Finite Group Databasebeta

← #153345

Subgroups — #153345

Order 512.

208 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 68 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#13634C211yesmaximal, characteristic
2C2×#1422C211yesmaximal, characteristic
3C2×#1420C211yesmaximal, characteristic
4#13618C211yesmaximal, characteristic
5#13236C211yesmaximal, characteristic
6#13169C211yesmaximal, characteristic
7#7478C211yesmaximal, characteristic
8C22×Q32C2211yescharacteristic
9C2×M64C2211yescharacteristic
10C2×M64C2211yescharacteristic
11C2×#368C2211yescharacteristic
12#2158C2211yescharacteristic
13C2×#366C2211yescharacteristic
14C2×#358—21no—
15C2×C4×C16C2211yescharacteristic
16#1969—21no—
17#1945—21no—
18#1424—21no—
19#1423—21no—
20#1422—22no—
21#1420—22no—
22#1416—21no—
23#1416—21no—
24#1382—21no—
25C22×Q16—21no—
26C2×Q32—21no—
27C2×Q32—21no—
28C2×Q32—21no—
29C2×Q32—21no—
30C2×Q32—21no—
31C2×Q32Q812yes—
32C22×C16D411yescharacteristic
33C22×C16C2311yescharacteristic
34C2×#108—21no—
35#427—21no—
36C2×#107—21no—
37#424—41no—
38C2×#103—41no—
39C2×C4×C8D411yescharacteristic
40#391—41no—
41M64—21no—
42M64—21no—
43M64—21no—
44M64D412yes—
45C2×C32—21no—
46C2×C32—21no—
47#368—21no—
48#368—21no—
49#367—21no—
50#366—21no—
51#366—21no—
52#366D411yescharacteristic
53#366D411yescharacteristic
54#358—42no—
55C4×C16—21no—
56C4×C16—21no—
57#339—41no—
58C22×Q8—41no—
59C2×Q16—22no—
60C2×Q16—41no—
61C2×Q16—41no—
62C2×Q16—41no—
63C2×Q16—41no—
64C2×Q16—41no—
65C2×M16—41no—
66C22×C8D811yescharacteristic
67C22×C8C2×D411yescharacteristic
68#117—41no—
69C2×C4⋊C4—41no—
70C2×C42D811yescharacteristic
71Q32—22no—
72Q32—22no—
73Q32—41no—
74Q32—41no—
75C2×C16—21no—
76C2×C16—21no—
77C2×C16—21no—
78C2×C16C2×D411yescharacteristic
79C2×C16C2×D411yescharacteristic
80C2×C16C2×Q811yescharacteristic
81C2×C16#4111yescharacteristic
82C2×C16#4111yescharacteristic
83C2×C16#4111yescharacteristic
84#108—41no—
85#108—41no—
86#107—21no—
87#107—21no—
88#107—41no—
89#107—41no—
90#106—41no—
91C4⋊C8—82no—
92#103—82no—
93C4×C8—21no—
94C4×C8—21no—
95#95—82no—
96C32—22no—
97C32—41no—
98C2×Q8—42no—
99C2×Q8—81no—
100C2×Q8—81no—
101C2×Q8—81no—
102C2×Q8—81no—
103C2×Q8—81no—
104C22×C4—21no—
105C22×C4—81no—
106C22×C4—81no—
107C22×C4C2×D811yescharacteristic
108Q16—42no—
109Q16—42no—
110Q16—81no—
111Q16—81no—
112M16—81no—
113M16—81no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—81no—
117C2×C8D1612yes—
118C2×C8#12111yescharacteristic
119C2×C8#12111yescharacteristic
120C2×C8C22⋊Q811yescharacteristic
121C2×C8#13511yescharacteristic
122C2×C8AGL(1,ℤ/8)11yescharacteristic
123C2×C8#13711yescharacteristic
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4—81no—
127C4⋊C4—81no—
128C22⋊C4—81no—
129C42—21no—
130C42—21no—
131C42—81no—
132C16—41no—
133C16#12111yescharacteristic
134C16C22⋊Q811yescharacteristic
135C16C22⋊Q811yescharacteristic
136C16#12611yescharacteristic
137C23#50911yescharacteristic
138Q8—82no—
139Q8—82no—
140Q8—161no—
141Q8—161no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4—82no—
148C2×C4—161no—
149C2×C4—161no—
150C2×C4—162no—
151C2×C4—161no—
152C2×C4#45911yescharacteristic
153C2×C4#46611yescharacteristic
154C2×C4#47411yescharacteristic
155C2×C4C2×D1611yescharacteristic
156C2×C4#50811yescharacteristic
157C2×C4#51011yescharacteristic
158C8—22no—
159C8—161no—
160C8#39611yescharacteristic
161C8#46411yescharacteristic
162C8#46811yescharacteristic
163C8#47811yescharacteristic
164C22—21no—
165C22—21no—
166C22#220611yescharacteristic
167C22#222011yescharacteristic
168C22#222811yescharacteristic
169C4—42no—
170C4—161no—
171C4—161no—
172C4—162no—
173C4—321no—
174C4#206711yescharacteristic
175C4#221111yescharacteristic
176C4#221511yescharacteristic
177C4#222411yescharacteristic
178C2—21no—
179C2—21no—
180C2#1347411yescharacteristic
181C2#1369111yescharacteristic
182C2#1369311yescharacteristic
subgroupG/H
center, socleC22#2206
commutator, Frattini, cosocleC22×C16C23
prime radicalC23#509
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G
LECSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#136342
C2×#14222
C2×#14202
#136182
#132362
#131692
#74782