Finite Group Databasebeta

← #13166095

Subgroups — #13166095

Order 1728.

348 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 46 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
1SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic
2C3×C3⋊#859C212yesmaximal
3#10593212—32nomaximal
4#10591630—31nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C2×#6296—41nomaximal
7Q8×C3⋊Dic3D311yescharacteristic
8#63919—31no—
9Dic3×SL(2,𝔽3)—32no—
10Dic3×SL(2,𝔽3)—61no—
11C3⋊#859D312yes—
12C3×#859—32no—
13C3×#859—62no—
14C3×#835—32no—
15#63573—32no—
16#63572—31no—
17C2×C3×C3⋊Dic3—42no—
18C2×C3×C3⋊Dic3—41no—
19C32×SL(2,𝔽3)Q811yescharacteristic, complemented
20#6296—44no—
21#5386—91no—
22#4791—92no—
23C2×C32×Q8C2×D311yescharacteristic
24C2×C3×SL(2,𝔽3)—22no—
25C2×C3×SL(2,𝔽3)—21no—
26C2×C3×SL(2,𝔽3)—41no—
27C2×C3×SL(2,𝔽3)C2×D312yes—
28C2×#627—61no—
29C2×#627—122no—
30C4⋊(C3⋊Dic3)—31no—
31C4×C3⋊Dic3—31no—
32C3×C4⋊Dic3—32no—
33C2×C3×C3⋊C8—32no—
34#3724—61no—
35C22×C33—41no—
36C3×C3⋊Dic3—44no—
37C3×C3⋊Dic3—42no—
38Q8×Dic3—32no—
39Q8×Dic3—61no—
40#888—92no—
41C4×SL(2,𝔽3)—91no—
42#859—32no—
43#859—62no—
44C3×#103—92no—
45#835—32no—
46#817—92no—
47Dic3⋊C8—92no—
48#808—92no—
49C32×Q8—31no—
50C32×Q8Dic611yescharacteristic, complemented
51C2×C32×C4—31no—
52C2×C3⋊Dic3—31no—
53C2×C3⋊Dic3—42no—
54C2×C3⋊Dic3S411yescharacteristic
55C2×C3×Dic3—62no—
56C2×C3×Dic3—122no—
57C2×C3×Dic3—122no—
58C2×C3×Dic3—241no—
59C2×C3×Dic3—242no—
60C3×SL(2,𝔽3)—22no—
61C3×SL(2,𝔽3)—21no—
62C3×SL(2,𝔽3)—41no—
63C3×SL(2,𝔽3)Dic612yescomplemented
64#627—128no—
65#627—122no—
66C3×C3⋊C8—62no—
67#475—271noSylow
68C2×C33—41no—
69C2×C33—41no—
70C2×C33—41no—
71C2×C3×Q8—21no—
72C2×C3×Q8D3212yes—
73C2×Dic6—182no—
74C2×Dic6—361no—
75C2×SL(2,𝔽3)—22no—
76C2×SL(2,𝔽3)—41no—
77C2×SL(2,𝔽3)D3211yescharacteristic
78C2×C3×C8—92no—
79C3×C4⋊C4—92no—
80C4⋊Dic3—32no—
81C4⋊Dic3—92no—
82C4⋊Dic3—181no—
83Dic3⋊C4—182no—
84C4×Dic3—92no—
85C4×Dic3—181no—
86C2×C3⋊C8—32no—
87C22×C32—42no—
88C22×C32—81no—
89C22×C32—82no—
90C22×C32—161no—
91C22×C32C2×S411yescharacteristic
92C32×C4—31no—
93C32×C4—31no—
94C3⋊Dic3—44no—
95C3⋊Dic3—61no—
96C3⋊Dic3GL(2,𝔽3)12yes—
97C3×Dic3—124no—
98C3×Dic3—124no—
99C3×Dic3—122no—
100C3×Dic3—242no—
101C3×Dic3—244no—
102C4⋊Q8—271no—
103C4×Q8—91no—
104#106—272no—
105C4⋊C8—271no—
106#103—92no—
107C33—41noSylow
108C3×Q8—21no—
109C3×Q8—32no—
110C3×Q8—61no—
111C3×Q8#62712yescomplemented
112C2×C3×C4—32no—
113C2×C3×C4—61no—
114C2×C3×C4—182no—
115C2×C3×C4—361no—
116C2×Dic3—32no—
117C2×Dic3—61no—
118C2×Dic3—62no—
119C2×Dic3—92no—
120C2×Dic3—122no—
121C2×Dic3—181no—
122C2×Dic3—242no—
123Dic6—364no—
124Dic6—364no—
125SL(2,𝔽3)—22no—
126SL(2,𝔽3)—41no—
127SL(2,𝔽3)#62711yescharacteristic, complemented
128C3×C8—182no—
129C3⋊C8—62no—
130C2×C32—42no—
131C2×C32—42no—
132C2×C32—42no—
133C2×C32—81no—
134C2×C32—81no—
135C2×C32—81no—
136C2×C32—82no—
137C2×C32—82no—
138C2×C32—82no—
139C2×C32—161no—
140C2×C32—161no—
141C2×C32—161no—
142C2×C32A4⋊Q811yescharacteristic
143C2×C32C2×GL(2,𝔽3)11yescharacteristic
144C2×C32#98411yescharacteristic
145C2×Q8—541no—
146C2×Q8#111311yescharacteristic
147C2×C8—92no—
148C4⋊C4—92no—
149C4⋊C4—271no—
150C4⋊C4—541no—
151C42—271no—
152C22×C3—21no—
153C22×C3—41no—
154C22×C3—82no—
155C22×C3—161no—
156C22×C3D3×S412yes—
157C3×C4—32no—
158C3×C4—32no—
159C3×C4—61no—
160C3×C4—61no—
161C3×C4—362no—
162C3×C4—362no—
163Dic3—34no—
164Dic3—62no—
165Dic3—124no—
166Dic3—122no—
167Dic3—182no—
168Dic3—244no—
169Dic3—361no—
170C32—42no—
171C32—81no—
172C32—82no—
173C32—161no—
174C32#538611yescharacteristic, complemented
175Q8—31no—
176Q8—1082no—
177Q8#629611yescharacteristic, complemented
178C2×C4—31no—
179C2×C4—91no—
180C2×C4—182no—
181C2×C4—271no—
182C8—182no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3—21no—
186C2×C3—41no—
187C2×C3—41no—
188C2×C3—41no—
189C2×C3—82no—
190C2×C3—82no—
191C2×C3—82no—
192C2×C3—161no—
193C2×C3—161no—
194C2×C3—161no—
195C2×C3#6419612yes—
196C2×C3D3×GL(2,𝔽3)12yes—
197C2×C3#6420412yes—
198C22#8910111yescharacteristic
199C4—31no—
200C4—31no—
201C4—92no—
202C4—362no—
203C4—541no—
204C3—21no—
205C3—41no—
206C3—82no—
207C3—161no—
208C3#1059321212yescomplemented
209C2#1172420611yescharacteristic
210C2#1172421111yescharacteristic
211C2#1172421211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, prime radical, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)Q8
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
19. C32×SL(2,𝔽3)Q8semidirect
50. C32×Q8Dic6semidirect
63. C3×SL(2,𝔽3)Dic6semidirect
111. C3×Q8#627semidirect
127. SL(2,𝔽3)#627semidirect
174. C32#5386semidirect
177. Q8#6296semidirect
208. C3#10593212semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×C3⋊Dic32
C3×C3⋊#8592
#105932123
#105916303
C2×#62964