Finite Group Databasebeta

← #10596646⋊C3 #13180670

Subgroups — #10596646⋊C3 #13180670

Order 1728.

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

normal subgroups only

This group has 322 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
1C2×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)×#157C212yesmaximal, complemented
3#10596646C311yesmaximal, characteristic, complemented
4C2×C32×SL(2,𝔽3)C411yescharacteristic, complemented
5C3×C22⋊#157—41nomaximal
6C3⋊D3×SL(2,𝔽3)C411yescharacteristic, complemented
7C3⋊D3×SL(2,𝔽3)C2211yescharacteristic
8C2×Q8×C3⋊D3C2×C311yescharacteristic
9Q8×#157—32no—
10Q8×#157C2×C312yescomplemented
11#64286—31no—
12#64286—31no—
13C2×D3×SL(2,𝔽3)—62no—
14C22×C3×C3⋊D3—41no—
15C2×C3×#157—42no—
16C32×SL(2,𝔽3)C2×C411yescharacteristic
17#519⋊C3—91nomaximal
18C2×C32×Q8C3×C411yescharacteristic, complemented
19Q8×C3⋊D3—31no—
20Q8×C3⋊D3—31no—
21Q8×C3⋊D3C3×C411yescharacteristic, complemented
22Q8×C3⋊D3C22×C311yescharacteristic
23C2×C4×C3⋊D3—31no—
24C2×#634—31no—
25C2×C3×SL(2,𝔽3)—22no—
26C2×C3×SL(2,𝔽3)—42no—
27C22⋊#157—31no—
28C22⋊#157A411yescharacteristic
29C4⋊#157—32no—
30C4⋊#157—32no—
31C4×#157—32no—
32C4×#157—32no—
33D3×SL(2,𝔽3)—62no—
34D3×SL(2,𝔽3)—62no—
35C22×C33—41no—
36C2×C3×C3⋊D3—41no—
37C2×C3×C3⋊D3—41no—
38C2×C3×C3⋊D3—82no—
39C3×#157—82no—
40C2×D3×Q8—62no—
41C22×SL(2,𝔽3)—91no—
42C4×SL(2,𝔽3)—92no—
43C22×C3⋊D3C2×A411yescharacteristic
44C22×C3×D3—242no—
45C2×#157—32no—
46C2×#157C2×A412yes—
47C32×Q8—31no—
48C32×Q8C2×C3×C411yescharacteristic
49C2×C32×C4—31no—
50C2×C3⋊Dic3—31no—
51C4×C3⋊D3—31no—
52C4×C3⋊D3—31no—
53C4×C3⋊D3—31no—
54C4×C3⋊D3—31no—
55#634—32no—
56#634—31no—
57#634—31no—
58C3×SL(2,𝔽3)—22no—
59C3×SL(2,𝔽3)—42no—
60#519—91noSylow
61C2×C33—41no—
62C2×C33—81no—
63C3×C3⋊D3—42no—
64C3×C3⋊D3—81no—
65C2×C3×Q8—22no—
66D3×Q8—62no—
67D3×Q8—62no—
68D3×Q8—182no—
69D3×Q8—182no—
70C2×C4×D3—182no—
71C2×Dic6—182no—
72C2×SL(2,𝔽3)—42no—
73C2×SL(2,𝔽3)—91no—
74C2×SL(2,𝔽3)—91no—
75C2×SL(2,𝔽3)#15711yescharacteristic, complemented
76C3×C22⋊C4—361no—
77C22×C32—82no—
78C22×C32—162no—
79C22×C32C4×A411yescharacteristic
80C2×C3⋊D3—22no—
81C2×C3⋊D3C4×A411yescharacteristic
82C2×C3⋊D3C22×A411yescharacteristic
83C2×C3×D3—242no—
84C2×C3×D3—242no—
85C2×C3×D3—248no—
86#157—22no—
87#157—62no—
88C32×C4—31no—
89C32×C4—31no—
90C3⋊Dic3—31no—
91C3⋊Dic3—31no—
92C22×Q8—91no—
93C4×Q8—92no—
94C4×Q8—272no—
95#117—271no—
96#117—271no—
97C33—41noSylow
98C23×C3—361no—
99C22×D3—62no—
100C3×Q8—22no—
101C3×Q8—62no—
102C2×C3×C4—62no—
103C2×C3×C4—362no—
104C2×Dic3—182no—
105C4×D3—182no—
106C4×D3—182no—
107C4×D3—182no—
108C4×D3—182no—
109Dic6—184no—
110Dic6—182no—
111Dic6—182no—
112SL(2,𝔽3)—42no—
113SL(2,𝔽3)C2×#15711yescharacteristic
114C2×C32—21no—
115C2×C32—82no—
116C2×C32—84no—
117C2×C32—164no—
118C2×C32—162no—
119C2×C32C2×C4×A411yescharacteristic
120C3⋊D3—21no—
121C3⋊D32−1+4⋊C312yes—
122C3×D3—244no—
123C3×D3—244no—
124C2×Q8—91no—
125C2×Q8—91no—
126C2×Q8—271no—
127C2×Q8—271no—
128C2×Q8—271no—
129C2×Q8C3×#15711yescharacteristic, complemented
130C22×C4—271no—
131C4⋊C4—272no—
132C4⋊C4—272no—
133C22⋊C4—91no—
134C22⋊C4—271no—
135C42—272no—
136C42—272no—
137C22×C3—22no—
138C22×C3—41no—
139C22×C3—162no—
140C22×C3—361no—
141C22×C3—361no—
142C22×C3—722no—
143C2×D3—62no—
144C2×D3—62no—
145C2×D3—68no—
146C3×C4—62no—
147C3×C4—62no—
148C3×C4—722no—
149Dic3—182no—
150Dic3—182no—
151C32—82no—
152C32—162no—
153C32#519⋊C311yescharacteristic, complemented
154C23—91no—
155Q8—31no—
156Q8—271no—
157Q8—271no—
158Q8—272no—
159Q8C2×C3×#15711yescharacteristic
160C2×C4—31no—
161C2×C4—92no—
162C2×C4—271no—
163C2×C4—271no—
164C2×C4—271no—
165C2×C4—271no—
166C2×C4—271no—
167C2×C4—272no—
168C2×C3—24no—
169C2×C3—22no—
170C2×C3—41no—
171C2×C3—81no—
172C2×C3—164no—
173C2×C3—162no—
174C2×C3—362no—
175C2×C3—721no—
176D3—64no—
177D3—64no—
178C22—91no—
179C22—91no—
180C22—182no—
181C22A4×#15711yescharacteristic
182C4—31no—
183C4—31no—
184C4—182no—
185C4—271no—
186C4—271no—
187C4—542no—
188C3—22no—
189C3—41no—
190C3—162no—
191C2—21no—
192C2—92no—
193C2—181no—
194C2C2×A4×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4×#157
commutator, hypocenterC32×Q8C2×C3×C4
FittingC2×C32×Q8C3×C4
socleC2×C32C2×C4×A4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
cosocleQ8×C3⋊D3C22×C3
hypercenterC22A4×#157
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×#157C2semidirect
3. #10596646C3semidirect
4. C2×C32×SL(2,𝔽3)C4semidirect
6. C3⋊D3×SL(2,𝔽3)C4semidirect
10. Q8×#157C2×C3semidirect
18. C2×C32×Q8C3×C4semidirect
21. Q8×C3⋊D3C3×C4semidirect
75. C2×SL(2,𝔽3)#157semidirect
129. C2×Q8C3×#157semidirect
153. C32#519⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
SL(2,𝔽3)×#1572
#105966463
C3×C22⋊#1574
#519⋊C39