Finite Group Databasebeta

← #13159123

Subgroups — #13159123

Order 1728.

720 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 160 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
1C3×#64097C211yesmaximal, characteristic
2Dic6×C3⋊Dic3C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic
5#11722052C211yesmaximal, characteristic
6#11722051C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10592490—31nomaximal
9#10590486—34nomaximal
10C2×C3×#634—21no—
11C2×C32×Dic6C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21Q8×C3⋊Dic3—31no—
22#64097D311yescharacteristic
23C3×#888—34no—
24Dic3×Dic6—34no—
25C32×#103—31no—
26#63662—31no—
27#63644—31no—
28#63641—31no—
29#63635—31no—
30#63635—34no—
31#63635—64no—
32C3×#810D314yes—
33#63573—34no—
34Dic3⋊(C3⋊C8)—34no—
35#63565—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×#634—42no—
43C2×C32×Dic3—21no—
44C32×Dic6—21no—
45C32×Dic6Q812yescomplemented
46C3⋊(C3⋊(C3⋊C8))—21no—
47C32×C3⋊C8—21no—
48#5094—91no—
49#4791—94no—
50C2×C32×Q8—31no—
51C2×#634—21no—
52C2×C3×Dic6—64no—
53C2×C3×Dic6C2×D314yes—
54C2×C32×C8—31no—
55C32×C4⋊C4—24no—
56C32×C4⋊C4C2×D314yes—
57C32×C4⋊C4C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—34no—
60C4⋊(C3⋊Dic3)—61no—
61C4⋊(C3⋊Dic3)—64no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63#3750—21no—
64C4×C3⋊Dic3—61no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×C4⋊Dic3—34no—
70C3×Dic3⋊C4—64no—
71C3×C4×Dic3—34no—
72C2×C3×C3⋊C8C2×D314yes—
73Dic3⋊Dic3—64no—
74Dic32—64no—
75C22×C33C2×D411yescharacteristic
76C33×C4—41no—
77C33×C4C2×Q811yescharacteristic
78C33×C4#4111yescharacteristic
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3—21no—
81C3×C3⋊Dic3—41no—
82C3×C3⋊Dic3SD1612yes—
83C32×Dic3—22no—
84C3×C4⋊Q8—91no—
85Q8×Dic3—94no—
86#888—34no—
87C4×Dic6—91no—
88C3×#103—34no—
89#835—94no—
90#817—94no—
91Dic3⋊C8—94no—
92#810—91no—
93#810C3⋊D311yescharacteristic
94#808—94no—
95#808—91no—
96#808—91no—
97#808—184no—
98C4⋊(C3⋊C8)—91no—
99C32×Q8—32no—
100C32×Q8—61no—
101C2×C32×C4—21no—
102C2×C32×C4—24no—
103C2×C32×C4—24no—
104C2×C32×C4—44no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—64no—
111C2×C3⋊Dic3—124no—
112C2×C3⋊Dic3#6711yescharacteristic
113C2×C3⋊Dic3#6711yescharacteristic
114#634—42no—
115C2×C3×Dic3—24no—
116C2×C3×Dic3—34no—
117C2×C3×Dic3—34no—
118C2×C3×Dic3—64no—
119C3×Dic6—24no—
120C3×Dic6—128no—
121C3×Dic6Dic618yescomplemented
122C32×C8—61no—
123C3⋊(C3⋊C8)—68no—
124C3⋊(C3⋊C8)—64no—
125C3⋊(C3⋊C8)—61no—
126C3×C3⋊C8—24no—
127#475—271noSylow
128C2×C33C22⋊Q811yescharacteristic
129C2×C33C2×SD1611yescharacteristic
130C2×C33#13711yescharacteristic
131C2×C3×Q8—34no—
132C2×C3×Q8—181no—
133C2×Dic6—64no—
134C2×Dic6C2×C3⋊D311yescharacteristic
135C2×C3×C8—34no—
136C3×C4⋊C4—24no—
137C3×C4⋊C4—91no—
138C3×C4⋊C4—181no—
139C3×C4⋊C4D3214yes—
140C3×C4⋊C4C2×C3⋊D311yescharacteristic
141C3×C42—91no—
142C4⋊Dic3—34no—
143C4⋊Dic3—94no—
144C4⋊Dic3—91no—
145C4⋊Dic3—184no—
146C4⋊Dic3—184no—
147Dic3⋊C4—64no—
148Dic3⋊C4—181no—
149C4×Dic3—34no—
150C4×Dic3—181no—
151C4×Dic3—184no—
152C2×C3⋊C8—94no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—184no—
155C2×C3⋊C8C2×C3⋊D311yescharacteristic
156C22×C32—24no—
157C22×C32D3×D414yes—
158C22×C32C2×#6711yescharacteristic
159C32×C4—24no—
160C32×C4—24no—
161C32×C4—41no—
162C32×C4—44no—
163C32×C4—62no—
164C32×C4—84no—
165C32×C4C2×Dic614yes—
166C32×C4#23714yes—
167C32×C4D3×Q811yescharacteristic
168C32×C4#23911yescharacteristic
169C3⋊Dic3—21no—
170C3⋊Dic3—41no—
171C3⋊Dic3—121no—
172C3⋊Dic3—124no—
173C3⋊Dic3—244no—
174C3⋊Dic3#21412yes—
175C3×Dic3—28no—
176C3×Dic3—38no—
177C3×Dic3—64no—
178C3×Dic3—124no—
179C4⋊Q8—91no—
180C4×Q8—271no—
181#106—271no—
182#106—271no—
183C4⋊C8—271no—
184#103—31no—
185#103—271no—
186C33#47511yescharacteristic, complemented, Sylow
187C3×Q8—38no—
188C3×Q8—64no—
189C3×Q8—362no—
190C2×C3×C4—24no—
191C2×C3×C4—21no—
192C2×C3×C4—24no—
193C2×C3×C4—44no—
194C2×C3×C4—64no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—181no—
198C2×C3×C4C2×D3214yes—
199C2×C3×C4C22×C3⋊D311yescharacteristic
200C2×Dic3—21no—
201C2×Dic3—34no—
202C2×Dic3—34no—
203C2×Dic3—64no—
204C2×Dic3—184no—
205C2×Dic3—181no—
206C2×Dic3—364no—
207Dic6—21no—
208Dic6—128no—
209Dic6#63412yescomplemented
210C3×C8—64no—
211C3⋊C8—21no—
212C3⋊C8—188no—
213C3⋊C8—184no—
214C3⋊C8—181no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32—24no—
218C2×C32#87814yes—
219C2×C32D3×SD1614yes—
220C2×C32#91814yes—
221C2×C32C2×#21411yescharacteristic
222C2×C32#94211yescharacteristic
223C2×C32#94611yescharacteristic
224C2×Q8—31no—
225C2×Q8—181no—
226C2×C8—31no—
227C2×C8—271no—
228C4⋊C4—91no—
229C4⋊C4—181no—
230C4⋊C4—271no—
231C4⋊C4—541no—
232C4⋊C4D3×C3⋊D311yescharacteristic
233C42—91no—
234C42—541no—
235C22×C3—24no—
236C22×C3D3×#6714yes—
237C22×C3D4×C3⋊D311yescharacteristic
238C3×C4—24no—
239C3×C4—24no—
240C3×C4—41no—
241C3×C4—44no—
242C3×C4—68no—
243C3×C4—84no—
244C3×C4—92no—
245C3×C4—181no—
246C3×C4—361no—
247C3×C4D3×Dic614yes—
248C3×C4#379614yes—
249C3×C4C2×#63411yescharacteristic
250C3×C4#383011yescharacteristic
251Dic3—22no—
252Dic3—38no—
253Dic3—64no—
254Dic3—124no—
255Dic3—361no—
256Dic3—364no—
257Dic3—724no—
258C32—24no—
259C32#479114yescomplemented
260C32#509411yescharacteristic, complemented
261Q8—32no—
262Q8—61no—
263Q8—362no—
264C2×C4—21no—
265C2×C4—61no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—181no—
269C2×C4—541no—
270C2×C4C2×D3×C3⋊D311yescharacteristic
271C8—61no—
272C8—541no—
273C2×C3—24no—
274C2×C3—24no—
275C2×C3—24no—
276C2×C3#6388714yes—
277C2×C3D3×#21414yes—
278C2×C3#6394814yes—
279C2×C3#6408711yescharacteristic
280C2×C3SD16×C3⋊D311yescharacteristic
281C2×C3#6412711yescharacteristic
282C22C3⋊D3×#6711yescharacteristic
283C4—41no—
284C4—62no—
285C4—92no—
286C4—181no—
287C4—361no—
288C4—1081no—
289C4D3×#63411yescharacteristic
290C4#8901511yescharacteristic
291C3—24no—
292C3#1059048614yescomplemented
293C3#1059249011yescharacteristic, complemented
294C2#1172348711yescharacteristic
295C2C3⋊D3×#21411yescharacteristic
296C2#1172358611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#475
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
45. C32×Dic6Q8semidirect
121. C3×Dic6Dic6semidirect
186. C33#475semidirect
209. Dic6#634semidirect
259. C32#4791semidirect
260. C32#5094semidirect
292. C3#10590486semidirect
293. C3#10592490semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
Dic6×C3⋊Dic32
#117221782
C32×#8102
#117220522
#117220512
#117220372
#105924903
#105904863