Finite Group Databasebeta

← #93229

Subgroups — #93229

Order 512.

227 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 26 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#10032C211yesmaximal, characteristic
2#7283C211yesmaximal, characteristic, complemented
3#7060C211yesmaximal, characteristic, complemented
4#3150C411yescharacteristic, complemented
5#3100—21no—
6#1955C2211yescharacteristic
7C4×#99—21no—
8#1480—21no—
9#1322—21no—
10#1275C411yescharacteristic, complemented
11C2×#135—41no—
12#532—21no—
13C2×C4×D4D411yescharacteristic
14#497—41no—
15#496—41no—
16#480—41no—
17#474—41no—
18#459—41no—
19#447—41no—
20C4×D8—42no—
21#431—21no—
22C2×C4⋊C8D411yescharacteristic
23C2×#99—41no—
24C2×C22⋊C8—41no—
25C2×#97C2×C411yescharacteristic
26C4×C22⋊C4—41no—
27C22⋊C16—41no—
28#342—22no—
29D4⋊C8—42no—
30C2×#41—41no—
31C22×D4D811yescharacteristic
32C23×C4—21no—
33#135—82no—
34#135—82no—
35C2×Q16—41no—
36C2×SD16—42no—
37C2×D8—41no—
38C2×D8—42no—
39C22×C8—21no—
40C22×C8—21no—
41C4⋊Q8—21no—
42#127—21no—
43#124—41no—
44#121—41no—
45#121—41no—
46C4×D4—41no—
47C4×D4—41no—
48C4×D4—41no—
49C4×D4—41no—
50C2×C4⋊C4SD1611yescharacteristic
51C2×C22⋊C4—21no—
52C2×C22⋊C4—42no—
53C2×C42—81no—
54C2×C42C22⋊C411yescharacteristic
55C2×C16—81no—
56#107—42no—
57C4⋊C8—41no—
58C4⋊C8—41no—
59#102—42no—
60#102—42no—
61#99—44no—
62C22⋊C8—41no—
63C22⋊C8—82no—
64#97—41no—
65C4×C8—22no—
66#95—81no—
67C24—21no—
68#41—82no—
69#41—82no—
70C2×Q8—41no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—42no—
75C2×D4—41no—
76C2×D4—81no—
77C22×C4—21no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—81no—
81C22×C4—81no—
82C22×C4—82no—
83C22×C4—81no—
84C22×C4#9911yescharacteristic
85C22×C4#10211yescharacteristic
86C22×C4#10411yescharacteristic
87Q16—82no—
88SD16—84no—
89D8—82no—
90D8—84no—
91C2×C8—22no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—42no—
95C2×C8—41no—
96C2×C8—81no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C22⋊C4—44no—
101C22⋊C4—81no—
102C22⋊C4—81no—
103C22⋊C4—81no—
104C22⋊C4—82no—
105C42—21no—
106C42—21no—
107C42—161no—
108C42—161no—
109C16—161no—
110C23—22no—
111C23—41no—
112C23—41no—
113C23—81no—
114C23—81no—
115C23—81no—
116C23#32711yescharacteristic
117Q8—82no—
118D4—42no—
119D4—82no—
120D4—81no—
121D4—82no—
122D4—82no—
123D4—162no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—81no—
132C2×C4—81no—
133C2×C4—81no—
134C2×C4—84no—
135C2×C4—81no—
136C2×C4—161no—
137C2×C4—162no—
138C2×C4—162no—
139C2×C4—161no—
140C2×C4—161no—
141C2×C4AΣL(1,𝔽16)11yescharacteristic
142C2×C4#35311yescharacteristic
143C8—42no—
144C8—82no—
145C8—81no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—42no—
150C22—82no—
151C22—81no—
152C22—81no—
153C22—161no—
154C22—161no—
155C22—161no—
156C22—161no—
157C22#134311yescharacteristic
158C4—21no—
159C4—41no—
160C4—41no—
161C4—81no—
162C4—164no—
163C4—161no—
164C4#140611yescharacteristic
165C4#140811yescharacteristic
166C2—41no—
167C2—82no—
168C2—321no—
169C2#739611yescharacteristic
170C2#740211yescharacteristic
171C2#740611yescharacteristic
subgroupG/H
centerC2×C4#353
commutatorC2×#97C2×C4
Frattini, cosocle#1955C22
socleC22#1343
prime radical#10032C2
LCSG ▷ C2×#97 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1955 ◁ G
LECSG ▷ #1955 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #7283C2semidirect
3. #7060C2semidirect
4. #3150C4semidirect
10. #1275C4semidirect

Maximal subgroups

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

subgroupindex
#100322
#72832
#70602