Finite Group Databasebeta

← #10596650⋊C3 #13180674

Subgroups — #10596650⋊C3 #13180674

Order 1728.

286 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 34 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⋊D3×#41⋊C3C211yesmaximal, characteristic
2SL(2,𝔽3)×#157C212yesmaximal, complemented
3#10596650C311yesmaximal, characteristic, complemented
4C32×#41⋊C3C411yescharacteristic, complemented
5C3×C4⋊#157—41nomaximal
6C3⋊D3×SL(2,𝔽3)C2211yescharacteristic
7C3⋊(C3⋊(#41⋊C3))C411yescharacteristic, complemented
8#41×C3⋊D3C2×C311yescharacteristic
9Q8×#157C2×C312yes—
10D4×#157—32no—
11#64286—31no—
12C2×C4⋊#157—31no—
13D3×#41⋊C3—62no—
14C2×C3×#157—42no—
15C3×C4×C3⋊D3—41no—
16C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
17#520⋊C3—91nomaximal
18C22×#157—32no—
19C32×#41C3×C411yescharacteristic, complemented
20#3832C3×C411yescharacteristic, complemented
21Q8×C3⋊D3C22×C311yescharacteristic
22#3830—31no—
23D4×C3⋊D3—31no—
24#3828—31no—
25C2×C4×C3⋊D3—31no—
26C3×#41⋊C3—22no—
27C3×#41⋊C3—42no—
28C22⋊#157—32no—
29C4⋊#157—32no—
30C4⋊#157—31no—
31C4⋊#157A411yescharacteristic
32C4×#157—32no—
33D3×SL(2,𝔽3)—62no—
34C3⋊(#41⋊C3)—62no—
35C2×C3×C3⋊D3—41no—
36C3×#157—82no—
37C33×C4—41no—
38C3×C3⋊Dic3—41no—
39D3×#41—62no—
40C2×#41⋊C3—91no—
41C4×SL(2,𝔽3)—92no—
42C22×C3⋊D3—31no—
43C2×#157—32no—
44C2×#157—122no—
45C2×#157C2×A412yes—
46C32×Q8C2×C3×C411yescharacteristic
47C32×D4—31no—
48C2×C32×C4—31no—
49#638—32no—
50C2×C3⋊Dic3—31no—
51C4⋊(C3⋊D3)—31no—
52C4×C3⋊D3—31no—
53C4×C3⋊D3—31no—
54C4×C3⋊D3—31no—
55C4×C3⋊D3C2×A411yescharacteristic
56#634—31no—
57C3×C4×D3—242no—
58C3×SL(2,𝔽3)—22no—
59C3×SL(2,𝔽3)—42no—
60#520—91noSylow
61C2×C33—41no—
62C3×C3⋊D3—41no—
63C3×C3⋊D3—41no—
64C3×#41—22no—
65#239—62no—
66D3×Q8—62no—
67#237—182no—
68D3×D4—182no—
69#235—182no—
70C2×C4×D3—182no—
71#41⋊C3—42no—
72#41⋊C3—91no—
73#41⋊C3#15711yescharacteristic, complemented
74C2×SL(2,𝔽3)—91no—
75C3×C4⋊C4—361no—
76C22×C32—31no—
77C2×C3⋊D3—31no—
78C2×C3⋊D3—61no—
79C2×C3⋊D3—61no—
80C2×C3⋊D3C22×A411yescharacteristic
81C2×C3×D3—242no—
82#157—22no—
83#157—62no—
84C32×C4—31no—
85C32×C4—82no—
86C32×C4—162no—
87C32×C4C4×A411yescharacteristic
88C3⋊Dic3—31no—
89C3⋊Dic3C4×A411yescharacteristic
90C3×Dic3—242no—
91C2×#41—91no—
92C4×Q8—92no—
93C4×D4—272no—
94#117—271no—
95C2×C4⋊C4—271no—
96C33—41noSylow
97C22×D3—182no—
98C3×Q8—22no—
99C3×D4—62no—
100C2×C3×C4—62no—
101C2×C3×C4—361no—
102C2×C3×C4—362no—
103#67—184no—
104C2×Dic3—182no—
105D12—182no—
106C4×D3—62no—
107C4×D3—182no—
108C4×D3—182no—
109C4×D3—182no—
110Dic6—182no—
111SL(2,𝔽3)—42no—
112SL(2,𝔽3)C2×#15711yescharacteristic, complemented
113C2×C32—61no—
114C2×C32—82no—
115C2×C32—162no—
116C2×C32C2×C4×A411yescharacteristic
117C3⋊D3—61no—
118C3⋊D3#99411yescharacteristic
119C3⋊D32−1+4⋊C311yescharacteristic
120C3×D3—242no—
121C3×D3—242no—
122#41—91no—
123#41—271no—
124#41—271no—
125#41C3×#15711yescharacteristic, complemented
126C2×Q8—91no—
127C2×D4—271no—
128C22×C4—271no—
129C22×C4—272no—
130C4⋊C4—91no—
131C4⋊C4—271no—
132C4⋊C4—272no—
133C22⋊C4—272no—
134C42—272no—
135C22×C3—62no—
136C22×C3—361no—
137C2×D3—62no—
138C2×D3—182no—
139C2×D3—362no—
140C2×D3—362no—
141C3×C4—22no—
142C3×C4—41no—
143C3×C4—62no—
144C3×C4—162no—
145C3×C4—361no—
146C3×C4—722no—
147Dic3—62no—
148Dic3—182no—
149C32—82no—
150C32—162no—
151C32#520⋊C311yescharacteristic, complemented
152C23—271no—
153Q8—271no—
154Q8C2×C3×#15711yescharacteristic
155D4—31no—
156D4—271no—
157D4—272no—
158C2×C4—31no—
159C2×C4—91no—
160C2×C4—92no—
161C2×C4—271no—
162C2×C4—271no—
163C2×C4—271no—
164C2×C4—271no—
165C2×C4—272no—
166C2×C4—1082no—
167C2×C3—22no—
168C2×C3—41no—
169C2×C3—122no—
170C2×C3—162no—
171C2×C3—361no—
172C2×C3—361no—
173D3—62no—
174D3—62no—
175D3—362no—
176C22—31no—
177C22—91no—
178C22—271no—
179C22—541no—
180C22—541no—
181C4—31no—
182C4—91no—
183C4—182no—
184C4—271no—
185C4—542no—
186C4A4×#15711yescharacteristic
187C3—22no—
188C3—41no—
189C3—162no—
190C2—61no—
191C2—91no—
192C2—91no—
193C2—541no—
194C2C2×A4×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4×#157
commutator, hypocenterC32×Q8C2×C3×C4
FittingC32×#41C3×C4
socleC2×C32C2×C4×A4
prime radicalC3⋊D3×#41⋊C3C2
cosocleQ8×C3⋊D3C22×C3
hypercenterC4A4×#157
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×#157C2semidirect
3. #10596650C3semidirect
4. C32×#41⋊C3C4semidirect
7. C3⋊(C3⋊(#41⋊C3))C4semidirect
16. C32×SL(2,𝔽3)C2×C4semidirect
19. C32×#41C3×C4semidirect
20. #3832C3×C4semidirect
73. #41⋊C3#157semidirect
112. SL(2,𝔽3)C2×#157semidirect
125. #41C3×#157semidirect
151. C32#520⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#41⋊C32
SL(2,𝔽3)×#1572
#105966503
C3×C4⋊#1574
#520⋊C39