Finite Group Databasebeta

← C3×#73068 #11920846

Subgroups — C3×#73068 #11920846

Order 1152.

277 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 68 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#73068C311yesmaximal, characteristic, direct product factor
4C3×#1476—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#1476—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×C22⋊C8—61no—
38C3×C4×C8—32no—
39C3×C4×C8—61no—
40C3×C4×C8—61no—
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—44no—
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—64no—
65C2×C3×C8—62no—
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—42no—
79C32×C4—42no—
80C2×#41C3×Dic311yescharacteristic
81C22×C8—61no—
82C4×Q8C3×Dic311yescharacteristic
83C4×Q8C2×C3×D311yescharacteristic
84C4×D4—31no—
85C4×D4—31no—
86#117—31no—
87C2×C42—31no—
88C4⋊C8—32no—
89C22⋊C8—61no—
90C4×C8—32no—
91C4×C8—61no—
92C4×C8—61no—
93C23×C3—31no—
94C3×Q8—31no—
95C3×Q8C2×C3⋊C811yescharacteristic
96C3×D4—32no—
97C3×D4—61no—
98C2×C3×C4—31no—
99C2×C3×C4—32no—
100C2×C3×C4—32no—
101C2×C3×C4—31no—
102C2×C3×C4—31no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4—62no—
107C2×C3×C4—62no—
108C2×C3×C4—81no—
109C2×C3×C4—81no—
110C2×C3×C4—81no—
111C2×C3×C4SL(2,ℤ/4)11yescharacteristic
112C2×C3×C4SL(2,ℤ/4)11yescharacteristic
113C2×C3×C4C2×S411yescharacteristic
114SL(2,𝔽3)—21no—
115SL(2,𝔽3)C2×C3×C811yescharacteristic, complemented
116C3×C8—64no—
117C3×C8—122no—
118C3⋊C8—44no—
119C2×C32—41no—
120C2×C32—41no—
121C2×C32—41no—
122#41—32no—
123#41C3×C3⋊C812yescomplemented
124C2×Q8C2×C3×Dic311yescharacteristic
125C2×D4—31no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4—31no—
129C2×C8—62no—
130C2×C8—64no—
131C2×C8—62no—
132C4⋊C4—31no—
133C4⋊C4—31no—
134C22⋊C4—61no—
135C42—31no—
136C42—31no—
137C42—31no—
138C42C3×S411yescharacteristic
139C22×C3—32no—
140C22×C3—41no—
141C22×C3—61no—
142C22×C3—61no—
143C22×C3—81no—
144C22×C3C2×SL(2,ℤ/4)11yescharacteristic
145C3×C4—31no—
146C3×C4—31no—
147C3×C4—42no—
148C3×C4—42no—
149C3×C4—42no—
150C3×C4—61no—
151C3×C4—61no—
152C3×C4—82no—
153C3×C4—82no—
154C3×C4—82no—
155C3×C4A4⋊C812yes—
156C3×C4GU(2,𝔽3)12yes—
157C3×C4#98512yes—
158C32—41noSylow
159C23—31no—
160Q8—31no—
161Q8C2×C3×C3⋊C811yescharacteristic
162D4—32no—
163D4—61no—
164C2×C4—31no—
165C2×C4—32no—
166C2×C4—32no—
167C2×C4—31no—
168C2×C4—31no—
169C2×C4—62no—
170C2×C4—62no—
171C2×C4C3×SL(2,ℤ/4)11yescharacteristic
172C2×C4C3×SL(2,ℤ/4)11yescharacteristic
173C2×C4C2×C3×S411yescharacteristic
174C8—64no—
175C8—122no—
176C2×C3—41no—
177C2×C3—41no—
178C2×C3—41no—
179C2×C3—62no—
180C2×C3—81no—
181C2×C3—81no—
182C2×C3—81no—
183C2×C3C2×A4⋊C811yescharacteristic
184C2×C3C2×GU(2,𝔽3)11yescharacteristic
185C2×C3#542411yescharacteristic
186C22—32no—
187C22—61no—
188C22—61no—
189C22C2×C3×SL(2,ℤ/4)11yescharacteristic
190C4—31no—
191C4—31no—
192C4—61no—
193C4—61no—
194C4C3×A4⋊C812yes—
195C4C3×GU(2,𝔽3)12yes—
196C4C3×#98512yes—
197C3—41no—
198C3—81no—
199C3#7306811yescharacteristic, direct product factor
200C2—62no—
201C2C2×C3×A4⋊C811yescharacteristic
202C2C2×C3×GU(2,𝔽3)11yescharacteristic
203C2C3×#542411yescharacteristic
subgroupG/H
center, hypercenterC3×C42S4
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
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C3×SL(2,𝔽3)⋊C8C2semidirect
3. #73068C3direct (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
115. SL(2,𝔽3)C2×C3×C8semidirect
123. #41C3×C3⋊C8semidirect
199. C3#73068direct (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
#730683
C3×#14763
C3×C4×C3⋊C84