Finite Group Databasebeta

← #10596204

Subgroups — #10596204

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×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64156C212yesmaximal, complemented
3#64155C212yesmaximal, complemented
4#64107C212yesmaximal, complemented
5#64106C211yesmaximal, characteristic, complemented
6#64104C211yesmaximal, characteristic
7#64103C212yesmaximal, complemented
8#64102C211yesmaximal, characteristic, complemented
9#64096C211yesmaximal, characteristic, complemented
10#64084C211yesmaximal, characteristic, complemented
11#64081C211yesmaximal, characteristic, complemented
12#5733—34nomaximal
13C2×C32×Q8C2212yes—
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4⋊(C3⋊D3)C2212yes—
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C4×C3⋊D3C2212yes—
18C2×#634C2211yescharacteristic
19C32×C4⋊C4C2211yescharacteristic
20C32×C4⋊C4C2211yescharacteristic
21C32×C4⋊C4C2212yes—
22C32×C42C2211yescharacteristic
23#3752C2211yescharacteristic
24#3752C2211yescharacteristic
25#3752C2212yes—
26#3752C2212yes—
27#3752C2212yes—
28#3752C2212yes—
29#3752C2212yes—
30C4⋊(C3⋊Dic3)C2212yes—
31#3750C2211yescharacteristic
32#3750C2211yescharacteristic
33#3750C2212yes—
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2212yes—
36C3×C4⋊Q8D314yescomplemented
37#947—38no—
38#946—38no—
39#898—38no—
40#897—34no—
41#895—34no—
42#894—38no—
43#893—34no—
44#887—34no—
45#875—34no—
46#872—34no—
47C22×C3⋊D3C2311yescharacteristic
48C22×C3⋊D3C2312yes—
49C32×Q8—42no—
50C2×C32×C4C2312yes—
51C2×C32×C4C2311yescharacteristic
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2312yes—
54C2×C32×C4C2311yescharacteristic
55C2×C3⋊Dic3C2311yescharacteristic
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2312yes—
58C2×C3⋊Dic3C2311yescharacteristic
59C4⋊(C3⋊D3)—22no—
60C4⋊(C3⋊D3)—42no—
61C4×C3⋊D3—22no—
62C4×C3⋊D3—42no—
63#634—22no—
64#542—91noSylow
65C2×C3×Q8C2×D318yes—
66C2×D12—34no—
67C2×D12—38no—
68C2×C4×D3—34no—
69C2×C4×D3—38no—
70C2×Dic6—34no—
71C3×C4⋊C4C2×D314yes—
72C3×C4⋊C4C2×D314yes—
73C3×C4⋊C4C2×D318yes—
74C3×C42C2×D314yes—
75(C2×D3)⋊C4—34no—
76(C2×D3)⋊C4—34no—
77(C2×D3)⋊C4—38no—
78(C2×D3)⋊C4—38no—
79(C2×D3)⋊C4—38no—
80(C2×D3)⋊C4—38no—
81(C2×D3)⋊C4—38no—
82C4⋊Dic3—38no—
83Dic3⋊C4—34no—
84Dic3⋊C4—34no—
85Dic3⋊C4—38no—
86C4×Dic3—34no—
87C4×Dic3—38no—
88C22×C32C2411yescharacteristic
89C2×C3⋊D3—21no—
90C2×C3⋊D3—22no—
91C2×C3⋊D3—22no—
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C2×C3⋊D3—22no—
95C32×C4—22no—
96C32×C4—22no—
97C32×C4—21no—
98C32×C4—21no—
99C32×C4#4112yes—
100C3⋊Dic3—22no—
101C3⋊Dic3—21no—
102C3⋊Dic3—21no—
103C3⋊Dic3—21no—
104C4⋊Q8C3⋊D311yescharacteristic, complemented
105#126—92no—
106#124—92no—
107#124—91no—
108#123—92no—
109C22⋊Q8—92no—
110C22⋊Q8—91no—
111#121—91no—
112C4×Q8—91no—
113C4×D4—91no—
114#117—91no—
115C22×D3—38no—
116C22×D3—34no—
117C3×Q8—48no—
118C2×C3×C4C22×D318yes—
119C2×C3×C4C22×D314yes—
120C2×C3×C4C22×D314yes—
121C2×C3×C4C22×D318yes—
122C2×C3×C4C22×D314yes—
123C2×Dic3—34no—
124C2×Dic3—34no—
125C2×Dic3—38no—
126C2×Dic3—34no—
127D12—68no—
128D12—128no—
129C4×D3—68no—
130C4×D3—128no—
131Dic6—68no—
132C2×C32C2×#4111yescharacteristic
133C2×C32H(5,𝔽2)11yescharacteristic
134C2×C322−1+411yescharacteristic
135C3⋊D3—41no—
136C3⋊D3—42no—
137C2×Q8—91no—
138C2×Q8C2×C3⋊D312yes—
139C2×D4—91no—
140C2×D4—92no—
141C22×C4—91no—
142C22×C4—92no—
143C4⋊C4—91no—
144C4⋊C4—91no—
145C4⋊C4—92no—
146C4⋊C4—92no—
147C4⋊C4C2×C3⋊D311yescharacteristic
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⋊C4—92no—
157C42—91no—
158C42—92no—
159C42C2×C3⋊D311yescharacteristic
160C22×C3C23×D314yes—
161C2×D3—68no—
162C2×D3—64no—
163C2×D3—64no—
164C2×D3—68no—
165C2×D3—68no—
166C2×D3—64no—
167C3×C4—28no—
168C3×C4—28no—
169C3×C4—24no—
170C3×C4—24no—
171C3×C4#23918yes—
172Dic3—64no—
173Dic3—68no—
174Dic3—64no—
175Dic3—64no—
176C32#54211yescharacteristic, complemented, Sylow
177C23—92no—
178C23—91no—
179Q8—42no—
180Q8—182no—
181D4—182no—
182D4—362no—
183C2×C4—91no—
184C2×C4—92no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—182no—
188C2×C4—362no—
189C2×C4C22×C3⋊D312yes—
190C2×C4C22×C3⋊D311yescharacteristic
191C2×C4C22×C3⋊D311yescharacteristic
192C2×C4C22×C3⋊D312yes—
193C2×C4C22×C3⋊D311yescharacteristic
194C2×C3#100414yes—
195C2×C3C2×#23914yes—
196C2×C3#100714yes—
197D3—128no—
198D3—124no—
199C22—182no—
200C22—181no—
201C22—181no—
202C22—182no—
203C22—182no—
204C22—181no—
205C22C23×C3⋊D311yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—21no—
210C4—181no—
211C4—181no—
212C4—182no—
213C4—181no—
214C4#383212yes—
215C3#573314yescomplemented
216C2—362no—
217C2—361no—
218C2#6436111yescharacteristic
219C2C2×#383211yescharacteristic
220C2#6436411yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊Q8C2
prime radical#64104C2
hypercenterC4⋊Q8C3⋊D3
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C4⋊Q8C2semidirect
2. #64156C2semidirect
3. #64155C2semidirect
4. #64107C2semidirect
5. #64106C2semidirect
7. #64103C2semidirect
8. #64102C2semidirect
9. #64096C2semidirect
10. #64084C2semidirect
11. #64081C2semidirect
36. C3×C4⋊Q8D3semidirect
104. C4⋊Q8C3⋊D3semidirect
176. C32#542semidirect
215. C3#5733semidirect

Maximal subgroups

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

subgroupindex
C32×C4⋊Q82
#641562
#641552
#641072
#641062
#641042
#641032
#641022
#640962
#640842
#640812
#57333