Finite Group Databasebeta

← #507603

Subgroups — #507603

Order 512.

415 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 144 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#21735C211yesmaximal, characteristic
2#21723C211yesmaximal, characteristic
3#13541C212yesmaximal
4#13454C212yesmaximal
5#13273C212yesmaximal
6#13181C212yesmaximal
7#13119C214yesmaximal
8#12881C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10C2×#499C2211yescharacteristic
11#2960—21no—
12#2330—21no—
13#2159C2214yes—
14C2×#364C2212yes—
15#1995—21no—
16#1992—21no—
17#1946—21no—
18#1946C2212yes—
19#1944C2212yes—
20#1942—21no—
21#1849C2211yescharacteristic
22#1845C2211yescharacteristic
23#1776C2212yes—
24C4×#107—21no—
25#1772—21no—
26#1772C2212yes—
27#1726C2211yescharacteristic
28#1387C2214yes—
29#1386C2214yes—
30#1383C2214yes—
31#1377C2212yes—
32#1375C2212yes—
33#533—21no—
34C2×#125—21no—
35C2×M32C2314yes—
36#501—22no—
37#501—22no—
38#499—22no—
39#499Q814yescomplemented
40#446—41no—
41#446—41no—
42C8×Q8—41no—
43C8×Q8—41no—
44C2×#108—21no—
45C2×#108C2312yes—
46#427—21no—
47#427C2312yes—
48C2×#107—21no—
49C2×#107—21no—
50C2×#107C2312yes—
51C2×#106C2312yes—
52#424—22no—
53#423—21no—
54#405—22no—
55C2×#97D411yescharacteristic
56C2×#97C2311yescharacteristic
57C2×#97C2311yescharacteristic
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61#389—22no—
62#389—42no—
63#387—42no—
64#384—21no—
65C4×C4⋊C4—22no—
66#376D411yescharacteristic
67#367—24no—
68#366—24no—
69#364—24no—
70#340—22no—
71#340—22no—
72#339—22no—
73#339—22no—
74C2×M16—22no—
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2411yescharacteristic
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81#125—22no—
82#125—24no—
83#125—42no—
84C22⋊Q8—41no—
85C22⋊Q8—41no—
86C4×Q8—41no—
87C4×Q8—41no—
88#117—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C42—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—21no—
104#107—21no—
105#107—24no—
106#107—24no—
107#107—41no—
108#107—41no—
109#107#4112yes—
110#107#4112yes—
111#106—22no—
112#106—22no—
113#106—22no—
114#106C2×Q814yes—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119#97—22no—
120#97—22no—
121#97—41no—
122#97—42no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—42no—
126C4×C8Q1614yes—
127C4×C8C2×Q812yes—
128C4×C8#4112yes—
129#95—21no—
130#95—21no—
131#95—42no—
132#95—42no—
133C2×Q8—41no—
134C2×Q8—41no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140C22×C4AGL(1,ℤ/8)11yescharacteristic
141C22×C4C22×D411yescharacteristic
142M16—42no—
143M16—42no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—42no—
150C2×C8C22⋊Q812yes—
151C2×C8C2×Q1612yes—
152C2×C8C2×Q1612yes—
153C2×C8AGL(1,ℤ/8)11yescharacteristic
154C2×C8AGL(1,ℤ/8)11yescharacteristic
155C2×C8C22×Q811yescharacteristic
156C2×C8C2×#4111yescharacteristic
157C2×C8C2×#4112yes—
158C2×C82−1+412yes—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—48no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—162no—
171C4⋊C4—162no—
172C22⋊C4—42no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—42no—
178C42—162no—
179C42C22⋊Q812yes—
180C42C2×Q1612yes—
181C16—44no—
182C23C2×AGL(1,ℤ/8)11yescharacteristic
183Q8—82no—
184Q8—82no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4C2×C22⋊Q811yescharacteristic
204C2×C4#52511yescharacteristic
205C2×C4#53611yescharacteristic
206C2×C4C22×Q1611yescharacteristic
207C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C4#57511yescharacteristic
209C8—22no—
210C8—41no—
211C8—82no—
212C8#47712yes—
213C8#47712yes—
214C8#55612yes—
215C8#55712yes—
216C22—21no—
217C22—21no—
218C22#307711yescharacteristic
219C22#310911yescharacteristic
220C22#323011yescharacteristic
221C4—42no—
222C4—42no—
223C4—41no—
224C4—41no—
225C4—84no—
226C4—82no—
227C4—82no—
228C4—162no—
229C4#258111yescharacteristic
230C4C2×#47711yescharacteristic
231C4#310411yescharacteristic
232C4#323511yescharacteristic
233C2—21no—
234C2—21no—
235C2#2927711yescharacteristic
236C2#3369211yescharacteristic
237C2#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

Complements

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

NH ≅ G/Ntype
39. #499Q8semidirect

Maximal subgroups

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

subgroupindex
#217352
#217232
#135412
#134542
#132732
#131812
#131192
#128812