Finite Group Databasebeta

← #145997

Subgroups — #145997

Order 512.

209 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 64 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#12036C211yesmaximal, characteristic
2#12018C211yesmaximal, characteristic
3#11972C211yesmaximal, characteristic
4#11940C211yesmaximal, characteristic
5#9799C211yesmaximal, characteristic
6#9634C211yesmaximal, characteristic
7#7245C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9#2162C2211yescharacteristic
10#2150C2211yescharacteristic
11#1966—21no—
12#1574—21no—
13#1567C2211yescharacteristic
14#1567C2211yescharacteristic
15#1564C2211yescharacteristic
16#1546—21no—
17C8×M16C2211yescharacteristic
18#1372—21no—
19#1371—21no—
20#1370—21no—
21#1367—21no—
22#1365—21no—
23#1365—21no—
24#1364—21no—
25#1364—21no—
26#1299—21no—
27C2×C4⋊Q8—21no—
28C2×M32—21no—
29#500—21no—
30#500—21no—
31#500—21no—
32#500—21no—
33#431—21no—
34#431D411yescharacteristic
35C2×#107—21no—
36C2×#107Q812yes—
37#423—21no—
38#423—21no—
39C2×#101—42no—
40C4×M16D411yescharacteristic
41C2×C4×C8C2311yescharacteristic
42C4⋊C16—21no—
43C4⋊C16—21no—
44C4⋊C16D411yescharacteristic
45C4⋊C16D411yescharacteristic
46C22⋊C16—21no—
47C4×C16—21no—
48#335—21no—
49#335—21no—
50#335—21no—
51#335—21no—
52#335—41no—
53#335D411yescharacteristic
54#335D411yescharacteristic
55#334—21no—
56#334—41no—
57#332—41no—
58#332—41no—
59#322—21no—
60C82—21no—
61C22×Q8—41no—
62C2×M16—21no—
63C2×M16—41no—
64C2×M16—41no—
65C22×C8C2×Q811yescharacteristic
66C22×C8#4111yescharacteristic
67C4⋊Q8—41no—
68C4⋊Q8—41no—
69C4⋊Q8—41no—
70C4⋊Q8—41no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—41no—
73C2×C42C2×D411yescharacteristic
74M32—81no—
75C2×C16—21no—
76C2×C16—21no—
77C2×C16—41no—
78#107—22no—
79#107—22no—
80#107—41no—
81#107—41no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84C4⋊C8—41no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C4⋊C8—41no—
88C4⋊C8D811yescharacteristic
89C4⋊C8D811yescharacteristic
90#101—84no—
91C22⋊C8—21no—
92C22⋊C8—41no—
93#97—21no—
94C4×C8—21no—
95C4×C8—21no—
96C4×C8—41no—
97C4×C8C2×D411yescharacteristic
98C4×C8C2×D411yescharacteristic
99C4×C8#4111yescharacteristic
100C4×C8#4111yescharacteristic
101C2×Q8—82no—
102C2×Q8—82no—
103C2×Q8—81no—
104C2×Q8—81no—
105C22×C4—42no—
106C22×C4C22⋊Q811yescharacteristic
107C22×C4#13511yescharacteristic
108C22×C4AGL(1,ℤ/8)11yescharacteristic
109M16—22no—
110M16—41no—
111M16—82no—
112M16—161no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—81no—
122C2×C8—81no—
123C2×C8#12111yescharacteristic
124C2×C8C22⋊Q811yescharacteristic
125C2×C8C22⋊Q811yescharacteristic
126C2×C8#12611yescharacteristic
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—81no—
130C4⋊C4—81no—
131C42#12111yescharacteristic
132C42#12111yescharacteristic
133C42C2×D811yescharacteristic
134C42#13711yescharacteristic
135C16—41no—
136C16—41no—
137C16—81no—
138C23#47811yescharacteristic
139Q8—162no—
140Q8—161no—
141Q8—161no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—84no—
149C2×C4—82no—
150C2×C4#39611yescharacteristic
151C2×C4#45911yescharacteristic
152C2×C4#46411yescharacteristic
153C2×C4#46611yescharacteristic
154C2×C4#46811yescharacteristic
155C2×C4#47411yescharacteristic
156C8—21no—
157C8—21no—
158C8—21no—
159C8—42no—
160C8—41no—
161C8—41no—
162C8—81no—
163C8—81no—
164C8—161no—
165C8—161no—
166C8#47111yescharacteristic
167C8#47311yescharacteristic
168C22—21no—
169C22—21no—
170C22—21no—
171C22#206711yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—162no—
176C4#206411yescharacteristic
177C4#207011yescharacteristic
178C4#221211yescharacteristic
179C4#221311yescharacteristic
180C4#222211yescharacteristic
181C4#222311yescharacteristic
182C2—41no—
183C2#1060411yescharacteristic
184C2#1347111yescharacteristic
185C2#1347811yescharacteristic
subgroupG/H
center, socleC22#2067
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#478
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#120362
#120182
#119722
#119402
#97992
#96342
#72452