Finite Group Databasebeta

← #10592900

Subgroups — #10592900

Order 576.

438 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 87 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#64367C211yesmaximal, characteristic, complemented
2C32×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#64145C211yesmaximal, characteristic, complemented
4#64141C211yesmaximal, characteristic, complemented
5#63664C211yesmaximal, characteristic, complemented
6#63652C211yesmaximal, characteristic, complemented
7#63651C211yesmaximal, characteristic, complemented
8#5201—34nomaximal
9C32×#41C2211yescharacteristic
10C2×C32×D4C2211yescharacteristic
11Q8×C3⋊D3—21no—
12#3830—21no—
13#3830—22no—
14#3828—21no—
15#3828C2211yescharacteristic
16C2×#634—21no—
17C2×#634C2211yescharacteristic
18C32×SD16—21no—
19C32×D8—21no—
20C32×M16C2211yescharacteristic
21#3757—22no—
22#3754—21no—
23#3753—21no—
24C4×C3⋊Dic3C2211yescharacteristic
25#3748C2211yescharacteristic
26#1010—34no—
27C3×AGL(1,ℤ/8)D314yescomplemented
28#936—34no—
29#932—34no—
30#837—34no—
31#825—34no—
32#824—34no—
33C23×C32—21no—
34C32×Q8D411yescharacteristic
35C32×D4—21no—
36C32×D4—21no—
37C32×D4—21no—
38C32×D4D411yescharacteristic
39C2×C32×C4—21no—
40C2×C32×C4C2311yescharacteristic
41#638—22no—
42#638—21no—
43C2×C3⋊Dic3—22no—
44C2×C3⋊Dic3—21no—
45C2×C3⋊Dic3D412yes—
46C4⋊(C3⋊D3)D411yescharacteristic
47C4×C3⋊D3—22no—
48C4×C3⋊D3—21no—
49#634—22no—
50#634—21no—
51#634—21no—
52#634D411yescharacteristic
53C32×C8—21no—
54C3⋊(C3⋊C8)—21no—
55#454—91noSylow
56C3×#41C2×D314yes—
57C2×C3×D4C2×D314yes—
58D3×Q8—64no—
59#237—68no—
60#237—64no—
61#235—34no—
62#235—64no—
63C2×Dic6—34no—
64C2×Dic6—64no—
65C3×SD16—24no—
66C3×D8—24no—
67C3×M16C2×D314yes—
68#217—68no—
69#214—64no—
70#213—64no—
71C4×Dic3—34no—
72#208—34no—
73C22×C32—21no—
74C22×C32—42no—
75C22×C32—41no—
76C22×C32C2×D411yescharacteristic
77C2×C3⋊D3—21no—
78C32×C4—21no—
79C32×C4C2×D411yescharacteristic
80C32×C4C2×D411yescharacteristic
81C3⋊Dic3—22no—
82C3⋊Dic3—21no—
83C3⋊Dic3—41no—
842−1+4—91no—
85AGL(1,ℤ/8)—91no—
86AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
87#124—91no—
88#104—91no—
89#104—91no—
90#101—91no—
91C23×C3—24no—
92C3×Q8#6714yes—
93C3×D4—24no—
94C3×D4—24no—
95C3×D4—24no—
96C3×D4#6714yes—
97C2×C3×C4—24no—
98C2×C3×C4C22×D314yes—
99#67—68no—
100#67—64no—
101C2×Dic3—38no—
102C2×Dic3—68no—
103C2×Dic3—64no—
104D12—34no—
105C4×D3—68no—
106C4×D3—64no—
107Dic6—34no—
108Dic6—68no—
109Dic6—64no—
110Dic6—64no—
111C3×C8—24no—
112C3⋊C8—64no—
113C2×C32—21no—
114C2×C32—41no—
115C2×C32—81no—
116C2×C32#12011yescharacteristic
117C3⋊D3—41no—
118#41—91no—
119#41—181no—
120#41—182no—
121#41—181no—
122#41C2×C3⋊D311yescharacteristic
123C2×Q8—91no—
124C2×Q8—181no—
125C2×Q8—181no—
126C2×D4C2×C3⋊D311yescharacteristic
127SD16—21no—
128SD16—181no—
129D8—21no—
130D8—181no—
131M16—91no—
132M16C2×C3⋊D311yescharacteristic
133C22⋊C4—182no—
134C42—91no—
135C22×C3—24no—
136C22×C3—48no—
137C22×C3—44no—
138C22×C3D3×D414yes—
139C2×D3—64no—
140C3×C4—24no—
141C3×C4D3×D414yes—
142C3×C4C2×#6714yes—
143Dic3—64no—
144Dic3—68no—
145Dic3—124no—
146C32#45411yescharacteristic, complemented, Sylow
147C23—21no—
148Q8—91no—
149Q8—182no—
150Q8—181no—
151Q8—181no—
152Q8#63811yescharacteristic
153D4—21no—
154D4—21no—
155D4—21no—
156D4—91no—
157D4—182no—
158D4—181no—
159D4#63811yescharacteristic
160C2×C4—21no—
161C2×C4—92no—
162C2×C4—182no—
163C2×C4—182no—
164C2×C4—181no—
165C2×C4—181no—
166C2×C4C22×C3⋊D311yescharacteristic
167C8—21no—
168C8—181no—
169C2×C3—24no—
170C2×C3—44no—
171C2×C3—84no—
172C2×C3#93714yes—
173D3—124no—
174C22—21no—
175C22—42no—
176C22—41no—
177C22—181no—
178C22D4×C3⋊D311yescharacteristic
179C4—21no—
180C4—182no—
181C4—181no—
182C4—361no—
183C4D4×C3⋊D311yescharacteristic
184C4C2×#63811yescharacteristic
185C3#520114yescomplemented
186C2—21no—
187C2—41no—
188C2—81no—
189C2—361no—
190C2#6414611yescharacteristic
subgroupG/H
centerC2#64146
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C22×C3⋊D3
FittingC32×AGL(1,ℤ/8)C2
socleC2×C32#120
hypercenterAGL(1,ℤ/8)C3⋊D3
hypocenterC32#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #64367C2semidirect
2. C32×AGL(1,ℤ/8)C2semidirect
3. #64145C2semidirect
4. #64141C2semidirect
5. #63664C2semidirect
6. #63652C2semidirect
7. #63651C2semidirect
27. C3×AGL(1,ℤ/8)D3semidirect
86. AGL(1,ℤ/8)C3⋊D3semidirect
146. C32#454semidirect
185. C3#5201semidirect

Maximal subgroups

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

subgroupindex
#643672
C32×AGL(1,ℤ/8)2
#641452
#641412
#636642
#636522
#636512
#52013