Finite Group Databasebeta

← #13160359

Subgroups — #13160359

Order 1728.

824 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 188 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×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C32×#924C211yesmaximal, characteristic
3#11722200C211yesmaximal, characteristic
4#11722053C212yesmaximal, complemented
5#11722037C212yesmaximal, complemented
6#10592742—31nomaximal
7#10591101—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C2×C32×Dic6C2211yescharacteristic
11C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
12C3×C4⋊(C3⋊Dic3)—21no—
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C32×#217—21no—
15C32×C4⋊Dic3C2211yescharacteristic
16C32×Dic3⋊C4—21no—
17C32×C22⋊Q8—31no—
18C2×C4⋊(C3⋊Dic3)D311yescharacteristic
19C2×C3×C4⋊Dic3—34no—
20C3×#924D314yes—
21#63662—32no—
22#63657—31no—
23#63657—34no—
24#63657—64no—
25#63635—32no—
26#63574—38no—
27#63565—38no—
28C23×C33D411yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4D411yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3—42no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C32×Dic6—22no—
37C3⋊(C3⋊(C3⋊C8))—22no—
38#5043—94no—
39#4971—91no—
40C2×C32×Q8—31no—
41C22×C32×C4—24no—
42C22×C32×C4C2×D314yes—
43C22×C32×C4C2×D311yescharacteristic
44C22×C3⋊Dic3—21no—
45C22×C3×Dic3—64no—
46C2×C3×Dic6C2×D314yes—
47C32×C4⋊C4—31no—
48C32×C4⋊C4—61no—
49C32×C22⋊C4—61no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)C2×D312yes—
52C2×C3⋊(C3⋊C8)—32no—
53C2×C3⋊(C3⋊C8)—38no—
54C2×C3⋊(C3⋊C8)—68no—
55C3×#217—24no—
56C3×C4⋊Dic3—38no—
57C3×C4⋊Dic3—64no—
58C3×C4⋊Dic3C2×D314yes—
59C3×Dic3⋊C4—24no—
60C22×C33—21no—
61C22×C33—21no—
62C22×C33SD1612yes—
63C22×C33C2×D411yescharacteristic
64C33×C4—21no—
65C33×C4#4112yes—
66C3×C3⋊Dic3—42no—
67C32×Dic3—41no—
68C32×Dic3—41no—
69C3×C22⋊Q8—34no—
70C2×C3×C4⋊C4—91no—
71C2×C4⋊Dic3—34no—
72#924C3⋊D311yescharacteristic
73#835—98no—
74#830—94no—
75#830—91no—
76#830—184no—
77#810—92no—
78#808—98no—
79#808—92no—
80C23×C32—24no—
81C23×C32#6714yes—
82C23×C32#6711yescharacteristic
83C32×Q8—62no—
84C2×C32×C4—22no—
85C2×C32×C4—24no—
86C2×C32×C4—24no—
87C2×C32×C4—28no—
88C2×C32×C4—48no—
89C2×C32×C4—61no—
90C2×C32×C4—61no—
91C2×C32×C4#6714yes—
92C2×C32×C4#6711yescharacteristic
93C2×C32×C4C22×D314yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—42no—
97C2×C3×Dic3—24no—
98C2×C3×Dic3—24no—
99C2×C3×Dic3—68no—
100C2×C3×Dic3—128no—
101C3×Dic6—28no—
102C3⋊(C3⋊C8)—62no—
103C3⋊(C3⋊C8)—68no—
104C3⋊(C3⋊C8)—128no—
105#483—271noSylow
106C2×C33—22no—
107C2×C33#12311yescharacteristic
108C2×C33C2×SD1611yescharacteristic
109C2×C33#13711yescharacteristic
110C2×C3×Q8—34no—
111C22×C3×C4—24no—
112C22×C3×C4—181no—
113C22×C3×C4D3214yes—
114C22×C3×C4C2×C3⋊D311yescharacteristic
115C22×Dic3—64no—
116C2×Dic6C2×C3⋊D311yescharacteristic
117C3×C4⋊C4—34no—
118C3×C4⋊C4—64no—
119C3×C4⋊C4—92no—
120C3×C4⋊C4—181no—
121C3×C22⋊C4—64no—
122#217—21no—
123C4⋊Dic3—38no—
124C4⋊Dic3—64no—
125C4⋊Dic3C2×C3⋊D311yescharacteristic
126Dic3⋊C4—21no—
127C2×C3⋊C8—98no—
128C2×C3⋊C8—92no—
129C2×C3⋊C8—188no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—28no—
133C22×C32—24no—
134C22×C32—24no—
135C22×C32—24no—
136C22×C32—44no—
137C22×C32—44no—
138C22×C32#21412yes—
139C22×C32#21518yes—
140C22×C32C2×#6714yes—
141C22×C32C2×#6711yescharacteristic
142C32×C4—21no—
143C32×C4—28no—
144C32×C4—24no—
145C32×C4—44no—
146C32×C4—121no—
147C32×C4—121no—
148C32×C4#23512yes—
149C32×C4#23718yes—
150C3⋊Dic3—42no—
151C3×Dic3—44no—
152C3×Dic3—44no—
153C3×Dic3—128no—
154C22⋊Q8—31no—
155C2×C4⋊C4—91no—
156#106—272no—
157#103—272no—
158C22⋊C8—271no—
159C33#48311yescharacteristic, complemented, Sylow
160C23×C3—24no—
161C23×C3#62514yes—
162C23×C3#63811yescharacteristic
163C3×Q8—68no—
164C2×C3×C4—28no—
165C2×C3×C4—22no—
166C2×C3×C4—24no—
167C2×C3×C4—24no—
168C2×C3×C4—48no—
169C2×C3×C4—64no—
170C2×C3×C4—64no—
171C2×C3×C4—182no—
172C2×C3×C4—362no—
173C2×C3×C4#62514yes—
174C2×C3×C4#63811yescharacteristic
175C2×C3×C4C2×D3214yes—
176C2×C3×C4C22×C3⋊D311yescharacteristic
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3—68no—
180C2×Dic3—128no—
181Dic6—22no—
182C3⋊C8—188no—
183C3⋊C8—182no—
184C3⋊C8—368no—
185C2×C32—22no—
186C2×C32—24no—
187C2×C32—24no—
188C2×C32—24no—
189C2×C32—28no—
190C2×C32—48no—
191C2×C32#92911yescharacteristic
192C2×C32C2×#21411yescharacteristic
193C2×C32#93514yes—
194C2×C32C2×#21514yes—
195C2×C32#94211yescharacteristic
196C2×C32#95114yes—
197C2×Q8—31no—
198C22×C4—181no—
199C22×C4D3×C3⋊D311yescharacteristic
200C2×C8—272no—
201C4⋊C4—31no—
202C4⋊C4—61no—
203C4⋊C4—92no—
204C4⋊C4—181no—
205C22⋊C4—61no—
206C22×C3—24no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—24no—
210C22×C3—24no—
211C22×C3—28no—
212C22×C3—44no—
213C22×C3—44no—
214C22×C3#371518yes—
215C22×C3#375512yes—
216C22×C3C2×#62514yes—
217C22×C3C2×#63811yescharacteristic
218C3×C4—21no—
219C3×C4—24no—
220C3×C4—28no—
221C3×C4—44no—
222C3×C4—124no—
223C3×C4—124no—
224C3×C4—362no—
225C3×C4#380418yes—
226C3×C4#383012yes—
227Dic3—41no—
228Dic3—41no—
229Dic3—128no—
230C32—24no—
231C32#497111yescharacteristic, complemented
232C32#504314yescomplemented
233C23#629011yescharacteristic
234Q8—62no—
235C2×C4—22no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—182no—
239C2×C4—362no—
240C2×C4#629011yescharacteristic
241C2×C4C2×D3×C3⋊D311yescharacteristic
242C8—542no—
243C2×C3—28no—
244C2×C3—22no—
245C2×C3—24no—
246C2×C3—24no—
247C2×C3—24no—
248C2×C3—48no—
249C2×C3C2×#371514yes—
250C2×C3#6382714yes—
251C2×C3#6396114yes—
252C2×C3#6414411yescharacteristic
253C2×C3C2×#375511yescharacteristic
254C2×C3#6416011yescharacteristic
255C22—21no—
256C22—21no—
257C22#8879312yes—
258C22C2×#629011yescharacteristic
259C4—21no—
260C4—121no—
261C4—121no—
262C4—362no—
263C4#8902812yes—
264C3—24no—
265C3#1059110114yescomplemented
266C3#1059274211yescharacteristic, complemented
267C2—22no—
268C2#1172337311yescharacteristic
269C2C2×#8879311yescharacteristic
270C2#1172360211yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#483
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722053C2semidirect
5. #11722037C2semidirect
159. C33#483semidirect
231. C32#4971semidirect
232. C32#5043semidirect
265. C3#10591101semidirect
266. C3#10592742semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C32×#9242
#117222002
#117220532
#117220372
#105927423
#105911013