Finite Group Databasebeta

← #508715

Subgroups — #508715

Order 512.

448 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 124 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#33249C211yesmaximal, characteristic
2#21737C211yesmaximal, characteristic
3#13536C212yesmaximal
4#13458C212yesmaximal
5#13415C212yesmaximal
6#13368C212yesmaximal
7#13141C214yesmaximal
8#12897C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10C2×#499C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#491—21no—
13#2960—21no—
14#2642—21no—
15C2×#365C2214yes—
16C2×#362C2214yes—
17#2141C2214yes—
18#1995—21no—
19#1971C2212yes—
20#1971C2212yes—
21#1969—21no—
22#1969—21no—
23#1945—21no—
24#1945C2212yes—
25#1849C2211yescharacteristic
26#1773—21no—
27#1773C2212yes—
28#1726C2211yescharacteristic
29#1386C2214yes—
30#1377C2212yes—
31#1376C2212yes—
32#1375C2212yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35#533—21no—
36C2×C4⋊Q8—21no—
37C2×#125—21no—
38C2×M32C2314yes—
39#501—24no—
40#499—42no—
41#494—42no—
42#491—44no—
43#446—42no—
44C8×Q8—42no—
45C2×#108—21no—
46C2×#108C2312yes—
47#427—21no—
48#427C2312yes—
49C2×#107—21no—
50C2×#106C2312yes—
51#424—22no—
52#423—21no—
53C2×#103—22no—
54C2×#103—22no—
55#405—22no—
56C2×#97D411yescharacteristic
57C2×#97C2311yescharacteristic
58C2×#97C2311yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8D411yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#400—44no—
63#391—22no—
64#391—22no—
65#376D411yescharacteristic
66#365—44no—
67#362—44no—
68#358—28no—
69#339—22no—
70#339—22no—
71C22×Q8—22no—
72C2×Q16—24no—
73C2×Q16—24no—
74C2×Q16—24no—
75C2×Q16—42no—
76C2×Q16—42no—
77C2×Q16—42no—
78C2×M16—22no—
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2411yescharacteristic
83C4⋊Q8—22no—
84C4⋊Q8—42no—
85C4⋊Q8—42no—
86#125—22no—
87#125—42no—
88#125—42no—
89C22⋊Q8—42no—
90C4×Q8—42no—
91#117—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42C2×D411yescharacteristic
98M32—24no—
99M32—24no—
100C2×C16—24no—
101#108—24no—
102#108—42no—
103#107—22no—
104#107—42no—
105#107#4114yes—
106#106—22no—
107#106—22no—
108#106—24no—
109C4⋊C8—42no—
110C4⋊C8—44no—
111#103—44no—
112#103—44no—
113#97—22no—
114#97—22no—
115#97—42no—
116#97SD1614yes—
117C4×C8—22no—
118C4×C8—22no—
119C4×C8—41no—
120C4×C8—42no—
121#95—22no—
122#95—44no—
123#95—44no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4AGL(1,ℤ/8)12yes—
136C22×C4C22×D411yescharacteristic
137Q16—48no—
138Q16—84no—
139M16—42no—
140M16—42no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—42no—
148C2×C8C2×SD1612yes—
149C2×C8C2×SD1612yes—
150C2×C8AGL(1,ℤ/8)12yes—
151C2×C8C2×#4112yes—
152C2×C8H(5,𝔽2)12yes—
153C2×C82−1+412yes—
154C4⋊C4—24no—
155C4⋊C4—44no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—84no—
160C4⋊C4—84no—
161C22⋊C4—42no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—42no—
168C42C2×SD1612yes—
169C16—44no—
170C23C2×AGL(1,ℤ/8)11yescharacteristic
171Q8—84no—
172Q8—82no—
173Q8—82no—
174Q8—82no—
175Q8—82no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—41no—
183C2×C4—84no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4#52511yescharacteristic
189C2×C4#53511yescharacteristic
190C2×C4#53611yescharacteristic
191C2×C4C22×SD1611yescharacteristic
192C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
193C2×C4#57511yescharacteristic
194C8—22no—
195C8—22no—
196C8—22no—
197C8—82no—
198C8#54214yes—
199C22—22no—
200C22#310911yescharacteristic
201C22#320811yescharacteristic
202C22#323511yescharacteristic
203C4—42no—
204C4—42no—
205C4—81no—
206C4—84no—
207C4—162no—
208C4—162no—
209C4#263211yescharacteristic
210C4#310311yescharacteristic
211C4#321311yescharacteristic
212C4#322711yescharacteristic
213C2—21no—
214C2—21no—
215C2#3006211yescharacteristic
216C2#3369312yes—
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 ◁ #1726 ◁ 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
#332492
#217372
#135362
#134582
#134152
#133682
#131412
#128972