Finite Group Databasebeta

← #102647

Subgroups — #102647

Order 512.

639 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 88 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#30296C211yesmaximal, characteristic
2#25811C212yesmaximal
3#10983C211yesmaximal, characteristic
4#7687C212yesmaximal
5#7026C211yesmaximal, characteristic
6C22×C4⋊Q8C2211yescharacteristic
7C2×#498—22no—
8C22×#101—22no—
9#2857C2212yes—
10#2084—22no—
11#2058—24no—
12#2057—22no—
13#1990—22no—
14#1853—24no—
15#1809—22no—
16C4×C22⋊C8C2212yes—
17#1728C2211yescharacteristic
18#1728C2211yescharacteristic
19#1272—24no—
20C23×Q8—21no—
21C22×M16—21no—
22C22×M16D412yes—
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—44no—
26C2×C4⋊Q8D412yes—
27C2×C22⋊Q8—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4D412yes—
30C22×C42C2311yescharacteristic
31#498—48no—
32C2×#107—22no—
33C2×#106—28no—
34C2×#106—22no—
35C2×#101—44no—
36C2×#101—44no—
37C2×#101—44no—
38C2×C22⋊C8—22no—
39C2×C4×C8—22no—
40#391—22no—
41#389—22no—
42#389—22no—
43#384D412yes—
44#380—22no—
45#380—22no—
46#340—24no—
47#340—28no—
48#336—22no—
49#336—22no—
50#336D414yes—
51C22×Q8—24no—
52C22×Q8—22no—
53C22×Q8—42no—
54C22×Q8—42no—
55C22×Q8—42no—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8—42no—
59C22×Q8—42no—
60C23×C4—22no—
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C23×C4#4111yescharacteristic
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—41no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16SD1614yes—
72C22×C8—22no—
73C22×C8—24no—
74C22×C8—41no—
75C4⋊Q8—28no—
76C4⋊Q8—44no—
77C4⋊Q8—44no—
78C4⋊Q8—48no—
79C22⋊Q8—84no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C42—42no—
92C2×C42C2×D412yes—
93C2×C42C2×D412yes—
94#107—82no—
95#106—416no—
96#106—82no—
97#101—416no—
98C22⋊C8—28no—
99C4×C8—44no—
100#95—42no—
101#95—42no—
102C24#12111yescharacteristic
103C2×Q8—44no—
104C2×Q8—44no—
105C2×Q8—44no—
106C2×Q8—44no—
107C2×Q8—44no—
108C2×Q8—44no—
109C2×Q8—44no—
110C2×Q8—82no—
111C2×Q8—82no—
112C2×Q8—82no—
113C2×Q8—82no—
114C2×Q8—82no—
115C2×Q8—82no—
116C2×Q8—82no—
117C2×Q8—82no—
118C2×Q8—81no—
119C2×Q8—81no—
120C2×Q8—81no—
121C2×Q8—81no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4#12012yes—
137C22×C4#12011yescharacteristic
138C22×C4#12112yes—
139C22×C4#12711yescharacteristic
140C22×C4C2×SD1612yes—
141C22×C4#13712yes—
142M16—44no—
143M16—44no—
144M16—82no—
145M16—82no—
146C2×C8—28no—
147C2×C8—24no—
148C2×C8—44no—
149C2×C8—42no—
150C2×C8—44no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—81no—
154C2×C8—81no—
155C4⋊C4—28no—
156C4⋊C4—44no—
157C4⋊C4—48no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—82no—
164C4⋊C4—84no—
165C22⋊C4—84no—
166C42—24no—
167C42—42no—
168C23—22no—
169C23—21no—
170C23—22no—
171C23—21no—
172C23#39211yescharacteristic
173C23#46112yes—
174Q8—84no—
175Q8—84no—
176Q8—84no—
177Q8—84no—
178Q8—164no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—48no—
192C2×C4—44no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4#44812yes—
203C2×C4#45311yescharacteristic
204C2×C4#45411yescharacteristic
205C2×C4#45614yes—
206C2×C4B(4,𝔽2)12yes—
207C2×C4#47114yes—
208C2×C4#49211yescharacteristic
209C2×C4#49711yescharacteristic
210C8—48no—
211C8—44no—
212C8—82no—
213C22—24no—
214C22—22no—
215C22—41no—
216C22—41no—
217C22—42no—
218C22#200512yes—
219C22#200611yescharacteristic
220C22#201311yescharacteristic
221C22#201912yes—
222C22#202111yescharacteristic
223C4—24no—
224C4—24no—
225C4—21no—
226C4—21no—
227C4—84no—
228C4—162no—
229C4—162no—
230C4#159612yes—
231C4#159912yes—
232C2—42no—
233C2#996611yescharacteristic
234C2#998811yescharacteristic
235C2#1011612yes—
236C2#1016712yes—
237C2#1033411yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#302962
#258112
#109832
#76872
#70262