Finite Group Databasebeta

← #134015

Subgroups — #134015

Order 512.

1124 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1124 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#12700C214yesmaximal
3#8551C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C22×C4⋊Q8C412yes—
6C22×C4⋊Q8C2212yes—
7C22×C4⋊Q8C2211yescharacteristic
8#2868—28no—
9C2×#384—24no—
10C2×C43C411yescharacteristic
11C2×#356—28no—
12#1784C2214yes—
13#1463—24no—
14C23×Q8C2×C412yes—
15C23×Q8C2×C411yescharacteristic
16C23×Q8C2311yescharacteristic
17C2×C4⋊Q8—24no—
18C2×C4⋊Q8—24no—
19C2×C4⋊Q8—24no—
20C2×C4⋊Q8—24no—
21C2×C4⋊Q8—24no—
22C2×C4⋊Q8—44no—
23C2×C4⋊Q8—48no—
24C2×C4⋊Q8—48no—
25C2×C4⋊Q8—44no—
26C2×C4⋊Q8D418yes—
27C22×C4⋊C4—24no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—24no—
30C22×C4⋊C4D414yes—
31C22×C42—22no—
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C412yes—
34C2×#101—28no—
35#407—24no—
36#384—232no—
37C43—24no—
38#356—416no—
39#324—48no—
40C22×Q8—22no—
41C22×Q8—22no—
42C22×Q8—42no—
43C22×Q8—42no—
44C22×Q8—44no—
45C22×Q8—44no—
46C22×Q8—44no—
47C22×Q8—44no—
48C22×Q8—44no—
49C22×Q8C22⋊C414yes—
50C22×Q8C2×D414yes—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4C22⋊C412yes—
54C23×C4C22⋊C412yes—
55C23×C4C22×C411yescharacteristic
56C23×C4C2×D412yes—
57C2×M16—44no—
58C4⋊Q8—216no—
59C4⋊Q8—48no—
60C4⋊Q8—416no—
61C4⋊Q8—416no—
62C4⋊Q8—416no—
63C4⋊Q8—416no—
64C4⋊Q8—48no—
65C4⋊Q8—48no—
66C2×C4⋊C4—28no—
67C2×C4⋊C4—216no—
68C2×C4⋊C4—28no—
69C2×C4⋊C4—216no—
70C2×C4⋊C4—28no—
71C2×C4⋊C4—44no—
72C2×C4⋊C4—416no—
73C2×C4⋊C4—416no—
74C2×C4⋊C4—44no—
75C2×C4⋊C4—44no—
76C2×C42—28no—
77C2×C42—216no—
78C2×C42—24no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—28no—
82C2×C42C22⋊C414yes—
83#101—416no—
84C22⋊C8—48no—
85C24C2×C22⋊C411yescharacteristic
86C2×Q8—28no—
87C2×Q8—28no—
88C2×Q8—44no—
89C2×Q8—44no—
90C2×Q8—44no—
91C2×Q8—44no—
92C2×Q8—44no—
93C2×Q8—44no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×Q8—88no—
99C2×Q8—816no—
100C2×Q8—88no—
101C22×C4—28no—
102C22×C4—216no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—28no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4#9914yes—
119C22×C4C2×C22⋊C412yes—
120C22×C4#12014yes—
121M16—88no—
122C2×C8—84no—
123C4⋊C4—416no—
124C4⋊C4—432no—
125C4⋊C4—432no—
126C4⋊C4—416no—
127C42—232no—
128C42—28no—
129C42—28no—
130C42—24no—
131C42#9918yes—
132C23—24no—
133C23—22no—
134C23#37911yescharacteristic
135C23C2×#9912yes—
136Q8—88no—
137Q8—88no—
138Q8—88no—
139Q8—88no—
140C2×C4—216no—
141C2×C4—28no—
142C2×C4—232no—
143C2×C4—28no—
144C2×C4—28no—
145C2×C4—24no—
146C2×C4—22no—
147C2×C4—28no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—28no—
151C2×C4—24no—
152C2×C4—44no—
153C2×C4—44no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—44no—
157C2×C4—84no—
158C2×C4—88no—
159C2×C4—84no—
160C2×C4#35618yescomplemented
161C2×C4C2×#9914yes—
162C8—164no—
163C22—28no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22#178112yes—
168C22#178111yescharacteristic
169C22C2×#35614yes—
170C4—216no—
171C4—24no—
172C4—28no—
173C4—44no—
174C4—84no—
175C4—82no—
176C2—22no—
177C2—24no—
178C2#854611yescharacteristic
179C2#1270012yes—
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
160. C2×C4#356semidirect

Maximal subgroups

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

subgroupindex
C2×#28682
#127002
#85512