Finite Group Databasebeta

← #508713

Subgroups — #508713

Order 512.

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

normal subgroups only

This group has 462 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#33255C211yesmaximal, characteristic
2#21736C211yesmaximal, characteristic
3#13536C212yesmaximal
4#13457C212yesmaximal
5#13415C212yesmaximal
6#13368C212yesmaximal
7#13141C214yesmaximal
8#12896C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10C2×#498C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#487—21no—
13#2961—21no—
14#2862—21no—
15C2×#365C2214yes—
16C2×#362C2214yes—
17#2141C2214yes—
18#1994—21no—
19#1971—21no—
20#1971—21no—
21#1971C2212yes—
22#1971C2212yes—
23#1946—21no—
24#1945C2212yes—
25#1849C2211yescharacteristic
26#1773C2212yes—
27#1772—21no—
28C4×#97C2211yescharacteristic
29#1386C2214yes—
30#1377C2212yes—
31#1376C2212yes—
32#1375C2212yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35#533—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×M32C2314yes—
39#501—24no—
40#498—42no—
41#494—42no—
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#365—44no—
67#362—44no—
68#358—28no—
69#339—22no—
70#339—22no—
71C22×Q8—22no—
72C22×Q8—41no—
73C2×Q16—24no—
74C2×Q16—24no—
75C2×Q16—24no—
76C2×Q16—42no—
77C2×Q16—42no—
78C2×Q16—42no—
79C2×M16—22no—
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2411yescharacteristic
84C4⋊Q8—22no—
85C4⋊Q8—42no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89#125—22no—
90C22⋊Q8—42no—
91C4×Q8—42no—
92#117—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C42—22no—
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42C2×D411yescharacteristic
101M32—24no—
102M32—24no—
103C2×C16—24no—
104#108—24no—
105#108—42no—
106#107—22no—
107#107#4114yes—
108#106—22no—
109#106—22no—
110#106—24no—
111#106—42no—
112C4⋊C8—42no—
113C4⋊C8—44no—
114#103—44no—
115#103—44no—
116#97—22no—
117#97—22no—
118#97—22no—
119#97—41no—
120#97—42no—
121C4×C8—22no—
122C4×C8—42no—
123C4×C8SD1614yes—
124#95—44no—
125#95—44no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—82no—
134C2×Q8—84no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4AGL(1,ℤ/8)12yes—
140C22×C4C22×D411yescharacteristic
141Q16—48no—
142Q16—84no—
143M16—42no—
144M16—42no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—42no—
152C2×C8C2×SD1612yes—
153C2×C8C2×SD1612yes—
154C2×C8AGL(1,ℤ/8)12yes—
155C2×C8C2×#4112yes—
156C2×C8H(5,𝔽2)12yes—
157C2×C82−1+412yes—
158C4⋊C4—24no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C22⋊C4—42no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—44no—
173C42—42no—
174C42C2×SD1612yes—
175C16—44no—
176C23C2×AGL(1,ℤ/8)11yescharacteristic
177Q8—82no—
178Q8—82no—
179Q8—82no—
180Q8—82no—
181Q8—164no—
182Q8—162no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4—41no—
190C2×C4—84no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4#52511yescharacteristic
196C2×C4#53511yescharacteristic
197C2×C4#53611yescharacteristic
198C2×C4C22×SD1611yescharacteristic
199C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
200C2×C4#57511yescharacteristic
201C8—22no—
202C8—22no—
203C8—22no—
204C8—82no—
205C8#54214yes—
206C22—22no—
207C22#310911yescharacteristic
208C22#320811yescharacteristic
209C22#323511yescharacteristic
210C4—24no—
211C4—42no—
212C4—81no—
213C4—84no—
214C4—162no—
215C4—162no—
216C4#263211yescharacteristic
217C4#310311yescharacteristic
218C4#321311yescharacteristic
219C4#322711yescharacteristic
220C2—21no—
221C2—21no—
222C2#3006211yescharacteristic
223C2#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
#135362
#134572
#134152
#133682
#131412
#128962