Finite Group Databasebeta

← C3×C4×#994 #11919686

Subgroups — C3×C4×#994 #11919686

Order 1152.

884 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 226 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×#994C211yesmaximal, characteristic
2C3×#520⋊C3C213yesmaximal, complemented
3C3×C4×#41⋊C3C213yesmaximal, complemented
4C3×C4×H(5,𝔽2)C311yesmaximal, characteristic, complemented
5C4×#994C313yesmaximal, direct product factor
6C3×#994C414yesdirect product factor
7C2×C3×#41⋊C3C2213yes—
8C32×C4×Q8—41nomaximal
9C3×C4×SL(2,𝔽3)C2211yescharacteristic, complemented
10C3×C4×SL(2,𝔽3)C2213yescomplemented
11C2×C3×H(5,𝔽2)C2×C311yescharacteristic
12C2×#994C2×C313yes—
13C3×#520—31no—
14C3×#520C2×C313yes—
15C3×#518—33no—
16C3×C4×#41—31no—
17C3×C4×#41C2×C313yes—
18C2×C3×C4×D4—33no—
19#520⋊C3C2×C319yescomplemented
20C4×#41⋊C3C2×C319yescomplemented
21C2×C32×Q8—41no—
22C3×#41⋊C3C2×C416yescomplemented
23C2×C3×SL(2,𝔽3)C2311yescharacteristic
24C32×C4⋊C4—43no—
25C32×C42—43no—
26C4×H(5,𝔽2)C3211yescharacteristic, complemented, Sylow
27C3×H(5,𝔽2)—34no—
28C3×H(5,𝔽2)C3×C414yescomplemented
29C2×C3×#41—31no—
30C2×C3×#41C22×C313yes—
31C22×C3×D4—33no—
32C23×C3×C4—32no—
33#994C3×C4112yesdirect product factor
34C2×#41⋊C3C22×C319yes—
35C3×C4×Q8—31no—
36C3×C4×Q8—43no—
37C3×C4×Q8A411yescharacteristic
38C3×C4×Q8C22×C311yescharacteristic
39C3×C4×Q8C22×C313yes—
40C3×C4×D4—36no—
41C3×C4×D4—33no—
42C3×C4×D4—33no—
43C3×C4×D4—33no—
44C3×C4×D4—36no—
45C3×C4×D4—33no—
46C3×#117—33no—
47C3×#117—33no—
48C2×C3×C4⋊C4—33no—
49C2×C3×C22⋊C4—36no—
50C2×C3×C42—33no—
51C4×SL(2,𝔽3)C22×C313yescomplemented
52C4×SL(2,𝔽3)C22×C319yescomplemented
53C32×Q8—44no—
54C2×C32×C4—43no—
55C2×C32×C4—43no—
56C2×C32×C4—41no—
57C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
58C2×H(5,𝔽2)C2×C3211yescharacteristic
59#520—31no—
60#520C2×C3213yes—
61#518—33no—
62C4×#41—31no—
63C4×#41C2×C3213yes—
64C2×C4×D4—33no—
65C24×C3—32no—
66C3×#41—34no—
67C3×#41—34no—
68C3×#41—36no—
69C3×#41C2×C3×C416yes—
70C2×C3×Q8—43no—
71C2×C3×Q8C2×A411yescharacteristic
72C2×C3×Q8C23×C311yescharacteristic
73C2×C3×D4—33no—
74C2×C3×D4—312no—
75C2×C3×D4—312no—
76C2×C3×D4—33no—
77C22×C3×C4—33no—
78C22×C3×C4—36no—
79C22×C3×C4—33no—
80C22×C3×C4—33no—
81C22×C3×C4—124no—
82#41⋊C3C2×C3×C4118yescomplemented
83C2×SL(2,𝔽3)C23×C313yes—
84C3×C4⋊C4—33no—
85C3×C4⋊C4—31no—
86C3×C4⋊C4—33no—
87C3×C4⋊C4—49no—
88C3×C4⋊C4C2×A413yes—
89C3×C22⋊C4—36no—
90C3×C22⋊C4—36no—
91C3×C42—33no—
92C3×C42—33no—
93C3×C42—31no—
94C3×C42—49no—
95C3×C42C2×A413yes—
96C22×C32—41no—
97C32×C4—46no—
98C32×C4—42no—
99C32×C4—83no—
100H(5,𝔽2)—34no—
101H(5,𝔽2)C32×C414yescomplemented
102C2×#41—31no—
103C2×#41C22×C3213yes—
104C22×D4—33no—
105C23×C4—32no—
106C4×Q8—31no—
107C4×Q8C3×A411yescharacteristic
108C4×Q8C22×C3211yescharacteristic
109C4×Q8C22×C3213yes—
110C4×D4—36no—
111C4×D4—33no—
112C4×D4—33no—
113C4×D4—33no—
114C4×D4—36no—
115C4×D4—33no—
116#117—33no—
117#117—33no—
118C2×C4⋊C4—33no—
119C2×C22⋊C4—36no—
120C2×C42—33no—
121C23×C3—38no—
122C23×C3—33no—
123C23×C3—122no—
124C23×C3—122no—
125C3×Q8—31no—
126C3×Q8—412no—
127C3×Q8C4×A414yes—
128C3×Q8C22×C3×C411yescharacteristic
129C3×D4—36no—
130C3×D4—36no—
131C3×D4—312no—
132C2×C3×C4—31no—
133C2×C3×C4—36no—
134C2×C3×C4—36no—
135C2×C3×C4—31no—
136C2×C3×C4—33no—
137C2×C3×C4—43no—
138C2×C3×C4—49no—
139C2×C3×C4—49no—
140C2×C3×C4—66no—
141C2×C3×C4—123no—
142C2×C3×C4—126no—
143C2×C3×C4C22×A413yes—
144C2×C3×C4C22×A413yes—
145C2×C3×C4C22×A411yescharacteristic
146SL(2,𝔽3)C22×C3×C413yescomplemented
147C2×C32—41no—
148C2×C32—41no—
149C2×C32—41no—
150C24—32no—
151#41—34no—
152#41—34no—
153#41—36no—
154#41C2×C32×C416yes—
155C2×Q8C2×C3×A411yescharacteristic
156C2×Q8C23×C3211yescharacteristic
157C2×D4—33no—
158C2×D4—312no—
159C2×D4—312no—
160C2×D4—33no—
161C22×C4—33no—
162C22×C4—36no—
163C22×C4—33no—
164C22×C4—33no—
165C22×C4—124no—
166C4⋊C4—33no—
167C4⋊C4—31no—
168C4⋊C4—33no—
169C4⋊C4C2×C3×A413yes—
170C22⋊C4—36no—
171C22⋊C4—36no—
172C42—31no—
173C42—33no—
174C42—33no—
175C42C2×C3×A413yes—
176C22×C3—36no—
177C22×C3—43no—
178C22×C3—63no—
179C22×C3—63no—
180C22×C3—128no—
181C22×C3C23×A411yescharacteristic
182C3×C4—23no—
183C3×C4—31no—
184C3×C4—31no—
185C3×C4—418no—
186C3×C4—46no—
187C3×C4—63no—
188C3×C4—61no—
189C3×C4—89no—
190C3×C4C2×C4×A416yes—
191C3×C4#99412yesdirect product factor
192C32—41noSylow
193C23—38no—
194C23—33no—
195C23—122no—
196C23—122no—
197Q8—31no—
198Q8C3×C4×A414yes—
199Q8C22×C32×C411yescharacteristic
200D4—36no—
201D4—36no—
202D4—312no—
203C2×C4—31no—
204C2×C4—36no—
205C2×C4—36no—
206C2×C4—31no—
207C2×C4—33no—
208C2×C4—66no—
209C2×C4—123no—
210C2×C4—126no—
211C2×C4C22×C3×A413yes—
212C2×C4C22×C3×A413yes—
213C2×C4C22×C3×A411yescharacteristic
214C2×C3—43no—
215C2×C3—43no—
216C2×C3—43no—
217C2×C3—66no—
218C2×C3C22×C4×A411yescharacteristic
219C2×C3C2×#99411yescharacteristic
220C2×C3#594511yescharacteristic
221C22—36no—
222C22—63no—
223C22—63no—
224C22—128no—
225C22C23×C3×A411yescharacteristic
226C4—23no—
227C4—31no—
228C4—31no—
229C4—63no—
230C4—61no—
231C4C2×C3×C4×A416yes—
232C4C3×#99412yesdirect product factor
233C3—43no—
234C3C4×#99411yescharacteristic, direct product factor
235C2—66no—
236C2C22×C3×C4×A411yescharacteristic
237C2C2×C3×#99411yescharacteristic
238C2C3×#594511yescharacteristic
subgroupG/H
centerC2×C3×C4C22×A4
commutator, hypocenterQ8C22×C32×C4
FrattiniC22C23×C3×A4
FittingC3×C4×H(5,𝔽2)C3
socleC22×C3C23×A4
prime radicalC2×C3×#994C2
cosocleC2×Q8C23×C32
hypercenterC3×C4×Q8A4
LCSG ▷ Q8
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. C3×C4×#41⋊C3C2semidirect
4. C3×C4×H(5,𝔽2)C3semidirect
5. C4×#994C3direct (N × H)
6. C3×#994C4direct (N × H)
9. C3×C4×SL(2,𝔽3)C22semidirect
10. C3×C4×SL(2,𝔽3)C22semidirect
19. #520⋊C3C2×C3semidirect
20. C4×#41⋊C3C2×C3semidirect
22. C3×#41⋊C3C2×C4semidirect
26. C4×H(5,𝔽2)C32semidirect
28. C3×H(5,𝔽2)C3×C4semidirect
33. #994C3×C4direct (N × H)
51. C4×SL(2,𝔽3)C22×C3semidirect
52. C4×SL(2,𝔽3)C22×C3semidirect
57. C3×SL(2,𝔽3)C22×C4semidirect
82. #41⋊C3C2×C3×C4semidirect
101. H(5,𝔽2)C32×C4semidirect
146. SL(2,𝔽3)C22×C3×C4semidirect
191. C3×C4#994direct (N × H)
232. C4C3×#994direct (N × H)
234. C3C4×#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
C3×#520⋊C32
C3×C4×#41⋊C32
C3×C4×H(5,𝔽2)3
C4×#9943
C32×C4×Q84