Finite Group Databasebeta

← #145814

Subgroups — #145814

Order 512.

339 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 80 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#30301C211yesmaximal, characteristic
2#12920C212yesmaximal
3#11270C211yesmaximal, characteristic
4#7259C212yesmaximal
5#7186C211yesmaximal, characteristic
6C22×#107C2211yescharacteristic
7#2629—21no—
8C2×#367—22no—
9C2×C22⋊C16C2212yes—
10#2083—22no—
11#2059—22no—
12#1809—21no—
13#1808C2212yes—
14C2×#343—21no—
15C2×#341C2211yescharacteristic
16#1736—21no—
17#1729C2211yescharacteristic
18#1380—22no—
19C22×M16—21no—
20C22×M16D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×#125—21no—
23C2×#117D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C16—22no—
26C2×#107—22no—
27C2×#107—21no—
28C2×#107—42no—
29C2×#107D412yes—
30C2×#107D412yes—
31C2×#106—21no—
32#398—42no—
33#396—42no—
34#389—22no—
35#384—21no—
36#380—22no—
37#367—44no—
38C22⋊C16—22no—
39C22⋊C16Q814yes—
40#343—42no—
41#341—21no—
42#341Q812yes—
43#340—22no—
44#340—22no—
45#340—42no—
46#336—22no—
47C23×C4—41no—
48C23×C4C2×D411yescharacteristic
49C2×M16—21no—
50C2×M16—22no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16Q1612yes—
59C22×C8—21no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8—41no—
63C22×C8C2×D412yes—
64C22×C8C2×Q812yes—
65C22×C8C2×Q811yescharacteristic
66C22×C8#4111yescharacteristic
67#125—81no—
68#125—81no—
69#117—41no—
70#117—41no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4D812yes—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—42no—
85C2×C42—21no—
86C2×C16—24no—
87C2×C16—42no—
88C2×C16—42no—
89#107—24no—
90#107—42no—
91#107—42no—
92#107—42no—
93#107—81no—
94#106—81no—
95#95—42no—
96#95—42no—
97C24AGL(1,ℤ/8)11yescharacteristic
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—81no—
107C22×C4—81no—
108C22×C4—162no—
109C22×C4#12011yescharacteristic
110C22×C4C22⋊Q811yescharacteristic
111C22×C4C4⋊Q811yescharacteristic
112C22×C4C2×D811yescharacteristic
113C22×C4C2×Q1611yescharacteristic
114C22×C4#13711yescharacteristic
115M16—42no—
116M16—42no—
117M16—42no—
118M16—81no—
119M16—81no—
120M16—81no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—42no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—42no—
130C2×C8—81no—
131C2×C8—81no—
132C2×C8C22⋊Q812yes—
133C2×C8C4⋊Q812yes—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—81no—
142C4⋊C4—84no—
143C4⋊C4—81no—
144C4⋊C4—162no—
145C4⋊C4—162no—
146C22⋊C4—81no—
147C22⋊C4—162no—
148C42—41no—
149C42—41no—
150C16—44no—
151C23—21no—
152C23—21no—
153C23—22no—
154C23—22no—
155C23#44711yescharacteristic
156C23#47711yescharacteristic
157C23#50111yescharacteristic
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—162no—
179C2×C4—162no—
180C2×C4—162no—
181C2×C4#39511yescharacteristic
182C2×C4#45111yescharacteristic
183C2×C4#45411yescharacteristic
184C2×C4#45511yescharacteristic
185C2×C4B(4,𝔽2)11yescharacteristic
186C2×C4#45811yescharacteristic
187C2×C4#47411yescharacteristic
188C2×C4#50011yescharacteristic
189C8—22no—
190C8—22no—
191C8—42no—
192C8—82no—
193C22—21no—
194C22—21no—
195C22—22no—
196C22—22no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22#159811yescharacteristic
201C22#160711yescharacteristic
202C22#205711yescharacteristic
203C22#219312yes—
204C22#219412yes—
205C4—21no—
206C4—21no—
207C4—82no—
208C4—162no—
209C4—162no—
210C4—162no—
211C4#160211yescharacteristic
212C4#161711yescharacteristic
213C4#206111yescharacteristic
214C4#206211yescharacteristic
215C2—41no—
216C2—41no—
217C2#1056911yescharacteristic
218C2#1216511yescharacteristic
219C2#1216711yescharacteristic
220C2#1228512yes—
221C2#1345612yes—
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC23×C8C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1729 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#303012
#129202
#112702
#72592
#71862