Finite Group Databasebeta

← #134038

Subgroups — #134038

Order 512.

1427 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1427 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×#2614C211yesmaximal, characteristic
2#11500C214yesmaximal, complemented
3#8553C212yesmaximal
4C24×Q8C411yescharacteristic
5C22×#123C412yes—
6C22×#123C2211yescharacteristic
7#2614—28no—
8C2×#395—22no—
9C2×#393—22no—
10C2×#381C411yescharacteristic
11C2×#381C2212yes—
12#1821—22no—
13#1462—22no—
14#1462C2214yes—
15#1317—28no—
16C23×Q8—24no—
17C23×Q8—22no—
18C23×Q8—44no—
19C23×Q8—48no—
20C23×D4C2×C411yescharacteristic
21C24×C4C2×C412yes—
22C24×C4C2×C411yescharacteristic
23C2×#123—24no—
24C2×#123—24no—
25C2×#123—24no—
26C2×#123—44no—
27C2×#123—48no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4D412yes—
30C22×C22⋊C4—22no—
31C22×C22⋊C4C2×C412yes—
32C22×C22⋊C4D412yes—
33C22×C22⋊C4C2311yescharacteristic
34#407—24no—
35#395—216no—
36#393—216no—
37#381—24no—
38#381—24no—
39#381D418yes—
40C2×#95—22no—
41C2×#95—24no—
42#343—44no—
43#323—28no—
44#323—44no—
45C25C22⋊C411yescharacteristic
46C25C22×C411yescharacteristic
47C22×Q8—28no—
48C22×Q8—416no—
49C22×Q8—416no—
50C22×Q8—416no—
51C22×Q8—44no—
52C22×Q8—48no—
53C22×Q8—464no—
54C22×Q8—44no—
55C22×Q8—48no—
56C22×D4—22no—
57C22×D4—42no—
58C22×D4—44no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—24no—
63C23×C4—28no—
64C23×C4—22no—
65C23×C4—44no—
66C23×C4—44no—
67C23×C4—44no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C412yes—
70C23×C4C22⋊C414yes—
71C23×C4C2×D412yes—
72C2×M16—44no—
73#123—416no—
74#123—48no—
75#123—416no—
76#123—48no—
77C2×C4⋊C4—28no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—48no—
82C2×C22⋊C4—28no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—48no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4C22⋊C414yes—
91C2×C22⋊C4C2×D414yes—
92C22⋊C8—44no—
93C22⋊C8—44no—
94#95—28no—
95#95—216no—
96C24—22no—
97C24—24no—
98C24—21no—
99C24—22no—
100C24—44no—
101C24—44no—
102C24—42no—
103C24C2×C22⋊C412yes—
104C24C2×C22⋊C411yescharacteristic
105C2×Q8—416no—
106C2×Q8—432no—
107C2×Q8—432no—
108C2×Q8—464no—
109C2×Q8—432no—
110C2×Q8—464no—
111C2×D4—44no—
112C2×D4—44no—
113C2×D4—44no—
114C2×D4—88no—
115C22×C4—28no—
116C22×C4—28no—
117C22×C4—28no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—24no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—416no—
128C22×C4—432no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—48no—
141C22×C4—42no—
142C22×C4#9914yes—
143C22×C4#10018yes—
144C22×C4#12014yes—
145M16—88no—
146C2×C8—84no—
147C4⋊C4—48no—
148C4⋊C4—416no—
149C22⋊C4—44no—
150C22⋊C4—48no—
151C22⋊C4—416no—
152C22⋊C4—44no—
153C22⋊C4—44no—
154C23—24no—
155C23—22no—
156C23—28no—
157C23—22no—
158C23—22no—
159C23—44no—
160C23—416no—
161C23—44no—
162C23—44no—
163C23—44no—
164C23—41no—
165C23—42no—
166C23—44no—
167C23—88no—
168C23—82no—
169C23—82no—
170C23#37911yescharacteristic
171C23C2×#9912yes—
172C23C2×#10014yes—
173Q8—464no—
174D4—88no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—28no—
178C2×C4—416no—
179C2×C4—44no—
180C2×C4—416no—
181C2×C4—416no—
182C2×C4—48no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—48no—
189C2×C4—44no—
190C2×C4—48no—
191C2×C4—41no—
192C2×C4—44no—
193C2×C4—42no—
194C2×C4—82no—
195C2×C4—88no—
196C2×C4—82no—
197C8—164no—
198C22—28no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—24no—
203C22—44no—
204C22—44no—
205C22—416no—
206C22—44no—
207C22—84no—
208C22—81no—
209C22—82no—
210C22—164no—
211C22#178111yescharacteristic
212C22#178312yes—
213C22#212314yes—
214C4—44no—
215C4—48no—
216C4—44no—
217C4—84no—
218C4—81no—
219C2—22no—
220C2—24no—
221C2—44no—
222C2—161no—
223C2#854711yescharacteristic
224C2#1269212yes—
subgroupG/H
center, socleC22#1781
commutatorC25C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11500C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26142
#115002
#85532