Finite Group Databasebeta

← #133962

Subgroups — #133962

Order 512.

884 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2640C211yesmaximal, characteristic
2#12856C214yesmaximal
3#8544C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic, complemented
5C22×#124C2211yescharacteristic
6#2640—28no—
7C2×#397—22no—
8C2×#393—22no—
9C2×#391C412yescomplemented
10C2×#390C2212yes—
11C2×#376C411yescharacteristic, complemented
12#1819—22no—
13#1819—22no—
14#1781C2214yes—
15#1393—28no—
16C23×Q8—21no—
17C23×Q8C2×C411yescharacteristic
18C23×D4C2311yescharacteristic
19C2×C4⋊Q8—24no—
20C2×C4⋊Q8—42no—
21C2×C4⋊Q8—42no—
22C2×#124—24no—
23C2×#124—42no—
24C2×#124—42no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C412yes—
27C22×C22⋊C4—24no—
28C22×C22⋊C4D412yes—
29C22×C22⋊C4D412yes—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C412yes—
32C2×#99—28no—
33#397—216no—
34#393—216no—
35#391—24no—
36#391—24no—
37#390—24no—
38#390D418yes—
39#379—24no—
40#376—24no—
41C2×#95—21no—
42C2×#95—24no—
43C2×#95—21no—
44#342—44no—
45#342—44no—
46C25C2×D412yes—
47C22×Q8—24no—
48C22×Q8—22no—
49C22×Q8—41no—
50C22×Q8—41no—
51C22×Q8—42no—
52C22×Q8—44no—
53C22×Q8—42no—
54C22×Q8—84no—
55C22×D4—44no—
56C22×D4—41no—
57C22×D4—41no—
58C22×D4C2×D414yes—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4C22⋊C412yes—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22×C411yescharacteristic
65C4⋊Q8—416no—
66#124—416no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4C22⋊C414yes—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—216no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—44no—
82C2×C22⋊C4—44no—
83C2×C22⋊C4—44no—
84C2×C22⋊C4—88no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—44no—
90C2×C42C22⋊C414yes—
91#99—416no—
92#95—24no—
93#95—24no—
94#95—216no—
95C24—22no—
96C24—22no—
97C24—44no—
98C24—44no—
99C24—44no—
100C24C2×C22⋊C411yescharacteristic
101C24#12014yes—
102C2×Q8—44no—
103C2×Q8—48no—
104C2×Q8—44no—
105C2×Q8—44no—
106C2×Q8—84no—
107C2×Q8—84no—
108C2×Q8—88no—
109C2×Q8—88no—
110C2×Q8—88no—
111C2×D4—28no—
112C2×D4—44no—
113C2×D4—44no—
114C2×D4—84no—
115C2×D4—84no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—24no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—48no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—44no—
137C22×C4—84no—
138C22×C4#9912yes—
139C22×C4#9912yes—
140C22×C4C2×C22⋊C412yes—
141C4⋊C4—44no—
142C4⋊C4—44no—
143C4⋊C4—48no—
144C22⋊C4—48no—
145C22⋊C4—48no—
146C22⋊C4—816no—
147C22⋊C4—88no—
148C22⋊C4—88no—
149C22⋊C4—88no—
150C42—28no—
151C42—44no—
152C42—44no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—24no—
157C23—28no—
158C23—44no—
159C23—44no—
160C23—44no—
161C23—44no—
162C23—82no—
163C23—82no—
164C23—82no—
165C23—82no—
166C23—816no—
167C23#37911yescharacteristic
168C23C2×#9911yescharacteristic
169C23C2×#9911yescharacteristic
170Q8—88no—
171Q8—816no—
172D4—88no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—48no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—88no—
193C2×C4—84no—
194C2×C4—84no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—164no—
198C2×C4—164no—
199C2×C4—164no—
200C2×C4#41014yes—
201C22—28no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—44no—
209C22—44no—
210C22—44no—
211C22—88no—
212C22—164no—
213C22—164no—
214C22#178111yescharacteristic
215C22#178212yes—
216C22#212812yes—
217C22#212912yes—
218C4—24no—
219C4—42no—
220C4—42no—
221C4—84no—
222C4—82no—
223C4—84no—
224C4—324no—
225C2—22no—
226C2—22no—
227C2—22no—
228C2—84no—
229C2#854411yescharacteristic
230C2#1269711yescharacteristic
231C2#1269811yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×C4⋊Q8C4semidirect
9. C2×#391C4semidirect
11. C2×#376C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
#128562
#85442