Finite Group Databasebeta

← #507661

Subgroups — #507661

Order 512.

419 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 146 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#33263C211yesmaximal, characteristic
2#21736C211yesmaximal, characteristic
3#13510C212yesmaximal
4#13454C212yesmaximal
5#13186C212yesmaximal
6#13181C212yesmaximal
7#13174C214yesmaximal
8#12659C211yesmaximal, characteristic
9#3162C2211yescharacteristic
10#2961—21no—
11#2859—21no—
12#2159C2214yes—
13C2×#365C2214yes—
14C2×#364C2212yes—
15C2×#347C2212yes—
16#1994—21no—
17#1946C2212yes—
18#1945—21no—
19#1944—21no—
20#1944C2212yes—
21#1943—21no—
22#1943C2212yes—
23#1849C2211yescharacteristic
24#1845C2211yescharacteristic
25#1844C2211yescharacteristic
26#1834—21no—
27#1773—21no—
28#1772C2212yes—
29C4×#97C2211yescharacteristic
30#1387C2214yes—
31#1386C2214yes—
32#1377C2212yes—
33#533—21no—
34C2×#125—41no—
35C2×M32C2314yes—
36#501—22no—
37#501—22no—
38#446—42no—
39#446—42no—
40C2×#108—21no—
41C2×#108C2312yes—
42#427—21no—
43#427C2312yes—
44C2×#107—21no—
45C2×#107C2312yes—
46C2×#106—21no—
47C2×#106C2312yes—
48#424—22no—
49#423—21no—
50#405—22no—
51C2×#97D411yescharacteristic
52C2×#97D411yescharacteristic
53C2×#97C2311yescharacteristic
54C2×#97C2311yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#389—22no—
58#389—22no—
59#389—41no—
60#389—41no—
61#384—21no—
62#384—21no—
63#382—41no—
64C43D411yescharacteristic
65#367—24no—
66#366—24no—
67#365—44no—
68#364Q818yescomplemented
69#347—42no—
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—84no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85C4×Q8—41no—
86C4×Q8—41no—
87#117—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42C2×D411yescharacteristic
101M32—24no—
102M32C2×Q818yes—
103C2×C16—24no—
104#108—22no—
105#108—22no—
106#108—42no—
107#107—21no—
108#107—21no—
109#107—24no—
110#107—42no—
111#107#4112yes—
112#107#4112yes—
113#106—22no—
114#106—22no—
115#106—22no—
116#106—22no—
117#106—42no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—44no—
121#97—22no—
122#97—22no—
123#97—22no—
124#97—21no—
125#97—21no—
126#97—42no—
127C4×C8—22no—
128C4×C8—42no—
129C4×C8C2×Q814yes—
130#95—42no—
131#95—42no—
132C2×Q8—41no—
133C2×Q8—41no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4AGL(1,ℤ/8)11yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140C22×C4C22×D411yescharacteristic
141M16—42no—
142M16—42no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—42no—
149C2×C8C4⋊Q812yes—
150C2×C8AGL(1,ℤ/8)12yes—
151C2×C8#13712yes—
152C2×C8#13712yes—
153C2×C8C22×Q812yes—
154C2×C8C2×#4112yes—
155C2×C8H(5,𝔽2)11yescharacteristic
156C2×C82−1+411yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—84no—
172C4⋊C4—84no—
173C22⋊C4—42no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—42no—
180C42—42no—
181C42—42no—
182C42C4⋊Q812yes—
183C16—28no—
184C23C2×AGL(1,ℤ/8)11yescharacteristic
185Q8—161no—
186Q8—161no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#52511yescharacteristic
206C2×C4C2×C4⋊Q811yescharacteristic
207C2×C4#53711yescharacteristic
208C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
209C2×C4C2×#13711yescharacteristic
210C2×C4C2×#13711yescharacteristic
211C8—22no—
212C8—41no—
213C8—82no—
214C8#50114yes—
215C8#55412yes—
216C8#55712yes—
217C22—21no—
218C22—21no—
219C22#310911yescharacteristic
220C22#316511yescharacteristic
221C22#325411yescharacteristic
222C4—24no—
223C4—41no—
224C4—41no—
225C4—81no—
226C4—82no—
227C4—82no—
228C4—162no—
229C4—162no—
230C4#258411yescharacteristic
231C4#311311yescharacteristic
232C4C2×#50111yescharacteristic
233C4#325411yescharacteristic
234C2—21no—
235C2—21no—
236C2#2931911yescharacteristic
237C2#3368811yescharacteristic
238C2#3368911yescharacteristic
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

Complements

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

NH ≅ G/Ntype
68. #364Q8semidirect

Maximal subgroups

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

subgroupindex
#332632
#217362
#135102
#134542
#131862
#131812
#131742
#126592