Finite Group Databasebeta

← #507455

Subgroups — #507455

Order 512.

396 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 150 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#33260C211yesmaximal, characteristic
2#21733C211yesmaximal, characteristic
3#13521C214yesmaximal
4#13288C211yesmaximal, characteristic
5#13287C211yesmaximal, characteristic
6#13116C214yesmaximal
7#12904C211yesmaximal, characteristic
8#12886C211yesmaximal, characteristic
9#12656C211yesmaximal, characteristic
10#3160C2211yescharacteristic
11#2961—21no—
12#2647—21no—
13C2×#365C2214yes—
14C2×#364C2212yes—
15C4×M32C2211yescharacteristic
16C4×M32C2211yescharacteristic
17#1995—21no—
18#1946—21no—
19#1945C2212yes—
20#1944—21no—
21#1943—21no—
22#1941C2212yes—
23#1941C2212yes—
24#1848C2211yescharacteristic
25#1846C2211yescharacteristic
26#1844C2211yescharacteristic
27#1836—21no—
28#1773—21no—
29#1772C2212yes—
30#1726C2211yescharacteristic
31#1386C2214yes—
32#1386C2214yes—
33#1383C2214yes—
34#1376C2212yes—
35#533—21no—
36C2×M32C2312yes—
37C2×M32C2312yes—
38#500Q812yes—
39#499—21no—
40#499Q812yes—
41#498—21no—
42#446—44no—
43C2×#108—21no—
44C2×#108C2312yes—
45#427—21no—
46#427C2312yes—
47C2×#107—21no—
48C2×#106—21no—
49C2×#106C2312yes—
50C2×#106C2312yes—
51#424—22no—
52#423—21no—
53#405—22no—
54C2×#97D411yescharacteristic
55C2×#97D411yescharacteristic
56C2×#97D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2311yescharacteristic
60#400—42no—
61#398—42no—
62#389—22no—
63#389—22no—
64#384—21no—
65#382—21no—
66#376D411yescharacteristic
67#365—28no—
68#364—24no—
69#346—21no—
70#346—21no—
71C4×C16—21no—
72C4×C16—21no—
73#340—22no—
74#340—22no—
75#339—22no—
76#339—22no—
77C2×M16—22no—
78C22×C8C2×D411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81C22×C8C2411yescharacteristic
82C4⋊Q8—22no—
83#125—22no—
84C22⋊Q8—42no—
85C4×Q8—42no—
86#117—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C42C2×D411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95M32—22no—
96M32—22no—
97M32—22no—
98M32#4114yes—
99C2×C16—22no—
100C2×C16—22no—
101#108—24no—
102#108—42no—
103#107—22no—
104#107—42no—
105#107C2×Q814yes—
106#106—22no—
107#106—22no—
108#106—24no—
109#106—24no—
110#106—42no—
111C4⋊C8—44no—
112C4⋊C8—42no—
113#97—22no—
114#97—22no—
115#97—42no—
116#97SD1614yes—
117C4×C8—22no—
118C4×C8—22no—
119C4×C8—42no—
120C4×C8C2×Q811yescharacteristic
121C4×C8C2×Q811yescharacteristic
122C4×C8#4111yescharacteristic
123C4×C8#4111yescharacteristic
124#95—22no—
125#95—42no—
126#95—42no—
127#95—42no—
128C2×Q8—42no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4AGL(1,ℤ/8)12yes—
134C22×C4C22×D411yescharacteristic
135M16—42no—
136M16—42no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—42no—
142C2×C8C22⋊Q811yescharacteristic
143C2×C8C22⋊Q811yescharacteristic
144C2×C8C2×SD1612yes—
145C2×C8C2×SD1612yes—
146C2×C8#13512yes—
147C2×C8#13712yes—
148C2×C8C22×Q811yescharacteristic
149C2×C8C2×#4111yescharacteristic
150C2×C8C2×#4112yes—
151C2×C82−1+412yes—
152C4⋊C4—24no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—162no—
160C4⋊C4—162no—
161C4⋊C4—162no—
162C22⋊C4—42no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—42no—
168C42C22⋊Q811yescharacteristic
169C42C22⋊Q811yescharacteristic
170C42C2×SD1612yes—
171C16—24no—
172C16—42no—
173C23C2×AGL(1,ℤ/8)11yescharacteristic
174Q8—162no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—44no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4#47612yes—
188C2×C4#47612yes—
189C2×C4C2×C22⋊Q811yescharacteristic
190C2×C4#53211yescharacteristic
191C2×C4#53611yescharacteristic
192C2×C4C22×SD1611yescharacteristic
193C2×C4C2×#13511yescharacteristic
194C2×C4C2×#13711yescharacteristic
195C8—22no—
196C8—22no—
197C8—82no—
198C8#47912yes—
199C8#47912yes—
200C8#55212yes—
201C8#55712yes—
202C22—22no—
203C22C2×#47611yescharacteristic
204C22#318211yescharacteristic
205C22#322811yescharacteristic
206C4—22no—
207C4—22no—
208C4—81no—
209C4—81no—
210C4—84no—
211C4—162no—
212C4—162no—
213C4#256511yescharacteristic
214C4C2×#47911yescharacteristic
215C4#316711yescharacteristic
216C4#322911yescharacteristic
217C2—21no—
218C2—21no—
219C2#2917611yescharacteristic
220C2#3363812yes—
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radicalC23C2×AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #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
#332602
#217332
#135212
#132882
#132872
#131162
#129042
#128862
#126562