Finite Group Databasebeta

← #13166087

Subgroups — #13166087

Order 1728.

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

normal subgroups only

This group has 364 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
1C2×C3×C3⋊#226C212yesmaximal
2SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic
3#10593201—32nomaximal
4#10591633—31nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C3×C3⋊#226—22no—
7#88815—41nomaximal
8C2×C3⋊#226D312yes—
9C2×C3×#226—32no—
10C2×C3×#226—62no—
11Q8×C3⋊Dic3D311yescharacteristic
12C2×C3×#216—32no—
13#63873—31no—
14Dic3×SL(2,𝔽3)—32no—
15Dic3×SL(2,𝔽3)—61no—
16#63572—31no—
17#63566—32no—
18C2×C3×C3⋊Dic3—41no—
19C2×C3×C3⋊Dic3—42no—
20C32×SL(2,𝔽3)D411yescharacteristic
21C4⋊#226—91no—
22#5182—92no—
23C2×C32×Q8C2×D311yescharacteristic
24C2×C3×Dic6—32no—
25C2×C3×SL(2,𝔽3)—22no—
26C2×C3×SL(2,𝔽3)—21no—
27C2×C3×SL(2,𝔽3)—41no—
28C2×C3×SL(2,𝔽3)C2×D312yes—
29C3⋊#226—22no—
30C3×#226—64no—
31C3×#226—62no—
32C4⋊(C3⋊Dic3)—31no—
33C4×C3⋊Dic3—31no—
34C3×#216—62no—
35C3×#216—62no—
36C2×C3×C3⋊C8—32no—
37#3724—61no—
38#3724—61no—
39Dic3⋊Dic3—122no—
40C22×C33—41no—
41C3×C3⋊Dic3—42no—
42C3×C3⋊Dic3—82no—
43C2×#226—32no—
44C2×#226—62no—
45C2×C3×Q16—92no—
46Q8×Dic3—32no—
47Q8×Dic3—61no—
48#944—92no—
49C2×#216—32no—
50C4×SL(2,𝔽3)—91no—
51#835—92no—
52#816—92no—
53Dic3⋊C8—92no—
54C32×Q8—31no—
55C32×Q8D1211yescharacteristic
56C2×C32×C4—31no—
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—42no—
59C2×C3⋊Dic3S411yescharacteristic
60C2×C3×Dic3—62no—
61C2×C3×Dic3—122no—
62C2×C3×Dic3—122no—
63C2×C3×Dic3—241no—
64C2×C3×Dic3—242no—
65C3×Dic6—62no—
66C3×Dic6—62no—
67C3×SL(2,𝔽3)—22no—
68C3×SL(2,𝔽3)—21no—
69C3×SL(2,𝔽3)—41no—
70C3×SL(2,𝔽3)#6712yes—
71C3×C3⋊C8—62no—
72#462—271noSylow
73C2×C33—41no—
74C2×C33—41no—
75C2×C33—41no—
76C2×C3×Q8—21no—
77C2×C3×Q8—92no—
78C2×C3×Q8D3212yes—
79C2×Dic6—32no—
80C2×SL(2,𝔽3)—22no—
81C2×SL(2,𝔽3)—41no—
82C2×SL(2,𝔽3)D3211yescharacteristic
83#226—64no—
84#226—62no—
85C3×Q16—182no—
86C3×Q16—182no—
87C2×C3×C8—92no—
88#216—62no—
89#216—62no—
90C4⋊Dic3—92no—
91C4⋊Dic3—181no—
92C4⋊Dic3—361no—
93Dic3⋊C4—182no—
94Dic3⋊C4—182no—
95C4×Dic3—92no—
96C4×Dic3—181no—
97C2×C3⋊C8—32no—
98C22×C32—42no—
99C22×C32—81no—
100C22×C32—82no—
101C22×C32—161no—
102C22×C32C2×S411yescharacteristic
103C32×C4—31no—
104C32×C4—31no—
105C3⋊Dic3—61no—
106C3⋊Dic3—82no—
107C3⋊Dic3#22612yescomplemented
108C3×Dic3—124no—
109C3×Dic3—122no—
110C3×Dic3—242no—
111C3×Dic3—242no—
112C3×Dic3—244no—
113C2×Q16—92no—
114C4⋊Q8—271no—
115C4×Q8—91no—
116C4⋊C8—271no—
117#103—272no—
118C33—41noSylow
119C3×Q8—21no—
120C3×Q8—32no—
121C3×Q8—61no—
122C3×Q8—182no—
123C3×Q8—182no—
124C3×Q8#62612yes—
125C2×C3×C4—32no—
126C2×C3×C4—61no—
127C2×C3×C4—182no—
128C2×C3×C4—361no—
129C2×Dic3—32no—
130C2×Dic3—61no—
131C2×Dic3—62no—
132C2×Dic3—92no—
133C2×Dic3—122no—
134C2×Dic3—181no—
135C2×Dic3—242no—
136Dic6—62no—
137Dic6—62no—
138SL(2,𝔽3)—22no—
139SL(2,𝔽3)—41no—
140SL(2,𝔽3)#62511yescharacteristic
141C3×C8—182no—
142C3⋊C8—62no—
143C2×C32—42no—
144C2×C32—42no—
145C2×C32—42no—
146C2×C32—81no—
147C2×C32—81no—
148C2×C32—81no—
149C2×C32—82no—
150C2×C32—82no—
151C2×C32—82no—
152C2×C32—161no—
153C2×C32—161no—
154C2×C32—161no—
155C2×C32C4⋊S411yescharacteristic
156C2×C32C2×#22611yescharacteristic
157C2×C32#98411yescharacteristic
158C2×Q8—92no—
159C2×Q8#111311yescharacteristic
160Q16—182no—
161Q16—182no—
162C2×C8—92no—
163C4⋊C4—271no—
164C4⋊C4—541no—
165C4⋊C4—541no—
166C42—271no—
167C22×C3—21no—
168C22×C3—41no—
169C22×C3—82no—
170C22×C3—161no—
171C22×C3D3×S412yes—
172C3×C4—32no—
173C3×C4—32no—
174C3×C4—61no—
175C3×C4—61no—
176C3×C4—362no—
177C3×C4—362no—
178Dic3—34no—
179Dic3—62no—
180Dic3—122no—
181Dic3—182no—
182Dic3—242no—
183Dic3—244no—
184Dic3—361no—
185C32—42no—
186C32—81no—
187C32—82no—
188C32—161no—
189C32C4⋊#22611yescharacteristic, complemented
190Q8—31no—
191Q8—182no—
192Q8—182no—
193Q8#629511yescharacteristic
194C2×C4—31no—
195C2×C4—91no—
196C2×C4—182no—
197C2×C4—271no—
198C8—182no—
199C2×C3—21no—
200C2×C3—21no—
201C2×C3—21no—
202C2×C3—41no—
203C2×C3—41no—
204C2×C3—41no—
205C2×C3—82no—
206C2×C3—82no—
207C2×C3—82no—
208C2×C3—161no—
209C2×C3—161no—
210C2×C3—161no—
211C2×C3#6419612yes—
212C2×C3D3×#22612yes—
213C2×C3#6420712yes—
214C22#8910111yescharacteristic
215C4—31no—
216C4—31no—
217C4—92no—
218C4—362no—
219C4—541no—
220C3—21no—
221C3—41no—
222C3—82no—
223C3—161no—
224C3#1059320112yescomplemented
225C2#1172420611yescharacteristic
226C2#1172420911yescharacteristic
227C2#1172421411yescharacteristic
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)D4
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
107. C3⋊Dic3#226semidirect
189. C32C4⋊#226semidirect
224. C3#10593201semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#2262
SL(2,𝔽3)×C3⋊Dic32
#105932013
#105916333
#888154