Finite Group Databasebeta

← #134127

Subgroups — #134127

Order 512.

1088 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2868C211yesmaximal, characteristic
2#11512C214yesmaximal
3#8551C212yesmaximal
4C22×C4⋊Q8—22no—
5C22×C4⋊Q8C411yescharacteristic
6C22×C4⋊Q8C2211yescharacteristic
7#2868—28no—
8C2×#384—22no—
9C2×#384C412yes—
10C2×#384C2212yes—
11C2×C43C411yescharacteristic
12#1784—22no—
13#1463—22no—
14#1463C2214yes—
15#1325—28no—
16C23×Q8—21no—
17C23×Q8D412yes—
18C2×C4⋊Q8—216no—
19C2×C4⋊Q8—24no—
20C2×C4⋊Q8—24no—
21C2×C4⋊Q8—42no—
22C2×C4⋊Q8—42no—
23C2×C4⋊Q8—42no—
24C2×C4⋊Q8—416no—
25C2×C4⋊Q8—42no—
26C22×C4⋊C4—24no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—24no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4D412yes—
32C22×C4⋊C4C2311yescharacteristic
33C22×C42—21no—
34C22×C42—21no—
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C412yes—
37C2×#101—44no—
38#407—24no—
39#384—216no—
40#384—24no—
41#384—24no—
42#384—24no—
43#384D418yes—
44C43—24no—
45#324—28no—
46#324—44no—
47C22×Q8—24no—
48C22×Q8—24no—
49C22×Q8—24no—
50C22×Q8—42no—
51C22×Q8—48no—
52C22×Q8—42no—
53C22×Q8—48no—
54C22×Q8—42no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22⋊C412yes—
60C23×C4C22×C411yescharacteristic
61C23×C4C2×D412yes—
62C2×M16—44no—
63C4⋊Q8—416no—
64C4⋊Q8—416no—
65C4⋊Q8—464no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—216no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—216no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—416no—
81C2×C4⋊C4—416no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4C22⋊C414yes—
85C2×C4⋊C4C2×D414yes—
86C2×C42—24no—
87C2×C42—28no—
88C2×C42—24no—
89C2×C42—28no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—28no—
95C2×C42C22⋊C414yes—
96#101—88no—
97C22⋊C8—44no—
98C22⋊C8—44no—
99C24C2×C22⋊C411yescharacteristic
100C2×Q8—48no—
101C2×Q8—416no—
102C2×Q8—416no—
103C2×Q8—48no—
104C2×Q8—48no—
105C2×Q8—84no—
106C2×Q8—88no—
107C2×Q8—88no—
108C2×Q8—84no—
109C2×Q8—84no—
110C22×C4—28no—
111C22×C4—28no—
112C22×C4—28no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—24no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—48no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4#9912yes—
135C22×C4#9912yes—
136C22×C4C2×C22⋊C412yes—
137C22×C4#12014yes—
138M16—88no—
139C2×C8—84no—
140C4⋊C4—44no—
141C4⋊C4—48no—
142C4⋊C4—44no—
143C4⋊C4—432no—
144C4⋊C4—432no—
145C4⋊C4—48no—
146C4⋊C4—44no—
147C4⋊C4—44no—
148C42—216no—
149C42—28no—
150C42—216no—
151C42—28no—
152C42—24no—
153C42#10118yescomplemented
154C23—22no—
155C23—22no—
156C23—22no—
157C23#37911yescharacteristic
158C23C2×#9911yescharacteristic
159C23C2×#9911yescharacteristic
160Q8—816no—
161Q8—816no—
162C2×C4—28no—
163C2×C4—28no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—232no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—28no—
178C2×C4—24no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—48no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—82no—
187C2×C4—84no—
188C2×C4—84no—
189C2×C4—82no—
190C2×C4#35314yes—
191C2×C4#35614yes—
192C2×C4C2×#10114yes—
193C8—164no—
194C22—28no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22#178111yescharacteristic
202C22#178412yes—
203C22C2×#35312yes—
204C22C2×#35612yes—
205C4—216no—
206C4—24no—
207C4—24no—
208C4—24no—
209C4—44no—
210C4—82no—
211C4—84no—
212C2—22no—
213C2—22no—
214C2—22no—
215C2#855111yescharacteristic
216C2#1269311yescharacteristic
217C2#1270011yescharacteristic
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×C4⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
153. C42#101semidirect

Maximal subgroups

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

subgroupindex
C2×#28682
#115122
#85512