Finite Group Databasebeta

← #157×#41⋊C3 #13180673

Subgroups — #157×#41⋊C3 #13180673

Order 1728.

298 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 38 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#41×#157C311yesmaximal, 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
17C4×#41⋊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—32no—
31C4×#157—31no—
32C4×#157A411yescharacteristic
33D3×SL(2,𝔽3)—62no—
34C3⋊(#41⋊C3)—62no—
35C2×C3×C3⋊D3—41no—
36C3×#157—44no—
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—64no—
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—
60C4×#41—91noSylow
61C2×C33—41no—
62C3×C3⋊D3—42no—
63C3×#41—22no—
64#239—62no—
65D3×Q8—62no—
66#237—182no—
67D3×D4—182no—
68#235—182no—
69C2×C4×D3—182no—
70#41⋊C3—42no—
71#41⋊C3—91no—
72#41⋊C3#15711yescharacteristic, direct product factor
73C2×SL(2,𝔽3)—91no—
74C3×C42—361no—
75C22×C32—31no—
76C2×C3⋊D3—31no—
77C2×C3⋊D3—62no—
78C2×C3⋊D3C22×A411yescharacteristic
79C2×C3×D3—242no—
80#157—62no—
81#157#41⋊C314yesdirect product factor
82C32×C4—31no—
83C32×C4—82no—
84C32×C4—162no—
85C32×C4C4×A411yescharacteristic
86C3⋊Dic3—31no—
87C3⋊Dic3C4×A411yescharacteristic
88C3×Dic3—242no—
89C2×#41—91no—
90C4×Q8—92no—
91C4×D4—272no—
92#117—271no—
93C2×C42—271no—
94C33—41noSylow
95C22×D3—182no—
96C3×Q8—22no—
97C3×D4—62no—
98C2×C3×C4—62no—
99C2×C3×C4—361no—
100C2×C3×C4—362no—
101#67—184no—
102C2×Dic3—182no—
103D12—182no—
104C4×D3—62no—
105C4×D3—182no—
106C4×D3—182no—
107C4×D3—182no—
108Dic6—182no—
109SL(2,𝔽3)—42no—
110SL(2,𝔽3)C2×#15711yescharacteristic, complemented
111C2×C32—61no—
112C2×C32—82no—
113C2×C32—162no—
114C2×C32C2×C4×A411yescharacteristic
115C3⋊D3—61no—
116C3⋊D3C2×#41⋊C312yes—
117C3×D3—244no—
118#41—91no—
119#41—271no—
120#41—271no—
121#41C3×#15711yescharacteristic, complemented
122C2×Q8—91no—
123C2×D4—271no—
124C22×C4—271no—
125C22×C4—272no—
126C4⋊C4—272no—
127C22⋊C4—272no—
128C42—91no—
129C42—271no—
130C42—272no—
131C22×C3—62no—
132C22×C3—361no—
133C2×D3—62no—
134C2×D3—182no—
135C2×D3—364no—
136C3×C4—22no—
137C3×C4—41no—
138C3×C4—62no—
139C3×C4—162no—
140C3×C4—361no—
141C3×C4—364no—
142Dic3—62no—
143Dic3—182no—
144C32—82no—
145C32—162no—
146C32C4×#41⋊C311yescharacteristic, complemented
147C23—271no—
148Q8—271no—
149Q8C2×C3×#15711yescharacteristic
150D4—31no—
151D4—271no—
152D4—272no—
153C2×C4—31no—
154C2×C4—92no—
155C2×C4—91no—
156C2×C4—271no—
157C2×C4—271no—
158C2×C4—271no—
159C2×C4—271no—
160C2×C4—272no—
161C2×C4—544no—
162C2×C3—22no—
163C2×C3—41no—
164C2×C3—122no—
165C2×C3—162no—
166C2×C3—362no—
167D3—64no—
168D3—362no—
169C22—31no—
170C22—91no—
171C22—271no—
172C22—542no—
173C4—31no—
174C4—94no—
175C4—91no—
176C4—271no—
177C4—542no—
178C4A4×#15711yescharacteristic
179C3—22no—
180C3—41no—
181C3—162no—
182C2—61no—
183C2—92no—
184C2—541no—
185C2C2×A4×#15711yescharacteristic
subgroupG/H
center, hypercenterC4A4×#157
commutator, hypocenterC32×Q8C2×C3×C4
FrattiniC2C2×A4×#157
FittingC32×#41C3×C4
socleC2×C32C2×C4×A4
prime radicalC3⋊D3×#41⋊C3C2
cosocleQ8×C3⋊D3C22×C3
LCSG ▷ C32×Q8
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×#157C2semidirect
3. #41×#157C3semidirect
4. C32×#41⋊C3C4semidirect
7. C3⋊(C3⋊(#41⋊C3))C4semidirect
16. C32×SL(2,𝔽3)C2×C4semidirect
19. C32×#41C3×C4semidirect
20. #3832C3×C4semidirect
72. #41⋊C3#157direct (N × H)
81. #157#41⋊C3direct (N × H)
110. SL(2,𝔽3)C2×#157semidirect
121. #41C3×#157semidirect
146. C32C4×#41⋊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
#41×#1573
C3×C4×#1574
C4×#41⋊C39