Finite Group Databasebeta

← #10593419

Subgroups — #10593419

Order 576.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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
1C2×#3774C211yesmaximal, characteristic
2#63902C211yesmaximal, characteristic, complemented
3C4×#625C211yesmaximal, characteristic, complemented
4#63777C211yesmaximal, characteristic, complemented
5#63742C211yesmaximal, characteristic, complemented
6#63739C211yesmaximal, characteristic, complemented
7#63739C211yesmaximal, characteristic, complemented
8C2×C3×C4×D3—21no—
9C2×#625C2211yescharacteristic
10C2×#625C2211yescharacteristic
11C2×D3×Dic3—21no—
12C2×#622C2211yescharacteristic
13C2×#622C2211yescharacteristic
14#3774C414yescomplemented
15C4×C3⋊Dic3C2211yescharacteristic
16C3×(C2×D3)⋊C4—21no—
17#3724C2211yescharacteristic
18#3724C2211yescharacteristic
19(C2×D3)⋊Dic3—21no—
20C4×#67—61no—
21#67⋊C4—61no—
22C22×C3×D3—21no—
23C22×C3×D3—21no—
24C2×C32×C4D411yescharacteristic
25C2×C3⋊Dic3D411yescharacteristic
26C2×C3⋊Dic3C2311yescharacteristic
27C2×C3×Dic3—21no—
28C2×C3×Dic3—21no—
29C3×C4×D3—42no—
30#625—21no—
31#625—21no—
32#625C2×C412yescomplemented
33#625C2×C412yescomplemented
34D3×Dic3—42no—
35#622—21no—
36#622C2×C412yescomplemented
37#442—91nomaximal, Sylow
38C22×C3×C4—61no—
39C2×#67—61no—
40C2×#67—61no—
41C22×Dic3—61no—
42C2×C4×D3—21no—
43C2×C4×D3—61no—
44C3×C22⋊C4—61no—
45#217—61no—
46(C2×D3)⋊C4—21no—
47(C2×D3)⋊C4—61no—
48Dic3⋊C4—61no—
49Dic3⋊C4—61no—
50C4×Dic3—61no—
51C4×Dic3—61no—
52C22×C32C2×D411yescharacteristic
53C2×C3×D3—22no—
54C2×C3×D3—22no—
55C2×C3×D3—41no—
56C2×C3×D3—41no—
57C2×C3×D3—41no—
58C2×C3×D3—41no—
59C32×C4—21no—
60C3⋊Dic3—21no—
61C3⋊Dic3C22×C411yescharacteristic
62C3⋊Dic3#4111yescharacteristic
63C3×Dic3—41no—
64C3×Dic3—41no—
65C2×D8—91no—
66C4×D4—91no—
67C4×D4—91no—
68#106—91no—
69#102—91no—
70#102—91no—
71#97—91no—
72C23×C3—61no—
73C23×C3—61no—
74C22×D3—21no—
75C22×D3—21no—
76C2×C3×C4—21no—
77C2×C3×C4—21no—
78C2×C3×C4—61no—
79C2×C3×C4—61no—
80C2×C3×C4—122no—
81#67—62no—
82#67—62no—
83#67—62no—
84#67—62no—
85C2×Dic3—21no—
86C2×Dic3—21no—
87C2×Dic3—61no—
88C2×Dic3—61no—
89C2×Dic3—61no—
90C2×Dic3—61no—
91C2×Dic3—122no—
92C4×D3—42no—
93C4×D3—122no—
94C2×C32C4×D411yescharacteristic
95C2×C32AGL(1,ℤ/8)11yescharacteristic
96C2×C32AGL(1,ℤ/8)11yescharacteristic
97C3×D3—42no—
98C3×D3—42no—
99C2×D4—91no—
100C2×D4—91no—
101C22×C4—61no—
102C22×C4—181no—
103D8—94no—
104C2×C8—91no—
105C2×C8—91no—
106C4⋊C4—91no—
107C4⋊C4—91no—
108C22⋊C4—61no—
109C22⋊C4—181no—
110C42—91no—
111C22×C3—21no—
112C22×C3—21no—
113C22×C3—62no—
114C22×C3—62no—
115C22×C3—121no—
116C22×C3—121no—
117C22×C3—121no—
118C22×C3—121no—
119C2×D3—22no—
120C2×D3—22no—
121C2×D3—41no—
122C2×D3—41no—
123C2×D3—41no—
124C2×D3—41no—
125C3×C4—22no—
126C3×C4—41no—
127C3×C4—121no—
128C3×C4—121no—
129Dic3—41no—
130Dic3—41no—
131Dic3—61no—
132Dic3—61no—
133Dic3—61no—
134Dic3—61no—
135Dic3—62no—
136Dic3—121no—
137C32#44211yescharacteristic, complemented, Sylow
138C23—61no—
139C23—61no—
140D4—92no—
141D4—92no—
142D4—181no—
143D4—181no—
144C2×C4—61no—
145C2×C4—61no—
146C2×C4—91no—
147C2×C4—91no—
148C2×C4—122no—
149C2×C4—362no—
150C2×C4GO+(4,𝔽2)11yescharacteristic
151C8—92no—
152C8—181no—
153C2×C3—21no—
154C2×C3—21no—
155C2×C3—21no—
156C2×C3—21no—
157C2×C3—21no—
158C2×C3—21no—
159C2×C3—122no—
160C2×C3—122no—
161D3—42no—
162D3—42no—
163C22—62no—
164C22—62no—
165C22—121no—
166C22—121no—
167C22—121no—
168C22—121no—
169C22C2×GO+(4,𝔽2)11yescharacteristic
170C4—21no—
171C4—91no—
172C4—91no—
173C4—121no—
174C4—121no—
175C4—181no—
176C3—21no—
177C3—21no—
178C2—122no—
179C2—122no—
180C2#6422411yescharacteristic
181C2C4×GO+(4,𝔽2)11yescharacteristic
182C2#6423611yescharacteristic
subgroupG/H
center, FrattiniC22C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C4
FittingC2×C32×C4D4
socleC22×C32C2×D4
prime radicalC2×#3774C2
cosocleC2×C3⋊Dic3C23
hypercenterC2×C4GO+(4,𝔽2)
hypocenterC32#442
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #63902C2semidirect
3. C4×#625C2semidirect
4. #63777C2semidirect
5. #63742C2semidirect
6. #63739C2semidirect
7. #63739C2semidirect
14. #3774C4semidirect
32. #625C2×C4semidirect
33. #625C2×C4semidirect
36. #622C2×C4semidirect
137. C32#442semidirect

Maximal subgroups

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

subgroupindex
C2×#37742
#639022
C4×#6252
#637772
#637422
#637392
#4429