Finite Group Databasebeta

← C3×SD16×SL(2,𝔽3) #11919606

Subgroups — C3×SD16×SL(2,𝔽3) #11919606

Order 1152.

358 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C3×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×Q8×SD16C311yesmaximal, characteristic, complemented
5SD16×SL(2,𝔽3)C313yesmaximal, direct product factor
6C22×C3×SL(2,𝔽3)—21no—
7C2×C32×SD16—41nomaximal
8C3×C4×SL(2,𝔽3)—21no—
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10C3×Q82C2×C311yescharacteristic, complemented
11C3×D4×Q8C2×C311yescharacteristic
12Q8×SL(2,𝔽3)C2×C313yescomplemented
13D4×SL(2,𝔽3)C2×C313yes—
14C3×#498—31no—
15C3×#476—31no—
16C3×#475—31no—
17C3×C8×Q8C2×C311yescharacteristic, complemented
18C3×C4×SD16—31no—
19C8×SL(2,𝔽3)C2×C313yescomplemented
20C2×C32×Q8—41no—
21C2×C32×D4—41no—
22C2×C3×SL(2,𝔽3)—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C32×SD16—44no—
25C2×C32×C8—41no—
26Q8×SD16C3211yescharacteristic, complemented, Sylow
27C22×C3×Q8—21no—
28C22×SL(2,𝔽3)—23no—
29C2×C3×SD16—43no—
30C2×C3×SD16A411yescharacteristic
31C3×C4⋊Q8—31no—
32C3×C4⋊Q8—31no—
33C3×C4⋊Q8—61no—
34C3×C22⋊Q8—61no—
35C3×C4×Q8—21no—
36C3×C4×Q8—31no—
37C3×C4×Q8C22×C311yescharacteristic
38C3×C4×D4—31no—
39C4×SL(2,𝔽3)—23no—
40C4×SL(2,𝔽3)C22×C313yes—
41C3×#106—32no—
42C3×#106—31no—
43C3×C4⋊C8—31no—
44C3×#103—31no—
45C3×#102—31no—
46C3×C4×C8—31no—
47C23×C32—81no—
48C32×Q8—42no—
49C32×Q8—81no—
50C32×D4—42no—
51C32×D4—81no—
52C2×C32×C4—41no—
53C2×C32×C4—81no—
54C3×SL(2,𝔽3)SD1611yescharacteristic, direct product factor
55C32×C8—42no—
56Q82C2×C3211yescharacteristic, complemented
57D4×Q8C2×C3211yescharacteristic
58#498—31no—
59#476—31no—
60#475—31no—
61C8×Q8C2×C3211yescharacteristic, complemented
62C4×SD16—31no—
63C2×C3×Q8—41no—
64C2×C3×Q8—43no—
65C2×C3×Q8—61no—
66C2×C3×Q8—61no—
67C2×C3×Q8—121no—
68C2×C3×Q8C3×D411yescharacteristic
69C2×C3×Q8C2×A411yescharacteristic
70C2×C3×D4—43no—
71C2×C3×D4C2×A411yescharacteristic
72C22×C3×C4—61no—
73C2×SL(2,𝔽3)—43no—
74C2×SL(2,𝔽3)C3×D413yes—
75C3×SD16—412no—
76C3×SD16SL(2,𝔽3)14yesdirect product factor
77C2×C3×C8—31no—
78C2×C3×C8—43no—
79C2×C3×C8C2×A411yescharacteristic
80C3×C4⋊C4—31no—
81C3×C4⋊C4—31no—
82C3×C4⋊C4—31no—
83C3×C4⋊C4—61no—
84C3×C4⋊C4—61no—
85C3×C4⋊C4—61no—
86C3×C22⋊C4—61no—
87C3×C42—31no—
88C3×C42—61no—
89C22×C32—41no—
90C22×C32—82no—
91C22×C32—161no—
92C22×C32—161no—
93C32×C4—41no—
94C32×C4—41no—
95C32×C4—82no—
96C22×Q8—21no—
97C2×SD16C3×A411yescharacteristic
98C4⋊Q8—31no—
99C4⋊Q8—31no—
100C4⋊Q8—61no—
101C22⋊Q8—61no—
102C4×Q8—21no—
103C4×Q8—31no—
104C4×Q8C22×C3211yescharacteristic
105C4×D4—31no—
106#106—32no—
107#106—31no—
108C4⋊C8—31no—
109#103—31no—
110#102—31no—
111C4×C8—31no—
112C23×C3—21no—
113C23×C3—83no—
114C3×Q8—21no—
115C3×Q8—31no—
116C3×Q8—46no—
117C3×Q8—83no—
118C3×Q8—121no—
119C3×Q8—121no—
120C3×Q8—241no—
121C3×Q8C3×SD1611yescharacteristic, complemented
122C3×Q8C2×SL(2,𝔽3)12yescomplemented
123C3×D4—21no—
124C3×D4—46no—
125C3×D4—83no—
126C3×D4C2×SL(2,𝔽3)12yes—
127C2×C3×C4—21no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—43no—
131C2×C3×C4—61no—
132C2×C3×C4—61no—
133C2×C3×C4—83no—
134C2×C3×C4—121no—
135C2×C3×C4—121no—
136C2×C3×C4C22×A411yescharacteristic
137SL(2,𝔽3)C3×SD1613yesdirect product factor
138C3×C8—46no—
139C3×C8—61no—
140C3×C8C2×SL(2,𝔽3)12yescomplemented
141C2×C32—41no—
142C2×C32—41no—
143C2×C32—41no—
144C2×C32—162no—
145C2×Q8—41no—
146C2×Q8—61no—
147C2×Q8—61no—
148C2×Q8—121no—
149C2×Q8C32×D411yescharacteristic
150C2×Q8C2×C3×A411yescharacteristic
151C2×D4C2×C3×A411yescharacteristic
152C22×C4—61no—
153SD16C3×SL(2,𝔽3)14yesdirect product factor
154C2×C8—31no—
155C2×C8C2×C3×A411yescharacteristic
156C4⋊C4—31no—
157C4⋊C4—31no—
158C4⋊C4—31no—
159C4⋊C4—61no—
160C4⋊C4—61no—
161C4⋊C4—61no—
162C22⋊C4—61no—
163C42—31no—
164C42—61no—
165C22×C3—22no—
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—43no—
169C22×C3—86no—
170C22×C3—163no—
171C22×C3—163no—
172C22×C3D4×A411yescharacteristic
173C3×C4—22no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—43no—
177C3×C4—43no—
178C3×C4—61no—
179C3×C4—86no—
180C3×C4—121no—
181C3×C4—121no—
182C3×C4C22×SL(2,𝔽3)11yescharacteristic
183C3×C42−1+4⋊C311yescharacteristic
184C32—41noSylow
185C23—21no—
186Q8—21no—
187Q8—31no—
188Q8—121no—
189Q8—121no—
190Q8—241no—
191Q8C32×SD1611yescharacteristic, complemented
192Q8C2×C3×SL(2,𝔽3)12yescomplemented
193D4—21no—
194D4C2×C3×SL(2,𝔽3)12yes—
195C2×C4—21no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4C22×C3×A411yescharacteristic
203C8—61no—
204C8C2×C3×SL(2,𝔽3)12yescomplemented
205C2×C3—42no—
206C2×C3—43no—
207C2×C3—43no—
208C2×C3—43no—
209C2×C3—166no—
210C2×C3D4×SL(2,𝔽3)11yescharacteristic
211C2×C3A4×SD1611yescharacteristic
212C2×C3#577⋊C311yescharacteristic
213C22—22no—
214C22—41no—
215C22—41no—
216C22C3×D4×A411yescharacteristic
217C4—22no—
218C4—31no—
219C4—31no—
220C4—61no—
221C4—121no—
222C4—121no—
223C4C22×C3×SL(2,𝔽3)11yescharacteristic
224C4C3×2−1+4⋊C311yescharacteristic
225C3—43no—
226C3SD16×SL(2,𝔽3)11yescharacteristic, direct product factor
227C2—42no—
228C2C3×D4×SL(2,𝔽3)11yescharacteristic
229C2C3×A4×SD1611yescharacteristic
230C2C3×#577⋊C311yescharacteristic
subgroupG/H
center, socleC22×C3D4×A4
commutator, cosocleC4×Q8C22×C32
FrattiniC2×C4C22×C3×A4
FittingC3×Q8×SD16C3
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC2×C3×SD16A4
hypocenterQ8C32×SD16
LCSG ▷ C4×Q8 ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
3. C3×C8×SL(2,𝔽3)C2semidirect
4. C3×Q8×SD16C3semidirect
5. SD16×SL(2,𝔽3)C3direct (N × H)
10. C3×Q82C2×C3semidirect
12. Q8×SL(2,𝔽3)C2×C3semidirect
17. C3×C8×Q8C2×C3semidirect
19. C8×SL(2,𝔽3)C2×C3semidirect
26. Q8×SD16C32semidirect
54. C3×SL(2,𝔽3)SD16direct (N × H)
56. Q82C2×C32semidirect
61. C8×Q8C2×C32semidirect
76. C3×SD16SL(2,𝔽3)direct (N × H)
121. C3×Q8C3×SD16semidirect
122. C3×Q8C2×SL(2,𝔽3)semidirect
137. SL(2,𝔽3)C3×SD16direct (N × H)
140. C3×C8C2×SL(2,𝔽3)semidirect
153. SD16C3×SL(2,𝔽3)direct (N × H)
191. Q8C32×SD16semidirect
192. Q8C2×C3×SL(2,𝔽3)semidirect
204. C8C2×C3×SL(2,𝔽3)semidirect
226. C3SD16×SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
C3×D4×SL(2,𝔽3)2
C3×C8×SL(2,𝔽3)2
C3×Q8×SD163
SD16×SL(2,𝔽3)3
C2×C32×SD164