Finite Group Databasebeta

← #153349

Subgroups — #153349

Order 512.

197 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 66 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#13636C211yesmaximal, characteristic
2#13630C211yesmaximal, characteristic
3C2×#1420C211yesmaximal, characteristic
4C2×#1420C211yesmaximal, characteristic
5#13244C211yesmaximal, characteristic
6#13173C211yesmaximal, characteristic
7#7476C211yesmaximal, characteristic
8C22×Q32C2211yescharacteristic
9C2×M64C2211yescharacteristic
10C2×M64C2211yescharacteristic
11C2×#368C2211yescharacteristic
12C2×#368C2211yescharacteristic
13C2×#368C2211yescharacteristic
14C2×#362—21no—
15#2108C2211yescharacteristic
16#1979—21no—
17#1953—21no—
18#1425—22no—
19#1422—22no—
20#1420—22no—
21#1420—22no—
22#1388—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
32#430—41no—
33C2×#108—21no—
34C2×#108—21no—
35C2×#108—21no—
36#419—41no—
37C2×#101—41no—
38#405D411yescharacteristic
39M64—21no—
40M64—21no—
41M64—21no—
42M64D412yes—
43C2×C32—21no—
44C2×C32—21no—
45#368—21no—
46#368—21no—
47#368—21no—
48#368—21no—
49#368—21no—
50#368D412yes—
51#362—42no—
52#346—21no—
53C4×C16—21no—
54#344—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×M16—41no—
65C2×M16D811yescharacteristic
66C22×C8C2×D411yescharacteristic
67#117D811yescharacteristic
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#108—41no—
86#108—41no—
87#108—82no—
88#103—82no—
89#101—82no—
90#97—21no—
91C4×C8—21no—
92C32—22no—
93C32—41no—
94C2×Q8—42no—
95C2×Q8—81no—
96C2×Q8—81no—
97C2×Q8—81no—
98C2×Q8—81no—
99C2×Q8—81no—
100C22×C4—81no—
101C22×C4C2×D811yescharacteristic
102Q16—42no—
103Q16—42no—
104Q16—81no—
105Q16—81no—
106M16—21no—
107M16—42no—
108M16—81no—
109M16—81no—
110M16—81no—
111M16—81no—
112M16—81no—
113M16D1612yes—
114C2×C8—21no—
115C2×C8—81no—
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—21no—
125C22⋊C4—21no—
126C42—21no—
127C16—41no—
128C16#12111yescharacteristic
129C16C22⋊Q811yescharacteristic
130C16C22⋊Q811yescharacteristic
131C16#12611yescharacteristic
132C23#51011yescharacteristic
133Q8—82no—
134Q8—82no—
135Q8—161no—
136Q8—161no—
137C2×C4—41no—
138C2×C4—82no—
139C2×C4—161no—
140C2×C4—161no—
141C2×C4#45911yescharacteristic
142C2×C4#46611yescharacteristic
143C2×C4#47411yescharacteristic
144C2×C4C2×D1611yescharacteristic
145C2×C4#50811yescharacteristic
146C2×C4#50911yescharacteristic
147C8—22no—
148C8—82no—
149C8—161no—
150C8—161no—
151C8#39611yescharacteristic
152C8#46411yescharacteristic
153C8#46811yescharacteristic
154C8#47811yescharacteristic
155C22—41no—
156C22#221111yescharacteristic
157C22#222011yescharacteristic
158C22#222411yescharacteristic
159C4—81no—
160C4—162no—
161C4#206711yescharacteristic
162C4#220611yescharacteristic
163C4#221511yescharacteristic
164C4#222811yescharacteristic
165C2—21no—
166C2—21no—
167C2—21no—
168C2#1347411yescharacteristic
subgroupG/H
center, socleC2#13474
commutator, Frattini, cosocleC22×C16C23
prime radicalC23#510
LCSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ 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
#136362
#136302
C2×#14202
#132442
#131732
#74762