Finite Group Databasebeta

← #13166094

Subgroups — #13166094

Order 1728.

443 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 50 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
2C3×C3⋊#859C212yesmaximal, complemented
3#10593209—32nomaximal
4#10591562—31nomaximal
5C2×C32×SL(2,𝔽3)C2211yescharacteristic
6C3⋊D3×SL(2,𝔽3)C412yescomplemented
7#88814—41nomaximal
8C2×Q8×C3⋊D3D311yescharacteristic
9C2×D3×SL(2,𝔽3)—32no—
10C2×D3×SL(2,𝔽3)—61no—
11#63849—31no—
12C2×#3712—31no—
13C3⋊#859D312yescomplemented
14C3×#859—32no—
15C3×#859—62no—
16C3×#835—32no—
17#63567—32no—
18C22×C3×C3⋊D3—41no—
19C2×C3×C3⋊Dic3—42no—
20C32×SL(2,𝔽3)C2×C411yescharacteristic
21#5420—91no—
22#4802—92no—
23C2×C32×Q8C2×D311yescharacteristic
24Q8×C3⋊D3—32no—
25Q8×C3⋊D3Dic312yescomplemented
26C2×C4×C3⋊D3—31no—
27C2×#634—31no—
28C2×C3×SL(2,𝔽3)—22no—
29C2×C3×SL(2,𝔽3)—21no—
30C2×C3×SL(2,𝔽3)—41no—
31C2×C3×SL(2,𝔽3)C2×D312yes—
32D3×SL(2,𝔽3)—34no—
33D3×SL(2,𝔽3)—62no—
34C3×C4⋊Dic3—32no—
35C2×C3×C3⋊C8—32no—
36#3722—61no—
37(C2×D3)⋊Dic3—122no—
38Dic32—61no—
39#3712—62no—
40C22×C33—41no—
41C2×C3×C3⋊D3—42no—
42C2×C3×C3⋊D3—82no—
43C3×C3⋊Dic3—82no—
44C2×D3×Q8—32no—
45C2×D3×Q8—61no—
46C22×SL(2,𝔽3)—91no—
47C2×ΓL(1,𝔽25)—92no—
48#892—92no—
49#859—32no—
50#859—62no—
51C3×#103—92no—
52#835—32no—
53#816—92no—
54#810—92no—
55C22×C3⋊D3S411yescharacteristic
56C22×C3×D3—122no—
57C22×C3×D3—241no—
58C32×Q8—31no—
59C32×Q8C2×Dic311yescharacteristic
60C2×C32×C4—31no—
61C2×C3⋊Dic3—31no—
62C2×C3⋊Dic3—42no—
63C4×C3⋊D3—32no—
64C4×C3⋊D3—32no—
65#634—32no—
66#634—61no—
67C2×C3×Dic3—62no—
68C2×C3×Dic3—122no—
69C2×C3×Dic3—242no—
70C3×SL(2,𝔽3)—22no—
71C3×SL(2,𝔽3)—21no—
72C3×SL(2,𝔽3)—41no—
73C3×SL(2,𝔽3)C4×D312yes—
74C3×C3⋊C8—62no—
75#419—271noSylow
76C2×C33—42no—
77C2×C33—41no—
78C3×C3⋊D3—82no—
79C2×C3×Q8—21no—
80C2×C3×Q8D3212yes—
81D3×Q8—34no—
82D3×Q8—62no—
83D3×Q8—94no—
84D3×Q8—182no—
85C2×C4×D3—92no—
86C2×C4×D3—181no—
87C2×Dic6—92no—
88C2×Dic6—181no—
89C2×SL(2,𝔽3)—22no—
90C2×SL(2,𝔽3)—41no—
91C2×SL(2,𝔽3)—92no—
92C2×SL(2,𝔽3)D3211yescharacteristic
93C2×C3×C8—92no—
94C3×C4⋊C4—92no—
95#217—361no—
96(C2×D3)⋊C4—182no—
97C4⋊Dic3—32no—
98C4×Dic3—182no—
99C2×C3⋊C8—32no—
100ΓL(1,𝔽25)—184no—
101C22×C32—42no—
102C22×C32—81no—
103C22×C32—82no—
104C22×C32—161no—
105C22×C32C2×S411yescharacteristic
106C2×C3⋊D3—22no—
107C2×C3⋊D3SL(2,ℤ/4)12yes—
108C2×C3×D3—124no—
109C2×C3×D3—242no—
110C2×C3×D3—242no—
111C2×C3×D3—242no—
112C2×C3×D3—244no—
113C32×C4—31no—
114C32×C4—31no—
115C3⋊Dic3—32no—
116C3⋊Dic3—82no—
117C3×Dic3—122no—
118C3×Dic3—242no—
119C3×Dic3—244no—
120C22×Q8—91no—
121C2×M16—271no—
122#117—271no—
123#103—92no—
124#103—272no—
125C33—41noSylow
126C23×C3—361no—
127C22×D3—32no—
128C22×D3—61no—
129C3×Q8—21no—
130C3×Q8—32no—
131C3×Q8—61no—
132C3×Q8D3×Dic312yes—
133C2×C3×C4—32no—
134C2×C3×C4—61no—
135C2×C3×C4—182no—
136C2×Dic3—62no—
137C2×Dic3—92no—
138C2×Dic3—122no—
139C2×Dic3—181no—
140C2×Dic3—242no—
141C4×D3—94no—
142C4×D3—94no—
143C4×D3—182no—
144C4×D3—182no—
145Dic6—94no—
146Dic6—182no—
147Dic6—182no—
148Dic6—182no—
149SL(2,𝔽3)—22no—
150SL(2,𝔽3)—41no—
151SL(2,𝔽3)#62411yescharacteristic
152C3×C8—182no—
153C3⋊C8—62no—
154C2×C32—42no—
155C2×C32—42no—
156C2×C32—42no—
157C2×C32—81no—
158C2×C32—82no—
159C2×C32—82no—
160C2×C32—82no—
161C2×C32—82no—
162C2×C32—161no—
163C2×C32—162no—
164C2×C32#98312yes—
165C2×C32C2×SL(2,ℤ/4)11yescharacteristic
166C3⋊D3—22no—
167C3×D3—244no—
168C3×D3—244no—
169C2×Q8—92no—
170C2×Q8—271no—
171C2×Q8—272no—
172C2×Q8#111311yescharacteristic
173C22×C4—271no—
174M16—542no—
175C2×C8—92no—
176C4⋊C4—92no—
177C22⋊C4—541no—
178C42—541no—
179C22×C3—21no—
180C22×C3—41no—
181C22×C3—82no—
182C22×C3—161no—
183C22×C3—362no—
184C22×C3—722no—
185C22×C3D3×S412yes—
186C2×D3—34no—
187C2×D3—62no—
188C2×D3—64no—
189C2×D3—62no—
190C2×D3—62no—
191C3×C4—32no—
192C3×C4—32no—
193C3×C4—61no—
194C3×C4—61no—
195C3×C4—362no—
196Dic3—94no—
197Dic3—122no—
198Dic3—182no—
199Dic3—242no—
200Dic3—244no—
201C32—42no—
202C32—81no—
203C32—82no—
204C32—161no—
205C32#542011yescharacteristic, complemented
206C23—91no—
207Q8—31no—
208Q8—272no—
209Q8—541no—
210Q8#629411yescharacteristic
211C2×C4—31no—
212C2×C4—182no—
213C2×C4—272no—
214C2×C4—272no—
215C2×C4—271no—
216C8—182no—
217C2×C3—22no—
218C2×C3—21no—
219C2×C3—42no—
220C2×C3—41no—
221C2×C3—82no—
222C2×C3—82no—
223C2×C3—82no—
224C2×C3—162no—
225C2×C3—161no—
226C2×C3—722no—
227C2×C3#6420112yes—
228C2×C3#6420212yes—
229C2×C3D3×SL(2,ℤ/4)12yes—
230D3—64no—
231D3—64no—
232C22—92no—
233C22—182no—
234C22#8910111yescharacteristic
235C4—31no—
236C4—31no—
237C4—272no—
238C4—362no—
239C3—21no—
240C3—41no—
241C3—82no—
242C3—161no—
243C3#1059320912yescomplemented
244C2—182no—
245C2#1172421012yes—
246C2#1172421511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#859C2semidirect
6. C3⋊D3×SL(2,𝔽3)C4semidirect
13. C3⋊#859D3semidirect
25. Q8×C3⋊D3Dic3semidirect
205. C32#5420semidirect
243. C3#10593209semidirect

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
C3×C3⋊#8592
#105932093
#105915623
#888144