Finite Group Databasebeta

← #133963

Subgroups — #133963

Order 512.

1046 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 1046 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×#2641C211yesmaximal, characteristic
2#12856C214yesmaximal, complemented
3#8546C212yesmaximal
4C22×#127C2211yescharacteristic
5C22×#125C411yescharacteristic, complemented
6#2641—28no—
7C2×#394—24no—
8C2×#390C412yescomplemented
9C2×#390C2212yes—
10C2×#376C411yescharacteristic, complemented
11#1781—24no—
12#1781C2214yes—
13#1393—28no—
14C23×D4C2×C411yescharacteristic
15C23×D4D414yes—
16C23×D4C2311yescharacteristic
17C2×#127—24no—
18C2×#127—44no—
19C2×#125—24no—
20C2×#125—44no—
21C22×C4⋊C4—22no—
22C22×C4⋊C4C2×C412yes—
23C22×C22⋊C4—24no—
24C22×C22⋊C4—24no—
25C22×C42C2×C411yescharacteristic
26C22×C42C2×C412yes—
27C2×#99—28no—
28C2×#99—48no—
29#394—232no—
30#390—24no—
31#390—24no—
32#390—24no—
33#390D418yes—
34#379—24no—
35#376—24no—
36C2×#95—22no—
37C25C22⋊C412yes—
38C25C2×D412yes—
39C22×D4—28no—
40C22×D4—28no—
41C22×D4—22no—
42C22×D4—44no—
43C22×D4—42no—
44C22×D4—44no—
45C22×D4—42no—
46C22×D4—44no—
47C22×D4—88no—
48C22×D4C2×D414yes—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4C22⋊C412yes—
52C23×C4C22×C411yescharacteristic
53#127—416no—
54#125—416no—
55C2×C4⋊C4—28no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—44no—
60C2×C4⋊C4—44no—
61C2×C4⋊C4C22⋊C414yes—
62C2×C22⋊C4—216no—
63C2×C22⋊C4—216no—
64C2×C22⋊C4—48no—
65C2×C22⋊C4—44no—
66C2×C22⋊C4—88no—
67C2×C22⋊C4—88no—
68C2×C42—24no—
69C2×C42—22no—
70C2×C42—22no—
71C2×C42—44no—
72C2×C42C22⋊C414yes—
73#99—416no—
74#99—816no—
75#95—28no—
76C24—24no—
77C24—22no—
78C24—24no—
79C24—44no—
80C24—48no—
81C24—48no—
82C24—44no—
83C24C2×C22⋊C411yescharacteristic
84C24#12014yes—
85C2×D4—28no—
86C2×D4—44no—
87C2×D4—44no—
88C2×D4—416no—
89C2×D4—416no—
90C2×D4—44no—
91C2×D4—44no—
92C2×D4—44no—
93C2×D4—88no—
94C2×D4—88no—
95C2×D4—88no—
96C2×D4—816no—
97C2×D4—816no—
98C22×C4—22no—
99C22×C4—24no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—44no—
105C22×C4—44no—
106C22×C4—44no—
107C22×C4—44no—
108C22×C4—48no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—84no—
112C22×C4#9914yes—
113C22×C4C2×C22⋊C412yes—
114C4⋊C4—44no—
115C4⋊C4—44no—
116C4⋊C4—416no—
117C22⋊C4—816no—
118C22⋊C4—816no—
119C22⋊C4—88no—
120C22⋊C4—816no—
121C42—28no—
122C42—44no—
123C42—44no—
124C23—24no—
125C23—22no—
126C23—24no—
127C23—28no—
128C23—48no—
129C23—48no—
130C23—48no—
131C23—42no—
132C23—44no—
133C23—48no—
134C23—84no—
135C23—84no—
136C23—816no—
137C23—816no—
138C23—84no—
139C23—84no—
140C23#37911yescharacteristic
141C23C2×#9912yes—
142D4—88no—
143D4—88no—
144D4—832no—
145C2×C4—28no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—28no—
149C2×C4—24no—
150C2×C4—44no—
151C2×C4—44no—
152C2×C4—44no—
153C2×C4—44no—
154C2×C4—48no—
155C2×C4—44no—
156C2×C4—44no—
157C2×C4—88no—
158C2×C4—84no—
159C2×C4—84no—
160C2×C4—164no—
161C2×C4—168no—
162C2×C4#41014yes—
163C22—28no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—48no—
169C22—88no—
170C22—88no—
171C22—82no—
172C22—84no—
173C22—168no—
174C22—168no—
175C22#178111yescharacteristic
176C22#178212yes—
177C22#212714yes—
178C4—24no—
179C4—44no—
180C4—84no—
181C4—84no—
182C4—324no—
183C2—22no—
184C2—24no—
185C2—84no—
186C2—162no—
187C2#854411yescharacteristic
188C2#1269612yes—
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC23×D4C23
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12856C2semidirect
5. C22×#125C4semidirect
8. C2×#390C4semidirect
10. C2×#376C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
#128562
#85462