Finite Group Databasebeta

← #135581

Subgroups — #135581

Order 512.

560 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 70 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#14274C211yesmaximal, characteristic
2#8551C212yesmaximal
3#7326C214yesmaximal
4C22×C4⋊Q8C2211yescharacteristic
5C2×#391—22no—
6C2×#382C411yescharacteristic
7C2×#376C411yescharacteristic
8#1784—22no—
9#1463—22no—
10#1463C2214yes—
11#1445—22no—
12#1445C412yes—
13#1445C2212yes—
14C23×Q8D412yes—
15C2×C4⋊Q8—24no—
16C2×C4⋊Q8—42no—
17C2×C4⋊Q8—41no—
18C2×C4⋊Q8—41no—
19C22×C4⋊C4—21no—
20C22×C4⋊C4C2×C411yescharacteristic
21C22×C4⋊C4D412yes—
22C22×C4⋊C4C2311yescharacteristic
23C22×C42—21no—
24C22×C42C2×C411yescharacteristic
25C2×#101—42no—
26C2×#101—42no—
27#407—24no—
28#391—48no—
29#382—41no—
30#382—41no—
31#376—42no—
32C2×#95—21no—
33C2×#95—21no—
34C2×#95—24no—
35C2×#95—24no—
36C2×#95C2×C412yes—
37C2×#95C2×C412yes—
38#324—28no—
39#324—44no—
40C22×Q8—22no—
41C22×Q8—22no—
42C22×Q8—24no—
43C22×Q8—42no—
44C22×Q8—44no—
45C22×Q8—44no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4C22⋊C411yescharacteristic
49C23×C4C22⋊C412yes—
50C23×C4C22⋊C411yescharacteristic
51C23×C4C22×C411yescharacteristic
52C23×C4C2×D412yes—
53C2×M16—44no—
54C4⋊Q8—48no—
55C4⋊Q8—48no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—44no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4#4114yes—
68C2×C42—22no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—22no—
72C2×C42—24no—
73C2×C42—42no—
74C2×C42—42no—
75#101—84no—
76#101—84no—
77C22⋊C8—44no—
78C22⋊C8—44no—
79#95—42no—
80#95—42no—
81#95—42no—
82#95—48no—
83#95—48no—
84#95—42no—
85C24C2×C22⋊C411yescharacteristic
86C2×Q8—44no—
87C2×Q8—44no—
88C2×Q8—416no—
89C2×Q8—44no—
90C2×Q8—44no—
91C2×Q8—82no—
92C2×Q8—82no—
93C2×Q8—88no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4—42no—
122C22×C4#9912yes—
123C22×C4#9911yescharacteristic
124C22×C4#9911yescharacteristic
125C22×C4#11712yes—
126C22×C4#12314yes—
127M16—84no—
128M16—84no—
129C2×C8—84no—
130C4⋊C4—44no—
131C4⋊C4—48no—
132C4⋊C4—44no—
133C4⋊C4—82no—
134C4⋊C4—82no—
135C42—24no—
136C42—24no—
137C42—48no—
138C23—22no—
139C23—21no—
140C23—21no—
141C23—22no—
142C23#38111yescharacteristic
143C23C2×#9911yescharacteristic
144C23C2×#9911yescharacteristic
145Q8—88no—
146Q8—88no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—48no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—82no—
172C2×C4—84no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—88no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4#35312yes—
181C2×C4#35412yes—
182C8—164no—
183C22—24no—
184C22—24no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22#181911yescharacteristic
192C22#183012yes—
193C22C2×#35311yescharacteristic
194C22C2×#35411yescharacteristic
195C22#213312yes—
196C4—22no—
197C4—22no—
198C4—42no—
199C4—44no—
200C4—42no—
201C4—84no—
202C4—82no—
203C4—162no—
204C2—22no—
205C2—21no—
206C2—22no—
207C2—21no—
208C2#875411yescharacteristic
209C2#1285611yescharacteristic
210C2#1286211yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ 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
#142742
#85512
#73262