5#7027C212yesmaximal
6#2859C2211yescharacteristic
7C4×#125C2211yescharacteristic
8#2100C412yes—
9#2099C412yes—
10#1991—21no—
11#1989—22no—
12#1917—22no—
13#1916—22no—
14#1835—21no—
15#1771—22no—
16#1767C2212yes—
17C4×C4⋊C8C2211yescharacteristic
18#1744C2212yes—
19C2×#330—21no—
20#1295—21no—
21#1271—22no—
22#1271—22no—
23C2×#125—21no—
24C2×#125D411yescharacteristic
25C2×C4⋊C8—22no—
26C2×C4⋊C8—22no—
27C2×C4⋊C8—21no—
28C2×#97—22no—
29C2×#97—22no—
30C2×C4×C8—21no—
31C2×C4×C8—22no—
32#389—22no—
33#389—22no—
34#389C2×C412yes—
35#387—22no—
36#384C2×C411yescharacteristic
37#384C2×C411yescharacteristic
38#382—21no—
39#382D411yescharacteristic
40C4×C4⋊C4—22no—
41C4×C4⋊C4D412yes—
42C43C2311yescharacteristic
43#340—22no—
44#340—22no—
45#340—22no—
46#340—22no—
47#336—22no—
48#336—22no—
49#336—22no—
50#336—22no—
51#336C2×C412yes—
52#336D412yes—
53#336D412yes—
54#330—42no—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—22no—
58C22×C8—42no—
59#125—21no—
60#125—21no—
61#125—22no—
62#125—44no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4C22⋊C412yes—
72C2×C4⋊C4C22⋊C412yes—
73C2×C42—22no—
74C2×C42C22×C411yescharacteristic
75C2×C42C2×D411yescharacteristic
76C2×C42C2×D412yes—
77C2×C42C2×D411yescharacteristic
78C2×C42#4112yes—
79C4⋊C8—24no—
80C4⋊C8—42no—
81C4⋊C8—42no—
82C4⋊C8—44no—
83#97—44no—
84#97—42no—
85#97—42no—
86C4×C8—44no—
87C4×C8—41no—
88C4×C8—41no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—42no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4C2×C22⋊C411yescharacteristic
98C22×C4C4×D412yes—
99C22×C4#12011yescharacteristic
100C22×C4#12112yes—
101C22×C4#12311yescharacteristic
102C2×C8—24no—
103C2×C8—42no—
104C2×C8—42no—
105C2×C8—42no—
106C2×C8—44no—
107C2×C8—42no—
108C2×C8—42no—
109C2×C8—42no—
110C2×C8—42no—
111C2×C8—42no—
112C2×C8—82no—
113C2×C8—82no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—82no—
126C4⋊C4—82no—
127C4⋊C4—82no—
128C42—22no—
129C42—22no—
130C42—22no—
131C42—42no—
132C42—41no—
133C42—42no—
134C42—42no—
135C42AGL(1,ℤ/8)11yescharacteristic
136C42#13711yescharacteristic
137C42#13712yes—
138C23#38611yescharacteristic
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—82no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—82no—
158C2×C4#41711yescharacteristic
159C2×C4#41911yescharacteristic
160C2×C4#44312yes—
161C2×C4#44412yes—
162C2×C4#45412yes—
163C2×C4#46812yes—
164C2×C4#47012yes—
165C8—44no—
166C8—82no—
167C8—82no—
168C8—84no—
169C22#187612yes—
170C22#188811yescharacteristic
171C22#189411yescharacteristic
172C22#190612yes—
173C22#191511yescharacteristic
174C4—22no—
175C4—22no—
176C4—21no—
177C4—21no—
178C4—22no—
179C4—22no—
180C4—22no—
181C4—41no—
182C4—82no—
183C4—82no—
184C4—161no—
185C4—161no—
186C2#907011yescharacteristic
187C2#920912yes—
188C2#926612yes—
189C2#940411yescharacteristic
190C2#944211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#292212
#232662
#85842
#76832
#70272