Finite Group Databasebeta

← #10593311

Subgroups — #10593311

Order 576.

266 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 45 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#64306C211yesmaximal, characteristic
2#64283C211yesmaximal, characteristic, complemented
3C2×#3712C211yesmaximal, characteristic, complemented
4#63728C212yesmaximal, complemented
5#63727C212yesmaximal, complemented
6C2×C4×C3⋊D3C2211yescharacteristic
7C3×#235—21no—
8#3811C412yes—
9#3805—21no—
10#3802C2211yescharacteristic
11C4×D32C412yes—
12#3798—21no—
13#3797C2211yescharacteristic
14#3796—21no—
15#3792—21no—
16#3788C2212yes—
17#3787—21no—
18C2×C3×C3⋊C8—21no—
19#3712—21no—
20#3712C2212yes—
21D3×#41—61no—
22C2×ΓL(1,𝔽25)—61no—
23C22×C3⋊D3C2×C411yescharacteristic
24C2×D32C2×C412yes—
25C2×C32×C4D411yescharacteristic
26C2×C3⋊Dic3D411yescharacteristic
27C4×C3⋊D3C2×C411yescharacteristic
28C4×C3⋊D3D411yescharacteristic
29C4×C3⋊D3D411yescharacteristic
30C4×C3⋊D3C2311yescharacteristic
31C3×#67—22no—
32C3×D12—21no—
33C3×C4×D3—22no—
34C3×Dic6—21no—
35#627—21no—
36#626—22no—
37#625—21no—
38#624C2×C412yes—
39D3×Dic3—22no—
40#622—22no—
41C3×C3⋊C8—22no—
42#413—91nomaximal, Sylow
43C3×#41—61no—
44#239—61no—
45D3×Q8—61no—
46#237—61no—
47#237—62no—
48D3×D4—61no—
49D3×D4—62no—
50#235—21no—
51#235—62no—
52C2×C4×D3—62no—
53C2×C4×D3—61no—
54C2×C4×D3—61no—
55C2×C3×C8—61no—
56C2×C3⋊C8—21no—
57ΓL(1,𝔽25)—62no—
58ΓL(1,𝔽25)—62no—
59C22×C32C22⋊C411yescharacteristic
60C2×C3⋊D3—41no—
61C2×C3⋊D3C22⋊C411yescharacteristic
62C2×C3⋊D3C22×C411yescharacteristic
63C2×C3×D3—22no—
64D32—42no—
65C32×C4C22⋊C411yescharacteristic
66C32×C4C2×D411yescharacteristic
67C3⋊Dic3C22⋊C411yescharacteristic
68C3⋊Dic3C2×D411yescharacteristic
69C3×Dic3—22no—
70C2×#41—91no—
71C2×M16—91no—
72C2×M16—91no—
73#101—92no—
74#100—92no—
75C22×D3—62no—
76C22×D3—61no—
77C22×D3—61no—
78C3×Q8—61no—
79C3×D4—61no—
80C3×D4—62no—
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—62no—
84#67—22no—
85#67—62no—
86#67—62no—
87C2×Dic3—61no—
88C2×Dic3—61no—
89C2×Dic3—62no—
90D12—21no—
91D12—62no—
92C4×D3—22no—
93C4×D3—62no—
94C4×D3—61no—
95C4×D3—61no—
96C4×D3—61no—
97C4×D3—61no—
98C4×D3—61no—
99C4×D3—61no—
100C4×D3—61no—
101C4×D3—61no—
102C4×D3—62no—
103Dic6—21no—
104Dic6—62no—
105C3×C8—62no—
106C3⋊C8—22no—
107C2×C32—21no—
108C2×C32C2×C22⋊C411yescharacteristic
109C3⋊D3—21no—
110C3⋊D3—21no—
111C3×D3—42no—
112#41—61no—
113#41—181no—
114#41—181no—
115#41—181no—
116C2×Q8—91no—
117C2×D4—91no—
118C2×D4—92no—
119C22×C4—92no—
120C22×C4—91no—
121M16—92no—
122M16—92no—
123M16—181no—
124M16—181no—
125C2×C8—61no—
126C2×C8—181no—
127C22×C3—21no—
128C22×C3—21no—
129C22×C3—62no—
130C2×D3—22no—
131C2×D3—61no—
132C2×D3—61no—
133C2×D3—61no—
134C2×D3—61no—
135C2×D3—124no—
136C2×D3—122no—
137C2×D3—122no—
138C3×C4—21no—
139C3×C4—21no—
140C3×C4—21no—
141C3×C4—21no—
142C3×C4—62no—
143Dic3—22no—
144Dic3—61no—
145Dic3—61no—
146Dic3—61no—
147Dic3—61no—
148C32#41311yescharacteristic, complemented, Sylow
149C23—92no—
150C23—91no—
151Q8—61no—
152Q8—181no—
153D4—62no—
154D4—61no—
155D4—182no—
156D4—181no—
157C2×C4—62no—
158C2×C4—92no—
159C2×C4—91no—
160C2×C4—91no—
161C2×C4—91no—
162C2×C4—91no—
163C2×C4—91no—
164C2×C4—182no—
165C2×C4GO+(4,𝔽2)11yescharacteristic
166C8—62no—
167C8—182no—
168C2×C3—22no—
169C2×C3—21no—
170C2×C3—21no—
171C2×C3—41no—
172C2×C3—122no—
173D3—42no—
174D3—62no—
175D3—121no—
176D3—121no—
177D3—121no—
178C22—62no—
179C22—91no—
180C22—91no—
181C22—361no—
182C22—362no—
183C22D32⋊C411yescharacteristic
184C4—62no—
185C4—91no—
186C4—91no—
187C4D32⋊C411yescharacteristic
188C4C2×GO+(4,𝔽2)11yescharacteristic
189C3—21no—
190C3—21no—
191C2—21no—
192C2—122no—
193C2—181no—
194C2—181no—
195C2C2×D32⋊C411yescharacteristic
subgroupG/H
centerC4D32⋊C4
commutatorC2×C3⋊D3C22×C4
FrattiniC4C2×GO+(4,𝔽2)
FittingC2×C32×C4D4
socleC2×C32C2×C22⋊C4
prime radical#64306C2
cosocleC4×C3⋊D3C23
hypercenterC2×C4GO+(4,𝔽2)
hypocenterC32#413
LCSG ▷ C2×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #64283C2semidirect
3. C2×#3712C2semidirect
4. #63728C2semidirect
5. #63727C2semidirect
148. C32#413semidirect

Maximal subgroups

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

subgroupindex
#643062
#642832
C2×#37122
#637282
#637272
#4139