Finite Group Databasebeta

← #10596106

Subgroups — #10596106

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
1C32×C22⋊Q8C211yesmaximal, characteristic, complemented
2#64155C211yesmaximal, characteristic, complemented
3Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C4×#638C211yesmaximal, characteristic
5C2×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6#64106C212yesmaximal, complemented
7C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
8#64098C212yesmaximal, complemented
9#64097C211yesmaximal, characteristic, complemented
10#64090C212yesmaximal
11#64087C212yesmaximal, complemented
12#5635—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C22×C3⋊Dic3C2212yes—
17C2×C4×C3⋊D3C2211yescharacteristic
18C2×C4×C3⋊D3C2212yes—
19C2×#634C2212yes—
20C32×C4⋊C4C2211yescharacteristic
21C32×C4⋊C4C2212yes—
22C32×C22⋊C4C2212yes—
23#3757C2211yescharacteristic
24#3752C2212yes—
25#3752C2211yescharacteristic
26C4⋊(C3⋊Dic3)C2212yes—
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28C4⋊(C3⋊Dic3)C2212yes—
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30#3750C2212yes—
31#3750C2212yes—
32#3750C2212yes—
33#3750C2211yescharacteristic
34C4×C3⋊Dic3C2212yes—
35C4×C3⋊Dic3C2211yescharacteristic
36C3×C22⋊Q8D314yescomplemented
37#946—34no—
38Q8×Dic3—34no—
39C4×#67—34no—
40C2×C4⋊Dic3—34no—
41#897—38no—
42D3×C4⋊C4—34no—
43#889—38no—
44#888—34no—
45#67⋊C4—38no—
46#878—38no—
47C23×C32C2311yescharacteristic
48C22×C3⋊D3C2311yescharacteristic
49C32×Q8—22no—
50C2×C32×C4—22no—
51C2×C32×C4C2312yes—
52C2×C32×C4C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55#638Q814yescomplemented
56C2×C3⋊Dic3—42no—
57C2×C3⋊Dic3C2312yes—
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2311yescharacteristic
60C2×C3⋊Dic3C2312yes—
61C2×C3⋊Dic3C2311yescharacteristic
62C4×C3⋊D3—22no—
63C4×C3⋊D3—42no—
64#634—42no—
65#554—91noSylow
66C2×C3×Q8C2×D314yes—
67C22×C3×C4C2×D314yes—
68C2×#67—34no—
69C22×Dic3—38no—
70C2×C4×D3—34no—
71C2×C4×D3—38no—
72C2×Dic6—38no—
73C3×C4⋊C4C2×D314yes—
74C3×C4⋊C4C2×D318yes—
75C3×C22⋊C4C2×D318yes—
76#217—34no—
77(C2×D3)⋊C4—38no—
78(C2×D3)⋊C4—34no—
79C4⋊Dic3—38no—
80C4⋊Dic3—34no—
81C4⋊Dic3—38no—
82C4⋊Dic3—34no—
83Dic3⋊C4—38no—
84Dic3⋊C4—38no—
85Dic3⋊C4—38no—
86Dic3⋊C4—34no—
87C4×Dic3—38no—
88C4×Dic3—34no—
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—21no—
101C3⋊Dic3—21no—
102C3⋊Dic3—22no—
103C3⋊Dic3—22no—
104C3⋊Dic3C2×Q812yes—
105C4⋊Q8—91no—
106#125—92no—
107C22⋊Q8—91no—
108C22⋊Q8—92no—
109C22⋊Q8—92no—
110C22⋊Q8C3⋊D311yescharacteristic, complemented
111C4×Q8—91no—
112C4×D4—91no—
113C4×D4—92no—
114C2×C4⋊C4—91no—
115C2×C4⋊C4—91no—
116C23×C3C22×D314yes—
117C22×D3—34no—
118C3×Q8—28no—
119C2×C3×C4—28no—
120C2×C3×C4C22×D318yes—
121C2×C3×C4C22×D318yes—
122C2×C3×C4C22×D314yes—
123C2×C3×C4C22×D314yes—
124#67—316no—
125C2×Dic3—34no—
126C2×Dic3—38no—
127C2×Dic3—34no—
128C2×Dic3—34no—
129C2×Dic3—38no—
130C2×Dic3—128no—
131C4×D3—68no—
132C4×D3—128no—
133Dic6—128no—
134C2×C32—22no—
135C2×C32C22×Q811yescharacteristic
136C2×C32C2×#4111yescharacteristic
137C2×C32H(5,𝔽2)11yescharacteristic
138C3⋊D3—22no—
139C2×Q8—92no—
140C2×Q8C2×C3⋊D311yescharacteristic
141C2×D4—91no—
142C22×C4—92no—
143C22×C4—91no—
144C22×C4—92no—
145C22×C4C2×C3⋊D311yescharacteristic
146C4⋊C4—91no—
147C4⋊C4—92no—
148C4⋊C4—92no—
149C4⋊C4—91no—
150C4⋊C4—91no—
151C4⋊C4—92no—
152C4⋊C4—92no—
153C4⋊C4—92no—
154C4⋊C4C2×C3⋊D311yescharacteristic
155C4⋊C4C2×C3⋊D312yes—
156C22⋊C4—91no—
157C22⋊C4—91no—
158C22⋊C4—92no—
159C22⋊C4C2×C3⋊D312yes—
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#23718yes—
173Dic3—38no—
174Dic3—64no—
175Dic3—64no—
176Dic3—68no—
177Dic3—68no—
178C32#55411yescharacteristic, complemented, Sylow
179C23—91no—
180C23C22×C3⋊D311yescharacteristic
181Q8—22no—
182Q8—362no—
183D4—94no—
184C2×C4—22no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—92no—
188C2×C4—91no—
189C2×C4—92no—
190C2×C4—182no—
191C2×C4—362no—
192C2×C4—362no—
193C2×C4C22×C3⋊D312yes—
194C2×C4C22×C3⋊D311yescharacteristic
195C2×C4C22×C3⋊D312yes—
196C2×C4C22×C3⋊D311yescharacteristic
197C2×C3—28no—
198C2×C3C2×#23714yes—
199C2×C3C2×D3×Q814yes—
200C2×C3#100914yes—
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#383012yes—
218C3#563514yescomplemented
219C2—22no—
220C2—182no—
221C2C2×#383011yescharacteristic
222C2C2×Q8×C3⋊D311yescharacteristic
223C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
prime radicalC2×#638C22
hypercenterC22⋊Q8C3⋊D3
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C22⋊Q8C2semidirect
2. #64155C2semidirect
3. Q8×C3⋊Dic3C2semidirect
5. C2×C4⋊(C3⋊Dic3)C2semidirect
6. #64106C2semidirect
7. C4⋊C4×C3⋊D3C2semidirect
8. #64098C2semidirect
9. #64097C2semidirect
11. #64087C2semidirect
36. C3×C22⋊Q8D3semidirect
55. #638Q8semidirect
110. C22⋊Q8C3⋊D3semidirect
178. C32#554semidirect
218. C3#5635semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊Q82
#641552
Q8×C3⋊Dic32
C4×#6382
C2×C4⋊(C3⋊Dic3)2
#641062
C4⋊C4×C3⋊D32
#640982
#640972
#640902
#640872
#56353