Finite Group Databasebeta

← #13166089

Subgroups — #13166089

Order 1728.

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

normal subgroups only

This group has 324 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#10593202—32nomaximal
4#10591631—31nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6#88815—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#63575—32no—
16#63572—31no—
17C2×C3×C3⋊Dic3—41no—
18C2×C3×C3⋊Dic3—42no—
19C32×SL(2,𝔽3)Q811yescharacteristic, complemented
20#5391—91no—
21#4793—92no—
22C2×C32×Q8C2×D311yescharacteristic
23C2×C3×SL(2,𝔽3)—22no—
24C2×C3×SL(2,𝔽3)—21no—
25C2×C3×SL(2,𝔽3)—41no—
26C2×C3×SL(2,𝔽3)C2×D312yes—
27C2×#627—61no—
28C4⋊(C3⋊Dic3)—31no—
29C4×C3⋊Dic3—31no—
30C3×C4⋊Dic3—32no—
31C2×C3×C3⋊C8—32no—
32#3724—61no—
33Dic3⋊Dic3—122no—
34C22×C33—41no—
35C3×C3⋊Dic3—42no—
36C3×C3⋊Dic3—82no—
37Q8×Dic3—32no—
38Q8×Dic3—61no—
39#888—92no—
40C4×SL(2,𝔽3)—91no—
41#859—32no—
42#859—62no—
43C3×#103—92no—
44#835—32no—
45#818—92no—
46Dic3⋊C8—92no—
47#807—92no—
48C32×Q8—31no—
49C32×Q8Dic611yescharacteristic
50C2×C32×C4—31no—
51C2×C3⋊Dic3—31no—
52C2×C3⋊Dic3—42no—
53C2×C3⋊Dic3S411yescharacteristic
54C2×C3×Dic3—62no—
55C2×C3×Dic3—122no—
56C2×C3×Dic3—122no—
57C2×C3×Dic3—241no—
58C2×C3×Dic3—242no—
59C3×SL(2,𝔽3)—22no—
60C3×SL(2,𝔽3)—21no—
61C3×SL(2,𝔽3)—41no—
62C3×SL(2,𝔽3)Dic612yescomplemented
63#627—122no—
64C3×C3⋊C8—62no—
65#477—271noSylow
66C2×C33—41no—
67C2×C33—41no—
68C2×C33—41no—
69C2×C3×Q8—21no—
70C2×C3×Q8D3212yes—
71C2×Dic6—182no—
72C2×SL(2,𝔽3)—22no—
73C2×SL(2,𝔽3)—41no—
74C2×SL(2,𝔽3)D3211yescharacteristic
75C2×C3×C8—92no—
76C3×C4⋊C4—92no—
77C4⋊Dic3—32no—
78C4⋊Dic3—92no—
79C4⋊Dic3—181no—
80C4⋊Dic3—361no—
81Dic3⋊C4—182no—
82C4×Dic3—92no—
83C4×Dic3—181no—
84C2×C3⋊C8—32no—
85C22×C32—42no—
86C22×C32—81no—
87C22×C32—82no—
88C22×C32—161no—
89C22×C32C2×S411yescharacteristic
90C32×C4—31no—
91C32×C4—31no—
92C3⋊Dic3—61no—
93C3⋊Dic3—82no—
94C3⋊Dic3#22612yes—
95C3×Dic3—124no—
96C3×Dic3—122no—
97C3×Dic3—242no—
98C3×Dic3—242no—
99C3×Dic3—244no—
100C4⋊Q8—271no—
101C4×Q8—91no—
102#107—272no—
103C4⋊C8—271no—
104#103—92no—
105C33—41noSylow
106C3×Q8—21no—
107C3×Q8—32no—
108C3×Q8—61no—
109C3×Q8#62712yes—
110C2×C3×C4—32no—
111C2×C3×C4—61no—
112C2×C3×C4—182no—
113C2×C3×C4—361no—
114C2×Dic3—32no—
115C2×Dic3—61no—
116C2×Dic3—62no—
117C2×Dic3—92no—
118C2×Dic3—122no—
119C2×Dic3—181no—
120C2×Dic3—242no—
121Dic6—364no—
122SL(2,𝔽3)—22no—
123SL(2,𝔽3)—41no—
124SL(2,𝔽3)#62711yescharacteristic, complemented
125C3×C8—182no—
126C3⋊C8—62no—
127C2×C32—42no—
128C2×C32—42no—
129C2×C32—42no—
130C2×C32—81no—
131C2×C32—81no—
132C2×C32—81no—
133C2×C32—82no—
134C2×C32—82no—
135C2×C32—82no—
136C2×C32—161no—
137C2×C32—161no—
138C2×C32—161no—
139C2×C32A4⋊Q811yescharacteristic
140C2×C32C2×#22611yescharacteristic
141C2×C32#98611yescharacteristic
142C2×Q8—541no—
143C2×Q8#111311yescharacteristic
144C2×C8—92no—
145C4⋊C4—92no—
146C4⋊C4—271no—
147C4⋊C4—541no—
148C42—271no—
149C22×C3—21no—
150C22×C3—41no—
151C22×C3—82no—
152C22×C3—161no—
153C22×C3D3×S412yes—
154C3×C4—32no—
155C3×C4—32no—
156C3×C4—61no—
157C3×C4—61no—
158C3×C4—362no—
159C3×C4—362no—
160Dic3—34no—
161Dic3—62no—
162Dic3—122no—
163Dic3—182no—
164Dic3—242no—
165Dic3—244no—
166Dic3—361no—
167C32—42no—
168C32—81no—
169C32—82no—
170C32—161no—
171C32#539111yescharacteristic, complemented
172Q8—31no—
173Q8—1082no—
174Q8#629611yescharacteristic
175C2×C4—31no—
176C2×C4—91no—
177C2×C4—182no—
178C2×C4—271no—
179C8—182no—
180C2×C3—21no—
181C2×C3—21no—
182C2×C3—21no—
183C2×C3—41no—
184C2×C3—41no—
185C2×C3—41no—
186C2×C3—82no—
187C2×C3—82no—
188C2×C3—82no—
189C2×C3—161no—
190C2×C3—161no—
191C2×C3—161no—
192C2×C3#6419912yes—
193C2×C3D3×#22612yes—
194C2×C3#6420412yes—
195C22#8910111yescharacteristic
196C4—31no—
197C4—31no—
198C4—92no—
199C4—362no—
200C4—541no—
201C3—21no—
202C3—41no—
203C3—82no—
204C3—161no—
205C3#1059320212yescomplemented
206C2#1172420811yescharacteristic
207C2#1172420911yescharacteristic
208C2#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
62. C3×SL(2,𝔽3)Dic6semidirect
124. SL(2,𝔽3)#627semidirect
171. C32#5391semidirect
205. C3#10593202semidirect

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
#105932023
#105916313
#888154