Finite Group Databasebeta

← #153347

Subgroups — #153347

Order 512.

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

normal subgroups only

This group has 202 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
1C2×#1425C211yesmaximal, characteristic
2#13628C211yesmaximal, characteristic
3C2×#1420C211yesmaximal, characteristic
4C2×#1416C211yesmaximal, characteristic
5#13244C211yesmaximal, characteristic
6#13171C211yesmaximal, characteristic
7#7478C211yesmaximal, characteristic
8C22×Q32C2211yescharacteristic
9C2×M64C2211yescharacteristic
10C22×C32C2211yescharacteristic
11C2×#368C2211yescharacteristic
12C2×#368C2211yescharacteristic
13C2×#366C2211yescharacteristic
14C2×#362—21no—
15#2108C2211yescharacteristic
16#1979—21no—
17#1952—21no—
18#1425—22no—
19C4⋊C32—22no—
20#1420—22no—
21#1416—22no—
22#1387—21no—
23C22×Q16—21no—
24C2×Q32—21no—
25C2×Q32—21no—
26C2×Q32—21no—
27C2×Q32—21no—
28C2×Q32—21no—
29C2×Q32Q812yes—
30C2×M32D411yescharacteristic
31C22×C16C2311yescharacteristic
32C2×#108—21no—
33C2×#108—21no—
34#428—41no—
35C2×#107—21no—
36#419—41no—
37C2×#101—41no—
38#405D411yescharacteristic
39M64—21no—
40M64—21no—
41C2×C32—21no—
42C2×C32—21no—
43C2×C32—21no—
44C2×C32D412yes—
45#368—21no—
46#368—21no—
47#368—21no—
48#368D412yes—
49#366—21no—
50#366—21no—
51#362—42no—
52#346—21no—
53C4×C16—21no—
54#343—41no—
55C22×Q8—41no—
56C2×Q16—22no—
57C2×Q16—41no—
58C2×Q16—41no—
59C2×Q16—41no—
60C2×Q16—41no—
61C2×Q16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16D811yescharacteristic
65C22×C8C2×D411yescharacteristic
66#117D811yescharacteristic
67C2×C4⋊C4—41no—
68Q32—22no—
69Q32—22no—
70Q32—41no—
71Q32—41no—
72M32—21no—
73M32—21no—
74C2×C16—21no—
75C2×C16C2×D411yescharacteristic
76C2×C16C2×D411yescharacteristic
77C2×C16C2×Q811yescharacteristic
78C2×C16#4111yescharacteristic
79C2×C16#4111yescharacteristic
80C2×C16#4111yescharacteristic
81#108—22no—
82#108—41no—
83#108—41no—
84#108—41no—
85#107—41no—
86#107—41no—
87#107—81no—
88#106—81no—
89#103—81no—
90#103—81no—
91#101—82no—
92#97—21no—
93C4×C8—21no—
94C32—22no—
95C32—41no—
96C2×Q8—42no—
97C2×Q8—81no—
98C2×Q8—81no—
99C2×Q8—81no—
100C2×Q8—81no—
101C2×Q8—81no—
102C22×C4—81no—
103C22×C4—81no—
104C22×C4C2×D811yescharacteristic
105Q16—42no—
106Q16—42no—
107Q16—81no—
108Q16—81no—
109M16—21no—
110M16—21no—
111M16—42no—
112M16—81no—
113M16—81no—
114M16—81no—
115C2×C8—21no—
116C2×C8—81no—
117C2×C8—81no—
118C2×C8#12111yescharacteristic
119C2×C8#12111yescharacteristic
120C2×C8C22⋊Q811yescharacteristic
121C2×C8#13511yescharacteristic
122C2×C8AGL(1,ℤ/8)11yescharacteristic
123C2×C8#13711yescharacteristic
124C4⋊C4—81no—
125C4⋊C4—81no—
126C4⋊C4D1611yescharacteristic
127C4⋊C4D1611yescharacteristic
128C22⋊C4—21no—
129C42—21no—
130C16—41no—
131C16#12111yescharacteristic
132C16C22⋊Q811yescharacteristic
133C16C22⋊Q811yescharacteristic
134C16#12611yescharacteristic
135C23#50811yescharacteristic
136Q8—82no—
137Q8—82no—
138Q8—161no—
139Q8—161no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—82no—
143C2×C4—161no—
144C2×C4—161no—
145C2×C4—161no—
146C2×C4—321no—
147C2×C4#45911yescharacteristic
148C2×C4#46611yescharacteristic
149C2×C4#47411yescharacteristic
150C2×C4C2×D1611yescharacteristic
151C2×C4#50911yescharacteristic
152C2×C4#51011yescharacteristic
153C8—41no—
154C8—82no—
155C8—161no—
156C8#39611yescharacteristic
157C8#46411yescharacteristic
158C8#46811yescharacteristic
159C8#47811yescharacteristic
160C22—21no—
161C22—21no—
162C22#221111yescharacteristic
163C22#221511yescharacteristic
164C22#222811yescharacteristic
165C4—41no—
166C4—41no—
167C4—162no—
168C4—321no—
169C4#206711yescharacteristic
170C4#220611yescharacteristic
171C4#222011yescharacteristic
172C4#222411yescharacteristic
173C2—21no—
174C2—21no—
175C2#1347411yescharacteristic
176C2#1370111yescharacteristic
177C2#1370211yescharacteristic
subgroupG/H
center, socleC22#2215
commutator, Frattini, cosocleC22×C16C23
prime radicalC23#508
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ 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
C2×#14252
#136282
C2×#14202
C2×#14162
#132442
#131712
#74782