Finite Group Databasebeta

← #508753

Subgroups — #508753

Order 512.

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

normal subgroups only

This group has 374 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#33261C211yesmaximal, characteristic
2#21737C211yesmaximal, characteristic
3#13542C212yesmaximal
4#13541C212yesmaximal
5#13508C212yesmaximal
6#13454C212yesmaximal
7#13118C214yesmaximal
8#12897C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10#2960—21no—
11#2647—21no—
12#2159C2214yes—
13#1995—21no—
14#1945—21no—
15#1945C2212yes—
16#1944C2212yes—
17#1943C2212yes—
18#1942—21no—
19#1942—21no—
20#1849C2211yescharacteristic
21#1846C2211yescharacteristic
22#1845C2211yescharacteristic
23#1836—21no—
24#1773—21no—
25#1773C2212yes—
26#1726C2211yescharacteristic
27#1387C2214yes—
28#1386C2214yes—
29#1383C2214yes—
30#1377C2212yes—
31#1376C2212yes—
32#1375C2212yes—
33#533—21no—
34C2×M32C2314yes—
35#501—22no—
36#501—22no—
37#446—41no—
38#446—41no—
39C8×Q8—41no—
40C8×Q8—41no—
41C2×#108—21no—
42C2×#108C2312yes—
43#427—21no—
44#427C2312yes—
45C2×#107—21no—
46C2×#107—21no—
47C2×#107C2312yes—
48C2×#106C2312yes—
49#424—22no—
50#423—21no—
51#405—22no—
52C2×#97D411yescharacteristic
53C2×#97C2311yescharacteristic
54C2×#97C2311yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#400—42no—
59#398—42no—
60#389—22no—
61#389—22no—
62#384—21no—
63#382—21no—
64#376D411yescharacteristic
65#367—24no—
66#366—24no—
67#340—22no—
68#340—22no—
69#339—22no—
70#339—22no—
71C2×M16—22no—
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22×C8C2411yescharacteristic
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78#125—22no—
79C22⋊Q8—41no—
80C22⋊Q8—41no—
81C4×Q8—41no—
82C4×Q8—41no—
83#117—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D411yescharacteristic
91C2×C42C2×D411yescharacteristic
92M32—24no—
93M32—24no—
94C2×C16—24no—
95#108—24no—
96#108—42no—
97#107—21no—
98#107—21no—
99#107—24no—
100#107—42no—
101#107—41no—
102#107—41no—
103#107#4112yes—
104#107#4112yes—
105#106—22no—
106#106—22no—
107#106—24no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112#97—22no—
113#97—22no—
114#97—42no—
115#97Q1614yes—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—22no—
119C4×C8—41no—
120C4×C8—42no—
121#95—21no—
122#95—21no—
123#95—42no—
124#95—42no—
125#95—42no—
126C2×Q8—41no—
127C2×Q8—41no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4AGL(1,ℤ/8)11yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134C22×C4C22×D411yescharacteristic
135M16—42no—
136M16—42no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—42no—
144C2×C8C2×Q1612yes—
145C2×C8C2×Q1612yes—
146C2×C8AGL(1,ℤ/8)11yescharacteristic
147C2×C8AGL(1,ℤ/8)11yescharacteristic
148C2×C8C2×#4112yes—
149C2×C82−1+412yes—
150C2×C82−1+412yes—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—162no—
162C4⋊C4—162no—
163C4⋊C4—162no—
164C22⋊C4—42no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—42no—
171C42C2×Q1612yes—
172C16—44no—
173C23C2×AGL(1,ℤ/8)11yescharacteristic
174Q8—82no—
175Q8—82no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4#52511yescharacteristic
195C2×C4#53611yescharacteristic
196C2×C4#53611yescharacteristic
197C2×C4C22×Q1611yescharacteristic
198C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
199C2×C4#57511yescharacteristic
200C8—22no—
201C8—22no—
202C8—21no—
203C8—21no—
204C8—82no—
205C8#54114yes—
206C22—21no—
207C22—21no—
208C22#310911yescharacteristic
209C22#323011yescharacteristic
210C22#323511yescharacteristic
211C4—42no—
212C4—41no—
213C4—41no—
214C4—81no—
215C4—82no—
216C4—82no—
217C4—162no—
218C4—162no—
219C4#263411yescharacteristic
220C4#310411yescharacteristic
221C4#323011yescharacteristic
222C4#323411yescharacteristic
223C2—21no—
224C2—21no—
225C2#3009511yescharacteristic
226C2#3369211yescharacteristic
227C2#3369311yescharacteristic
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
#332612
#217372
#135422
#135412
#135082
#134542
#131182
#128972