Finite Group Databasebeta

← #507709

Subgroups — #507709

Order 512.

471 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 130 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#33255C211yesmaximal, characteristic
2#21736C211yesmaximal, characteristic
3#13488C212yesmaximal
4#13368C212yesmaximal
5#13272C212yesmaximal
6#13262C212yesmaximal
7#13154C214yesmaximal
8#12659C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10C2×#498C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#487—21no—
13#2961—21no—
14#2862—21no—
15C2×#364C2212yes—
16C2×#362C2214yes—
17#2141C2214yes—
18C2×#347C2212yes—
19#1994—21no—
20#1971—21no—
21#1971—21no—
22#1971C2212yes—
23#1971C2212yes—
24#1946C2212yes—
25#1945—21no—
26#1849C2211yescharacteristic
27#1773—21no—
28#1772C2212yes—
29C4×#97C2211yescharacteristic
30#1386C2214yes—
31#1383C2214yes—
32#1377C2212yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35#533—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×M32C2314yes—
39#501—24no—
40#498—24no—
41#494—24no—
42#487—44no—
43#446—44no—
44C2×#108—21no—
45C2×#108C2312yes—
46#427—21no—
47#427C2312yes—
48C2×#106—21no—
49C2×#106C2312yes—
50#424—22no—
51#423—21no—
52C2×#103—22no—
53C2×#103—22no—
54#405—22no—
55C2×#97D411yescharacteristic
56C2×#97D411yescharacteristic
57C2×#97C2311yescharacteristic
58C2×#97C2311yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61#391—22no—
62#391—22no—
63#391—42no—
64#382—42no—
65C43D411yescharacteristic
66#364—24no—
67#362—44no—
68#358—28no—
69#347—42no—
70#339—22no—
71#339—22no—
72C22×Q8—22no—
73C22×Q8—41no—
74C2×Q16—24no—
75C2×Q16—24no—
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—42no—
79C2×Q16—42no—
80C2×Q16—42no—
81C2×M16—22no—
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22×C8C2411yescharacteristic
86C4⋊Q8—22no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91#125—22no—
92C22⋊Q8—42no—
93C4×Q8—42no—
94#117—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C42—22no—
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D411yescharacteristic
102C2×C42C2×D411yescharacteristic
103M32—24no—
104M32—24no—
105C2×C16—24no—
106#108—22no—
107#108—22no—
108#108—42no—
109#107—22no—
110#107#4114yes—
111#106—22no—
112#106—22no—
113#106—24no—
114#106—42no—
115C4⋊C8—42no—
116C4⋊C8—44no—
117#103—44no—
118#103—44no—
119#97—22no—
120#97—22no—
121#97—22no—
122#97—21no—
123#97—21no—
124#97—42no—
125C4×C8—22no—
126C4×C8—42no—
127C4×C8#4114yes—
128#95—44no—
129#95—44no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—82no—
138C2×Q8—84no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4AGL(1,ℤ/8)12yes—
144C22×C4C22×D411yescharacteristic
145Q16—44no—
146Q16—44no—
147Q16—84no—
148M16—42no—
149M16—42no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—42no—
156C2×C8#12412yes—
157C2×C8AGL(1,ℤ/8)12yes—
158C2×C8AGL(1,ℤ/8)12yes—
159C2×C8#13712yes—
160C2×C8C2×#4112yes—
161C2×C8C2×#4112yes—
162C2×C8H(5,𝔽2)11yescharacteristic
163C2×C82−1+411yescharacteristic
164C4⋊C4—24no—
165C4⋊C4—44no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C22⋊C4—42no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—44no—
179C42—42no—
180C42#12412yes—
181C16—44no—
182C23C2×AGL(1,ℤ/8)11yescharacteristic
183Q8—82no—
184Q8—82no—
185Q8—82no—
186Q8—82no—
187Q8—164no—
188Q8—162no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—41no—
196C2×C4—84no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#52511yescharacteristic
202C2×C4C2×#12411yescharacteristic
203C2×C4#53711yescharacteristic
204C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
205C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
206C2×C4C2×#13711yescharacteristic
207C8—22no—
208C8—41no—
209C8—82no—
210C8#48914yes—
211C8#55612yes—
212C8#55912yes—
213C22—22no—
214C22#310911yescharacteristic
215C22#313811yescharacteristic
216C22#325411yescharacteristic
217C4—24no—
218C4—42no—
219C4—81no—
220C4—84no—
221C4—162no—
222C4—162no—
223C4#259211yescharacteristic
224C4#311211yescharacteristic
225C4C2×#48911yescharacteristic
226C4#325311yescharacteristic
227C2—21no—
228C2—21no—
229C2#2935411yescharacteristic
230C2#3368912yes—
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC23C2×AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332552
#217362
#134882
#133682
#132722
#132622
#131542
#126592