Finite Group Databasebeta

← C22×C3×#994 #11919872

Subgroups — C22×C3×#994 #11919872

Order 1152.

2960 conjugacy classes of subgroups; only the 558 normal subgroups are shown.

normal subgroups only

This group has 2960 conjugacy classes of subgroups — too many to draw. Only its 558 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×#994C2112yesmaximal, direct product factor
2C22×C3×#41⋊C3C213yesmaximal, complemented
3C22×C3×H(5,𝔽2)C311yesmaximal, characteristic, complemented
4C22×#994C313yesmaximal, direct product factor
5C22×C32×Q8—41nomaximal
6C3×#994C22116yesdirect product factor
7C2×C3×#41⋊C3C22118yescomplemented
8C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
9C2×C3×H(5,𝔽2)—312no—
10C2×C3×H(5,𝔽2)C2×C3112yescomplemented
11C22×C3×#41—31no—
12C22×C3×#41C2×C313yes—
13C23×C3×D4—33no—
14C2×#994C2×C3136yesdirect product factor
15C22×#41⋊C3C2×C319yescomplemented
16C2×C32×Q8—412no—
17C22×C32×C4—43no—
18C3×#41⋊C3C23112yescomplemented
19C2×C3×SL(2,𝔽3)C2313yescomplemented
20C22×H(5,𝔽2)C3211yescharacteristic, complemented, Sylow
21C25×C3—32no—
22C3×H(5,𝔽2)—348no—
23C3×H(5,𝔽2)—332no—
24C3×H(5,𝔽2)C22×C3116yescomplemented
25C2×C3×#41—312no—
26C2×C3×#41—312no—
27C2×C3×#41—318no—
28C2×C3×#41C22×C3118yes—
29C22×C3×Q8—43no—
30C22×C3×Q8A411yescharacteristic
31C22×C3×Q8C22×C311yescharacteristic
32C22×C3×D4—33no—
33C22×C3×D4—336no—
34C22×C3×D4—336no—
35C22×C3×D4—33no—
36C23×C3×C4—33no—
37#994C22×C3148yesdirect product factor
38C2×#41⋊C3C22×C3154yescomplemented
39C22×SL(2,𝔽3)C22×C313yescomplemented
40C23×C32—41no—
41C32×Q8—416no—
42C2×C32×C4—418no—
43C3×SL(2,𝔽3)C2411yescharacteristic, complemented
44C2×H(5,𝔽2)—312no—
45C2×H(5,𝔽2)C2×C32112yescomplemented
46C22×#41—31no—
47C22×#41C2×C3213yes—
48C23×D4—33no—
49C24×C3—324no—
50C24×C3—33no—
51C24×C3—128no—
52C3×#41—324no—
53C3×#41—348no—
54C3×#41—316no—
55C3×#41—336no—
56C3×#41C23×C3112yes—
57C2×C3×Q8—33no—
58C2×C3×Q8—436no—
59C2×C3×Q8C2×A4112yes—
60C2×C3×Q8C23×C313yes—
61C2×C3×D4—318no—
62C2×C3×D4—318no—
63C2×C3×D4—3144no—
64C2×C3×D4—348no—
65C2×C3×D4—336no—
66C22×C3×C4—31no—
67C22×C3×C4—318no—
68C22×C3×C4—318no—
69C22×C3×C4—49no—
70C22×C3×C4C2×A413yes—
71#41⋊C3C23×C3136yescomplemented
72C2×SL(2,𝔽3)C23×C319yescomplemented
73C22×C32—44no—
74C22×C32—43no—
75C32×C4—412no—
76C25—32no—
77H(5,𝔽2)—348no—
78H(5,𝔽2)—332no—
79H(5,𝔽2)C22×C32116yescomplemented
80C2×#41—312no—
81C2×#41—312no—
82C2×#41—318no—
83C2×#41C22×C32118yes—
84C22×Q8C3×A411yescharacteristic
85C22×Q8C22×C3211yescharacteristic
86C22×D4—33no—
87C22×D4—336no—
88C22×D4—336no—
89C22×D4—33no—
90C23×C4—33no—
91C23×C3—332no—
92C23×C3—318no—
93C23×C3—43no—
94C23×C3—612no—
95C23×C3—1248no—
96C23×C3C22×A411yescharacteristic
97C3×Q8—33no—
98C3×Q8—32no—
99C3×Q8—448no—
100C3×Q8C22×A4116yes—
101C3×Q8C24×C311yescharacteristic
102C3×D4—312no—
103C3×D4—336no—
104C3×D4—348no—
105C2×C3×C4—33no—
106C2×C3×C4—33no—
107C2×C3×C4—336no—
108C2×C3×C4—312no—
109C2×C3×C4—454no—
110C2×C3×C4C22×A4118yes—
111SL(2,𝔽3)C24×C313yescomplemented
112C2×C32—46no—
113C2×C32—41no—
114C24—324no—
115C24—33no—
116C24—128no—
117#41—324no—
118#41—316no—
119#41—348no—
120#41—336no—
121#41C23×C32112yes—
122C2×Q8—33no—
123C2×Q8C2×C3×A4112yes—
124C2×Q8C23×C3213yes—
125C2×D4—318no—
126C2×D4—318no—
127C2×D4—3144no—
128C2×D4—348no—
129C2×D4—336no—
130C22×C4—31no—
131C22×C4—318no—
132C22×C4—318no—
133C22×C4C2×C3×A413yes—
134C22×C3—312no—
135C22×C3—412no—
136C22×C3—49no—
137C22×C3—636no—
138C22×C3—1232no—
139C22×C3#99414yesdirect product factor
140C22×C3C23×A413yes—
141C3×C4—33no—
142C3×C4—31no—
143C3×C4—436no—
144C3×C4C23×A4112yes—
145C32—41noSylow
146C23—332no—
147C23—318no—
148C23—612no—
149C23—1248no—
150C23C22×C3×A411yescharacteristic
151Q8—33no—
152Q8—32no—
153Q8C22×C3×A4116yes—
154Q8C24×C3211yescharacteristic
155D4—312no—
156D4—336no—
157D4—348no—
158C2×C4—33no—
159C2×C4—33no—
160C2×C4—312no—
161C2×C4—336no—
162C2×C4C22×C3×A4118yes—
163C2×C3—418no—
164C2×C3—43no—
165C2×C3—612no—
166C2×C3C2×#99416yesdirect product factor
167C2×C3C24×A411yescharacteristic
168C22—312no—
169C22—636no—
170C22—1232no—
171C22C3×#99414yesdirect product factor
172C22C23×C3×A413yes—
173C4—31no—
174C4—33no—
175C4C23×C3×A4112yes—
176C3—43no—
177C3C22×#99411yescharacteristic, direct product factor
178C2—612no—
179C2C2×C3×#99416yesdirect product factor
180C2C24×C3×A411yescharacteristic
subgroupG/H
center, socleC23×C3C22×A4
commutator, cosocle, hypocenterQ8C24×C32
FrattiniC2C24×C3×A4
FittingC22×C3×H(5,𝔽2)C3
hypercenterC22×C3×Q8A4
LCSG ▷ Q8
UCSC1 ◁ C23×C3 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#994C2direct (N × H)
2. C22×C3×#41⋊C3C2semidirect
3. C22×C3×H(5,𝔽2)C3semidirect
4. C22×#994C3direct (N × H)
6. C3×#994C22direct (N × H)
7. C2×C3×#41⋊C3C22semidirect
8. C22×C3×SL(2,𝔽3)C22semidirect
10. C2×C3×H(5,𝔽2)C2×C3semidirect
14. C2×#994C2×C3direct (N × H)
15. C22×#41⋊C3C2×C3semidirect
18. C3×#41⋊C3C23semidirect
19. C2×C3×SL(2,𝔽3)C23semidirect
20. C22×H(5,𝔽2)C32semidirect
24. C3×H(5,𝔽2)C22×C3semidirect
37. #994C22×C3direct (N × H)
38. C2×#41⋊C3C22×C3semidirect
39. C22×SL(2,𝔽3)C22×C3semidirect
43. C3×SL(2,𝔽3)C24semidirect
45. C2×H(5,𝔽2)C2×C32semidirect
71. #41⋊C3C23×C3semidirect
72. C2×SL(2,𝔽3)C23×C3semidirect
79. H(5,𝔽2)C22×C32semidirect
111. SL(2,𝔽3)C24×C3semidirect
139. C22×C3#994direct (N × H)
166. C2×C3C2×#994direct (N × H)
171. C22C3×#994direct (N × H)
177. C3C22×#994direct (N × H)
179. C2C2×C3×#994direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9942
C22×C3×#41⋊C32
C22×C3×H(5,𝔽2)3
C22×#9943
C22×C32×Q84