Finite Group Databasebeta

← #93221

Subgroups — #93221

Order 512.

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

normal subgroups only

This group has 216 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#10026C211yesmaximal, characteristic
2#7283C211yesmaximal, characteristic, complemented
3#7105C211yesmaximal, characteristic, complemented
4#3245C411yescharacteristic, complemented
5#3109—21no—
6#1955C2211yescharacteristic
7C4×#99—21no—
8#1530—21no—
9#1377—21no—
10#1290C411yescharacteristic, complemented
11#532—21no—
12C2×C4×D4D411yescharacteristic
13C2×M32—41no—
14#481—41no—
15#476—41no—
16#474—41no—
17#466—41no—
18#459—41no—
19#459—41no—
20#447—41no—
21#442—42no—
22#432—21no—
23C2×C4⋊C8C2×C411yescharacteristic
24C2×#99—41no—
25C2×C22⋊C8—41no—
26C2×#97D411yescharacteristic
27C4×C22⋊C4—41no—
28#342—22no—
29#329—42no—
30C2×#41—41no—
31C22×D4D811yescharacteristic
32C23×C4—21no—
33C2×D8—42no—
34C2×D8—41no—
35C22×C8—21no—
36C22×C8—21no—
37C4⋊Q8—21no—
38#127—21no—
39#124—41no—
40#121—41no—
41#121—41no—
42C4×D4—41no—
43C4×D4—41no—
44C4×D4—41no—
45C4×D4—41no—
46C2×C4⋊C4SD1611yescharacteristic
47C2×C22⋊C4—21no—
48C2×C22⋊C4—42no—
49C2×C42—81no—
50C2×C42C22⋊C411yescharacteristic
51M32—42no—
52M32—81no—
53C2×C16—81no—
54#107—41no—
55#106—42no—
56C4⋊C8—41no—
57C4⋊C8—41no—
58#102—42no—
59#102—42no—
60#102—42no—
61#99—44no—
62C22⋊C8—41no—
63C22⋊C8—81no—
64C22⋊C8—81no—
65#97—22no—
66#97—41no—
67#95—81no—
68C24—21no—
69#41—162no—
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—81no—
83C22×C4—82no—
84C22×C4#9911yescharacteristic
85C22×C4#10211yescharacteristic
86C22×C4#10411yescharacteristic
87D8—84no—
88D8—161no—
89C2×C8—21no—
90C2×C8—21no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—42no—
94C2×C8—41no—
95C2×C8—81no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C22⋊C4—44no—
100C22⋊C4—81no—
101C22⋊C4—81no—
102C22⋊C4—82no—
103C22⋊C4—81no—
104C42—21no—
105C42—21no—
106C42—161no—
107C42—161no—
108C16—82no—
109C23—22no—
110C23—41no—
111C23—41no—
112C23—81no—
113C23—81no—
114C23—81no—
115C23#32711yescharacteristic
116Q8—161no—
117D4—42no—
118D4—81no—
119D4—82no—
120D4—81no—
121D4—81no—
122D4—161no—
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—162no—
140C2×C4AΣL(1,𝔽16)11yescharacteristic
141C2×C4#35311yescharacteristic
142C8—41no—
143C8—41no—
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#140211yescharacteristic
165C4#140411yescharacteristic
166C2—41no—
167C2—82no—
168C2—321no—
169C2#739611yescharacteristic
170C2#739811yescharacteristic
171C2#740811yescharacteristic
subgroupG/H
center, socleC22#1343
commutatorC2×C4⋊C8C2×C4
Frattini, cosocle#1955C22
prime radical#10026C2
LCSG ▷ C2×C4⋊C8 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ 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. #7105C2semidirect
4. #3245C4semidirect
10. #1290C4semidirect

Maximal subgroups

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

subgroupindex
#100262
#72832
#71052