Finite Group Databasebeta

← C3×#73087 #11920865

Subgroups — C3×#73087 #11920865

Order 1152.

287 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 44 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#5386C211yesmaximal, characteristic, complemented
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#73087C311yesmaximal, characteristic, direct product factor
5C3×#1651—31nomaximal
6C22×C3×SL(2,𝔽3)—21no—
7C2×C3×#214—41nomaximal
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#859—21no—
10C3×D4×Q8D311yescharacteristic
11D4×SL(2,𝔽3)—21no—
12D4×SL(2,𝔽3)C2×C311yescharacteristic
13#5386C2×C311yescharacteristic, complemented
14C3×#498—31no—
15C3×#475—31no—
16C3×#461—31no—
17SL(2,𝔽3)⋊C8C2×C311yescharacteristic, complemented
18C3×#332—31no—
19C3×Q8⋊C8—31no—
20C3×D4⋊C8—31no—
21C2×C32×D4—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)—41no—
24C2×C3×SL(2,𝔽3)D411yescharacteristic
25C3×#214—44no—
26C2×C3×C3⋊C8—41no—
27#1651—31noSylow
28C22×C3×Q8—21no—
29C22×SL(2,𝔽3)—22no—
30C22×SL(2,𝔽3)—21no—
31C2×C3×SD16—61no—
32C3×C4⋊Q8—31no—
33C3×C4⋊Q8—31no—
34C3×C22⋊Q8—61no—
35C3×C4×Q8C2×D311yescharacteristic
36C3×C4×D4—31no—
37C2×#214—41no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)C22×C311yescharacteristic
40#859—21no—
41C3×#106—61no—
42C3×#106—62no—
43C3×C4⋊C8—31no—
44C3×C4⋊C8—31no—
45C3×#103—61no—
46C3×#103—61no—
47C3×C4×C8—31no—
48C23×C32—81no—
49C32×D4—42no—
50C32×D4—81no—
51C2×C32×C4—41no—
52C2×C3×Dic3—81no—
53C3×Dic6—42no—
54C3×Dic6—81no—
55C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
56C3×C3⋊C8—42no—
57D4×Q8C3×D311yescharacteristic
58#498—31no—
59#475—31no—
60#461—31no—
61#332—31no—
62Q8⋊C8—31no—
63D4⋊C8—31no—
64C2×C3×Q8—41no—
65C2×C3×Q8—61no—
66C2×C3×Q8—61no—
67C2×C3×Q8—121no—
68C2×C3×Q8#6711yescharacteristic
69C2×C3×D4—41no—
70C2×C3×D4—81no—
71C2×C3×D4S411yescharacteristic
72C22×C3×C4—61no—
73C2×Dic6—41no—
74C2×SL(2,𝔽3)—21no—
75C2×SL(2,𝔽3)—42no—
76C2×SL(2,𝔽3)—41no—
77C2×SL(2,𝔽3)C3×D411yescharacteristic
78C3×SD16—122no—
79C2×C3×C8—61no—
80C2×C3×C8—61no—
81C2×C3×C8—61no—
82C3×C4⋊C4—31no—
83C3×C4⋊C4—32no—
84C3×C4⋊C4—31no—
85C3×C4⋊C4—61no—
86C3×C4⋊C4—61no—
87C3×C22⋊C4—61no—
88C3×C42—31no—
89#214—44no—
90C2×C3⋊C8—41no—
91C22×C32—41no—
92C22×C32—82no—
93C22×C32—161no—
94C22×C32—161no—
95C32×C4—41no—
96C32×C4—41no—
97C3×Dic3—82no—
98C22×Q8—21no—
99C2×SD16—61no—
100C4⋊Q8—31no—
101C4⋊Q8—31no—
102C22⋊Q8—61no—
103C4×Q8C2×C3×D311yescharacteristic
104C4×D4—31no—
105#106—61no—
106#106—62no—
107C4⋊C8—31no—
108C4⋊C8—31no—
109#103—61no—
110#103—61no—
111C4×C8—31no—
112C23×C3—21no—
113C23×C3—81no—
114C23×C3—82no—
115C3×Q8—31no—
116C3×Q8—121no—
117C3×Q8—121no—
118C3×Q8—121no—
119C3×Q8—121no—
120C3×Q8#21411yescharacteristic, complemented
121C3×D4—21no—
122C3×D4—42no—
123C3×D4—81no—
124C3×D4—82no—
125C3×D4—82no—
126C3×D4GL(2,𝔽3)12yes—
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—41no—
130C2×C3×C4—61no—
131C2×C3×C4—81no—
132C2×C3×C4—121no—
133C2×C3×C4—121no—
134C2×C3×C4—121no—
135C2×C3×C4C2×S411yescharacteristic
136C2×Dic3—81no—
137Dic6—42no—
138Dic6—81no—
139SL(2,𝔽3)—21no—
140SL(2,𝔽3)C3×SD1611yescharacteristic, complemented
141C3×C8—62no—
142C3×C8—121no—
143C3×C8—121no—
144C3⋊C8—42no—
145C2×C32—41no—
146C2×C32—41no—
147C2×C32—41no—
148C2×C32—162no—
149C2×Q8—41no—
150C2×Q8—61no—
151C2×Q8—61no—
152C2×Q8—121no—
153C2×Q8C3×#6711yescharacteristic
154C2×D4C3×S411yescharacteristic
155C22×C4—61no—
156SD16—122no—
157C2×C8—61no—
158C2×C8—61no—
159C2×C8—61no—
160C4⋊C4—31no—
161C4⋊C4—32no—
162C4⋊C4—31no—
163C4⋊C4—61no—
164C4⋊C4—61no—
165C22⋊C4—61no—
166C42—31no—
167C22×C3—22no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—82no—
172C22×C3—84no—
173C22×C3—81no—
174C22×C3—161no—
175C22×C3—161no—
176C22×C3—162no—
177C22×C3—162no—
178C22×C3GL(2,ℤ/4)11yescharacteristic
179C3×C4—31no—
180C3×C4—31no—
181C3×C4—41no—
182C3×C4—41no—
183C3×C4—61no—
184C3×C4—81no—
185C3×C4—81no—
186C3×C4—121no—
187C3×C4—241no—
188C3×C4C2×GL(2,𝔽3)11yescharacteristic
189C3×C4#98311yescharacteristic
190Dic3—82no—
191C32—41noSylow
192C23—21no—
193Q8—31no—
194Q8—121no—
195Q8—121no—
196Q8—121no—
197Q8—121no—
198Q8C3×#21411yescharacteristic, complemented
199D4—21no—
200D4C3×GL(2,𝔽3)12yes—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—61no—
204C2×C4—121no—
205C2×C4—121no—
206C2×C4—121no—
207C2×C4C2×C3×S411yescharacteristic
208C8—62no—
209C8—121no—
210C8—121no—
211C2×C3—42no—
212C2×C3—41no—
213C2×C3—41no—
214C2×C3—41no—
215C2×C3—81no—
216C2×C3—81no—
217C2×C3—81no—
218C2×C3—162no—
219C2×C3—164no—
220C2×C3#541411yescharacteristic
221C2×C3#542111yescharacteristic
222C2×C3#543011yescharacteristic
223C22—22no—
224C22—41no—
225C22—41no—
226C22C3×GL(2,ℤ/4)11yescharacteristic
227C4—31no—
228C4—31no—
229C4—61no—
230C4—121no—
231C4—241no—
232C4C2×C3×GL(2,𝔽3)11yescharacteristic
233C4C3×#98311yescharacteristic
234C3—41no—
235C3—81no—
236C3#7308711yescharacteristic, direct product factor
237C2—42no—
238C2C3×#541411yescharacteristic
239C2C3×#542111yescharacteristic
240C2C3×#543011yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator, cosocleC4×SL(2,𝔽3)C22×C3
FrattiniC2×C4C2×C3×S4
FittingC3×D4×Q8D3
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C3×SD16
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#5386C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
4. #73087C3direct (N × H)
13. #5386C2×C3semidirect
17. SL(2,𝔽3)⋊C8C2×C3semidirect
55. C3×SL(2,𝔽3)SD16semidirect
120. C3×Q8#214semidirect
140. SL(2,𝔽3)C3×SD16semidirect
198. Q8C3×#214semidirect
236. C3#73087direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
C3×#53862
C3×SL(2,𝔽3)⋊C82
#730873
C3×#16513
C2×C3×#2144