Finite Group Databasebeta

← C22×C3×#131⋊C3 #11919780

Subgroups — C22×C3×#131⋊C3 #11919780

Order 1152.

848 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 238 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2C2×C3×#131⋊C3C216yesmaximal, direct product factor
3C22×C3×#131C311yesmaximal, characteristic, complemented
4C22×#131⋊C3C313yesmaximal, direct product factor
5C2×C3×#41⋊C3C413yes—
6C2×C3×#41⋊C3C2213yes—
7C22×C3×SL(2,𝔽3)C411yescharacteristic
8C22×C32×C8—41nomaximal
9C3×#131⋊C3C2214yesdirect product factor
10C22×C3×#41C2×C311yescharacteristic
11C22×#41⋊C3C2×C313yes—
12C2×C3×#131—36no—
13C2×C3×#131C2×C316yescomplemented
14C22×C3×M16—31no—
15C23×C3×C8—31no—
16C2×#131⋊C3C2×C3118yesdirect product factor
17C22×C32×C4—41no—
18C3×#41⋊C3C2×C413yes—
19C3×#41⋊C3C2311yescharacteristic
20C2×C3×SL(2,𝔽3)C2×C413yes—
21C2×C32×C8—46no—
22C22×#131C3211yescharacteristic, complemented, Sylow
23C2×C3×#41—33no—
24C2×C3×#41—33no—
25C2×C3×#41C3×C413yes—
26C2×C3×#41C22×C313yes—
27C22×C3×Q8C3×C411yescharacteristic
28C22×C3×D4—31no—
29C23×C3×C4—31no—
30C2×#41⋊C3C3×C419yes—
31C2×#41⋊C3C22×C319yes—
32C22×SL(2,𝔽3)C3×C413yes—
33C3×#131—38no—
34C3×#131—312no—
35C3×#131C22×C314yescomplemented
36C2×C3×M16—36no—
37C2×C3×M16—36no—
38C22×C3×C8—31no—
39C22×C3×C8—36no—
40C22×C3×C8—36no—
41C22×C3×C8—43no—
42C22×C3×C8A411yescharacteristic
43#131⋊C3C22×C3112yesdirect product factor
44C23×C32—41no—
45C2×C32×C4—43no—
46C2×C32×C4—43no—
47C3×SL(2,𝔽3)C22×C411yescharacteristic
48C32×C8—44no—
49C22×#41C2×C3211yescharacteristic
50C2×#131—36no—
51C2×#131C2×C3216yescomplemented
52C22×M16—31no—
53C23×C8—31no—
54C24×C3—31no—
55C3×#41—32no—
56C3×#41—36no—
57C3×#41—33no—
58C3×#41—36no—
59C3×#41—33no—
60C3×#41C2×C3×C413yes—
61C3×#41C23×C311yescharacteristic
62C2×C3×Q8—33no—
63C2×C3×Q8C2×C3×C413yes—
64C2×C3×D4—33no—
65C2×C3×D4—33no—
66C2×C3×D4—36no—
67C22×C3×C4—31no—
68C22×C3×C4—33no—
69C22×C3×C4—33no—
70C22×C3×C4—33no—
71C22×C3×C4—33no—
72C22×C3×C4—43no—
73C22×C3×C4C2×A411yescharacteristic
74#41⋊C3C2×C3×C419yes—
75#41⋊C3C23×C313yes—
76C2×SL(2,𝔽3)C2×C3×C419yes—
77C3×M16—312no—
78C3×M16—34no—
79C2×C3×C8—36no—
80C2×C3×C8—312no—
81C2×C3×C8—34no—
82C2×C3×C8—418no—
83C2×C3×C8C2×A416yes—
84C22×C32—44no—
85C22×C32—43no—
86C32×C4—43no—
87C32×C4—41no—
88C2×#41—33no—
89C2×#41—33no—
90C2×#41C32×C413yes—
91C2×#41C22×C3213yes—
92C22×Q8C32×C411yescharacteristic
93C22×D4—31no—
94C23×C4—31no—
95#131—38no—
96#131—312no—
97#131C22×C3214yescomplemented
98C2×M16—36no—
99C2×M16—36no—
100C22×C8—31no—
101C22×C8—36no—
102C22×C8—36no—
103C22×C8C3×A411yescharacteristic
104C23×C3—33no—
105C23×C3—33no—
106C23×C3—43no—
107C23×C3—64no—
108C23×C3C4×A411yescharacteristic
109C3×Q8—33no—
110C3×Q8—32no—
111C3×Q8C22×C3×C411yescharacteristic
112C3×D4—33no—
113C3×D4—31no—
114C3×D4—36no—
115C3×D4—36no—
116C2×C3×C4—33no—
117C2×C3×C4—33no—
118C2×C3×C4—33no—
119C2×C3×C4—36no—
120C2×C3×C4—33no—
121C2×C3×C4—33no—
122C2×C3×C4—31no—
123C2×C3×C4—49no—
124C2×C3×C4—49no—
125C2×C3×C4C4×A413yes—
126C2×C3×C4C22×A413yes—
127SL(2,𝔽3)C22×C3×C413yes—
128C3×C8—34no—
129C3×C8—412no—
130C3×C8C22×A414yes—
131C2×C32—41no—
132C2×C32—46no—
133C24—31no—
134#41—32no—
135#41—36no—
136#41—33no—
137#41—36no—
138#41—33no—
139#41C2×C32×C413yes—
140#41C23×C3211yescharacteristic
141C2×Q8—33no—
142C2×Q8C2×C32×C413yes—
143C2×D4—33no—
144C2×D4—33no—
145C2×D4—36no—
146C22×C4—31no—
147C22×C4—33no—
148C22×C4—33no—
149C22×C4—33no—
150C22×C4—33no—
151C22×C4C2×C3×A411yescharacteristic
152M16—312no—
153M16—34no—
154C2×C8—36no—
155C2×C8—34no—
156C2×C8—312no—
157C2×C8C2×C3×A416yes—
158C22×C3—31no—
159C22×C3—33no—
160C22×C3—49no—
161C22×C3—412no—
162C22×C3—66no—
163C22×C3—66no—
164C22×C3#131⋊C314yesdirect product factor
165C22×C3C2×C4×A413yes—
166C3×C4—33no—
167C3×C4—31no—
168C3×C4—49no—
169C3×C4—43no—
170C3×C4C2×C4×A413yes—
171C3×C4C23×A411yescharacteristic
172C32—41noSylow
173C23—33no—
174C23—33no—
175C23—64no—
176C23C3×C4×A411yescharacteristic
177Q8—33no—
178Q8—32no—
179Q8C22×C32×C411yescharacteristic
180D4—33no—
181D4—31no—
182D4—36no—
183D4—36no—
184C2×C4—33no—
185C2×C4—33no—
186C2×C4—33no—
187C2×C4—33no—
188C2×C4—36no—
189C2×C4—31no—
190C2×C4—33no—
191C2×C4C3×C4×A413yes—
192C2×C4C22×C3×A413yes—
193C8—34no—
194C8C22×C3×A414yes—
195C2×C3—43no—
196C2×C3—418no—
197C2×C3—61no—
198C2×C3—63no—
199C2×C3C2×#131⋊C316yesdirect product factor
200C2×C3C22×C4×A411yescharacteristic
201C22—33no—
202C22—31no—
203C22—66no—
204C22—66no—
205C22C3×#131⋊C314yesdirect product factor
206C22C2×C3×C4×A413yes—
207C4—31no—
208C4—33no—
209C4C2×C3×C4×A413yes—
210C4C23×C3×A411yescharacteristic
211C3—43no—
212C3C22×#131⋊C311yescharacteristic, direct product factor
213C2—63no—
214C2—61no—
215C2C2×C3×#131⋊C316yesdirect product factor
216C2C22×C3×C4×A411yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C8A4
commutator, hypocenterQ8C22×C32×C4
FrattiniC4C23×C3×A4
FittingC22×C3×#131C3
socleC23×C3C4×A4
prime radicalC22×C3×#41⋊C3C2
cosocle#41C23×C32
LCSG ▷ Q8
UCSC1 ◁ C22×C3×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#131⋊C3C2direct (N × H)
3. C22×C3×#131C3semidirect
4. C22×#131⋊C3C3direct (N × H)
9. C3×#131⋊C3C22direct (N × H)
13. C2×C3×#131C2×C3semidirect
16. C2×#131⋊C3C2×C3direct (N × H)
22. C22×#131C32semidirect
35. C3×#131C22×C3semidirect
43. #131⋊C3C22×C3direct (N × H)
51. C2×#131C2×C32semidirect
97. #131C22×C32semidirect
164. C22×C3#131⋊C3direct (N × H)
199. C2×C3C2×#131⋊C3direct (N × H)
205. C22C3×#131⋊C3direct (N × H)
212. C3C22×#131⋊C3direct (N × H)
215. C2C2×C3×#131⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3×#131⋊C32
C22×C3×#1313
C22×#131⋊C33
C22×C32×C84