Finite Group Databasebeta

← #146136

Subgroups — #146136

Order 512.

182 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 56 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#12029C211yesmaximal, characteristic
2#11642C211yesmaximal, characteristic
3#8665C211yesmaximal, characteristic
4#8619C211yesmaximal, characteristic
5#7291C211yesmaximal, characteristic
6#7223C211yesmaximal, characteristic
7#7219C211yesmaximal, characteristic
8C2×#446C2211yescharacteristic
9#2163C2211yescharacteristic
10#2162C2211yescharacteristic
11#1837C2211yescharacteristic
12#1801—21no—
13#1567C2211yescharacteristic
14#1566C2211yescharacteristic
15#1564C2211yescharacteristic
16#1487—21no—
17#1368—22no—
18#1338—22no—
19#1301—21no—
20#1292—21no—
21C2×C4×Q8D411yescharacteristic
22#446—21no—
23#446—21no—
24#446—21no—
25#446—21no—
26#431—21no—
27#423—21no—
28#423—21no—
29#423—21no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8D411yescharacteristic
32C2×#101—41no—
33C2×#97—21no—
34C2×C4×C8C2311yescharacteristic
35C4⋊C16—21no—
36C4⋊C16—21no—
37C22⋊C16—21no—
38C22⋊C16—21no—
39#346D412yes—
40C4×C16—21no—
41#343—41no—
42#341—21no—
43#341—21no—
44#335—21no—
45#335—21no—
46#335—21no—
47#334—21no—
48#334—21no—
49#334—21no—
50Q8⋊C8—42no—
51C22×Q8SD1611yescharacteristic
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C22×C8—21no—
56C22×C8#4111yescharacteristic
57C22×C8#4111yescharacteristic
58C4×Q8—21no—
59C4×Q8—21no—
60C4×Q8—21no—
61C4×Q8—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4SD1611yescharacteristic
64C2×C42—21no—
65C2×C42C2×D411yescharacteristic
66C2×C16—22no—
67C2×C16—41no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70C4⋊C8—41no—
71C4⋊C8—41no—
72C4⋊C8—41no—
73C4⋊C8—41no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76C4⋊C8—41no—
77C4⋊C8—41no—
78C4⋊C8—41no—
79#101—82no—
80C22⋊C8—41no—
81#97—41no—
82#97—41no—
83C4×C8—41no—
84C4×C8C2×D411yescharacteristic
85C4×C8#4111yescharacteristic
86C4×C8#4111yescharacteristic
87C4×C8#4111yescharacteristic
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—41no—
91C2×Q8—41no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4#12311yescharacteristic
95C22×C4C2×SD1611yescharacteristic
96C22×C4#13711yescharacteristic
97M16—81no—
98M16—81no—
99M16—81no—
100M16—81no—
101M16—81no—
102M16—81no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—41no—
106C2×C8—81no—
107C2×C8—81no—
108C2×C8—81no—
109C2×C8—81no—
110C2×C8—81no—
111C2×C8#12311yescharacteristic
112C2×C8#12411yescharacteristic
113C2×C8#12611yescharacteristic
114C2×C8#12611yescharacteristic
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C42—41no—
120C42—41no—
121C42#12111yescharacteristic
122C42#12311yescharacteristic
123C42#13511yescharacteristic
124C42AGL(1,ℤ/8)11yescharacteristic
125C16—42no—
126C16—81no—
127C23#48311yescharacteristic
128Q8—41no—
129Q8—41no—
130Q8—81no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—81no—
140C2×C4#39711yescharacteristic
141C2×C4#46511yescharacteristic
142C2×C4#46811yescharacteristic
143C2×C4#48111yescharacteristic
144C2×C4#48211yescharacteristic
145C2×C4#48511yescharacteristic
146C8—21no—
147C8—21no—
148C8—21no—
149C8—21no—
150C8—81no—
151C8—161no—
152C8—161no—
153C8—161no—
154C8—161no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22#207711yescharacteristic
159C4—21no—
160C4—21no—
161C4—21no—
162C4—21no—
163C4—81no—
164C4—81no—
165C4#208011yescharacteristic
166C4#208211yescharacteristic
167C4#223811yescharacteristic
168C4#223911yescharacteristic
169C2—41no—
170C2#1064311yescharacteristic
171C2#1350111yescharacteristic
172C2#1350311yescharacteristic
subgroupG/H
center, socleC22#2077
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#483
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ 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
#120292
#116422
#86652
#86192
#72912
#72232
#72192