Finite Group Databasebeta

← C3×#73021 #11920799

Subgroups — C3×#73021 #11920799

Order 1152.

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

normal subgroups only

This group has 251 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×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
2C3×#5386C211yesmaximal, characteristic, complemented
3C3×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73021C311yesmaximal, characteristic, direct product factor
5C3×#1666—31nomaximal
6C2×C3×GL(2,𝔽3)—21no—
7C2×C3×#204—41nomaximal
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#859—21no—
10C4⋊GL(2,𝔽3)C2×C311yescharacteristic
11#5386C2×C311yescharacteristic, complemented
12C3×#492—31no—
13C3×#475—31no—
14C3×#460—31no—
15C3×C8×Q8D311yescharacteristic, complemented
16C8×SL(2,𝔽3)—21no—
17C8×SL(2,𝔽3)C2×C311yescharacteristic, complemented
18C3×#334—31no—
19C3×#332—31no—
20C3×#331—31no—
21C2×C3×D12—41no—
22C2×C3×Dic6—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3×GL(2,𝔽3)—41no—
25C2×C32×C8—41no—
26C3×#204—44no—
27#1666—31noSylow
28C2×GL(2,𝔽3)—21no—
29C2×C3×SD16—61no—
30C2×C3×SD16—61no—
31C2×C3×SD16—61no—
32C3×C4⋊Q8—31no—
33C3×#127—31no—
34C3×C4×Q8C2×D311yescharacteristic
35C2×#204—41no—
36C4×SL(2,𝔽3)—21no—
37C4×SL(2,𝔽3)C22×C311yescharacteristic
38#859—21no—
39C3×#106—61no—
40C3×C4⋊C8—31no—
41C3×C4⋊C8—31no—
42C3×C4⋊C8—31no—
43C3×#103—61no—
44C3×#102—61no—
45C3×C4×C8—31no—
46C22×C3×D3—81no—
47C2×C32×C4—41no—
48C2×C3×Dic3—81no—
49C3×D12—42no—
50C3×D12—81no—
51C3×Dic6—42no—
52C3×Dic6—81no—
53C3×SL(2,𝔽3)SD1611yescharacteristic, complemented
54C32×C8—42no—
55#492—31no—
56#475—31no—
57#460—31no—
58C8×Q8C3×D311yescharacteristic, complemented
59#334—31no—
60#332—31no—
61#331—31no—
62C2×C3×Q8—61no—
63C2×C3×Q8D1211yescharacteristic
64C2×C3×D4—61no—
65C2×C3×D4—61no—
66C2×C3×D4—61no—
67C2×D12—41no—
68C2×Dic6—41no—
69C2×SL(2,𝔽3)—21no—
70C2×SL(2,𝔽3)C3×D411yescharacteristic
71GL(2,𝔽3)—41no—
72C3×SD16—122no—
73C3×SD16—121no—
74C3×SD16—121no—
75C3×SD16—122no—
76C2×C3×C8—31no—
77C2×C3×C8—41no—
78C2×C3×C8—61no—
79C2×C3×C8—61no—
80C2×C3×C8—81no—
81C2×C3×C8S411yescharacteristic
82C3×C4⋊C4—31no—
83C3×C4⋊C4—61no—
84C3×C4⋊C4—61no—
85C3×C42—31no—
86#204—44no—
87C22×C32—41no—
88C2×C3×D3—82no—
89C2×C3×D3—161no—
90C2×C3×D3—161no—
91C32×C4—41no—
92C32×C4—41no—
93C3×Dic3—82no—
94C2×SD16—61no—
95C2×SD16—61no—
96C2×SD16—61no—
97C4⋊Q8—31no—
98#127—31no—
99C4×Q8C2×C3×D311yescharacteristic
100#106—61no—
101C4⋊C8—31no—
102C4⋊C8—31no—
103C4⋊C8—31no—
104#103—61no—
105#102—61no—
106C4×C8—31no—
107C23×C3—121no—
108C22×D3—81no—
109C3×Q8—31no—
110C3×Q8—121no—
111C3×Q8—121no—
112C3×Q8#20411yescharacteristic, complemented
113C3×D4—121no—
114C3×D4—121no—
115C3×D4—121no—
116C3×D4—121no—
117C3×D4—241no—
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—41no—
121C2×C3×C4—81no—
122C2×C3×C4—121no—
123C2×C3×C4C2×S411yescharacteristic
124C2×Dic3—81no—
125D12—42no—
126D12—81no—
127Dic6—42no—
128Dic6—81no—
129SL(2,𝔽3)—21no—
130SL(2,𝔽3)C3×SD1611yescharacteristic, complemented
131C3×C8—42no—
132C3×C8—61no—
133C3×C8—82no—
134C3×C8—121no—
135C3×C8—121no—
136C3×C8GL(2,𝔽3)12yescomplemented
137C2×C32—41no—
138C2×C32—41no—
139C2×C32—41no—
140C3×D3—162no—
141C2×Q8—61no—
142C2×Q8C3×D1211yescharacteristic
143C2×D4—61no—
144C2×D4—61no—
145C2×D4—61no—
146SD16—122no—
147SD16—121no—
148SD16—121no—
149SD16—122no—
150C2×C8—31no—
151C2×C8—61no—
152C2×C8—61no—
153C2×C8C3×S411yescharacteristic
154C4⋊C4—31no—
155C4⋊C4—61no—
156C4⋊C4—61no—
157C42—31no—
158C22×C3—41no—
159C22×C3—81no—
160C22×C3—241no—
161C22×C3—241no—
162C22×C3—241no—
163C22×C3C4⋊S411yescharacteristic
164C2×D3—82no—
165C2×D3—161no—
166C2×D3—161no—
167C3×C4—31no—
168C3×C4—31no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—61no—
172C3×C4—81no—
173C3×C4—81no—
174C3×C4—241no—
175C3×C4C2×GL(2,𝔽3)11yescharacteristic
176C3×C4#98611yescharacteristic
177Dic3—82no—
178C32—41noSylow
179C23—121no—
180Q8—31no—
181Q8—121no—
182Q8—121no—
183Q8C3×#20411yescharacteristic, complemented
184D4—121no—
185D4—121no—
186D4—121no—
187D4—121no—
188D4—241no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—121no—
192C2×C4C2×C3×S411yescharacteristic
193C8—61no—
194C8—121no—
195C8—121no—
196C8C3×GL(2,𝔽3)12yescomplemented
197C2×C3—41no—
198C2×C3—41no—
199C2×C3—41no—
200C2×C3—81no—
201C2×C3—81no—
202C2×C3—81no—
203C2×C3—481no—
204C2×C3C4⋊GL(2,𝔽3)11yescharacteristic
205C2×C3#540111yescharacteristic
206C2×C3#540611yescharacteristic
207D3—162no—
208C22—241no—
209C22—241no—
210C22—241no—
211C22C3×C4⋊S411yescharacteristic
212C4—31no—
213C4—31no—
214C4—61no—
215C4—241no—
216C4C2×C3×GL(2,𝔽3)11yescharacteristic
217C4C3×#98611yescharacteristic
218C3—41no—
219C3—81no—
220C3#7302111yescharacteristic, direct product factor
221C2—481no—
222C2C3×C4⋊GL(2,𝔽3)11yescharacteristic
223C2C3×#540111yescharacteristic
224C2C3×#540611yescharacteristic
subgroupG/H
center, socleC22×C3C4⋊S4
commutator, cosocleC4×SL(2,𝔽3)C22×C3
FrattiniC2×C4C2×C3×S4
FittingC3×C8×Q8D3
prime radicalC3×C4⋊GL(2,𝔽3)C2
hypercenterC2×C3×C8S4
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×C8

Complements

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

NH ≅ G/Ntype
2. C3×#5386C2semidirect
3. C3×C8×SL(2,𝔽3)C2semidirect
4. #73021C3direct (N × H)
11. #5386C2×C3semidirect
15. C3×C8×Q8D3semidirect
17. C8×SL(2,𝔽3)C2×C3semidirect
53. C3×SL(2,𝔽3)SD16semidirect
58. C8×Q8C3×D3semidirect
112. C3×Q8#204semidirect
130. SL(2,𝔽3)C3×SD16semidirect
136. C3×C8GL(2,𝔽3)semidirect
183. Q8C3×#204semidirect
196. C8C3×GL(2,𝔽3)semidirect
220. C3#73021direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4⋊GL(2,𝔽3)2
C3×#53862
C3×C8×SL(2,𝔽3)2
#730213
C3×#16663
C2×C3×#2044