Finite Group Databasebeta

← #10596153

Subgroups — #10596153

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×#124C211yesmaximal, characteristic, complemented
2#64155C211yesmaximal, characteristic, complemented
3Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#64147C211yesmaximal, characteristic
5D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#64095C212yesmaximal, complemented
7#64094C212yesmaximal, complemented
8#64088C212yesmaximal, complemented
9#64087C212yesmaximal, complemented
10#64081C211yesmaximal, characteristic, complemented
11#64078C211yesmaximal, characteristic, complemented
12#5682—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C22×C3⋊Dic3C2212yes—
17C2×C4×C3⋊D3C2211yescharacteristic
18C2×#634C2212yes—
19C32×C22⋊C4C2212yes—
20C32×C22⋊C4C2212yes—
21C32×C42C2211yescharacteristic
22#3757C2212yes—
23#3757C2212yes—
24#3752C2211yescharacteristic
25#3752C2211yescharacteristic
26#3752C2212yes—
27C4⋊(C3⋊Dic3)C2212yes—
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2212yes—
31#3750C2211yescharacteristic
32#3750C2211yescharacteristic
33#3750C2212yes—
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2212yes—
36C3×#124D314yescomplemented
37#946—34no—
38Q8×Dic3—34no—
39#938—34no—
40D4×Dic3—34no—
41#886—38no—
42#885—38no—
43#879—38no—
44#878—38no—
45#872—34no—
46#869—34no—
47C23×C32C2312yes—
48C22×C3⋊D3C2311yescharacteristic
49C32×Q8—22no—
50C32×D4—22no—
51C2×C32×C4C2312yes—
52C2×C32×C4C2311yescharacteristic
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55#638—42no—
56C2×C3⋊Dic3—42no—
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2312yes—
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2312yes—
62C4×C3⋊D3—22no—
63#634—42no—
64#542—91noSylow
65C2×C3×Q8C2×D314yes—
66C2×C3×D4C2×D314yes—
67C2×#67—38no—
68C22×Dic3—38no—
69C2×C4×D3—34no—
70C2×Dic6—38no—
71C3×C22⋊C4C2×D318yes—
72C3×C22⋊C4C2×D318yes—
73C3×C42C2×D314yes—
74#217—38no—
75#217—38no—
76(C2×D3)⋊C4—34no—
77(C2×D3)⋊C4—34no—
78(C2×D3)⋊C4—38no—
79C4⋊Dic3—38no—
80C4⋊Dic3—34no—
81C4⋊Dic3—34no—
82C4⋊Dic3—38no—
83Dic3⋊C4—34no—
84Dic3⋊C4—34no—
85Dic3⋊C4—38no—
86C4×Dic3—34no—
87C4×Dic3—38no—
88C22×C32—22no—
89C22×C32—22no—
90C22×C32—22no—
91C22×C32C2411yescharacteristic
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C2×C3⋊D3—21no—
95C32×C4—22no—
96C32×C4—21no—
97C32×C4—21no—
98C32×C4#4112yes—
99C3⋊Dic3—21no—
100C3⋊Dic3—22no—
101C3⋊Dic3—22no—
102C3⋊Dic3—21no—
103C3⋊Dic3—21no—
104C4⋊Q8—91no—
105#126—92no—
106#124—92no—
107#124C3⋊D311yescharacteristic, complemented
108#123—92no—
109C22⋊Q8—91no—
110C22⋊Q8—92no—
111#121—91no—
112C4×Q8—91no—
113C4×D4—91no—
114#117—91no—
115C23×C3C22×D318yes—
116C22×D3—34no—
117C3×Q8—28no—
118C3×D4—28no—
119C2×C3×C4C22×D318yes—
120C2×C3×C4C22×D314yes—
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—38no—
129C2×Dic3—128no—
130C4×D3—68no—
131Dic6—128no—
132C2×C32—42no—
133C2×C32C2×#4111yescharacteristic
134C2×C32H(5,𝔽2)11yescharacteristic
135C2×C322−1+411yescharacteristic
136C3⋊D3—41no—
137C2×Q8—92no—
138C2×Q8C2×C3⋊D311yescharacteristic
139C2×D4—92no—
140C2×D4C2×C3⋊D311yescharacteristic
141C22×C4—92no—
142C22×C4—91no—
143C4⋊C4—91no—
144C4⋊C4—91no—
145C4⋊C4—91no—
146C4⋊C4—91no—
147C4⋊C4—92no—
148C4⋊C4—92no—
149C4⋊C4—92no—
150C22⋊C4—92no—
151C22⋊C4—92no—
152C22⋊C4—91no—
153C22⋊C4—91no—
154C22⋊C4—92no—
155C22⋊C4C2×C3⋊D312yes—
156C22⋊C4C2×C3⋊D312yes—
157C42—91no—
158C42—92no—
159C42C2×C3⋊D311yescharacteristic
160C22×C3—28no—
161C22×C3—28no—
162C22×C3—28no—
163C22×C3C23×D314yes—
164C2×D3—64no—
165C2×D3—64no—
166C2×D3—64no—
167C3×C4—28no—
168C3×C4—24no—
169C3×C4—24no—
170C3×C4#23718yes—
171Dic3—64no—
172Dic3—68no—
173Dic3—68no—
174Dic3—64no—
175Dic3—64no—
176C32#54211yescharacteristic, complemented, Sylow
177C23—91no—
178C23C22×C3⋊D312yes—
179Q8—22no—
180Q8—362no—
181D4—22no—
182D4—362no—
183C2×C4—91no—
184C2×C4—91no—
185C2×C4—92no—
186C2×C4—91no—
187C2×C4—92no—
188C2×C4—182no—
189C2×C4—362no—
190C2×C4C22×C3⋊D312yes—
191C2×C4C22×C3⋊D311yescharacteristic
192C2×C4C22×C3⋊D311yescharacteristic
193C2×C4C22×C3⋊D311yescharacteristic
194C2×C3—48no—
195C2×C3C2×#23714yes—
196C2×C3#100914yes—
197C2×C3#101014yes—
198D3—124no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—181no—
203C22—181no—
204C22—181no—
205C22C23×C3⋊D311yescharacteristic
206C4—22no—
207C4—21no—
208C4—21no—
209C4—181no—
210C4—181no—
211C4—182no—
212C4—182no—
213C4—181no—
214C4#383012yes—
215C3#568214yescomplemented
216C2—42no—
217C2—361no—
218C2C2×#383011yescharacteristic
219C2#6436611yescharacteristic
220C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#124C2
prime radical#64147C2
hypercenter#124C3⋊D3
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C32×#124C2semidirect
2. #64155C2semidirect
3. Q8×C3⋊Dic3C2semidirect
5. D4×C3⋊Dic3C2semidirect
6. #64095C2semidirect
7. #64094C2semidirect
8. #64088C2semidirect
9. #64087C2semidirect
10. #64081C2semidirect
11. #64078C2semidirect
36. C3×#124D3semidirect
107. #124C3⋊D3semidirect
176. C32#542semidirect
215. C3#5682semidirect

Maximal subgroups

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

subgroupindex
C32×#1242
#641552
Q8×C3⋊Dic32
#641472
D4×C3⋊Dic32
#640952
#640942
#640882
#640872
#640812
#640782
#56823