Finite Group Databasebeta

← #145784

Subgroups — #145784

Order 512.

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

normal subgroups only

This group has 365 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#30300C211yesmaximal, characteristic
2#12920C212yesmaximal
3#11331C211yesmaximal, characteristic
4#7259C212yesmaximal
5#7186C211yesmaximal, characteristic
6C2×#428—21no—
7C22×#107C2211yescharacteristic
8#2628—21no—
9C2×#367—22no—
10C2×C22⋊C16C2212yes—
11#2085—22no—
12#2057—22no—
13#1809—21no—
14#1808C2212yes—
15C2×#343—21no—
16C2×#341C2211yescharacteristic
17#1729C2211yescharacteristic
18#1380—22no—
19C22×M16—21no—
20C22×M16D411yescharacteristic
21C23×C8C2311yescharacteristic
22C2×C4⋊Q8—21no—
23C2×#117D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C16—22no—
26#428—42no—
27#428—42no—
28C2×#107—22no—
29C2×#107—21no—
30C2×#107—42no—
31C2×#107—41no—
32C2×#107D412yes—
33C2×#107D412yes—
34C2×#106—21no—
35C2×#106—41no—
36#400—41no—
37#399—41no—
38#395—41no—
39#393—41no—
40#389—22no—
41#382—21no—
42#380—22no—
43#367—44no—
44C22⋊C16—22no—
45C22⋊C16Q814yescomplemented
46#343—42no—
47#341—21no—
48#341Q812yescomplemented
49#340—22no—
50#340—22no—
51#336—22no—
52C22×Q8—41no—
53C23×C4—41no—
54C23×C4C2×D411yescharacteristic
55C2×M16—21no—
56C2×M16—22no—
57C2×M16—22no—
58C2×M16—22no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16Q1612yes—
65C22×C8—21no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C2×D412yes—
70C22×C8C2×Q812yes—
71C22×C8C2×Q811yescharacteristic
72C22×C8#4111yescharacteristic
73C4⋊Q8—81no—
74C4⋊Q8—81no—
75#117—41no—
76#117—41no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4D811yescharacteristic
88C2×C4⋊C4D811yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—42no—
91C2×C42—21no—
92C2×C16—24no—
93C2×C16—42no—
94C2×C16—42no—
95#107—24no—
96#107—42no—
97#107—44no—
98#107—42no—
99#107—42no—
100#107—81no—
101#106—44no—
102#106—81no—
103#95—41no—
104#95—41no—
105#95—42no—
106#95—42no—
107C24AGL(1,ℤ/8)11yescharacteristic
108C2×Q8—82no—
109C2×Q8—162no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—81no—
120C22×C4—81no—
121C22×C4—162no—
122C22×C4#12011yescharacteristic
123C22×C4C22⋊Q811yescharacteristic
124C22×C4C4⋊Q811yescharacteristic
125C22×C4C2×D811yescharacteristic
126C22×C4C2×Q1611yescharacteristic
127C22×C4#13711yescharacteristic
128M16—24no—
129M16—42no—
130M16—42no—
131M16—42no—
132M16—42no—
133M16—81no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—42no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—42no—
145C2×C8C22⋊Q812yes—
146C2×C8C4⋊Q812yes—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—81no—
156C4⋊C4—82no—
157C4⋊C4—81no—
158C4⋊C4—162no—
159C22⋊C4—81no—
160C22⋊C4—162no—
161C42—41no—
162C42—41no—
163C16—44no—
164C23—21no—
165C23—21no—
166C23—22no—
167C23—22no—
168C23#44711yescharacteristic
169C23#47711yescharacteristic
170C23#50111yescharacteristic
171Q8—164no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—162no—
194C2×C4—162no—
195C2×C4—162no—
196C2×C4#39511yescharacteristic
197C2×C4#45111yescharacteristic
198C2×C4#45311yescharacteristic
199C2×C4#45611yescharacteristic
200C2×C4B(4,𝔽2)11yescharacteristic
201C2×C4#45811yescharacteristic
202C2×C4#47411yescharacteristic
203C2×C4#50011yescharacteristic
204C8—22no—
205C8—22no—
206C8—44no—
207C8—42no—
208C22—21no—
209C22—21no—
210C22—22no—
211C22—22no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22#159511yescharacteristic
216C22#161011yescharacteristic
217C22#205711yescharacteristic
218C22#219312yes—
219C22#219412yes—
220C4—21no—
221C4—21no—
222C4—81no—
223C4—81no—
224C4—162no—
225C4—162no—
226C4—162no—
227C4#160511yescharacteristic
228C4#161411yescharacteristic
229C4#206011yescharacteristic
230C4#206211yescharacteristic
231C2—41no—
232C2—41no—
233C2#1056711yescharacteristic
234C2#1210911yescharacteristic
235C2#1211111yescharacteristic
236C2#1229512yes—
237C2#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

Complements

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

NH ≅ G/Ntype
45. C22⋊C16Q8semidirect
48. #341Q8semidirect

Maximal subgroups

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

subgroupindex
#303002
#129202
#113312
#72592
#71862