Finite Group Databasebeta

← #10592572⋊C3 #13142219

Subgroups — #10592572⋊C3 #13142219

Order 1728.

421 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 116 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×#3746⋊C3C211yesmaximal, characteristic
2#64096⋊C3C212yesmaximal
3H(3,𝔽3)×#107C211yesmaximal, characteristic
4C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
5#63637⋊C3C212yesmaximal
6#10592572C311yesmaximal, characteristic, complemented
7C3×#4873—31nomaximal
8C2×#634⋊C3C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4⋊C4×H(3,𝔽3)C2212yes—
11#3750⋊C3—22no—
12C4×(C3⋊Dic3)⋊C3—22no—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#3746⋊C3C414yescomplemented
16C2×#3746C2×C311yescharacteristic
17#64096C2×C312yes—
18C2×C3×Dic12—31no—
19C3×#887—32no—
20C32×#107—21no—
21C32×#107D311yescharacteristic
22C32×#107C2×C311yescharacteristic
23C8×C3⋊Dic3C2×C311yescharacteristic
24#63637C2×C312yes—
25C3×C8×Dic3—31no—
26C3×#810—32no—
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×(C3⋊Dic3)⋊C3—22no—
30C2×(C3⋊Dic3)⋊C3D411yescharacteristic
31C2×(C3⋊Dic3)⋊C3D411yescharacteristic
32#634⋊C3—22no—
33#634⋊C3C2×C414yes—
34C8×H(3,𝔽3)C2×C412yes—
35(C3⋊(C3⋊C8))⋊C3—21no—
36C3×C4×Q16—91no—
37#4873—31no—
38#4873—91no—
39C2×#634C22×C312yes—
40C2×C3×Dic6—32no—
41C2×C32×C8—21no—
42C2×C32×C8C2×D311yescharacteristic
43C2×C32×C8C22×C311yescharacteristic
44C32×C4⋊C4—22no—
45C32×C4⋊C4C2×D312yes—
46C32×C4⋊C4C22×C312yes—
47#3750—22no—
48C4×C3⋊Dic3—22no—
49C4×C3⋊Dic3C22×C311yescharacteristic
50C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
51#3746C3×C414yescomplemented
52C3×Dic3⋊C4—62no—
53C3×C4×Dic3—31no—
54C3×C4×Dic3—62no—
55C2×C3×C3⋊C8—31no—
56C3×Dic12—34no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C4×H(3,𝔽3)—42no—
59C4×H(3,𝔽3)C22×C411yescharacteristic
60C4×H(3,𝔽3)#4111yescharacteristic
61(C3⋊Dic3)⋊C3—24no—
62(C3⋊Dic3)⋊C3—21no—
63(C3⋊Dic3)⋊C3Q1612yescomplemented
64C2×C3×Q16—91no—
65C3×C4×Q8—92no—
66C2×Dic12—31no—
67C2×Dic12—91no—
68#887—32no—
69#887—92no—
70C3×#107—31no—
71C3×#107—31no—
72C3×#107—61no—
73C3×#107C3×D311yescharacteristic
74C3×#103—92no—
75C3×C4×C8—91no—
76C8×Dic3—31no—
77C8×Dic3—91no—
78#810—32no—
79#810—92no—
80C2×C32×C4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—21no—
83C2×C32×C4—42no—
84C2×C32×C4C22×D311yescharacteristic
85C2×C32×C4C23×C311yescharacteristic
86C2×C3⋊Dic3—22no—
87C2×C3⋊Dic3C3×D411yescharacteristic
88C2×C3⋊Dic3C3×D411yescharacteristic
89#634—22no—
90#634C2×C3×C414yes—
91C2×C3×Dic3—31no—
92C2×C3×Dic3—31no—
93C2×C3×Dic3—62no—
94C3×Dic6—34no—
95C3×Dic6—62no—
96C32×C8—22no—
97C32×C8C4×D312yes—
98C32×C8C2×C3×C412yes—
99C3⋊(C3⋊C8)—21no—
100C3×C3⋊C8—61no—
101C4×Q16—91noSylow
102C2×H(3,𝔽3)C4×D411yescharacteristic
103C2×H(3,𝔽3)C2×Q1611yescharacteristic
104C2×H(3,𝔽3)#13511yescharacteristic
105C2×C3×Q8—92no—
106C2×Dic6—32no—
107C2×Dic6—92no—
108C3×Q16—94no—
109C2×C3×C8—31no—
110C2×C3×C8—31no—
111C2×C3×C8—61no—
112C2×C3×C8—91no—
113C2×C3×C8C2×C3×D311yescharacteristic
114C3×C4⋊C4—32no—
115C3×C4⋊C4—32no—
116C3×C4⋊C4—62no—
117C3×C4⋊C4—182no—
118C3×C4⋊C4C2×C3×D312yes—
119C3×C42—91no—
120C3×C42—182no—
121Dic3⋊C4—62no—
122Dic3⋊C4—182no—
123C4×Dic3—31no—
124C4×Dic3—62no—
125C4×Dic3—91no—
126C4×Dic3—182no—
127C2×C3⋊C8—31no—
128C2×C3⋊C8—91no—
129Dic12—34no—
130Dic12—94no—
131C22×C32—21no—
132C22×C32D3×D411yescharacteristic
133C22×C32C2×C3×D411yescharacteristic
134C32×C4—21no—
135C32×C4—21no—
136C32×C4—42no—
137C32×C4—42no—
138C32×C4—82no—
139C32×C4C2×C4×D311yescharacteristic
140C32×C4#23911yescharacteristic
141C32×C4C22×C3×C411yescharacteristic
142C32×C4C3×#4111yescharacteristic
143C3⋊Dic3—24no—
144C3⋊Dic3—21no—
145C3⋊Dic3C3×Q1612yescomplemented
146C3×Dic3—32no—
147C3×Dic3—64no—
148C3×Dic3—61no—
149C2×Q16—91no—
150C4×Q8—92no—
151#107(C3⋊D3)⋊C311yescharacteristic
152#103—92no—
153C4×C8—91no—
154H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
155C3×Q8—94no—
156C3×Q8—182no—
157C2×C3×C4—22no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—62no—
161C2×C3×C4—62no—
162C2×C3×C4—61no—
163C2×C3×C4—91no—
164C2×C3×C4—91no—
165C2×C3×C4—122no—
166C2×C3×C4—182no—
167C2×C3×C4C22×C3×D311yescharacteristic
168C2×Dic3—31no—
169C2×Dic3—31no—
170C2×Dic3—62no—
171C2×Dic3—91no—
172C2×Dic3—91no—
173C2×Dic3—182no—
174Dic6—34no—
175Dic6—62no—
176Dic6—94no—
177Dic6—182no—
178C3×C8—32no—
179C3×C8—32no—
180C3×C8—62no—
181C3×C8—181no—
182C3×C8C3×C4×D312yes—
183C3⋊C8—61no—
184C3⋊C8—181no—
185C2×C32—21no—
186C2×C32—21no—
187C2×C32—21no—
188C2×C32#89311yescharacteristic
189C2×C32#91211yescharacteristic
190C2×C32D3×Q1611yescharacteristic
191C2×C32C3×C4×D411yescharacteristic
192C2×C32C2×C3×Q1611yescharacteristic
193C2×C32C3×#13511yescharacteristic
194C2×Q8—92no—
195Q16—94no—
196C2×C8—91no—
197C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
198C4⋊C4—182no—
199C4⋊C4C2×(C3⋊D3)⋊C312yes—
200C42—91no—
201C42—182no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—61no—
205C22×C3C3×D3×D411yescharacteristic
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—31no—
209C3×C4—31no—
210C3×C4—42no—
211C3×C4—61no—
212C3×C4—61no—
213C3×C4—92no—
214C3×C4—122no—
215C3×C4—122no—
216C3×C4—181no—
217C3×C4—184no—
218C3×C4—242no—
219C3×C4C2×C3×C4×D311yescharacteristic
220C3×C4C3×#23911yescharacteristic
221Dic3—32no—
222Dic3—64no—
223Dic3—61no—
224Dic3—92no—
225Dic3—184no—
226Dic3—181no—
227C32—21no—
228C32#487311yescharacteristic, complemented
229C32C3×C4×Q1611yescharacteristic, complemented
230Q8—94no—
231Q8—182no—
232C2×C4—22no—
233C2×C4—91no—
234C2×C4—91no—
235C2×C4—182no—
236C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
237C8—181no—
238C8C4×(C3⋊D3)⋊C312yes—
239C2×C3—31no—
240C2×C3—31no—
241C2×C3—31no—
242C2×C3—31no—
243C2×C3—31no—
244C2×C3—31no—
245C2×C3—61no—
246C2×C3—61no—
247C2×C3—61no—
248C2×C3C3×#89311yescharacteristic
249C2×C3C3×#91211yescharacteristic
250C2×C3C3×D3×Q1611yescharacteristic
251C22D4×(C3⋊D3)⋊C311yescharacteristic
252C4—42no—
253C4—92no—
254C4—181no—
255C4—184no—
256C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
257C4#8872211yescharacteristic
258C3—31no—
259C3—31no—
260C3—61no—
261C3C3×#487311yescharacteristic
262C2#64102⋊C311yescharacteristic
263C2#64121⋊C311yescharacteristic
264C2Q16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C32×C4C23×C3
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×C4×Q16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
6. #10592572C3semidirect
15. #3746⋊C3C4semidirect
51. #3746C3×C4semidirect
63. (C3⋊Dic3)⋊C3Q16semidirect
145. C3⋊Dic3C3×Q16semidirect
154. H(3,𝔽3)C4×Q16semidirect
228. C32#4873semidirect
229. C32C3×C4×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#3746⋊C32
#64096⋊C32
H(3,𝔽3)×#1072
C8×(C3⋊Dic3)⋊C32
#63637⋊C32
#105925723
C3×#48733