Finite Group Databasebeta

← #102646

Subgroups — #102646

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#30250C211yesmaximal, characteristic
2#25811C212yesmaximal
3#11382C211yesmaximal, characteristic
4#7685C212yesmaximal
5#7026C211yesmaximal, characteristic
6C22×C4⋊Q8C2211yescharacteristic
7C2×#498—22no—
8C2×#470—22no—
9#2857C2212yes—
10#2096—22no—
11#2058—24no—
12#2016—22no—
13#1990—22no—
14#1853—24no—
15C4×C22⋊C8C2212yes—
16C2×#343—22no—
17#1728C2211yescharacteristic
18#1728C2211yescharacteristic
19#1271—24no—
20C23×Q8—21no—
21C22×Q16—22no—
22C22×M16—21no—
23C22×M16D412yes—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—44no—
27C2×C4⋊Q8D412yes—
28C2×C22⋊Q8—22no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4D412yes—
31C22×C42C2311yescharacteristic
32#498—48no—
33#470—44no—
34#470—44no—
35C2×#106—28no—
36C2×#106—22no—
37C2×#103—24no—
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#343—44no—
47#340—28no—
48#336—22no—
49#336—22no—
50#336D414yes—
51C22×Q8—22no—
52C22×Q8—44no—
53C22×Q8—42no—
54C22×Q8—84no—
55C23×C4—22no—
56C23×C4C2×D411yescharacteristic
57C23×C4C2×D411yescharacteristic
58C23×C4#4111yescharacteristic
59C2×Q16—44no—
60C2×Q16—82no—
61C2×Q16—82no—
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—41no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16SD1614yes—
70C22×C8—22no—
71C22×C8—24no—
72C22×C8—41no—
73C4⋊Q8—28no—
74C4⋊Q8—44no—
75C4⋊Q8—44no—
76C4⋊Q8—48no—
77C22⋊Q8—44no—
78C22⋊Q8—44no—
79C2×C4⋊C4—24no—
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—42no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C42—42no—
93C2×C42C2×D412yes—
94C2×C42C2×D412yes—
95#106—416no—
96#106—44no—
97#103—48no—
98C22⋊C8—28no—
99C4×C8—44no—
100#95—42no—
101#95—42no—
102C24#12111yescharacteristic
103C2×Q8—48no—
104C2×Q8—44no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—44no—
108C2×Q8—84no—
109C2×Q8—816no—
110C2×Q8—84no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—84no—
124C22×C4#12012yes—
125C22×C4#12011yescharacteristic
126C22×C4#12112yes—
127C22×C4#12711yescharacteristic
128C22×C4C2×SD1612yes—
129C22×C4AGL(1,ℤ/8)12yes—
130Q16—88no—
131M16—44no—
132M16—44no—
133M16—82no—
134M16—82no—
135C2×C8—28no—
136C2×C8—24no—
137C2×C8—44no—
138C2×C8—42no—
139C2×C8—44no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—81no—
143C2×C8—81no—
144C4⋊C4—28no—
145C4⋊C4—44no—
146C4⋊C4—44no—
147C4⋊C4—44no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C4⋊C4—44no—
151C4⋊C4—44no—
152C4⋊C4—82no—
153C4⋊C4—82no—
154C4⋊C4—82no—
155C22⋊C4—84no—
156C42—24no—
157C42—42no—
158C23—22no—
159C23—21no—
160C23—22no—
161C23—21no—
162C23#39211yescharacteristic
163C23#46012yes—
164Q8—816no—
165Q8—84no—
166Q8—84no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—84no—
181C2×C4—84no—
182C2×C4—84no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4#45012yes—
190C2×C4#45511yescharacteristic
191C2×C4#45614yes—
192C2×C4#45611yescharacteristic
193C2×C4B(4,𝔽2)12yes—
194C2×C4#47114yes—
195C2×C4#49211yescharacteristic
196C2×C4#49611yescharacteristic
197C8—48no—
198C8—44no—
199C8—82no—
200C22—24no—
201C22—22no—
202C22—41no—
203C22—41no—
204C22—42no—
205C22#200512yes—
206C22#200711yescharacteristic
207C22#201411yescharacteristic
208C22#201912yes—
209C22#202111yescharacteristic
210C4—24no—
211C4—24no—
212C4—21no—
213C4—21no—
214C4—84no—
215C4—162no—
216C4—162no—
217C4#160112yes—
218C4#160412yes—
219C2—42no—
220C2#996411yescharacteristic
221C2#999011yescharacteristic
222C2#1011912yes—
223C2#1017212yes—
224C2#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
#302502
#258112
#113822
#76852
#70262