Finite Group Databasebeta

← #10596114

Subgroups — #10596114

Order 576.

648 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 183 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#64156C211yesmaximal, characteristic, complemented
3Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#64139C211yesmaximal, characteristic
5#64135C211yesmaximal, characteristic, complemented
6#64107C212yesmaximal, complemented
7#64106C212yesmaximal, complemented
8#64102C211yesmaximal, characteristic, complemented
9#64097C211yesmaximal, characteristic, complemented
10#64094C212yesmaximal, complemented
11#64092C212yesmaximal, complemented
12#5643—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C4×C3⋊D3C2212yes—
18C2×#634C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20C32×C4⋊C4C2212yes—
21C32×C22⋊C4C2212yes—
22#3757C2211yescharacteristic
23#3757C2211yescharacteristic
24#3752C2212yes—
25#3752C2212yes—
26#3752C2212yes—
27#3752C2212yes—
28C4⋊(C3⋊Dic3)C2212yes—
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2211yescharacteristic
31C4⋊(C3⋊Dic3)C2211yescharacteristic
32#3750C2212yes—
33C4×C3⋊Dic3C2212yes—
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2211yescharacteristic
36C3×C22⋊Q8D314yescomplemented
37#947—34no—
38Q8×Dic3—34no—
39#930—34no—
40#926—34no—
41#898—38no—
42#897—38no—
43#893—34no—
44#888—34no—
45#885—38no—
46#883—38no—
47C23×C32C2311yescharacteristic
48C22×C3⋊D3C2312yes—
49C32×Q8—22no—
50C2×C32×C4—22no—
51C2×C32×C4C2312yes—
52C2×C32×C4C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55#638—42no—
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2312yes—
59C2×C3⋊Dic3C2311yescharacteristic
60C2×C3⋊Dic3C2311yescharacteristic
61C4⋊(C3⋊D3)—22no—
62C4×C3⋊D3—42no—
63#634—42no—
64#542—91noSylow
65C2×C3×Q8C2×D314yes—
66C22×C3×C4C2×D314yes—
67C2×#67—38no—
68C2×D12—34no—
69C2×C4×D3—38no—
70C2×Dic6—38no—
71C3×C4⋊C4C2×D314yes—
72C3×C4⋊C4C2×D318yes—
73C3×C22⋊C4C2×D318yes—
74#217—34no—
75#217—34no—
76(C2×D3)⋊C4—38no—
77(C2×D3)⋊C4—38no—
78(C2×D3)⋊C4—38no—
79(C2×D3)⋊C4—38no—
80C4⋊Dic3—38no—
81C4⋊Dic3—34no—
82C4⋊Dic3—34no—
83C4⋊Dic3—34no—
84Dic3⋊C4—38no—
85C4×Dic3—38no—
86C4×Dic3—34no—
87C4×Dic3—34no—
88C22×C32—21no—
89C22×C32—21no—
90C22×C32—21no—
91C22×C32C2411yescharacteristic
92C2×C3⋊D3—22no—
93C2×C3⋊D3—22no—
94C2×C3⋊D3—22no—
95C32×C4—21no—
96C32×C4—22no—
97C32×C4—22no—
98C32×C4#4112yes—
99C3⋊Dic3—21no—
100C3⋊Dic3—22no—
101C3⋊Dic3—21no—
102C3⋊Dic3—21no—
103C3⋊Dic3—21no—
104C4⋊Q8—91no—
105#126—92no—
106#124—91no—
107#124—92no—
108#123—92no—
109C22⋊Q8—92no—
110C22⋊Q8C3⋊D311yescharacteristic, complemented
111#121—91no—
112C4×Q8—91no—
113C4×D4—91no—
114#117—91no—
115C23×C3C22×D314yes—
116C22×D3—38no—
117C3×Q8—28no—
118C2×C3×C4—28no—
119C2×C3×C4C22×D318yes—
120C2×C3×C4C22×D318yes—
121C2×C3×C4C22×D314yes—
122C2×C3×C4C22×D314yes—
123#67—128no—
124C2×Dic3—34no—
125C2×Dic3—34no—
126C2×Dic3—38no—
127C2×Dic3—34no—
128C2×Dic3—34no—
129D12—68no—
130C4×D3—128no—
131Dic6—128no—
132C2×C32—41no—
133C2×C32C2×#4111yescharacteristic
134C2×C32H(5,𝔽2)11yescharacteristic
135C2×C322−1+411yescharacteristic
136C3⋊D3—42no—
137C2×Q8—92no—
138C2×Q8C2×C3⋊D311yescharacteristic
139C2×D4—92no—
140C2×D4—91no—
141C22×C4—92no—
142C22×C4C2×C3⋊D311yescharacteristic
143C4⋊C4—91no—
144C4⋊C4—91no—
145C4⋊C4—91no—
146C4⋊C4—92no—
147C4⋊C4—92no—
148C4⋊C4C2×C3⋊D311yescharacteristic
149C4⋊C4C2×C3⋊D312yes—
150C22⋊C4—91no—
151C22⋊C4—91no—
152C22⋊C4—92no—
153C22⋊C4—92no—
154C22⋊C4—92no—
155C22⋊C4—92no—
156C22⋊C4C2×C3⋊D312yes—
157C42—91no—
158C42—91no—
159C42—92no—
160C22×C3—24no—
161C22×C3—24no—
162C22×C3—24no—
163C22×C3C23×D314yes—
164C2×D3—68no—
165C2×D3—68no—
166C2×D3—68no—
167C3×C4—24no—
168C3×C4—28no—
169C3×C4—28no—
170C3×C4#23718yes—
171Dic3—64no—
172Dic3—64no—
173Dic3—64no—
174Dic3—68no—
175Dic3—64no—
176C32#54211yescharacteristic, complemented, Sylow
177C23—92no—
178C23C22×C3⋊D311yescharacteristic
179Q8—22no—
180Q8—362no—
181D4—182no—
182D4—362no—
183C2×C4—22no—
184C2×C4—91no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—91no—
188C2×C4—92no—
189C2×C4—362no—
190C2×C4C22×C3⋊D312yes—
191C2×C4C22×C3⋊D311yescharacteristic
192C2×C4C22×C3⋊D312yes—
193C2×C4C22×C3⋊D311yescharacteristic
194C2×C3—44no—
195C2×C3C2×#23714yes—
196C2×C3#100714yes—
197C2×C3#100914yes—
198D3—128no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—182no—
203C22—182no—
204C22—182no—
205C22C23×C3⋊D311yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—181no—
210C4—181no—
211C4—181no—
212C4—182no—
213C4—181no—
214C4#383012yes—
215C3#564314yescomplemented
216C2—41no—
217C2—362no—
218C2C2×#383011yescharacteristic
219C2#6436411yescharacteristic
220C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
prime radical#64139C2
hypercenterC22⋊Q8C3⋊D3
hypocenterC32#542
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. #64156C2semidirect
3. Q8×C3⋊Dic3C2semidirect
5. #64135C2semidirect
6. #64107C2semidirect
7. #64106C2semidirect
8. #64102C2semidirect
9. #64097C2semidirect
10. #64094C2semidirect
11. #64092C2semidirect
36. C3×C22⋊Q8D3semidirect
110. C22⋊Q8C3⋊D3semidirect
176. C32#542semidirect
215. C3#5643semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊Q82
#641562
Q8×C3⋊Dic32
#641392
#641352
#641072
#641062
#641022
#640972
#640942
#640922
#56433