Finite Group Databasebeta

← C3×#71951 #11798741

Subgroups — C3×#71951 #11798741

Order 1152.

626 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C32×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2C3×#5062C213yesmaximal, complemented
3C3×#4542C213yesmaximal, complemented
4C3×#2202—31nomaximal
5#71951C311yesmaximal, characteristic, direct product factor
6C3×#1007C2211yescharacteristic, complemented
7C32×#124—23no—
8C32×#124C2213yes—
9C32×#120—23no—
10C3×#932—23no—
11C3×#836C2213yescomplemented
12C3×#805—23no—
13C3×H(3,𝔽4)—21no—
14C3×H(3,𝔽4)D311yescharacteristic, complemented
15C3×H(3,𝔽4)C2×C311yescharacteristic, complemented
16C3×#454—33no—
17#5062C2×C313yescomplemented
18C3×#355—33no—
19#4542C2×C313yescomplemented
20C24×C32—21no—
21C2×C32×Q8—21no—
22C2×C32×Q8C2311yescharacteristic
23C2×C32×D4—23no—
24C2×C32×D4D413yes—
25C3×#239—41no—
26C3×D3×Q8—41no—
27C3×#235—23no—
28C32×C22⋊C4—23no—
29C32×C22⋊C4—26no—
30C32×C42D413yes—
31C3×#214—43no—
32C3×#213—43no—
33C3×#208—23no—
34#2202—31noSylow
35C3×2−1+4—31no—
36#1007C22×C311yescharacteristic, complemented
37C3×AGL(1,ℤ/8)—63no—
38C3×#124—26no—
39C3×#124—23no—
40C3×#124—23no—
41C3×#124—23no—
42C3×#124C2×D313yes—
43C3×#124C22×C313yes—
44C3×#120—26no—
45C3×#120—23no—
46C3×#120—23no—
47#932—23no—
48C3×#104—63no—
49C3×#101—33no—
50#836C22×C313yescomplemented
51#805—23no—
52C23×C32—23no—
53C23×C32—82no—
54C23×C32—81no—
55C32×Q8—41no—
56C32×Q8—81no—
57C32×D4—43no—
58C32×D4—83no—
59C2×C32×C4—23no—
60C2×C32×C4C2×D413yes—
61C3×#67—41no—
62C3×D12—23no—
63C3×C4×D3—43no—
64C3×Dic6—23no—
65C3×C3⋊C8—43no—
66H(3,𝔽4)C3×D311yescharacteristic, complemented
67#454—33no—
68#355—33no—
69C24×C3—22no—
70C24×C3—21no—
71C24×C3—21no—
72C3×#41—63no—
73C3×#41—121no—
74C2×C3×Q8—22no—
75C2×C3×Q8—21no—
76C2×C3×Q8—21no—
77C2×C3×Q8—21no—
78C2×C3×Q8—121no—
79C2×C3×Q8C22×D311yescharacteristic
80C2×C3×Q8C23×C311yescharacteristic
81C2×C3×D4—26no—
82C2×C3×D4—23no—
83C2×C3×D4—23no—
84C2×C3×D4—23no—
85C2×C3×D4#6713yes—
86C2×C3×D4C3×D413yes—
87#239—41no—
88D3×Q8—41no—
89#235—23no—
90C3×SD16—123no—
91C3×D8—123no—
92C3×M16—63no—
93C3×C22⋊C4—26no—
94C3×C22⋊C4—23no—
95C3×C22⋊C4—23no—
96C3×C22⋊C4—212no—
97C3×C22⋊C4—26no—
98C3×C22⋊C4—26no—
99C3×C42—23no—
100C3×C42#6713yes—
101C3×C42C3×D413yes—
102#214—43no—
103#213—43no—
104#208—23no—
105C22×C32—46no—
106C22×C32—43no—
107C22×C32—84no—
108C22×C32#12011yescharacteristic
109C2×C3×D3—41no—
110C32×C4—23no—
111C32×C4—43no—
112C3×Dic3—41no—
1132−1+4—31no—
114AGL(1,ℤ/8)—63no—
115#124—23no—
116#124C2×C3×D313yes—
117#120—23no—
118#104—63no—
119#101—33no—
120C23×C3—26no—
121C23×C3—23no—
122C23×C3—23no—
123C23×C3—84no—
124C23×C3—82no—
125C23×C3—82no—
126C23×C3—82no—
127C23×C3—81no—
128C23×C3—81no—
129C3×Q8—41no—
130C3×Q8—41no—
131C3×Q8—63no—
132C3×Q8—81no—
133C3×Q8—82no—
134C3×Q8—81no—
135C3×Q8—81no—
136C3×D4—43no—
137C3×D4—43no—
138C3×D4—63no—
139C3×D4—83no—
140C3×D4—86no—
141C3×D4—83no—
142C3×D4—83no—
143C3×D4—121no—
144C2×C3×C4—26no—
145C2×C3×C4—23no—
146C2×C3×C4—23no—
147C2×C3×C4—23no—
148C2×C3×C4—123no—
149C2×C3×C4C2×#6713yes—
150C2×C3×C4C2×C3×D413yes—
151#67—41no—
152D12—23no—
153C4×D3—43no—
154Dic6—23no—
155C3×C8—123no—
156C3⋊C8—43no—
157C2×C32—21no—
158C2×C32—83no—
159C2×C32B(4,𝔽2)11yescharacteristic
160C3×D3—81no—
161C24—21no—
162#41—63no—
163#41—121no—
164C2×Q8—21no—
165C2×Q8—121no—
166C2×Q8C22×C3×D311yescharacteristic
167C2×D4—23no—
168C2×D4C3×#6713yes—
169SD16—123no—
170D8—123no—
171M16—63no—
172C22⋊C4—23no—
173C22⋊C4—26no—
174C42C3×#6713yes—
175C22×C3—21no—
176C22×C3—412no—
177C22×C3—46no—
178C22×C3—46no—
179C22×C3—46no—
180C22×C3—43no—
181C22×C3—43no—
182C22×C3—88no—
183C22×C3—84no—
184C22×C3—84no—
185C22×C3—121no—
186C22×C3#95311yescharacteristic
187C22×C3C3×#12011yescharacteristic
188C2×D3—41no—
189C3×C4—23no—
190C3×C4—23no—
191C3×C4—43no—
192C3×C4—46no—
193C3×C4—43no—
194C3×C4—43no—
195C3×C4—121no—
196Dic3—41no—
197C32#220211yescharacteristic, complemented, Sylow
198C23—23no—
199C23—82no—
200C23—81no—
201Q8—41no—
202Q8—63no—
203Q8—81no—
204D4—43no—
205D4—63no—
206D4—83no—
207D4—121no—
208C2×C4—23no—
209C2×C4—123no—
210C2×C4C2×C3×#6713yes—
211C8—123no—
212C2×C3—22no—
213C2×C3—21no—
214C2×C3—21no—
215C2×C3—21no—
216C2×C3—86no—
217C2×C3—83no—
218C2×C3—83no—
219C2×C3—241no—
220C2×C3#524411yescharacteristic
221C2×C3C3×B(4,𝔽2)11yescharacteristic
222D3—81no—
223C22—46no—
224C22—43no—
225C22—84no—
226C22—121no—
227C22C3×#95311yescharacteristic
228C4—23no—
229C4—43no—
230C4—121no—
231C3—21no—
232C3#7195111yescharacteristic, direct product factor
233C3C3×#220211yescharacteristic, complemented
234C2—21no—
235C2—83no—
236C2—241no—
237C2C3×#524411yescharacteristic
subgroupG/H
centerC2×C3#5244
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC2×Q8C22×C3×D3
FittingC32×H(3,𝔽4)C2
socleC2×C32B(4,𝔽2)
hypercenterC3×H(3,𝔽4)D3
hypocenterC3C3×#2202
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C32×H(3,𝔽4)C2semidirect
2. C3×#5062C2semidirect
3. C3×#4542C2semidirect
5. #71951C3direct (N × H)
6. C3×#1007C22semidirect
11. C3×#836C22semidirect
14. C3×H(3,𝔽4)D3semidirect
15. C3×H(3,𝔽4)C2×C3semidirect
17. #5062C2×C3semidirect
19. #4542C2×C3semidirect
36. #1007C22×C3semidirect
50. #836C22×C3semidirect
66. H(3,𝔽4)C3×D3semidirect
197. C32#2202semidirect
232. C3#71951direct (N × H)
233. C3C3×#2202semidirect

Maximal subgroups

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

subgroupindex
C32×H(3,𝔽4)2
C3×#50622
C3×#45422
C3×#22023
#719513