Finite Group Databasebeta

← #13159252

Subgroups — #13159252

Order 1728.

660 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C32×#890C211yesmaximal, characteristic
2#11723454C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic
5#11722057C211yesmaximal, characteristic
6#11722053C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
8#10592764—31nomaximal
9#10590413—34nomaximal
10C33×C4⋊C4C2211yescharacteristic
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88807—21no—
20#88806—21no—
21C32×#125—31no—
22#64079—31no—
23C3×#890D314yes—
24#63869—34no—
25#63635—31no—
26#63635—31no—
27#63635—34no—
28#63635—64no—
29#63634—31no—
30#63634D311yescharacteristic
31C4×C3⋊(C3⋊C8)—31no—
32C3×#807—34no—
33#63575—34no—
34#63574—34no—
35Dic3×C3⋊C8—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3×C3⋊Dic3—21no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3D411yescharacteristic
42C2×C32×Dic3D411yescharacteristic
43C3⋊(C3⋊(C3⋊C8))Q812yes—
44C3×C3⋊(C3⋊C8)Q812yes—
45#5065—94no—
46#4878—91no—
47C32×C4⋊C4—24no—
48C32×C4⋊C4—31no—
49C32×C4⋊C4—61no—
50C32×C4⋊C4—61no—
51C32×C4⋊C4C2×D314yes—
52C32×C4⋊C4C2×D311yescharacteristic
53C32×C42—31no—
54C4⋊(C3⋊Dic3)—31no—
55C4⋊(C3⋊Dic3)—34no—
56C4⋊(C3⋊Dic3)—64no—
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58#3750—61no—
59#3750—61no—
60C2×C3⋊(C3⋊C8)—31no—
61C2×C3⋊(C3⋊C8)—34no—
62C2×C3⋊(C3⋊C8)—64no—
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×C4⋊Dic3—24no—
65C3×C4⋊Dic3—34no—
66C3×C4⋊Dic3C2×D314yes—
67C3×Dic3⋊C4—24no—
68C3×C4×Dic3C2×D314yes—
69C2×C3×C3⋊C8—34no—
70#3724—64no—
71Dic3⋊Dic3—64no—
72C22×C33C2×D411yescharacteristic
73C33×C4—41no—
74C33×C4C2×Q811yescharacteristic
75C33×C4C2×Q811yescharacteristic
76C3⋊(C3⋊Dic3)—41no—
77C3×C3⋊Dic3—41no—
78C32×Dic3—21no—
79C32×Dic3—21no—
80C32×Dic3—41no—
81C3×#125—34no—
82#890—91no—
83#890C3⋊D311yescharacteristic
84#870—94no—
85C3×#107—91no—
86#818—91no—
87C8×Dic3—91no—
88#808—94no—
89#808—94no—
90#808—91no—
91#808—91no—
92#808—184no—
93#807—34no—
94#807—94no—
95C4×C3⋊C8—94no—
96C2×C32×C4—21no—
97C2×C32×C4—24no—
98C2×C32×C4—24no—
99C2×C32×C4—31no—
100C2×C32×C4—31no—
101C2×C32×C4—44no—
102C2×C32×C4—61no—
103C2×C32×C4C22×D314yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—61no—
107C2×C3⋊Dic3—64no—
108C2×C3⋊Dic3—124no—
109C2×C3×Dic3—24no—
110C2×C3×Dic3—64no—
111C2×C3×Dic3#6714yes—
112C2×C3×Dic3#6714yes—
113C3⋊(C3⋊C8)—32no—
114C3⋊(C3⋊C8)—38no—
115C3⋊(C3⋊C8)—68no—
116C3⋊(C3⋊C8)Dic612yes—
117C3×C3⋊C8—38no—
118#499—271noSylow
119C2×C33C4⋊Q811yescharacteristic
120C2×C33#13511yescharacteristic
121C2×C33#13511yescharacteristic
122C2×C3×C8—91no—
123C3×C4⋊C4—24no—
124C3×C4⋊C4—34no—
125C3×C4⋊C4—64no—
126C3×C4⋊C4—64no—
127C3×C4⋊C4—91no—
128C3×C4⋊C4D3214yes—
129C3×C4⋊C4C2×C3⋊D311yescharacteristic
130C3×C42—34no—
131C4⋊Dic3—21no—
132C4⋊Dic3—34no—
133C4⋊Dic3—94no—
134C4⋊Dic3—91no—
135C4⋊Dic3—184no—
136C4⋊Dic3—181no—
137C4⋊Dic3C2×C3⋊D311yescharacteristic
138Dic3⋊C4—21no—
139Dic3⋊C4—184no—
140Dic3⋊C4—184no—
141Dic3⋊C4—181no—
142C4×Dic3C2×C3⋊D311yescharacteristic
143C2×C3⋊C8—34no—
144C2×C3⋊C8—94no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—184no—
147C22×C32—24no—
148C22×C32D3×D411yescharacteristic
149C22×C32C2×#6714yes—
150C32×C4—24no—
151C32×C4—24no—
152C32×C4—41no—
153C32×C4—44no—
154C32×C4—61no—
155C32×C4—61no—
156C32×C4—84no—
157C32×C4—121no—
158C32×C4C2×Dic611yescharacteristic
159C32×C4D3×Q814yes—
160C32×C4D3×Q814yes—
161C32×C4D3×Q811yescharacteristic
162C3⋊Dic3—41no—
163C3⋊Dic3—121no—
164C3⋊Dic3—124no—
165C3⋊Dic3—244no—
166C3×Dic3—24no—
167C3×Dic3—24no—
168C3×Dic3—44no—
169C3×Dic3—124no—
170#125—31no—
171#125—271no—
172#107—91no—
173#107—271no—
174#106—271no—
175#106—271no—
176C4×C8—271no—
177C33#49911yescharacteristic, complemented, Sylow
178C2×C3×C4—24no—
179C2×C3×C4—21no—
180C2×C3×C4—24no—
181C2×C3×C4—34no—
182C2×C3×C4—34no—
183C2×C3×C4—44no—
184C2×C3×C4—64no—
185C2×C3×C4—181no—
186C2×C3×C4C2×D3214yes—
187C2×C3×C4C22×C3⋊D311yescharacteristic
188C2×Dic3—21no—
189C2×Dic3—64no—
190C2×Dic3—184no—
191C2×Dic3—181no—
192C2×Dic3—364no—
193C2×Dic3#63811yescharacteristic
194C2×Dic3#63811yescharacteristic
195C3×C8—92no—
196C3⋊C8—38no—
197C3⋊C8—98no—
198C3⋊C8—92no—
199C3⋊C8—188no—
200C2×C32—24no—
201C2×C32—24no—
202C2×C32—24no—
203C2×C32#88811yescharacteristic
204C2×C32#91211yescharacteristic
205C2×C32#91911yescharacteristic
206C2×C32#94414yes—
207C2×C32#95014yes—
208C2×C32#95014yes—
209C2×C8—91no—
210C2×C8—271no—
211C4⋊C4—31no—
212C4⋊C4—61no—
213C4⋊C4—61no—
214C4⋊C4—91no—
215C4⋊C4—271no—
216C4⋊C4—541no—
217C4⋊C4—541no—
218C4⋊C4D3×C3⋊D311yescharacteristic
219C42—31no—
220C22×C3—24no—
221C22×C3D3×#6714yes—
222C22×C3C2×#63811yescharacteristic
223C3×C4—24no—
224C3×C4—24no—
225C3×C4—41no—
226C3×C4—44no—
227C3×C4—64no—
228C3×C4—64no—
229C3×C4—84no—
230C3×C4—124no—
231C3×C4—361no—
232C3×C4D3×Dic614yes—
233C3×C4#379714yes—
234C3×C4Q8×C3⋊D311yescharacteristic
235C3×C4Q8×C3⋊D311yescharacteristic
236Dic3—21no—
237Dic3—21no—
238Dic3—41no—
239Dic3—124no—
240Dic3—361no—
241Dic3—364no—
242Dic3—724no—
243C32—24no—
244C32#487811yescharacteristic, complemented
245C32#506514yescomplemented
246C2×C4—21no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—61no—
250C2×C4—181no—
251C2×C4—541no—
252C2×C4C2×D3×C3⋊D311yescharacteristic
253C8—92no—
254C8—272no—
255C2×C3—24no—
256C2×C3—24no—
257C2×C3—24no—
258C2×C3#6383914yes—
259C2×C3#6393314yes—
260C2×C3#6394914yes—
261C2×C3#6415311yescharacteristic
262C2×C3#6415911yescharacteristic
263C2×C3#6415911yescharacteristic
264C22D3×#63811yescharacteristic
265C4—41no—
266C4—61no—
267C4—61no—
268C4—121no—
269C4—361no—
270C4—1081no—
271C4C3⋊D3×Dic611yescharacteristic
272C4#8901911yescharacteristic
273C3—24no—
274C3#1059041314yescomplemented
275C3#1059276411yescharacteristic, complemented
276C2#1172339311yescharacteristic
277C2#1172355611yescharacteristic
278C2#1172358211yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#499
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
177. C33#499semidirect
244. C32#4878semidirect
245. C32#5065semidirect
274. C3#10590413semidirect
275. C3#10592764semidirect

Maximal subgroups

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

subgroupindex
C32×#8902
#117234542
#117221782
C3×#636342
#117220572
#117220532
Dic3×C3⋊(C3⋊C8)2
#105927643
#105904133