Finite Group Databasebeta

← C2×D3×#41⋊C3 #10596619

Subgroups — C2×D3×#41⋊C3 #10596619

Order 576.

453 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 77 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×#41⋊C3C211yesmaximal, characteristic, complemented
2D3×#41⋊C3C214yesmaximal, direct product factor
3C2×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
5C2×D3×#41C311yesmaximal, characteristic, complemented
6C22×#41⋊C3—31nomaximal
7C2×C3×C4×D3—41nomaximal
8C3×#41⋊C3C2212yescomplemented
9C2×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
10D3×SL(2,𝔽3)C2212yescomplemented
11C3⋊(#41⋊C3)C2212yescomplemented
12C2×C3×#41C2×C311yescharacteristic, complemented
13D3×#41—34no—
14D3×#41C2×C314yescomplemented
15C2×#239C2×C311yescharacteristic, complemented
16C2×D3×Q8C2×C311yescharacteristic
17C2×#237—31no—
18C2×D3×D4—31no—
19C2×#235—31no—
20C22×C4×D3—31no—
21C2×#41⋊C3—21no—
22C2×#41⋊C3—34no—
23C2×#41⋊C3—31no—
24C2×#41⋊C3D311yescharacteristic, direct product factor
25C22×SL(2,𝔽3)—31no—
26C22×C3×D3—41no—
27C2×C32×C4—41no—
28C2×C3×Dic3—41no—
29C3×C4×D3—44no—
30C3×SL(2,𝔽3)C2311yescharacteristic, complemented
31C22×#41—31noSylow
32C23×D3—31no—
33C3×#41—32no—
34C3×#41C22×C312yescomplemented
35C2×C3×Q8C22×C311yescharacteristic
36C2×C3×D4—31no—
37C22×C3×C4—31no—
38C22×C3×C4—121no—
39C2×#67—32no—
40C22×Dic3—31no—
41#239—32no—
42#239C22×C312yescomplemented
43D3×Q8—32no—
44D3×Q8C22×C312yes—
45#237—34no—
46#237—34no—
47D3×D4—34no—
48D3×D4—34no—
49#235—34no—
50#235—34no—
51C2×D12—31no—
52C2×C4×D3—31no—
53C2×C4×D3—31no—
54C2×C4×D3—31no—
55C2×C4×D3—34no—
56C2×C4×D3—34no—
57C2×C4×D3A411yescharacteristic
58C2×Dic6—31no—
59#41⋊C3—22no—
60#41⋊C3—32no—
61#41⋊C3C2×D312yesdirect product factor
62C2×SL(2,𝔽3)—21no—
63C2×SL(2,𝔽3)—32no—
64C2×SL(2,𝔽3)C2×D311yescharacteristic, complemented
65C22×C32—41no—
66C2×C3×D3—44no—
67C2×C3×D3—42no—
68C32×C4—42no—
69C3×Dic3—42no—
70C2×#41—34no—
71C2×#41—31no—
72C2×#41—91no—
73C2×#41—91no—
74C2×#41—94no—
75C2×#41C3×D311yescharacteristic, complemented
76C22×Q8—31no—
77C22×D4—91no—
78C23×C4—91no—
79C23×C3—31no—
80C23×C3—121no—
81C22×D3—31no—
82C22×D3—34no—
83C22×D3—64no—
84C22×D3C2×A411yescharacteristic
85C3×Q8—31no—
86C3×Q8C23×C311yescharacteristic
87C3×D4—32no—
88C3×D4—32no—
89C2×C3×C4—31no—
90C2×C3×C4—32no—
91C2×C3×C4—32no—
92C2×C3×C4—41no—
93C2×C3×C4—81no—
94C2×C3×C4—124no—
95C2×C3×C4—121no—
96C2×C3×C4C2×A411yescharacteristic
97#67—38no—
98C2×Dic3—31no—
99C2×Dic3—34no—
100C2×Dic3C2×A411yescharacteristic
101D12—32no—
102D12—32no—
103C4×D3—32no—
104C4×D3—32no—
105C4×D3—34no—
106C4×D3—32no—
107C4×D3—32no—
108C4×D3C2×A414yes—
109Dic6—32no—
110Dic6—32no—
111SL(2,𝔽3)—21no—
112SL(2,𝔽3)C22×D311yescharacteristic, complemented
113C2×C32—42no—
114C2×C32—41no—
115C3×D3—44no—
116C24—91no—
117#41—32no—
118#41—32no—
119#41—94no—
120#41—94no—
121#41—94no—
122#41—94no—
123#41—92no—
124#41C2×C3×D312yescomplemented
125C2×Q8—32no—
126C2×Q8—91no—
127C2×Q8—92no—
128C2×Q8C2×C3×D311yescharacteristic
129C2×D4—31no—
130C2×D4—91no—
131C2×D4—94no—
132C2×D4—94no—
133C2×D4—92no—
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—91no—
137C22×C4—94no—
138C22×C4—94no—
139C22×C4—91no—
140C22×C4—91no—
141C22×C4—91no—
142C22×C3—32no—
143C22×C3—41no—
144C22×C3—62no—
145C22×C3—81no—
146C22×C3—122no—
147C22×C3—124no—
148C22×C3C22×A411yescharacteristic
149C2×D3—32no—
150C2×D3—62no—
151C2×D3—68no—
152C2×D3#41⋊C314yesdirect product factor
153C2×D3C22×A412yes—
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—42no—
157C3×C4—82no—
158C3×C4—122no—
159C3×C4C22×A412yes—
160Dic3—32no—
161Dic3C22×A412yes—
162C32—41noSylow
163C23—31no—
164C23—31no—
165C23—94no—
166C23—91no—
167C23—184no—
168Q8—31no—
169Q8—92no—
170Q8—92no—
171Q8C22×C3×D311yescharacteristic
172D4—32no—
173D4—32no—
174D4—92no—
175D4—92no—
176D4—98no—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—32no—
180C2×C4—32no—
181C2×C4—34no—
182C2×C4—92no—
183C2×C4—92no—
184C2×C4—92no—
185C2×C4—92no—
186C2×C4—94no—
187C2×C4—91no—
188C2×C4—94no—
189C2×C4D3×A411yescharacteristic
190C2×C3—42no—
191C2×C3—41no—
192C2×C3—62no—
193C2×C3—82no—
194C2×C3—81no—
195C2×C3—124no—
196C2×C3C2×#41⋊C312yescomplemented
197C2×C3C23×A411yescharacteristic
198D3—62no—
199D3C2×#41⋊C314yesdirect product factor
200C22—34no—
201C22—32no—
202C22—32no—
203C22—62no—
204C22—92no—
205C22—188no—
206C22—182no—
207C22C2×D3×A411yescharacteristic
208C4—31no—
209C4—31no—
210C4—32no—
211C4—92no—
212C4C2×D3×A412yes—
213C3—41no—
214C3—81no—
215C3C22×#41⋊C311yescharacteristic, complemented
216C2—34no—
217C2—62no—
218C2—182no—
219C2D3×#41⋊C312yesdirect product factor
220C2C22×D3×A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×A4
commutator, cosocle, hypocenterC3×Q8C23×C3
FrattiniC2C22×D3×A4
FittingC2×C3×#41C2×C3
socleC22×C3C22×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C3C2semidirect
2. D3×#41⋊C3C2direct (N × H)
3. C2×D3×SL(2,𝔽3)C2semidirect
4. C2×C3⋊(#41⋊C3)C2semidirect
5. C2×D3×#41C3semidirect
8. C3×#41⋊C3C22semidirect
9. C2×C3×SL(2,𝔽3)C22semidirect
10. D3×SL(2,𝔽3)C22semidirect
11. C3⋊(#41⋊C3)C22semidirect
12. C2×C3×#41C2×C3semidirect
14. D3×#41C2×C3semidirect
15. C2×#239C2×C3semidirect
24. C2×#41⋊C3D3direct (N × H)
30. C3×SL(2,𝔽3)C23semidirect
34. C3×#41C22×C3semidirect
42. #239C22×C3semidirect
61. #41⋊C3C2×D3direct (N × H)
64. C2×SL(2,𝔽3)C2×D3semidirect
75. C2×#41C3×D3semidirect
112. SL(2,𝔽3)C22×D3semidirect
124. #41C2×C3×D3semidirect
152. C2×D3#41⋊C3direct (N × H)
196. C2×C3C2×#41⋊C3semidirect
199. D3C2×#41⋊C3direct (N × H)
215. C3C22×#41⋊C3semidirect
219. C2D3×#41⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C32
D3×#41⋊C32
C2×D3×SL(2,𝔽3)2
C2×C3⋊(#41⋊C3)2
C2×D3×#413
C22×#41⋊C33
C2×C3×C4×D34