Finite Group Databasebeta

← C22×C3×#997 #11919898

Subgroups — C22×C3×#997 #11919898

Order 1152.

2608 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 2608 conjugacy classes of subgroups — too many to draw. Only its 112 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×#997C213yesmaximal, direct product factor
2C22×C3×H(5,𝔽2)C311yesmaximal, characteristic, complemented
3C22×#997C313yesmaximal, direct product factor
4C23×C3×A4—43nomaximal
5C3×#997C2211yescharacteristic, direct product factor
6C22×C3×SL(2,𝔽3)—42nomaximal
7C2×C3×H(5,𝔽2)—36no—
8C2×C3×H(5,𝔽2)—39no—
9C2×C3×H(5,𝔽2)C2×C313yescomplemented
10C22×C3×#41—32no—
11C23×C3×D4—33no—
12C2×#997C2×C319yesdirect product factor
13C22×C3×A4—412no—
14C22×C3×A4—49no—
15C2×C3×SL(2,𝔽3)—46no—
16C22×H(5,𝔽2)C3211yescharacteristic, complemented, Sylow
17C25×C3—31no—
18C25×C3A413yescomplemented
19C23×A4—49no—
20C3×H(5,𝔽2)—36no—
21C3×H(5,𝔽2)—318no—
22C3×H(5,𝔽2)—336no—
23C3×H(5,𝔽2)—36no—
24C3×H(5,𝔽2)—34no—
25C3×H(5,𝔽2)—36no—
26C3×H(5,𝔽2)—39no—
27C3×H(5,𝔽2)C22×C311yescharacteristic, complemented
28C2×C3×#41—318no—
29C2×C3×#41—318no—
30C2×C3×#41—36no—
31C2×C3×#41—36no—
32C22×C3×Q8A412yescomplemented
33C22×C3×D4—36no—
34C22×C3×D4—39no—
35C22×C3×D4—336no—
36C22×C3×D4—39no—
37C22×C3×D4—39no—
38C22×C3×D4—39no—
39C23×C3×C4—33no—
40#997C22×C313yesdirect product factor
41C22×SL(2,𝔽3)—46no—
42C23×C32—161no—
43C2×C3×A4—418no—
44C2×C3×A4—43no—
45C3×SL(2,𝔽3)—42no—
46C2×H(5,𝔽2)—36no—
47C2×H(5,𝔽2)—39no—
48C2×H(5,𝔽2)C2×C3213yescomplemented
49C22×#41—32no—
50C23×D4—33no—
51C24×C3—39no—
52C24×C3—33no—
53C24×C3—33no—
54C24×C3—39no—
55C24×C3—412no—
56C24×C3—124no—
57C24×C3C2×A419yescomplemented
58C22×A4—436no—
59C22×A4—427no—
60C3×#41—336no—
61C3×#41—318no—
62C3×#41—318no—
63C3×#41—336no—
64C3×#41—312no—
65C3×#41—36no—
66C3×#41—32no—
67C2×C3×Q8—36no—
68C2×C3×Q8C2×A416yescomplemented
69C2×C3×D4—336no—
70C2×C3×D4—318no—
71C2×C3×D4—318no—
72C2×C3×D4—336no—
73C2×C3×D4—336no—
74C2×C3×D4—336no—
75C2×C3×D4—318no—
76C2×C3×D4—39no—
77C2×C3×D4—336no—
78C2×C3×D4—39no—
79C2×C3×D4—39no—
80C2×C3×D4—33no—
81C22×C3×C4—39no—
82C22×C3×C4—318no—
83C22×C3×C4—39no—
84C22×C3×C4—32no—
85C2×SL(2,𝔽3)—418no—
86C22×C32—164no—
87C22×C32—163no—
88C3×A4—43no—
89C25—31no—
90C25C3×A413yescomplemented
91H(5,𝔽2)—36no—
92H(5,𝔽2)—318no—
93H(5,𝔽2)—336no—
94H(5,𝔽2)—36no—
95H(5,𝔽2)—34no—
96H(5,𝔽2)—36no—
97H(5,𝔽2)—39no—
98H(5,𝔽2)C22×C3211yescharacteristic, complemented
99C2×#41—318no—
100C2×#41—318no—
101C2×#41—36no—
102C2×#41—36no—
103C22×Q8C3×A412yescomplemented
104C22×D4—36no—
105C22×D4—39no—
106C22×D4—336no—
107C22×D4—39no—
108C22×D4—39no—
109C22×D4—39no—
110C23×C4—33no—
111C23×C3—36no—
112C23×C3—36no—
113C23×C3—39no—
114C23×C3—31no—
115C23×C3—39no—
116C23×C3—39no—
117C23×C3—39no—
118C23×C3—418no—
119C23×C3—612no—
120C23×C3—126no—
121C23×C3—1218no—
122C23×C3—1218no—
123C23×C3—163no—
124C23×C3C22×A413yescomplemented
125C23×C3C22⋊A411yescharacteristic
126C2×A4—454no—
127C2×A4—49no—
128C3×Q8—36no—
129C3×Q8—34no—
130C3×Q8C22×A412yescomplemented
131C3×D4—336no—
132C3×D4—336no—
133C3×D4—318no—
134C3×D4—36no—
135C2×C3×C4—318no—
136C2×C3×C4—39no—
137C2×C3×C4—318no—
138C2×C3×C4—33no—
139C2×C3×C4—36no—
140C2×C3×C4—36no—
141SL(2,𝔽3)—46no—
142C2×C32—166no—
143C2×C32—161no—
144C24—33no—
145C24—39no—
146C24—33no—
147C24—39no—
148C24—412no—
149C24—124no—
150C24C2×C3×A419yescomplemented
151#41—318no—
152#41—318no—
153#41—336no—
154#41—312no—
155#41—336no—
156#41—32no—
157#41—36no—
158C2×Q8—36no—
159C2×Q8C2×C3×A416yescomplemented
160C2×D4—318no—
161C2×D4—318no—
162C2×D4—336no—
163C2×D4—336no—
164C2×D4—336no—
165C2×D4—39no—
166C2×D4—336no—
167C2×D4—39no—
168C2×D4—318no—
169C2×D4—336no—
170C2×D4—33no—
171C2×D4—39no—
172C22×C4—39no—
173C22×C4—318no—
174C22×C4—39no—
175C22×C4—32no—
176C22×C3—39no—
177C22×C3—33no—
178C22×C3—43no—
179C22×C3—618no—
180C22×C3—618no—
181C22×C3—129no—
182C22×C3—121no—
183C22×C3—129no—
184C22×C3—126no—
185C22×C3—126no—
186C22×C3—1612no—
187C22×C3—169no—
188C22×C3#99714yesdirect product factor
189C22×C3C2×C22⋊A413yes—
190A4—49no—
191C3×C4—36no—
192C3×C4—32no—
193C32—161noSylow
194C23—31no—
195C23—39no—
196C23—36no—
197C23—36no—
198C23—39no—
199C23—39no—
200C23—39no—
201C23—418no—
202C23—612no—
203C23—126no—
204C23—1218no—
205C23—1218no—
206C23C22×C3×A413yescomplemented
207C23C3×C22⋊A411yescharacteristic
208Q8—36no—
209Q8—34no—
210Q8C22×C3×A412yescomplemented
211D4—36no—
212D4—318no—
213D4—336no—
214D4—336no—
215C2×C4—33no—
216C2×C4—318no—
217C2×C4—39no—
218C2×C4—318no—
219C2×C4—36no—
220C2×C4—36no—
221C2×C3—63no—
222C2×C3—69no—
223C2×C3—1618no—
224C2×C3—163no—
225C2×C3C2×#99716yesdirect product factor
226C2×C3C22×C22⋊A411yescharacteristic
227C22—33no—
228C22—39no—
229C22—43no—
230C22—618no—
231C22—618no—
232C22—129no—
233C22—121no—
234C22—129no—
235C22—126no—
236C22—126no—
237C22C3×#99714yesdirect product factor
238C22C2×C3×C22⋊A413yes—
239C4—32no—
240C4—36no—
241C3—163no—
242C3C22×#99711yescharacteristic, direct product factor
243C2—63no—
244C2—69no—
245C2C2×C3×#99716yesdirect product factor
246C2C22×C3×C22⋊A411yescharacteristic
subgroupG/H
center, socle, hypercenterC23×C3C22⋊A4
commutator, cosocle, hypocenterH(5,𝔽2)C22×C32
FrattiniC2C22×C3×C22⋊A4
FittingC22×C3×H(5,𝔽2)C3
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#997C2direct (N × H)
2. C22×C3×H(5,𝔽2)C3semidirect
3. C22×#997C3direct (N × H)
5. C3×#997C22direct (N × H)
9. C2×C3×H(5,𝔽2)C2×C3semidirect
12. C2×#997C2×C3direct (N × H)
16. C22×H(5,𝔽2)C32semidirect
18. C25×C3A4semidirect
27. C3×H(5,𝔽2)C22×C3semidirect
32. C22×C3×Q8A4semidirect
40. #997C22×C3direct (N × H)
48. C2×H(5,𝔽2)C2×C32semidirect
57. C24×C3C2×A4semidirect
68. C2×C3×Q8C2×A4semidirect
90. C25C3×A4semidirect
98. H(5,𝔽2)C22×C32semidirect
103. C22×Q8C3×A4semidirect
124. C23×C3C22×A4semidirect
130. C3×Q8C22×A4semidirect
150. C24C2×C3×A4semidirect
159. C2×Q8C2×C3×A4semidirect
188. C22×C3#997direct (N × H)
206. C23C22×C3×A4semidirect
210. Q8C22×C3×A4semidirect
225. C2×C3C2×#997direct (N × H)
237. C22C3×#997direct (N × H)
242. C3C22×#997direct (N × H)
245. C2C2×C3×#997direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9972
C22×C3×H(5,𝔽2)3
C22×#9973
C23×C3×A44
C22×C3×SL(2,𝔽3)4