Finite Group Databasebeta

← #10595982

Subgroups — #10595982

Order 576.

684 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 203 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×C32×C4⋊C4C211yesmaximal, characteristic, complemented
2C4×#638C211yesmaximal, characteristic
3C4×#638C212yesmaximal
4#64133C212yesmaximal, complemented
5#64133C211yesmaximal, characteristic, complemented
6#64106C211yesmaximal, characteristic, complemented
7#64105C212yesmaximal, complemented
8C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
9#64098C212yesmaximal, complemented
10#64097C211yesmaximal, characteristic, complemented
11#64096C211yesmaximal, characteristic, complemented
12#5511—34nomaximal
13C22×C32×C4C2211yescharacteristic
14C22×C32×C4C2212yes—
15C2×#638C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C4×C3⋊D3C2212yes—
18C2×#634C2211yescharacteristic
19C2×#634C2212yes—
20C32×C4⋊C4C2211yescharacteristic
21C32×C4⋊C4C2211yescharacteristic
22C32×C4⋊C4C2212yes—
23#3757C2212yes—
24#3757C2211yescharacteristic
25#3752C2211yescharacteristic
26#3752C2212yes—
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28C4⋊(C3⋊Dic3)C2212yes—
29#3750C2212yes—
30#3750C2212yes—
31#3750C2211yescharacteristic
32#3750C2212yes—
33#3750C2212yes—
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2212yes—
36C2×C3×C4⋊C4D314yescomplemented
37C4×#67—34no—
38C4×#67—38no—
39#924—38no—
40#924—34no—
41#897—34no—
42#896—38no—
43D3×C4⋊C4—34no—
44#889—38no—
45#888—34no—
46#887—34no—
47C23×C32C2311yescharacteristic
48C22×C3⋊D3C2311yescharacteristic
49C2×C32×C4—22no—
50C2×C32×C4—42no—
51C2×C32×C4C2312yes—
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2312yes—
54C2×C32×C4C2311yescharacteristic
55#638Q814yescomplemented
56C2×C3⋊Dic3C2312yes—
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2312yes—
60C2×C3⋊Dic3C2311yescharacteristic
61C4×C3⋊D3—22no—
62C4×C3⋊D3—42no—
63#634—22no—
64#634—42no—
65#554—91noSylow
66C22×C3×C4C2×D314yes—
67C22×C3×C4C2×D318yes—
68C2×#67—34no—
69C2×C4×D3—34no—
70C2×C4×D3—38no—
71C2×Dic6—34no—
72C2×Dic6—38no—
73C3×C4⋊C4C2×D314yes—
74C3×C4⋊C4C2×D314yes—
75C3×C4⋊C4C2×D318yes—
76#217—38no—
77#217—34no—
78(C2×D3)⋊C4—34no—
79(C2×D3)⋊C4—38no—
80C4⋊Dic3—34no—
81C4⋊Dic3—38no—
82Dic3⋊C4—38no—
83Dic3⋊C4—38no—
84Dic3⋊C4—34no—
85Dic3⋊C4—38no—
86Dic3⋊C4—38no—
87C4×Dic3—34no—
88C4×Dic3—38no—
89C22×C32—21no—
90C22×C32—21no—
91C22×C32C2×Q812yes—
92C22×C32C2411yescharacteristic
93C2×C3⋊D3—21no—
94C2×C3⋊D3—21no—
95C2×C3⋊D3C2×Q812yes—
96C32×C4—21no—
97C32×C4—22no—
98C32×C4—22no—
99C32×C4#4112yes—
100C3⋊Dic3—22no—
101C3⋊Dic3—21no—
102C3⋊Dic3—21no—
103C3⋊Dic3—22no—
104C3⋊Dic3C2×Q812yes—
105C4⋊Q8—91no—
106#125—92no—
107C22⋊Q8—92no—
108C22⋊Q8—91no—
109C22⋊Q8—92no—
110C22⋊Q8—91no—
111C4×Q8—91no—
112C4×D4—92no—
113C4×D4—91no—
114C2×C4⋊C4—91no—
115C2×C4⋊C4C3⋊D311yescharacteristic, complemented
116C23×C3C22×D314yes—
117C22×D3—34no—
118C2×C3×C4—28no—
119C2×C3×C4—48no—
120C2×C3×C4C22×D318yes—
121C2×C3×C4C22×D314yes—
122C2×C3×C4C22×D318yes—
123C2×C3×C4C22×D314yes—
124#67—316no—
125C2×Dic3—34no—
126C2×Dic3—38no—
127C2×Dic3—34no—
128C2×Dic3—34no—
129C2×Dic3—38no—
130C4×D3—68no—
131C4×D3—128no—
132Dic6—68no—
133Dic6—128no—
134C2×C32—22no—
135C2×C32C22×Q811yescharacteristic
136C2×C32C2×#4111yescharacteristic
137C2×C32H(5,𝔽2)11yescharacteristic
138C3⋊D3—22no—
139C2×Q8—91no—
140C2×Q8—92no—
141C2×D4—91no—
142C22×C4—91no—
143C22×C4—92no—
144C22×C4C2×C3⋊D311yescharacteristic
145C22×C4C2×C3⋊D312yes—
146C4⋊C4—92no—
147C4⋊C4—91no—
148C4⋊C4—91no—
149C4⋊C4—92no—
150C4⋊C4—92no—
151C4⋊C4—92no—
152C4⋊C4—92no—
153C4⋊C4C2×C3⋊D311yescharacteristic
154C4⋊C4C2×C3⋊D312yes—
155C4⋊C4C2×C3⋊D311yescharacteristic
156C22⋊C4—92no—
157C22⋊C4—91no—
158C22⋊C4—92no—
159C22⋊C4—91no—
160C42—92no—
161C42—91no—
162C22×C3—24no—
163C22×C3—24no—
164C22×C3D3×Q818yes—
165C22×C3C23×D314yes—
166C2×D3—38no—
167C2×D3—64no—
168C2×D3—64no—
169C3×C4—24no—
170C3×C4—28no—
171C3×C4—28no—
172C3×C4#23518yes—
173Dic3—38no—
174Dic3—68no—
175Dic3—64no—
176Dic3—64no—
177Dic3—68no—
178C32#55411yescharacteristic, complemented, Sylow
179C23—91no—
180C23C22×C3⋊D311yescharacteristic
181Q8—182no—
182Q8—362no—
183D4—94no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—91no—
187C2×C4—92no—
188C2×C4—91no—
189C2×C4—91no—
190C2×C4—92no—
191C2×C4—182no—
192C2×C4—362no—
193C2×C4C22×C3⋊D312yes—
194C2×C4C22×C3⋊D311yescharacteristic
195C2×C4C22×C3⋊D312yes—
196C2×C4C22×C3⋊D311yescharacteristic
197C2×C3—28no—
198C2×C3C2×#23514yes—
199C2×C3#100414yes—
200C2×C3C2×D3×Q814yes—
201D3—68no—
202C22—21no—
203C22—21no—
204C22—92no—
205C22—181no—
206C22—181no—
207C22Q8×C3⋊D312yes—
208C22C23×C3⋊D311yescharacteristic
209C4—21no—
210C4—22no—
211C4—22no—
212C4—92no—
213C4—181no—
214C4—182no—
215C4—182no—
216C4—181no—
217C4#382812yes—
218C3#551114yescomplemented
219C2—22no—
220C2—182no—
221C2C2×#382811yescharacteristic
222C2#6436111yescharacteristic
223C2C2×Q8×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC2×C32×C4⋊C4C2
prime radicalC2×#638C22
hypercenterC2×C4⋊C4C3⋊D3
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4⋊C4C2semidirect
4. #64133C2semidirect
5. #64133C2semidirect
6. #64106C2semidirect
7. #64105C2semidirect
8. C4⋊C4×C3⋊D3C2semidirect
9. #64098C2semidirect
10. #64097C2semidirect
11. #64096C2semidirect
36. C2×C3×C4⋊C4D3semidirect
55. #638Q8semidirect
115. C2×C4⋊C4C3⋊D3semidirect
178. C32#554semidirect
218. C3#5511semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4⋊C42
C4×#6382
#641332
#641062
#641052
C4⋊C4×C3⋊D32
#640982
#640972
#640962
#55113