Finite Group Databasebeta

← #10596135

Subgroups — #10596135

Order 576.

708 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 175 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×#123C211yesmaximal, characteristic, complemented
2#64149C211yesmaximal, characteristic
3#64144C211yesmaximal, characteristic, complemented
4C4×#638C212yesmaximal, complemented
5#64103C212yesmaximal, complemented
6#64096C212yesmaximal, complemented
7#64094C212yesmaximal, complemented
8#64094C212yesmaximal, complemented
9#64090C212yesmaximal, complemented
10#5664—34nomaximal
11C2×C32×D4C2211yescharacteristic
12C22×C32×C4C2211yescharacteristic
13C2×#638C2212yes—
14C2×#638C2212yes—
15C22×C3⋊Dic3C2211yescharacteristic
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C4×C3⋊D3C2212yes—
18C2×#634C2212yes—
19C32×C4⋊C4C2212yes—
20C32×C22⋊C4C2212yes—
21C32×C22⋊C4C2211yescharacteristic
22#3757C2211yescharacteristic
23#3757C2212yes—
24#3752C2212yes—
25#3752C2212yes—
26#3752C2212yes—
27#3750C2212yes—
28#3750C2212yes—
29C4×C3⋊Dic3C2212yes—
30C4×C3⋊Dic3C2212yes—
31C4×C3⋊Dic3C2211yescharacteristic
32C3×#123D314yescomplemented
33#940—34no—
34#935—34no—
35C4×#67—38no—
36#894—38no—
37#887—38no—
38#885—38no—
39#885—38no—
40#67⋊C4—38no—
41C23×C32C2311yescharacteristic
42C23×C32C2311yescharacteristic
43C22×C3⋊D3C2312yes—
44C32×D4—41no—
45C2×C32×C4—41no—
46C2×C32×C4C2311yescharacteristic
47C2×C32×C4C2312yes—
48C2×C32×C4C2312yes—
49#638—24no—
50#638—24no—
51C2×C3⋊Dic3—41no—
52C2×C3⋊Dic3C2312yes—
53C2×C3⋊Dic3C2312yes—
54C2×C3⋊Dic3C2312yes—
55C4⋊(C3⋊D3)—41no—
56C4×C3⋊D3—42no—
57#634—24no—
58#559—91noSylow
59C2×C3×D4C2×D314yes—
60C22×C3×C4C2×D314yes—
61C2×#67—38no—
62C2×#67—38no—
63C22×Dic3—34no—
64C2×D12—34no—
65C2×C4×D3—38no—
66C2×Dic6—38no—
67C3×C4⋊C4C2×D318yes—
68C3×C22⋊C4C2×D318yes—
69C3×C22⋊C4C2×D314yes—
70#217—34no—
71#217—38no—
72(C2×D3)⋊C4—38no—
73(C2×D3)⋊C4—38no—
74(C2×D3)⋊C4—38no—
75Dic3⋊C4—38no—
76Dic3⋊C4—38no—
77C4×Dic3—38no—
78C4×Dic3—38no—
79C4×Dic3—34no—
80C22×C32—22no—
81C22×C32—22no—
82C22×C32—21no—
83C22×C32—21no—
84C22×C32C2411yescharacteristic
85C2×C3⋊D3—22no—
86C2×C3⋊D3—22no—
87C2×C3⋊D3—22no—
88C32×C4—22no—
89C32×C4—22no—
90C32×C4—21no—
91C3⋊Dic3—22no—
92C3⋊Dic3—22no—
93C3⋊Dic3#4114yes—
94#127—91no—
95#124—92no—
96#124—92no—
97#123—91no—
98#123—92no—
99#123C3⋊D311yescharacteristic, complemented
100C4×Q8—92no—
101C4×D4—92no—
102C4×D4—92no—
103C23×C3C22×D314yes—
104C23×C3C22×D314yes—
105C22×D3—38no—
106C3×D4—44no—
107C2×C3×C4—44no—
108C2×C3×C4C22×D314yes—
109C2×C3×C4C22×D318yes—
110C2×C3×C4C22×D318yes—
111#67—616no—
112#67—616no—
113C2×Dic3—38no—
114C2×Dic3—38no—
115C2×Dic3—38no—
116C2×Dic3—124no—
117D12—124no—
118C4×D3—128no—
119Dic6—616no—
120C2×C32—41no—
121C2×C32—41no—
122C2×C32C2×#4112yes—
123C2×C32H(5,𝔽2)11yescharacteristic
124C3⋊D3—42no—
125C2×Q8—92no—
126C2×D4—92no—
127C2×D4—92no—
128C2×D4—91no—
129C2×D4C2×C3⋊D311yescharacteristic
130C22×C4—91no—
131C22×C4—92no—
132C22×C4C2×C3⋊D311yescharacteristic
133C4⋊C4—92no—
134C4⋊C4—92no—
135C4⋊C4C2×C3⋊D312yes—
136C22⋊C4—92no—
137C22⋊C4—91no—
138C22⋊C4—92no—
139C22⋊C4—92no—
140C22⋊C4—92no—
141C22⋊C4C2×C3⋊D312yes—
142C22⋊C4C2×C3⋊D311yescharacteristic
143C42—92no—
144C42—92no—
145C42—91no—
146C22×C3—24no—
147C22×C3—24no—
148C22×C3—28no—
149C22×C3—28no—
150C22×C3C23×D314yes—
151C2×D3—68no—
152C2×D3—68no—
153C2×D3—68no—
154C3×C4—28no—
155C3×C4—24no—
156C3×C4—28no—
157Dic3—316no—
158Dic3—68no—
159Dic3—68no—
160C32#55911yescharacteristic, complemented, Sylow
161C23—92no—
162C23C22×C3⋊D311yescharacteristic
163C23C22×C3⋊D311yescharacteristic
164Q8—184no—
165D4—41no—
166D4—184no—
167D4—184no—
168D4—361no—
169C2×C4—41no—
170C2×C4—92no—
171C2×C4—92no—
172C2×C4—92no—
173C2×C4—362no—
174C2×C4—361no—
175C2×C4C22×C3⋊D312yes—
176C2×C4C22×C3⋊D312yes—
177C2×C4C22×C3⋊D311yescharacteristic
178C2×C3—44no—
179C2×C3—44no—
180C2×C3#100414yes—
181C2×C3D3×#4118yes—
182D3—128no—
183C22—21no—
184C22—21no—
185C22—22no—
186C22—22no—
187C22—182no—
188C22—182no—
189C22—182no—
190C22C23×C3⋊D311yescharacteristic
191C4—22no—
192C4—21no—
193C4—22no—
194C4—94no—
195C4—182no—
196C4—182no—
197C3#566414yescomplemented
198C2—41no—
199C2—41no—
200C2—362no—
201C2#6436111yescharacteristic
202C2#41×C3⋊D312yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#123C2
prime radical#64149C2
hypercenter#123C3⋊D3
hypocenterC32#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C32×#123C2semidirect
3. #64144C2semidirect
4. C4×#638C2semidirect
5. #64103C2semidirect
6. #64096C2semidirect
7. #64094C2semidirect
8. #64094C2semidirect
9. #64090C2semidirect
32. C3×#123D3semidirect
99. #123C3⋊D3semidirect
160. C32#559semidirect
197. C3#5664semidirect

Maximal subgroups

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

subgroupindex
C32×#1232
#641492
#641442
C4×#6382
#641032
#640962
#640942
#640902
#56643