Finite Group Databasebeta

← #134136

Subgroups — #134136

Order 512.

644 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 90 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#35640C211yesmaximal, characteristic
2#11514C214yesmaximal
3#8554C212yesmaximal
4C2×#544C411yescharacteristic
5C22×#125C2211yescharacteristic
6#2583—22no—
7#2552—24no—
8#2550—24no—
9#2502—22no—
10#2317C2212yes—
11#2296C412yes—
12#2274C411yescharacteristic
13#1821—22no—
14#1463—22no—
15#1463C2214yes—
16#1325—24no—
17#1324—24no—
18#544—44no—
19C2×C4⋊Q8—22no—
20C2×#125—22no—
21C2×#125—24no—
22C2×#125—44no—
23C2×C22⋊Q8—22no—
24C2×#117C2×C412yes—
25C2×#117C2×C412yes—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4D412yes—
28C22×C4⋊C4D412yes—
29C22×C4⋊C4C2311yescharacteristic
30C22×C42C2×C411yescharacteristic
31#407—24no—
32#398—24no—
33#398—28no—
34#395—24no—
35#391—24no—
36#389—28no—
37#389—22no—
38#389—22no—
39#389—24no—
40#385—22no—
41#385—22no—
42#385—24no—
43#384—24no—
44#384D414yes—
45#382D414yes—
46C4×C4⋊C4—24no—
47C4×C4⋊C4—24no—
48C4×C22⋊C4—22no—
49#343—44no—
50#324—28no—
51#324—44no—
52C22×Q8—22no—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C412yes—
56C23×C4C22×C411yescharacteristic
57C23×C4C2×D412yes—
58C2×M16—44no—
59C4⋊Q8—48no—
60#125—416no—
61#125—48no—
62C22⋊Q8—44no—
63C22⋊Q8—44no—
64#117—28no—
65#117—44no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—28no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4C22⋊C412yes—
82C2×C4⋊C4C22⋊C412yes—
83C2×C4⋊C4C2×D414yes—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—24no—
89C2×C42—42no—
90C2×C42C22⋊C414yes—
91C22⋊C8—44no—
92C22⋊C8—44no—
93#95—24no—
94#95—24no—
95#95—24no—
96#95—28no—
97C24C2×C22⋊C411yescharacteristic
98C2×Q8—44no—
99C2×Q8—84no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—82no—
114C22×C4—82no—
115C22×C4—81no—
116C22×C4—81no—
117C22×C4#9914yes—
118C22×C4C2×C22⋊C412yes—
119C22×C4#12014yes—
120M16—88no—
121C2×C8—84no—
122C4⋊C4—28no—
123C4⋊C4—28no—
124C4⋊C4—44no—
125C4⋊C4—48no—
126C4⋊C4—48no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—44no—
130C4⋊C4—48no—
131C4⋊C4—84no—
132C4⋊C4—88no—
133C4⋊C4—84no—
134C22⋊C4—28no—
135C22⋊C4—44no—
136C42—24no—
137C42—44no—
138C42—44no—
139C42—84no—
140C42#10118yes—
141C23—22no—
142C23—41no—
143C23—41no—
144C23#37911yescharacteristic
145C23C2×#9912yes—
146Q8—164no—
147C2×C4—28no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—28no—
153C2×C4—44no—
154C2×C4—44no—
155C2×C4—44no—
156C2×C4—44no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4C2×#10114yes—
172C8—164no—
173C22—21no—
174C22—21no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—44no—
179C22#178111yescharacteristic
180C22#178412yes—
181C4—24no—
182C4—48no—
183C4—44no—
184C4—44no—
185C4—82no—
186C4—81no—
187C4—81no—
188C4—82no—
189C2—22no—
190C2—42no—
191C2#855111yescharacteristic
192C2#1270811yescharacteristic
193C2#1270911yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#356402
#115142
#85542