Finite Group Databasebeta

← #97559

Subgroups — #97559

Order 512.

803 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 803 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#35455C211yesmaximal, characteristic
2#8702C214yesmaximal, complemented
3#8304C212yesmaximal
4C2×#543C411yescharacteristic
5C22×#123C2211yescharacteristic
6#2561—22no—
7#2503—22no—
8#2488—28no—
9#2308C412yes—
10C2×#381C2212yes—
11#2273C411yescharacteristic
12#1779C2214yes—
13C2×#324—22no—
14C2×#323—22no—
15#1280—28no—
16C23×D4C2×C411yescharacteristic
17C24×C4C2311yescharacteristic
18#543—44no—
19C2×#123—24no—
20C2×#123—44no—
21C2×C22⋊Q8—22no—
22C2×C22⋊Q8—24no—
23C22×C4⋊C4D412yes—
24C22×C22⋊C4C2×C411yescharacteristic
25C22×C22⋊C4C2×C412yes—
26C22×C22⋊C4C2×C412yes—
27C22×C22⋊C4D412yes—
28C2×C22⋊C8—28no—
29C2×C22⋊C8—24no—
30#400—28no—
31#397—24no—
32#393—24no—
33#391—24no—
34#388—28no—
35#386—24no—
36#386—24no—
37#382—24no—
38#381—22no—
39#381—24no—
40#381—24no—
41#379—22no—
42C4×C22⋊C4—22no—
43C4×C22⋊C4—24no—
44C2×#95D418yes—
45#324—44no—
46#323—44no—
47C25C22⋊C411yescharacteristic
48C25C22×C411yescharacteristic
49C22×Q8—22no—
50C22×D4—22no—
51C22×D4—42no—
52C22×D4—44no—
53C23×C4—42no—
54C23×C4—44no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C412yes—
57C23×C4C22⋊C414yes—
58C23×C4C22⋊C414yes—
59C23×C4C2×D412yes—
60C23×C4C2×D414yes—
61C22×C8—44no—
62#123—416no—
63C22⋊Q8—416no—
64C22⋊Q8—44no—
65C22⋊Q8—44no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—42no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—28no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—44no—
82C2×C22⋊C4—44no—
83C2×C42—24no—
84C22⋊C8—44no—
85C22⋊C8—44no—
86C22⋊C8—416no—
87#95—24no—
88#95—28no—
89#95—28no—
90#95—28no—
91C24—22no—
92C24—24no—
93C24—21no—
94C24—22no—
95C24—44no—
96C24—44no—
97C24—42no—
98C24C2×C22⋊C411yescharacteristic
99C24C2×C22⋊C412yes—
100C2×Q8—44no—
101C2×Q8—84no—
102C2×D4—44no—
103C2×D4—44no—
104C2×D4—44no—
105C2×D4—88no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—28no—
111C22×C4—24no—
112C22×C4—21no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4—42no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—48no—
123C22×C4—48no—
124C22×C4—88no—
125C22×C4#9914yes—
126C22×C4#12014yes—
127C2×C8—84no—
128C2×C8—88no—
129C4⋊C4—48no—
130C4⋊C4—44no—
131C4⋊C4—48no—
132C22⋊C4—28no—
133C22⋊C4—48no—
134C22⋊C4—44no—
135C22⋊C4—44no—
136C22⋊C4—44no—
137C22⋊C4—44no—
138C22⋊C4—88no—
139C42—84no—
140C23—28no—
141C23—22no—
142C23—22no—
143C23—22no—
144C23—28no—
145C23—24no—
146C23—44no—
147C23—41no—
148C23—42no—
149C23—44no—
150C23—82no—
151C23—82no—
152C23—88no—
153C23—82no—
154C23—82no—
155C23—164no—
156C23#37911yescharacteristic
157C23C2×#9912yes—
158C23#41014yes—
159Q8—164no—
160D4—88no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—28no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—44no—
169C2×C4—42no—
170C2×C4—48no—
171C2×C4—48no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—88no—
176C2×C4—82no—
177C2×C4—164no—
178C2×C4—164no—
179C8—164no—
180C22—22no—
181C22—22no—
182C22—28no—
183C22—22no—
184C22—24no—
185C22—44no—
186C22—44no—
187C22—44no—
188C22—88no—
189C22—82no—
190C22—84no—
191C22—81no—
192C22—164no—
193C22#178111yescharacteristic
194C22#178212yes—
195C22#179114yes—
196C4—44no—
197C4—44no—
198C4—84no—
199C4—84no—
200C4—81no—
201C2—44no—
202C2—42no—
203C2—161no—
204C2#853912yes—
205C2#854014yes—
206C2#854411yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC25C22×C4
Frattini, cosocleC24×C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C25 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8702C2semidirect

Maximal subgroups

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

subgroupindex
#354552
#87022
#83042