Finite Group Databasebeta

← C3×#73020 #11920798

Subgroups — C3×#73020 #11920798

Order 1152.

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

normal subgroups only

This group has 227 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×C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#73020C311yesmaximal, characteristic, direct product factor
5C3×#1588—31nomaximal
6C2×C3×GL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3×#859C411yescharacteristic
9C3×(C2×D3)⋊C8—41nomaximal
10C4×GL(2,𝔽3)C2×C311yescharacteristic
11C3×C8×Q8D311yescharacteristic, complemented
12C3×C4×SD16—31no—
13C3×#436—31no—
14C8×SL(2,𝔽3)—21no—
15C8×SL(2,𝔽3)C2×C311yescharacteristic, complemented
16SL(2,𝔽3)⋊C8C2×C311yescharacteristic, complemented
17C3×#335—31no—
18C3×Q8⋊C8—31no—
19C3×D4⋊C8—31no—
20C3×#322—31no—
21C2×C3×C4×D3—41no—
22C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
23C3×GL(2,𝔽3)—21no—
24C2×C32×C8—41no—
25C2×C3×C3⋊C8—41no—
26#1588—31noSylow
27C2×GL(2,𝔽3)C3×C411yescharacteristic
28C2×C3×SD16—31no—
29C2×C3×M16—61no—
30C3×C4×Q8C2×D311yescharacteristic
31C3×C4×D4—31no—
32C4×SL(2,𝔽3)—21no—
33C4×SL(2,𝔽3)C22×C311yescharacteristic
34#859C3×C411yescharacteristic
35C3×#106—31no—
36C3×C4⋊C8—31no—
37C3×C4⋊C8—31no—
38C3×#103—31no—
39C3×#102—31no—
40C3×C22⋊C8—61no—
41C3×C4×C8—31no—
42C3×C4×C8—31no—
43C3×C4×C8—31no—
44(C2×D3)⋊C8—41no—
45C22×C3×D3—41no—
46C2×C32×C4—41no—
47C2×C3×Dic3—41no—
48C3×C4×D3—81no—
49C3×C4×D3—81no—
50C3×SL(2,𝔽3)M1611yescharacteristic, complemented
51C32×C8—81no—
52C3×C3⋊C8—81no—
53C8×Q8C3×D311yescharacteristic, complemented
54C4×SD16—31no—
55#436—31no—
56#335—31no—
57Q8⋊C8—31no—
58D4⋊C8—31no—
59#322—31no—
60C2×C3×Q8C4×D311yescharacteristic
61C2×C3×D4—31no—
62C22×C3×C4—61no—
63C2×C4×D3—41no—
64C2×SL(2,𝔽3)—21no—
65C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
66GL(2,𝔽3)—21no—
67C3×SD16—61no—
68C3×SD16—61no—
69C3×M16—122no—
70C2×C3×C8—31no—
71C2×C3×C8—31no—
72C2×C3×C8—31no—
73C2×C3×C8—41no—
74C2×C3×C8—61no—
75C2×C3×C8—61no—
76C2×C3×C8—81no—
77C2×C3×C8S411yescharacteristic
78C3×C4⋊C4—31no—
79C3×C4⋊C4—31no—
80C3×C22⋊C4—61no—
81C3×C42—31no—
82C2×C3⋊C8—41no—
83C22×C32—41no—
84C2×C3×D3—42no—
85C2×C3×D3—81no—
86C2×C3×D3—81no—
87C32×C4—41no—
88C32×C4—41no—
89C3×Dic3—81no—
90C2×SD16—31no—
91C2×M16—61no—
92C4×Q8C2×C3×D311yescharacteristic
93C4×D4—31no—
94#106—31no—
95C4⋊C8—31no—
96C4⋊C8—31no—
97#103—31no—
98#102—31no—
99C22⋊C8—61no—
100C4×C8—31no—
101C4×C8—31no—
102C4×C8—31no—
103C23×C3—61no—
104C22×D3—41no—
105C3×Q8—31no—
106C3×Q8ΓL(1,𝔽25)11yescharacteristic
107C3×D4—61no—
108C3×D4—61no—
109C2×C3×C4—31no—
110C2×C3×C4—31no—
111C2×C3×C4—41no—
112C2×C3×C4—61no—
113C2×C3×C4—81no—
114C2×C3×C4—121no—
115C2×C3×C4—121no—
116C2×C3×C4C2×S411yescharacteristic
117C2×Dic3—41no—
118C4×D3—81no—
119C4×D3—81no—
120SL(2,𝔽3)—21no—
121SL(2,𝔽3)C3×M1611yescharacteristic, complemented
122C3×C8—21no—
123C3×C8—61no—
124C3×C8—61no—
125C3×C8—62no—
126C3×C8—61no—
127C3×C8—81no—
128C3×C8—82no—
129C3×C8—121no—
130C3⋊C8—81no—
131C2×C32—41no—
132C2×C32—41no—
133C2×C32—41no—
134C3×D3—82no—
135C2×Q8C3×C4×D311yescharacteristic
136C2×D4—31no—
137C22×C4—61no—
138SD16—61no—
139SD16—61no—
140M16—122no—
141C2×C8—31no—
142C2×C8—31no—
143C2×C8—31no—
144C2×C8—61no—
145C2×C8—61no—
146C2×C8C3×S411yescharacteristic
147C4⋊C4—31no—
148C4⋊C4—31no—
149C22⋊C4—61no—
150C42—31no—
151C22×C3—41no—
152C22×C3—81no—
153C22×C3—121no—
154C22×C3—121no—
155C22×C3—121no—
156C22×C3C4×S411yescharacteristic
157C2×D3—42no—
158C2×D3—81no—
159C2×D3—81no—
160C3×C4—31no—
161C3×C4—31no—
162C3×C4—41no—
163C3×C4—41no—
164C3×C4—61no—
165C3×C4—81no—
166C3×C4—81no—
167C3×C4—121no—
168C3×C4#98311yescharacteristic
169C3×C4#98611yescharacteristic
170Dic3—81no—
171C32—41noSylow
172C23—61no—
173Q8—31no—
174Q8C3×ΓL(1,𝔽25)11yescharacteristic
175D4—61no—
176D4—61no—
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—61no—
180C2×C4—121no—
181C2×C4—121no—
182C2×C4C2×C3×S411yescharacteristic
183C8—21no—
184C8—61no—
185C8—61no—
186C8—62no—
187C8—61no—
188C8—121no—
189C2×C3—41no—
190C2×C3—41no—
191C2×C3—41no—
192C2×C3—81no—
193C2×C3—81no—
194C2×C3—81no—
195C2×C3—241no—
196C2×C3#539411yescharacteristic
197C2×C3#540011yescharacteristic
198C2×C3#540411yescharacteristic
199D3—82no—
200C22—121no—
201C22—121no—
202C22—121no—
203C22C3×C4×S411yescharacteristic
204C4—31no—
205C4—31no—
206C4—61no—
207C4—121no—
208C4C3×#98311yescharacteristic
209C4C3×#98611yescharacteristic
210C3—41no—
211C3—81no—
212C3#7302011yescharacteristic, direct product factor
213C2—241no—
214C2C3×#539411yescharacteristic
215C2C3×#540011yescharacteristic
216C2C3×#540411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×S4
commutatorC2×SL(2,𝔽3)C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC3×C8×Q8D3
socleC22×C3C4×S4
prime radicalC2×C3×GL(2,𝔽3)C4
cosocleC4×SL(2,𝔽3)C22×C3
hypercenterC2×C3×C8S4
hypocenterSL(2,𝔽3)C3×M16
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×C8×SL(2,𝔽3)C2semidirect
3. C3×SL(2,𝔽3)⋊C8C2semidirect
4. #73020C3direct (N × H)
11. C3×C8×Q8D3semidirect
15. C8×SL(2,𝔽3)C2×C3semidirect
16. SL(2,𝔽3)⋊C8C2×C3semidirect
50. C3×SL(2,𝔽3)M16semidirect
53. C8×Q8C3×D3semidirect
121. SL(2,𝔽3)C3×M16semidirect
212. C3#73020direct (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×C8×SL(2,𝔽3)2
C3×SL(2,𝔽3)⋊C82
#730203
C3×#15883
C3×(C2×D3)⋊C84