Finite Group Databasebeta

← C3×#73071 #11920849

Subgroups — C3×#73071 #11920849

Order 1152.

259 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 64 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×#41⋊C3C211yesmaximal, characteristic
2C3×SL(2,𝔽3)⋊C8C212yesmaximal, complemented
3#73071C311yesmaximal, characteristic, direct product factor
4C3×#1477—31nomaximal
5C2×C3×#41⋊C3C411yescharacteristic
6C3×C4×SL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C2211yescharacteristic
8C3×C4⋊(C3⋊C8)—41nomaximal
9C3×C4×#41D311yescharacteristic
10C4×#41⋊C3—21no—
11C4×#41⋊C3C2×C311yescharacteristic
12C3×#431—31no—
13C2×C3×C4⋊C8—31no—
14SL(2,𝔽3)⋊C8C2×C312yescomplemented
15C3×Q8⋊C8—32no—
16C3×D4⋊C8—32no—
17C3×#41⋊C3C812yescomplemented
18C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
19C32×C42—41no—
20C2×C3×C3⋊C8—42no—
21#1477—31noSylow
22C2×C3×#41Dic311yescharacteristic
23C2×#41⋊C3—21no—
24C2×#41⋊C3C3×C411yescharacteristic
25C22×C3×C8—61no—
26C3×C4×Q8Dic311yescharacteristic
27C3×C4×Q8C2×D311yescharacteristic
28C3×C4×D4—31no—
29C3×C4×D4—31no—
30C3×#117—31no—
31C2×C3×C42—31no—
32C4×SL(2,𝔽3)—21no—
33C4×SL(2,𝔽3)—21no—
34C4×SL(2,𝔽3)C3×C411yescharacteristic
35C4×SL(2,𝔽3)C22×C311yescharacteristic
36C3×C4⋊C8—32no—
37C3×C4⋊C8—61no—
38C3×C4⋊C8—61no—
39C3×C22⋊C8—61no—
40C3×C4×C8—32no—
41C4⋊(C3⋊C8)—41no—
42C2×C32×C4—41no—
43C2×C32×C4—41no—
44C2×C32×C4—41no—
45C3×SL(2,𝔽3)C2×C811yescharacteristic, complemented
46C3×C3⋊C8—82no—
47C4×#41C3×D311yescharacteristic
48#431—31no—
49C2×C4⋊C8—31no—
50Q8⋊C8—32no—
51D4⋊C8—32no—
52C3×#41—32no—
53C3×#41C3⋊C812yescomplemented
54C2×C3×Q8C2×Dic311yescharacteristic
55C2×C3×D4—31no—
56C22×C3×C4—31no—
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59#41⋊C3—22no—
60#41⋊C3C3×C812yescomplemented
61C2×SL(2,𝔽3)—21no—
62C2×SL(2,𝔽3)C2×C3×C411yescharacteristic
63C2×C3×C8—62no—
64C2×C3×C8—62no—
65C2×C3×C8—122no—
66C3×C4⋊C4—31no—
67C3×C4⋊C4—31no—
68C3×C22⋊C4—61no—
69C3×C42—31no—
70C3×C42—31no—
71C3×C42—31no—
72C3×C42—41no—
73C3×C42—81no—
74C3×C42S411yescharacteristic
75C2×C3⋊C8—42no—
76C22×C32—41no—
77C32×C4—42no—
78C32×C4—41no—
79C32×C4—41no—
80C32×C4—81no—
81C2×#41C3×Dic311yescharacteristic
82C22×C8—61no—
83C4×Q8C3×Dic311yescharacteristic
84C4×Q8C2×C3×D311yescharacteristic
85C4×D4—31no—
86C4×D4—31no—
87#117—31no—
88C2×C42—31no—
89C4⋊C8—32no—
90C4⋊C8—61no—
91C4⋊C8—61no—
92C22⋊C8—61no—
93C4×C8—32no—
94C23×C3—31no—
95C3×Q8—31no—
96C3×Q8C2×C3⋊C811yescharacteristic
97C3×D4—32no—
98C3×D4—61no—
99C2×C3×C4—31no—
100C2×C3×C4—32no—
101C2×C3×C4—32no—
102C2×C3×C4—31no—
103C2×C3×C4—31no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4—41no—
107C2×C3×C4—61no—
108C2×C3×C4—61no—
109C2×C3×C4—81no—
110C2×C3×C4—81no—
111C2×C3×C4—81no—
112C2×C3×C4—121no—
113C2×C3×C4SL(2,ℤ/4)11yescharacteristic
114C2×C3×C4SL(2,ℤ/4)11yescharacteristic
115C2×C3×C4C2×S411yescharacteristic
116SL(2,𝔽3)—21no—
117SL(2,𝔽3)C2×C3×C811yescharacteristic, complemented
118C3×C8—122no—
119C3×C8—122no—
120C3⋊C8—82no—
121C2×C32—41no—
122C2×C32—41no—
123C2×C32—41no—
124#41—32no—
125#41C3×C3⋊C812yescomplemented
126C2×Q8C2×C3×Dic311yescharacteristic
127C2×D4—31no—
128C22×C4—31no—
129C22×C4—31no—
130C22×C4—31no—
131C2×C8—62no—
132C2×C8—62no—
133C2×C8—122no—
134C4⋊C4—31no—
135C4⋊C4—31no—
136C22⋊C4—61no—
137C42—31no—
138C42—31no—
139C42—31no—
140C42C3×S411yescharacteristic
141C22×C3—32no—
142C22×C3—41no—
143C22×C3—61no—
144C22×C3—61no—
145C22×C3—81no—
146C22×C3C2×SL(2,ℤ/4)11yescharacteristic
147C3×C4—21no—
148C3×C4—31no—
149C3×C4—31no—
150C3×C4—41no—
151C3×C4—42no—
152C3×C4—41no—
153C3×C4—61no—
154C3×C4—61no—
155C3×C4—81no—
156C3×C4—82no—
157C3×C4—81no—
158C3×C4—82no—
159C3×C4—81no—
160C3×C4A4⋊C812yes—
161C3×C4#98411yescharacteristic
162C3×C4#98611yescharacteristic
163C32—41noSylow
164C23—31no—
165Q8—31no—
166Q8C2×C3×C3⋊C811yescharacteristic
167D4—32no—
168D4—61no—
169C2×C4—31no—
170C2×C4—32no—
171C2×C4—32no—
172C2×C4—31no—
173C2×C4—31no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—121no—
177C2×C4C3×SL(2,ℤ/4)11yescharacteristic
178C2×C4C3×SL(2,ℤ/4)11yescharacteristic
179C2×C4C2×C3×S411yescharacteristic
180C8—122no—
181C8—122no—
182C2×C3—41no—
183C2×C3—41no—
184C2×C3—41no—
185C2×C3—62no—
186C2×C3—81no—
187C2×C3—81no—
188C2×C3—81no—
189C2×C3C2×A4⋊C811yescharacteristic
190C2×C3#542311yescharacteristic
191C2×C3#542611yescharacteristic
192C22—32no—
193C22—61no—
194C22—61no—
195C22C2×C3×SL(2,ℤ/4)11yescharacteristic
196C4—21no—
197C4—31no—
198C4—31no—
199C4—61no—
200C4—61no—
201C4C3×A4⋊C812yes—
202C4C3×#98411yescharacteristic
203C4C3×#98611yescharacteristic
204C3—41no—
205C3—81no—
206C3#7307111yescharacteristic, direct product factor
207C2—62no—
208C2C2×C3×A4⋊C811yescharacteristic
209C2C3×#542311yescharacteristic
210C2C3×#542611yescharacteristic
subgroupG/H
centerC2×C3×C4C2×S4
commutator, hypocenterSL(2,𝔽3)C2×C3×C8
FrattiniC2×C4C2×C3×S4
FittingC3×C4×#41D3
socleC22×C3C2×SL(2,ℤ/4)
prime radicalC2×C3×#41⋊C3C4
cosocleC4×SL(2,𝔽3)C22×C3
hypercenterC3×C42S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C3×SL(2,𝔽3)⋊C8C2semidirect
3. #73071C3direct (N × H)
14. SL(2,𝔽3)⋊C8C2×C3semidirect
17. C3×#41⋊C3C8semidirect
45. C3×SL(2,𝔽3)C2×C8semidirect
53. C3×#41C3⋊C8semidirect
60. #41⋊C3C3×C8semidirect
117. SL(2,𝔽3)C2×C3×C8semidirect
125. #41C3×C3⋊C8semidirect
206. C3#73071direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
C3×SL(2,𝔽3)⋊C82
#730713
C3×#14773
C3×C4⋊(C3⋊C8)4