Finite Group Databasebeta

← #13139004

Subgroups — #13139004

Order 1728.

445 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88530C211yesmaximal, characteristic
2#11721083C212yesmaximal
3#63645⋊C3C211yesmaximal, characteristic
4C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
5#11720326C212yesmaximal
6#10590818—31nomaximal
7Dic3×Dic12—31nomaximal
8C2×#6229C2212yes—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10#88530C414yescomplemented
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#88449—22no—
15#88447—22no—
16C2×C3×Dic12—31no—
17C2×C3×Dic12—31no—
18C2×C3×Dic12—61no—
19#63843—32no—
20Dic3×Dic6—32no—
21#63645D311yescharacteristic
22C8×C3⋊Dic3D311yescharacteristic
23C3×#818—31no—
24C3×C8×Dic3—31no—
25#63566—32no—
26#63565—32no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—22no—
29#6229—22no—
30#6229C2×C414yes—
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3D411yescharacteristic
33C2×(C3⋊Dic3)⋊C3D411yescharacteristic
34C8×H(3,𝔽3)C2×C412yes—
35(C3⋊(C3⋊C8))⋊C3—21no—
36Dic3×Q16—91no—
37#4873—91no—
38C4×Dic12—91no—
39C2×C3×Dic6—32no—
40C2×C3×Dic6—32no—
41C2×C3×Dic6—62no—
42C2×C32×C8—21no—
43C2×C32×C8C2×D311yescharacteristic
44C2×C32×C8C2×D311yescharacteristic
45C4⋊(C3⋊Dic3)C2×D312yes—
46C4×C3⋊Dic3C2×D311yescharacteristic
47C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
48C3×C4⋊Dic3—32no—
49C3×C4×Dic3—31no—
50C2×C3×C3⋊C8—31no—
51C3×Dic12—34no—
52C3×Dic12—34no—
53C3×Dic12—64no—
54Dic3⋊Dic3—62no—
55Dic3⋊Dic3—62no—
56Dic32—62no—
57Dic32—62no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C4×H(3,𝔽3)C22×C411yescharacteristic
60C4×H(3,𝔽3)#4111yescharacteristic
61#1084—24no—
62(C3⋊Dic3)⋊C3—21no—
63(C3⋊Dic3)⋊C3—42no—
64(C3⋊Dic3)⋊C3Q1612yescomplemented
65C2×C3×Q16—91no—
66Q8×Dic3—92no—
67C2×Dic12—31no—
68C2×Dic12—31no—
69C2×Dic12—61no—
70#887—92no—
71C4×Dic6—92no—
72C3×#107—91no—
73C3×C4×C8—91no—
74#835—92no—
75#818—31no—
76#818—91no—
77#816—92no—
78C8×Dic3—31no—
79C8×Dic3—91no—
80#810—92no—
81C2×C32×C4—21no—
82C2×C32×C4C22×D311yescharacteristic
83C2×C32×C4C22×D311yescharacteristic
84C2×C3⋊Dic3—22no—
85C2×C3⋊Dic3D1211yescharacteristic
86C2×C3⋊Dic3D1211yescharacteristic
87C2×C3×Dic3—31no—
88C2×C3×Dic3—31no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—122no—
93C3×Dic6—34no—
94C3×Dic6—34no—
95C3×Dic6—64no—
96C3×Dic6—62no—
97C3×Dic6—62no—
98C3×Dic6—122no—
99C32×C8—22no—
100C32×C8C4×D312yes—
101C32×C8C4×D312yes—
102C3⋊(C3⋊C8)—21no—
103C3×C3⋊C8—61no—
104C4×Q16—271noSylow
105C2×H(3,𝔽3)C4×D411yescharacteristic
106C2×H(3,𝔽3)C2×Q1611yescharacteristic
107C2×H(3,𝔽3)#13511yescharacteristic
108C2×C3×Q8—92no—
109C2×Dic6—32no—
110C2×Dic6—32no—
111C2×Dic6—62no—
112C3×Q16—94no—
113C2×C3×C8—31no—
114C2×C3×C8—31no—
115C2×C3×C8—61no—
116C2×C3×C8—91no—
117C2×C3×C8D3211yescharacteristic
118C3×C4⋊C4—92no—
119C3×C42—91no—
120C4⋊Dic3—32no—
121C4⋊Dic3—92no—
122C4⋊Dic3—182no—
123Dic3⋊C4—182no—
124Dic3⋊C4—182no—
125C4×Dic3—31no—
126C4×Dic3—91no—
127C4×Dic3—182no—
128C4×Dic3—182no—
129C4×Dic3—182no—
130C2×C3⋊C8—31no—
131C2×C3⋊C8—91no—
132Dic12—34no—
133Dic12—34no—
134Dic12—64no—
135C22×C32—21no—
136C22×C32C2×D1211yescharacteristic
137C22×C32D3×D411yescharacteristic
138C32×C4—21no—
139C32×C4—21no—
140C32×C4C2×C4×D311yescharacteristic
141C32×C4C2×C4×D311yescharacteristic
142C32×C4#23511yescharacteristic
143C32×C4#23911yescharacteristic
144C3⋊Dic3—21no—
145C3⋊Dic3—42no—
146C3⋊Dic3Dic1212yescomplemented
147C3×Dic3—32no—
148C3×Dic3—64no—
149C3×Dic3—64no—
150C3×Dic3—61no—
151C3×Dic3—124no—
152C3×Dic3—122no—
153C2×Q16—91no—
154C4×Q8—272no—
155#107—91no—
156#103—272no—
157C4×C8—91no—
158H(3,𝔽3)C4×Q1611yescharacteristic, complemented, Sylow
159C3×Q8—94no—
160C3×Q8—182no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—61no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4—182no—
167C2×C3×C4—182no—
168C2×C3×C4C2×D3211yescharacteristic
169C2×Dic3—31no—
170C2×Dic3—31no—
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—91no—
175C2×Dic3—91no—
176C2×Dic3—122no—
177C2×Dic3—182no—
178Dic6—34no—
179Dic6—34no—
180Dic6—64no—
181Dic6—62no—
182Dic6—62no—
183Dic6—122no—
184C3×C8—32no—
185C3×C8—32no—
186C3×C8—62no—
187C3×C8—181no—
188C3×C8#62412yes—
189C3⋊C8—61no—
190C3⋊C8—181no—
191C2×C32—21no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32C4×D1211yescharacteristic
195C2×C32#89311yescharacteristic
196C2×C32#90411yescharacteristic
197C2×C32C2×Dic1211yescharacteristic
198C2×C32#91211yescharacteristic
199C2×C32D3×Q1611yescharacteristic
200C2×Q8—92no—
201Q16—94no—
202C2×C8—91no—
203C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
204C4⋊C4—92no—
205C4⋊C4—542no—
206C42—91no—
207C42—542no—
208C22×C3—31no—
209C22×C3—31no—
210C22×C3—61no—
211C22×C3D3×D1211yescharacteristic
212C3×C4—31no—
213C3×C4—31no—
214C3×C4—31no—
215C3×C4—31no—
216C3×C4—61no—
217C3×C4—61no—
218C3×C4—92no—
219C3×C4—181no—
220C3×C4—184no—
221C3×C4—362no—
222C3×C4#379911yescharacteristic
223C3×C4C2×#62411yescharacteristic
224Dic3—32no—
225Dic3—64no—
226Dic3—64no—
227Dic3—61no—
228Dic3—92no—
229Dic3—124no—
230Dic3—122no—
231Dic3—181no—
232Dic3—362no—
233C32—21no—
234C32C4×Dic1211yescharacteristic, complemented
235C32#487311yescharacteristic, complemented
236Q8—94no—
237Q8—182no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—182no—
241C2×C4—182no—
242C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C8—181no—
244C8#619912yes—
245C2×C3—31no—
246C2×C3—31no—
247C2×C3—31no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—31no—
251C2×C3—61no—
252C2×C3—61no—
253C2×C3—61no—
254C2×C3D3×Dic1211yescharacteristic
255C2×C3#6380711yescharacteristic
256C2×C3#6389411yescharacteristic
257C22#8865411yescharacteristic
258C4—92no—
259C4—181no—
260C4—184no—
261C4—362no—
262C4#8865211yescharacteristic
263C4C2×#619911yescharacteristic
264C3—31no—
265C3—31no—
266C3—61no—
267C3#1059080711yescharacteristic
268C2#1172085111yescharacteristic
269C2#1172085511yescharacteristic
270C2#1172114911yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×Q16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
10. #88530C4semidirect
64. (C3⋊Dic3)⋊C3Q16semidirect
146. C3⋊Dic3Dic12semidirect
158. H(3,𝔽3)C4×Q16semidirect
234. C32C4×Dic12semidirect
235. C32#4873semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
#117210832
#63645⋊C32
C8×(C3⋊Dic3)⋊C32
#117203262
#105908183
Dic3×Dic123