Finite Group Databasebeta

← #146133

Subgroups — #146133

Order 512.

190 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 54 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, complemented
2#11635C211yesmaximal, characteristic
3#8678C211yesmaximal, characteristic
4#8636C211yesmaximal, characteristic
5#7295C211yesmaximal, characteristic, complemented
6#7221C211yesmaximal, characteristic, complemented
7#7221C211yesmaximal, characteristic, complemented
8#2931C2211yescharacteristic
9#2163C2211yescharacteristic
10#2163C2211yescharacteristic
11#1835C2211yescharacteristic
12#1566C2211yescharacteristic
13#1566C2211yescharacteristic
14#1564C2211yescharacteristic
15#1560—21no—
16#1517—21no—
17#1368—22no—
18#1330—22no—
19#1309—21no—
20#1286—21no—
21C2×#131—21no—
22#532D411yescharacteristic
23#436—21no—
24#436—21no—
25C8×D4—21no—
26C8×D4—21no—
27#431—21no—
28#423—21no—
29#423—21no—
30#423D411yescharacteristic
31#408—21no—
32C2×C4×C8C2311yescharacteristic
33#382—21no—
34C4⋊C16—21no—
35C4⋊C16—21no—
36C22⋊C16—21no—
37C22⋊C16—21no—
38#346—21no—
39#346D412yes—
40#340—21no—
41#340—21no—
42#335—21no—
43#334—21no—
44#334—21no—
45#334—21no—
46#334—21no—
47#334—21no—
48#333—41no—
49#331—41no—
50#327—42no—
51C2×#41—21no—
52#131—82no—
53C2×M16—21no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C22×C8—41no—
58C22×C8#4111yescharacteristic
59C22×C8#4111yescharacteristic
60C4⋊Q8SD1611yescharacteristic
61#127SD1611yescharacteristic
62#124—21no—
63#121—21no—
64#121—21no—
65C4×D4—21no—
66C4×D4—21no—
67C2×C4⋊C4—41no—
68C2×C42C2×D411yescharacteristic
69C2×C16—22no—
70C2×C16—41no—
71C4⋊C8—21no—
72C4⋊C8—21no—
73C4⋊C8—41no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76C4⋊C8—41no—
77C4⋊C8—41no—
78C22⋊C8—41no—
79C22⋊C8—41no—
80C22⋊C8—41no—
81C4×C8—41no—
82C4×C8C2×D411yescharacteristic
83C4×C8#4111yescharacteristic
84C4×C8#4111yescharacteristic
85C4×C8#4111yescharacteristic
86#95—41no—
87#95—41no—
88#41—81no—
89#41—81no—
90C2×Q8—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—41no—
95C22×C4—41no—
96C22×C4—81no—
97C22×C4#12311yescharacteristic
98C22×C4#13511yescharacteristic
99C22×C4AGL(1,ℤ/8)11yescharacteristic
100M16—82no—
101M16—81no—
102M16—161no—
103M16—161no—
104C2×C8—41no—
105C2×C8—41no—
106C2×C8—41no—
107C2×C8—82no—
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—81no—
118C4⋊C4—81no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C42#12111yescharacteristic
122C42#12311yescharacteristic
123C42C2×SD1611yescharacteristic
124C42#13711yescharacteristic
125C16—42no—
126C16—81no—
127C23—41no—
128C23#48211yescharacteristic
129Q8—81no—
130D4—81no—
131D4—81no—
132D4—81no—
133D4—82no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4—161no—
142C2×C4—161no—
143C2×C4—161no—
144C2×C4#39711yescharacteristic
145C2×C4#46511yescharacteristic
146C2×C4#46811yescharacteristic
147C2×C4#48111yescharacteristic
148C2×C4#48311yescharacteristic
149C2×C4#48511yescharacteristic
150C8—21no—
151C8—21no—
152C8—21no—
153C8—21no—
154C8—81no—
155C8—161no—
156C8—161no—
157C8—161no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22#207711yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—81no—
171C4—321no—
172C4#208011yescharacteristic
173C4#208111yescharacteristic
174C2—41no—
175C2—161no—
176C2#1064111yescharacteristic
177C2#1350211yescharacteristic
178C2#1350411yescharacteristic
subgroupG/H
center, socleC22#2077
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#532D4
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #12029C2semidirect
5. #7295C2semidirect
6. #7221C2semidirect
7. #7221C2semidirect

Maximal subgroups

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

subgroupindex
#120292
#116352
#86782
#86362
#72952
#72212