Finite Group Databasebeta

← #13160367

Subgroups — #13160367

Order 1728.

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

normal subgroups only

This group has 920 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
1#11723965C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic
4#11722055C212yesmaximal, complemented
5#11722042C212yesmaximal, complemented
6#10592421—31nomaximal
7#10591135—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C2×#6308C2211yescharacteristic
10C22×C3×C3⋊Dic3—21no—
11#88864—21no—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13#88857—21no—
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C2×C32×C3⋊C8C2212yes—
17C2×C4⋊(C3⋊Dic3)D311yescharacteristic
18#64133—31no—
19#64133—34no—
20#64133—64no—
21C2×C3×C4⋊Dic3—34no—
22C32×C22⋊C8—31no—
23#63645—32no—
24#63643—32no—
25C3×#830D314yes—
26#63575—38no—
27#63567—38no—
28C23×C33D411yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4D411yescharacteristic
31C2×C33×C4C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)—21no—
34#6308—22no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—42no—
37C32×C3⋊C8—22no—
38#4970—91no—
39#4722—94no—
40C22×C32×C4—24no—
41C22×C32×C4C2×D314yes—
42C22×C32×C4C2×D311yescharacteristic
43C22×C3⋊Dic3—21no—
44C2×#634—31no—
45C2×#634—34no—
46C2×#634—64no—
47C22×C3×Dic3—64no—
48C2×C32×C8—32no—
49#3757—68no—
50#3757—64no—
51#3757—61no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)—31no—
54C4⋊(C3⋊Dic3)—34no—
55C4⋊(C3⋊Dic3)—64no—
56C4⋊(C3⋊Dic3)C2×D312yes—
57#3750—68no—
58#3750—64no—
59#3750—61no—
60C3×C4⋊Dic3—38no—
61C3×C4⋊Dic3—64no—
62C2×C3×C3⋊C8C2×D318yes—
63C22×C33—21no—
64C22×C33—21no—
65C22×C33Q1612yes—
66C22×C33C2×D411yescharacteristic
67C33×C4—21no—
68C33×C4#4112yes—
69C3⋊(C3⋊Dic3)—41no—
70C3⋊(C3⋊Dic3)—41no—
71C3×C3⋊Dic3—42no—
72C2×C3×C4⋊C4—91no—
73C2×C4⋊Dic3—34no—
74#924—94no—
75#924—91no—
76#924—184no—
77C3×C22⋊C8—34no—
78#830C3⋊D311yescharacteristic
79#818—98no—
80#816—98no—
81#810—92no—
82#807—92no—
83C23×C32—24no—
84C23×C32D1214yes—
85C23×C32#6711yescharacteristic
86C2×C32×C4—22no—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4—28no—
90C2×C32×C4—48no—
91C2×C32×C4D1214yes—
92C2×C32×C4#6711yescharacteristic
93C2×C32×C4C22×D314yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—42no—
97C2×C3⋊Dic3—68no—
98C2×C3⋊Dic3—68no—
99C2×C3⋊Dic3—64no—
100C2×C3⋊Dic3—64no—
101C2×C3⋊Dic3—61no—
102C2×C3⋊Dic3—61no—
103#634—62no—
104#634—68no—
105#634—128no—
106C2×C3×Dic3—68no—
107C2×C3×Dic3—128no—
108C32×C8—62no—
109C3×C3⋊C8—28no—
110#484—271noSylow
111C2×C33—22no—
112C2×C33#12311yescharacteristic
113C2×C33C2×Q1611yescharacteristic
114C2×C33AGL(1,ℤ/8)11yescharacteristic
115C22×C3×C4—24no—
116C22×C3×C4—181no—
117C22×C3×C4D3214yes—
118C22×C3×C4C2×C3⋊D311yescharacteristic
119C22×Dic3—64no—
120C2×Dic6—94no—
121C2×Dic6—91no—
122C2×Dic6—184no—
123C2×C3×C8—38no—
124C3×C4⋊C4—92no—
125C3×C4⋊C4—181no—
126#217—188no—
127#217—184no—
128#217—181no—
129C4⋊Dic3—38no—
130C4⋊Dic3—64no—
131C4⋊Dic3—94no—
132C4⋊Dic3—91no—
133C4⋊Dic3—184no—
134Dic3⋊C4—188no—
135Dic3⋊C4—184no—
136Dic3⋊C4—181no—
137C2×C3⋊C8C2×C3⋊D312yes—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—28no—
141C22×C32—24no—
142C22×C32—24no—
143C22×C32—24no—
144C22×C32—44no—
145C22×C32—44no—
146C22×C32Dic1218yes—
147C22×C32#21612yes—
148C22×C32C2×D1214yes—
149C22×C32C2×#6711yescharacteristic
150C32×C4—21no—
151C32×C4—28no—
152C32×C4—24no—
153C32×C4—44no—
154C32×C4#23512yes—
155C32×C4#23718yes—
156C3⋊Dic3—42no—
157C3⋊Dic3—128no—
158C3⋊Dic3—128no—
159C3⋊Dic3—124no—
160C3⋊Dic3—124no—
161C3⋊Dic3—121no—
162C3⋊Dic3—121no—
163C3×Dic3—128no—
164C22⋊Q8—271no—
165C2×C4⋊C4—91no—
166#107—272no—
167#103—272no—
168C22⋊C8—31no—
169C33#48411yescharacteristic, complemented, Sylow
170C23×C3—24no—
171C23×C3#62614yes—
172C23×C3C4⋊(C3⋊D3)11yescharacteristic
173C2×C3×C4—28no—
174C2×C3×C4—22no—
175C2×C3×C4—24no—
176C2×C3×C4—24no—
177C2×C3×C4—48no—
178C2×C3×C4—182no—
179C2×C3×C4—362no—
180C2×C3×C4#62614yes—
181C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
182C2×C3×C4C2×D3214yes—
183C2×C3×C4C22×C3⋊D311yescharacteristic
184C2×Dic3—68no—
185C2×Dic3—128no—
186C2×Dic3—188no—
187C2×Dic3—188no—
188C2×Dic3—184no—
189C2×Dic3—184no—
190C2×Dic3—181no—
191C2×Dic3—181no—
192Dic6—188no—
193Dic6—182no—
194Dic6—368no—
195C3×C8—68no—
196C3⋊C8—22no—
197C2×C32—22no—
198C2×C32—24no—
199C2×C32—24no—
200C2×C32—24no—
201C2×C32—28no—
202C2×C32—48no—
203C2×C32#88614yes—
204C2×C32C2×Dic1214yes—
205C2×C32#90814yes—
206C2×C32#92911yescharacteristic
207C2×C32#93211yescharacteristic
208C2×C32C2×#21611yescharacteristic
209C2×Q8—271no—
210C22×C4—181no—
211C22×C4D3×C3⋊D311yescharacteristic
212C2×C8—32no—
213C4⋊C4—92no—
214C4⋊C4—181no—
215C4⋊C4—271no—
216C4⋊C4—541no—
217C22⋊C4—541no—
218C22×C3—24no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—24no—
222C22×C3—24no—
223C22×C3—28no—
224C22×C3—44no—
225C22×C3—44no—
226C22×C3#371918yes—
227C22×C3#374612yes—
228C22×C3C2×#62614yes—
229C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
230C3×C4—21no—
231C3×C4—24no—
232C3×C4—28no—
233C3×C4—44no—
234C3×C4—362no—
235C3×C4#380418yes—
236C3×C4#383012yes—
237Dic3—128no—
238Dic3—368no—
239Dic3—368no—
240Dic3—364no—
241Dic3—361no—
242Dic3—361no—
243Dic3—364no—
244C32—24no—
245C32#472214yescomplemented
246C32#497011yescharacteristic, complemented
247C23#629211yescharacteristic
248Q8—542no—
249C2×C4—22no—
250C2×C4—182no—
251C2×C4—362no—
252C2×C4—541no—
253C2×C4—541no—
254C2×C4#629211yescharacteristic
255C2×C4C2×D3×C3⋊D311yescharacteristic
256C8—62no—
257C2×C3—28no—
258C2×C3—22no—
259C2×C3—24no—
260C2×C3—24no—
261C2×C3—24no—
262C2×C3—48no—
263C2×C3#6383014yes—
264C2×C3C2×#371914yes—
265C2×C3#6396614yes—
266C2×C3#6409511yescharacteristic
267C2×C3C2×#374611yescharacteristic
268C2×C3#6411711yescharacteristic
269C22—21no—
270C22—21no—
271C22#8880012yes—
272C22C2×#629211yescharacteristic
273C4—21no—
274C4—362no—
275C4—1081no—
276C4—1081no—
277C4#8902812yes—
278C3—24no—
279C3#1059113514yescomplemented
280C3#1059242111yescharacteristic, complemented
281C2—22no—
282C2#1172337011yescharacteristic
283C2C2×#8880011yescharacteristic
284C2#1172360811yescharacteristic
subgroupG/H
centerC22C2×#6292
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#484
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. #11722055C2semidirect
5. #11722042C2semidirect
169. C33#484semidirect
245. C32#4722semidirect
246. C32#4970semidirect
279. C3#10591135semidirect
280. C3#10592421semidirect

Maximal subgroups

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

subgroupindex
#117239652
C2×C3×C4⋊(C3⋊Dic3)2
C32×#8302
#117220552
#117220422
#105924213
#105911353