Finite Group Databasebeta

← C3×#4751 #10591848

Subgroups — C3×#4751 #10591848

Order 576.

319 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×C3×#239C211yesmaximal, characteristic
2C3×D3×M16C212yesmaximal, complemented
3C32×#101C211yesmaximal, characteristic, complemented
4C3×#836C211yesmaximal, characteristic, complemented
5C3×#823C212yesmaximal, complemented
6C3×#413—31nomaximal
7#4751C311yesmaximal, characteristic, direct product factor
8C2×C32×Q8C2211yescharacteristic
9C3×#239—22no—
10C3×#239—22no—
11C2×C3×D12C412yes—
12C2×C3×D12C2211yescharacteristic
13C2×C3×C4×D3C412yes—
14C2×C3×C4×D3C2211yescharacteristic
15C32×M16C2212yes—
16C3×#208C2212yes—
17C3×ΓL(1,𝔽25)—22no—
18C3×D3×C8—22no—
19C2×C3×#41—31no—
20C2×#239C2×C311yescharacteristic
21C2×C3×M16—32no—
22D3×M16C2×C312yescomplemented
23C3×#101—21no—
24C3×#101—31no—
25C3×#101D311yescharacteristic, complemented
26C3×#101C2×C311yescharacteristic, complemented
27C3×#100—32no—
28#836C2×C311yescharacteristic, complemented
29#823C2×C312yescomplemented
30C22×C3×D3C2×C412yes—
31C22×C3×D3C2×C411yescharacteristic
32C32×Q8—22no—
33C2×C32×C4C2×C412yes—
34C2×C32×C4C2311yescharacteristic
35C2×C3×Dic3C2×C411yescharacteristic
36C3×D12—24no—
37C3×D12—22no—
38C3×C4×D3—22no—
39C3×C4×D3—22no—
40C3×C4×D3D412yes—
41C3×C4×D3D412yes—
42C32×C8—22no—
43C3×C3⋊C8—22no—
44#413—31noSylow
45C3×#41—62no—
46C3×#41—62no—
47C2×C3×Q8—21no—
48C2×C3×Q8C2×D311yescharacteristic
49C2×C3×Q8C22×C311yescharacteristic
50C2×C3×D4—32no—
51C2×C3×D4—31no—
52C22×C3×C4—31no—
53C22×C3×C4—32no—
54#239—22no—
55#239—22no—
56C2×D12C3×C412yes—
57C2×D12C22×C311yescharacteristic
58C2×C4×D3C3×C412yes—
59C2×C4×D3C22×C311yescharacteristic
60C3×M16—22no—
61C3×M16—32no—
62C3×M16—62no—
63C3×M16C2×D312yes—
64C3×M16C22×C312yes—
65C2×C3×C8—62no—
66#208C22×C312yes—
67ΓL(1,𝔽25)—22no—
68D3×C8—22no—
69C22×C32C22×C411yescharacteristic
70C2×C3×D3—22no—
71C2×C3×D3—41no—
72C2×C3×D3—42no—
73C2×C3×D3C22⋊C412yes—
74C32×C4—22no—
75C32×C4C2×D412yes—
76C3×Dic3C22⋊C411yescharacteristic
77C3×Dic3C22⋊C411yescharacteristic
78C2×#41—31no—
79C2×M16—32no—
80#101—31no—
81#101C3×D311yescharacteristic, complemented
82#100—32no—
83C23×C3—31no—
84C23×C3—32no—
85C22×D3C2×C3×C412yes—
86C22×D3C2×C3×C411yescharacteristic
87C3×Q8—22no—
88C3×Q8—22no—
89C3×Q8—42no—
90C3×D4—64no—
91C3×D4—62no—
92C2×C3×C4—22no—
93C2×C3×C4—21no—
94C2×C3×C4—31no—
95C2×C3×C4—32no—
96C2×C3×C4—32no—
97C2×C3×C4—62no—
98C2×C3×C4—62no—
99C2×C3×C4C4×D312yes—
100C2×C3×C4C2×C3×C412yes—
101C2×C3×C4C22×D311yescharacteristic
102C2×C3×C4C23×C311yescharacteristic
103C2×Dic3C2×C3×C411yescharacteristic
104D12—24no—
105D12—22no—
106C4×D3—22no—
107C4×D3—22no—
108C4×D3C3×D412yes—
109C4×D3C3×D412yes—
110C3×C8—22no—
111C3×C8—22no—
112C3×C8—42no—
113C3×C8—62no—
114C3⋊C8—22no—
115C2×C32—21no—
116C2×C32C2×C22⋊C411yescharacteristic
117C3×D3—22no—
118C3×D3—42no—
119#41—62no—
120#41—62no—
121C2×Q8C2×C3×D311yescharacteristic
122C2×D4—32no—
123C2×D4—31no—
124C22×C4—31no—
125C22×C4—32no—
126M16—32no—
127M16—62no—
128M16C2×C3×D312yes—
129C2×C8—62no—
130C22×C3—21no—
131C22×C3—32no—
132C22×C3—62no—
133C22×C3—121no—
134C22×C3—122no—
135C22×C3C2×C4×D311yescharacteristic
136C22×C3C22×C3×C411yescharacteristic
137C2×D3—22no—
138C2×D3—41no—
139C2×D3—42no—
140C2×D3C3×C22⋊C412yes—
141C3×C4—22no—
142C3×C4—22no—
143C3×C4—22no—
144C3×C4—31no—
145C3×C4—31no—
146C3×C4—42no—
147C3×C4D3×D412yes—
148C3×C4C2×C3×D412yes—
149Dic3C3×C22⋊C411yescharacteristic
150Dic3C3×C22⋊C411yescharacteristic
151C32#41311yescharacteristic, complemented, Sylow
152C23—31no—
153C23—32no—
154Q8—22no—
155D4—64no—
156D4—62no—
157C2×C4—31no—
158C2×C4—32no—
159C2×C4—32no—
160C2×C4—62no—
161C2×C4—62no—
162C2×C4C3×C4×D312yes—
163C2×C4C22×C3×D311yescharacteristic
164C8—22no—
165C8—62no—
166C2×C3—21no—
167C2×C3—21no—
168C2×C3—21no—
169C2×C3—41no—
170C2×C3—62no—
171C2×C3—122no—
172C2×C3D3×C22⋊C411yescharacteristic
173C2×C3C2×C3×C22⋊C411yescharacteristic
174D3—22no—
175D3—42no—
176C22—32no—
177C22—62no—
178C22—121no—
179C22—122no—
180C22C2×C3×C4×D311yescharacteristic
181C4—22no—
182C4—31no—
183C4—31no—
184C4C3×D3×D412yes—
185C3—21no—
186C3#475111yescharacteristic, direct product factor
187C3C3×#41311yescharacteristic, complemented
188C2—21no—
189C2—62no—
190C2—122no—
191C2C3×D3×C22⋊C411yescharacteristic
subgroupG/H
centerC2×C3D3×C22⋊C4
commutatorC22×C3C22×C3×C4
FrattiniC2×C4C22×C3×D3
FittingC32×#101C2
socleC2×C32C2×C22⋊C4
prime radicalC2×C3×#239C2
cosocleC2×C3×C4C23×C3
hypercenterC3×#101D3
hypocenterC3C3×#413
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
2. C3×D3×M16C2semidirect
3. C32×#101C2semidirect
4. C3×#836C2semidirect
5. C3×#823C2semidirect
7. #4751C3direct (N × H)
22. D3×M16C2×C3semidirect
25. C3×#101D3semidirect
26. C3×#101C2×C3semidirect
28. #836C2×C3semidirect
29. #823C2×C3semidirect
81. #101C3×D3semidirect
151. C32#413semidirect
186. C3#4751direct (N × H)
187. C3C3×#413semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2392
C3×D3×M162
C32×#1012
C3×#8362
C3×#8232
C3×#4133
#47513