Finite Group Databasebeta

← C22×C3⋊(#131⋊C3) #11923373

Subgroups — C22×C3⋊(#131⋊C3) #11923373

Order 1152.

761 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 151 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⋊C3)C216yesmaximal, direct product factor
3C22×#948C311yesmaximal, characteristic, complemented
4C22×#131⋊C3—31nomaximal
5C2×C3×#41⋊C3C413yes—
6C2×C3×#41⋊C3C2213yes—
7C22×C3×SL(2,𝔽3)C411yescharacteristic
8C22×C3×C3⋊C8—41nomaximal
9C3⋊(#131⋊C3)C2214yesdirect product factor
10C22×C3×#41C2×C311yescharacteristic
11C22×#41⋊C3—21no—
12C22×#41⋊C3D311yescharacteristic
13C2×#948—36no—
14C2×#948C2×C316yescomplemented
15C22×#208—31no—
16C23×C3⋊C8—31no—
17C2×#131⋊C3—36no—
18C22×C32×C4—41no—
19C3×#41⋊C3C2×C413yes—
20C3×#41⋊C3C2311yescharacteristic
21C2×C3×SL(2,𝔽3)C2×C413yes—
22C2×C3×C3⋊C8—46no—
23C22×#131—31noSylow
24C2×C3×#41—33no—
25C2×C3×#41—33no—
26C2×C3×#41C3×C413yes—
27C2×C3×#41C22×C313yes—
28C22×C3×Q8C3×C411yescharacteristic
29C22×C3×D4—31no—
30C23×C3×C4—31no—
31C2×#41⋊C3—23no—
32C2×#41⋊C3—23no—
33C2×#41⋊C3Dic313yes—
34C2×#41⋊C3C2×D313yes—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)Dic311yescharacteristic
37C22×C3×C8—121no—
38#948—38no—
39#948—312no—
40#948C22×C314yescomplemented
41C2×#208—36no—
42C2×#208—36no—
43C22×C3⋊C8—31no—
44C22×C3⋊C8—36no—
45C22×C3⋊C8—36no—
46C22×C3⋊C8A411yescharacteristic
47#131⋊C3—34no—
48C23×C32—41no—
49C2×C32×C4—43no—
50C2×C32×C4—43no—
51C3×SL(2,𝔽3)C22×C411yescharacteristic
52C3×C3⋊C8—44no—
53C22×#41C3×D311yescharacteristic
54C2×#131—36no—
55C2×#131—96no—
56C22×M16—91no—
57C23×C8—91no—
58C24×C3—31no—
59C3×#41—32no—
60C3×#41—36no—
61C3×#41—33no—
62C3×#41—36no—
63C3×#41—33no—
64C3×#41C2×C3×C413yes—
65C3×#41C23×C311yescharacteristic
66C2×C3×Q8—33no—
67C2×C3×Q8C2×C3×C413yes—
68C2×C3×D4—33no—
69C2×C3×D4—33no—
70C2×C3×D4—36no—
71C22×C3×C4—31no—
72C22×C3×C4—33no—
73C22×C3×C4—33no—
74C22×C3×C4—33no—
75C22×C3×C4—33no—
76C22×C3×C4—41no—
77C22×C3×C4—81no—
78C22×C3×C4C2×A411yescharacteristic
79#41⋊C3—23no—
80#41⋊C3—21no—
81#41⋊C3C2×Dic313yes—
82#41⋊C3C22×D311yescharacteristic
83C2×SL(2,𝔽3)—23no—
84C2×SL(2,𝔽3)C2×Dic313yes—
85C2×C3×C8—126no—
86#208—312no—
87#208—34no—
88C2×C3⋊C8—36no—
89C2×C3⋊C8—312no—
90C2×C3⋊C8—34no—
91C2×C3⋊C8C2×A416yes—
92C22×C32—44no—
93C22×C32—43no—
94C32×C4—43no—
95C32×C4—41no—
96C2×#41—33no—
97C2×#41—33no—
98C2×#41C3×Dic313yes—
99C2×#41C2×C3×D313yes—
100C22×Q8C3×Dic311yescharacteristic
101C22×D4—31no—
102C23×C4—31no—
103#131—34no—
104#131—98no—
105#131—912no—
106C2×M16—96no—
107C2×M16—96no—
108C22×C8—31no—
109C22×C8—96no—
110C22×C8—96no—
111C22×C8—91no—
112C23×C3—33no—
113C23×C3—33no—
114C23×C3—41no—
115C23×C3—64no—
116C23×C3—81no—
117C23×C3C4×A411yescharacteristic
118C3×Q8—33no—
119C3×Q8—32no—
120C3×Q8C22×C3×C411yescharacteristic
121C3×D4—33no—
122C3×D4—31no—
123C3×D4—36no—
124C3×D4—36no—
125C2×C3×C4—33no—
126C2×C3×C4—33no—
127C2×C3×C4—33no—
128C2×C3×C4—36no—
129C2×C3×C4—33no—
130C2×C3×C4—33no—
131C2×C3×C4—31no—
132C2×C3×C4—43no—
133C2×C3×C4—43no—
134C2×C3×C4—83no—
135C2×C3×C4—83no—
136C2×C3×C4C4×A413yes—
137C2×C3×C4C22×A413yes—
138SL(2,𝔽3)—21no—
139SL(2,𝔽3)C22×Dic311yescharacteristic
140C3×C8—124no—
141C3⋊C8—34no—
142C3⋊C8C22×A414yes—
143C2×C32—41no—
144C2×C32—46no—
145C24—31no—
146#41—32no—
147#41—36no—
148#41—33no—
149#41—36no—
150#41—33no—
151#41C2×C3×Dic313yes—
152#41C22×C3×D311yescharacteristic
153C2×Q8—33no—
154C2×Q8C2×C3×Dic313yes—
155C2×D4—33no—
156C2×D4—33no—
157C2×D4—36no—
158C22×C4—31no—
159C22×C4—33no—
160C22×C4—33no—
161C22×C4—33no—
162C22×C4—33no—
163C22×C4D3×A411yescharacteristic
164M16—912no—
165M16—94no—
166C2×C8—36no—
167C2×C8—94no—
168C2×C8—912no—
169C2×C8—96no—
170C22×C3—31no—
171C22×C3—33no—
172C22×C3—44no—
173C22×C3—43no—
174C22×C3—66no—
175C22×C3—66no—
176C22×C3—84no—
177C22×C3—83no—
178C22×C3#131⋊C314yescomplemented
179C22×C3C2×C4×A413yes—
180C3×C4—33no—
181C3×C4—31no—
182C3×C4—43no—
183C3×C4—41no—
184C3×C4—83no—
185C3×C4—81no—
186C3×C4C2×C4×A413yes—
187C3×C4C23×A411yescharacteristic
188C32—41noSylow
189C23—33no—
190C23—33no—
191C23—64no—
192C23Dic3×A411yescharacteristic
193Q8—33no—
194Q8—32no—
195Q8C22×C3×Dic311yescharacteristic
196D4—33no—
197D4—31no—
198D4—36no—
199D4—36no—
200C2×C4—33no—
201C2×C4—33no—
202C2×C4—33no—
203C2×C4—33no—
204C2×C4—36no—
205C2×C4—31no—
206C2×C4—33no—
207C2×C4Dic3×A413yes—
208C2×C4C2×D3×A413yes—
209C8—34no—
210C8—94no—
211C2×C3—41no—
212C2×C3—46no—
213C2×C3—61no—
214C2×C3—63no—
215C2×C3—81no—
216C2×C3—86no—
217C2×C3C2×#131⋊C316yescomplemented
218C2×C3C22×C4×A411yescharacteristic
219C22—33no—
220C22—31no—
221C22—66no—
222C22—66no—
223C22C3⋊(#131⋊C3)14yesdirect product factor
224C22C2×Dic3×A413yes—
225C4—31no—
226C4—33no—
227C4C2×Dic3×A413yes—
228C4C22×D3×A411yescharacteristic
229C3—41no—
230C3—81no—
231C3C22×#131⋊C311yescharacteristic, complemented
232C2—63no—
233C2—61no—
234C2C2×C3⋊(#131⋊C3)16yesdirect product factor
235C2C22×Dic3×A411yescharacteristic
subgroupG/H
center, hypercenterC22×C4D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FrattiniC4C22×D3×A4
FittingC22×C3×#41C2×C3
socleC23×C3C4×A4
prime radicalC22×C3×#41⋊C3C2
cosocleC3×#41C23×C3
LCSG ▷ C3×Q8
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊(#131⋊C3)C2direct (N × H)
3. C22×#948C3semidirect
9. C3⋊(#131⋊C3)C22direct (N × H)
14. C2×#948C2×C3semidirect
40. #948C22×C3semidirect
178. C22×C3#131⋊C3semidirect
217. C2×C3C2×#131⋊C3semidirect
223. C22C3⋊(#131⋊C3)direct (N × H)
231. C3C22×#131⋊C3semidirect
234. C2C2×C3⋊(#131⋊C3)direct (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⋊C3)2
C22×#9483
C22×#131⋊C33
C22×C3×C3⋊C84