Finite Group Databasebeta

← C22×C3⋊#859 #11923049

Subgroups — C22×C3⋊#859 #11923049

Order 1152.

1168 conjugacy classes of subgroups; only the 241 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 241 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊#859C216yesmaximal, direct product factor
3C22×#859—33nomaximal
4C22×#835—31nomaximal
5C23×C3⋊Dic3—41nomaximal
6C22×C3×SL(2,𝔽3)C414yescomplemented
7C22×C3×SL(2,𝔽3)C2213yes—
8C3⋊#859C2214yesdirect product factor
9C23×C3×Q8D311yescharacteristic
10C23×SL(2,𝔽3)D313yes—
11C22×C4⋊Dic3—31no—
12C23×C3⋊C8—31no—
13C2×#859—318no—
14C2×#835—36no—
15C2×#835—36no—
16C24×C32—41no—
17C22×C3⋊Dic3—46no—
18C22×C3⋊Dic3—48no—
19C2×C3×SL(2,𝔽3)C2×C416yescomplemented
20C2×C3×SL(2,𝔽3)C2311yescharacteristic
21C22×#103—91noSylow
22C22×C3×Q8—34no—
23C22×C3×Q8—33no—
24C22×C3×Q8Dic314yescomplemented
25C22×C3×Q8C2×D313yes—
26C23×C3×C4—31no—
27C23×Dic3—61no—
28C23×Dic3—123no—
29C22×SL(2,𝔽3)Dic3112yescomplemented
30C22×SL(2,𝔽3)C2×D319yes—
31C2×C4⋊Dic3—36no—
32C2×C4⋊Dic3—63no—
33C22×C3⋊C8—36no—
34C22×C3⋊C8—64no—
35#859—312no—
36#835—312no—
37#835—34no—
38C23×C32—44no—
39C23×C32—44no—
40C23×C32—44no—
41C23×C32—43no—
42C2×C3⋊Dic3—44no—
43C2×C3⋊Dic3—424no—
44C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
45C23×Q8C3⋊D311yescharacteristic
46C23×C8—91no—
47C22×C4⋊C4—91no—
48C2×#103—96no—
49C2×#103—96no—
50C24×C3—43no—
51C24×C3S411yescharacteristic
52C2×C3×Q8—33no—
53C2×C3×Q8—312no—
54C2×C3×Q8—36no—
55C2×C3×Q8—66no—
56C2×C3×Q8—61no—
57C2×C3×Q8C2×Dic316yescomplemented
58C2×C3×Q8C22×D311yescharacteristic
59C22×C3×C4—34no—
60C22×C3×C4—34no—
61C22×C3×C4—33no—
62C22×C3×C4—33no—
63C22×Dic3—66no—
64C22×Dic3—1218no—
65C22×Dic3—1224no—
66C22×Dic3—124no—
67C2×SL(2,𝔽3)C2×Dic3118yescomplemented
68C2×SL(2,𝔽3)C22×D313yes—
69C4⋊Dic3—34no—
70C4⋊Dic3—66no—
71C2×C3⋊C8—34no—
72C2×C3⋊C8—612no—
73C22×C32—416no—
74C22×C32—46no—
75C22×C32—46no—
76C22×C32—46no—
77C22×C32—41no—
78C3⋊Dic3—48no—
79C22×Q8—33no—
80C22×Q8—34no—
81C22×Q8C3⋊Dic314yescomplemented
82C22×Q8C2×C3⋊D313yes—
83C23×C4—31no—
84C23×C4—181no—
85C22×C8—96no—
86C22×C8—184no—
87C2×C4⋊C4—96no—
88C2×C4⋊C4—183no—
89#103—912no—
90#103—94no—
91C23×C3—49no—
92C23×C3—412no—
93C23×C3—412no—
94C23×C3—412no—
95C23×C3#22614yes—
96C23×C3GL(2,𝔽3)14yes—
97C23×C3SL(2,ℤ/4)14yes—
98C23×C3C2×S413yes—
99C3×Q8—36no—
100C3×Q8—31no—
101C3×Q8—63no—
102C3×Q8—64no—
103C3×Q8C22×Dic311yescharacteristic, complemented
104C2×C3×C4—36no—
105C2×C3×C4—36no—
106C2×C3×C4—312no—
107C2×C3×C4—31no—
108C2×C3×C4—33no—
109C2×Dic3—64no—
110C2×Dic3—1212no—
111C2×Dic3—1272no—
112C2×Dic3—1212no—
113SL(2,𝔽3)C22×Dic313yescomplemented
114C3⋊C8—64no—
115C2×C32—41no—
116C2×C32—41no—
117C2×C32—41no—
118C2×C32—412no—
119C24C3⋊S411yescharacteristic
120C2×Q8—33no—
121C2×Q8—312no—
122C2×Q8—36no—
123C2×Q8—66no—
124C2×Q8—61no—
125C2×Q8C2×C3⋊Dic316yescomplemented
126C2×Q8C22×C3⋊D311yescharacteristic
127C22×C4—33no—
128C22×C4—33no—
129C22×C4—34no—
130C22×C4—34no—
131C22×C4—186no—
132C22×C4—364no—
133C2×C8—94no—
134C2×C8—1812no—
135C4⋊C4—94no—
136C4⋊C4—186no—
137C22×C3—418no—
138C22×C3—418no—
139C22×C3—448no—
140C22×C3—418no—
141C22×C3—43no—
142C22×C3#859116yescomplemented
143C22×C3C2×#22616yes—
144C22×C3C2×GL(2,𝔽3)16yes—
145C22×C3C2×SL(2,ℤ/4)16yes—
146C22×C3C22×S411yescharacteristic
147C3×C4—31no—
148C3×C4—31no—
149C3×C4—36no—
150Dic3—1224no—
151Dic3—124no—
152C32—41noSylow
153C23C3⋊#22614yes—
154C23C3⋊GL(2,𝔽3)14yes—
155C23C3⋊SL(2,ℤ/4)14yes—
156C23C2×C3⋊S413yes—
157Q8—36no—
158Q8—31no—
159Q8—63no—
160Q8—64no—
161Q8C22×C3⋊Dic311yescharacteristic, complemented
162C2×C4—36no—
163C2×C4—36no—
164C2×C4—312no—
165C2×C4—31no—
166C2×C4—33no—
167C2×C4—184no—
168C2×C4—3612no—
169C8—184no—
170C2×C3—436no—
171C2×C3—43no—
172C2×C3—43no—
173C2×C3—43no—
174C2×C3C2×#859112yescomplemented
175C2×C3C22×#22611yescharacteristic
176C2×C3C22×GL(2,𝔽3)11yescharacteristic
177C2×C3C22×SL(2,ℤ/4)11yescharacteristic
178C22C3⋊#859116yesdirect product factor
179C22C2×C3⋊#22616yes—
180C22C2×C3⋊GL(2,𝔽3)16yes—
181C22C2×C3⋊SL(2,ℤ/4)16yes—
182C22C22×C3⋊S411yescharacteristic
183C4—31no—
184C4—31no—
185C4—36no—
186C4—364no—
187C3—43no—
188C3C22×#85911yescharacteristic, complemented
189C2C2×C3⋊#859112yesdirect product factor
190C2C22×C3⋊#22611yescharacteristic
191C2C22×C3⋊GL(2,𝔽3)11yescharacteristic
192C2C22×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC24C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C22×C3⋊S4
FittingC23×C3×Q8D3
socleC24×C3S4
prime radicalC23×C3×SL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#859C2direct (N × H)
6. C22×C3×SL(2,𝔽3)C4semidirect
8. C3⋊#859C22direct (N × H)
19. C2×C3×SL(2,𝔽3)C2×C4semidirect
24. C22×C3×Q8Dic3semidirect
29. C22×SL(2,𝔽3)Dic3semidirect
44. C3×SL(2,𝔽3)C22×C4semidirect
57. C2×C3×Q8C2×Dic3semidirect
67. C2×SL(2,𝔽3)C2×Dic3semidirect
81. C22×Q8C3⋊Dic3semidirect
103. C3×Q8C22×Dic3semidirect
113. SL(2,𝔽3)C22×Dic3semidirect
125. C2×Q8C2×C3⋊Dic3semidirect
142. C22×C3#859semidirect
161. Q8C22×C3⋊Dic3semidirect
174. C2×C3C2×#859semidirect
178. C22C3⋊#859direct (N × H)
188. C3C22×#859semidirect
189. C2C2×C3⋊#859direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×C3⋊#8592
C22×#8593
C22×#8353
C23×C3⋊Dic34