Finite Group Databasebeta

← #13160214

Subgroups — #13160214

Order 1728.

608 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 100 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×#64121C211yesmaximal, characteristic, complemented
2#11723339C211yesmaximal, characteristic, complemented
3#11722197C211yesmaximal, characteristic, complemented
4C32×#826C211yesmaximal, characteristic, complemented
5#11722015C211yesmaximal, characteristic, complemented
6#11722007C211yesmaximal, characteristic, complemented
7C3⋊D3×C3⋊C16C211yesmaximal, characteristic, complemented
8#10592611—31nomaximal
9#10591032—34nomaximal
10C3×#3830—21no—
11#89014—21no—
12C33×D8C2211yescharacteristic
13#88853C2211yescharacteristic
14C3×#3754—21no—
15C3×#3746C2211yescharacteristic
16C3×C8×C3⋊D3C2211yescharacteristic
17C32×D24C2211yescharacteristic
18#88794—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
20C32×C3⋊C16C2211yescharacteristic
21#64121—31no—
22#64121D311yescharacteristic, complemented
23C3×#912—34no—
24#63807—34no—
25C32×D16—31no—
26#63654—31no—
27#63654—31no—
28#63654—34no—
29#63654—64no—
30#63628—31no—
31C16×C3⋊D3—31no—
32C3×#826D314yescomplemented
33#63551—34no—
34#63547—34no—
35D3×C3⋊C16—34no—
36C33×D4—21no—
37#6308—21no—
38C3×#638—41no—
39C2×C3×C3⋊Dic3—41no—
40C3×C4×C3⋊D3D411yescharacteristic
41C3×#634—21no—
42C32×D12—21no—
43#6290—41no—
44D3×C3⋊Dic3—41no—
45C33×C8C2311yescharacteristic
46C3×C3⋊(C3⋊C8)D411yescharacteristic
47#5194—91no—
48#4912—94no—
49#3830—21no—
50#3830—61no—
51C3×#237—64no—
52#3795—64no—
53C32×D8—24no—
54C32×D8—31no—
55C32×D8C2×D314yes—
56C32×D8C2×D311yescharacteristic
57#3754—21no—
58#3754—61no—
59#3746—31no—
60#3746—34no—
61#3746—64no—
62#3746C2×D311yescharacteristic
63C8×C3⋊D3C2×D311yescharacteristic
64C3×#214—64no—
65C3×Dic12—34no—
66C3×D24C2×D314yes—
67C3×D3×C8—34no—
68#3716—64no—
69C32×C16—31no—
70C3⋊(C3⋊C16)—31no—
71C3⋊(C3⋊C16)—34no—
72C3⋊(C3⋊C16)—64no—
73C3×C3⋊C16C2×D314yes—
74C22×C33—41no—
75C2×C3×C3⋊D3D811yescharacteristic
76C2×C32×D3—41no—
77C33×C4C2×D411yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3D811yescharacteristic
81C3×#135—91no—
82#912—34no—
83#912—94no—
84#904—91no—
85C3×D16—34no—
86#829—91no—
87#828—91no—
88#827—94no—
89#827—94no—
90#827—91no—
91#827—184no—
92#826C3⋊D311yescharacteristic, complemented
93C2×C3⋊C16—91no—
94Dic24—94no—
95D3×C16—94no—
96C32×D4—21no—
97C32×D4—24no—
98C32×D4—44no—
99C32×D4—61no—
100#638—41no—
101#638—121no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3—121no—
104C4×C3⋊D3#6711yescharacteristic
105#634—21no—
106#634—61no—
107#634—64no—
108#634—124no—
109C3×#67—124no—
110C2×C3×Dic3—124no—
111C3×D12—24no—
112C3×C4×D3—34no—
113C3×Dic6—64no—
114#625—124no—
115D3×Dic3—124no—
116C32×C8—24no—
117C32×C8C22×D314yes—
118C32×C8C22×D311yescharacteristic
119C3⋊(C3⋊C8)#6711yescharacteristic
120C3×C3⋊C8—34no—
121#508—271noSylow
122C2×C33—81no—
123C2×C33C2×D811yescharacteristic
124C3×C3⋊D3—21no—
125C32×D3—81no—
126C3×#41—181no—
127#237—64no—
128#237—184no—
129#235—181no—
130C3×Q16—91no—
131C3×SD16—181no—
132C3×D8—24no—
133C3×D8—34no—
134C3×D8D3214yes—
135C3×D8C2×C3⋊D311yescharacteristic
136C2×C3×C8—91no—
137#214—64no—
138#214—184no—
139Dic12—34no—
140Dic12—94no—
141Dic12—91no—
142Dic12—184no—
143D24C2×C3⋊D311yescharacteristic
144#204—181no—
145D3×C8—34no—
146C3×C16—34no—
147C3⋊C16—94no—
148C3⋊C16—91no—
149C3⋊C16—184no—
150C3⋊C16C2×C3⋊D311yescharacteristic
151C22×C32—41no—
152C22×C32—44no—
153C22×C32—84no—
154C22×C32—121no—
155C2×C3⋊D3#21311yescharacteristic
156C2×C3×D3—34no—
157C2×C3×D3—44no—
158C32×C4—24no—
159C32×C4D3×D414yes—
160C32×C4C2×#6711yescharacteristic
161C3⋊Dic3—41no—
162C3⋊Dic3—121no—
163C3⋊Dic3—124no—
164C3⋊Dic3—244no—
165C3⋊Dic3#21311yescharacteristic
166C3×Dic3—34no—
167C3×Dic3—124no—
168#135—91no—
169#135—271no—
170Q32—271no—
171SD32—271no—
172SD32—271no—
173D16—31no—
174C2×C16—271no—
175C33#50811yescharacteristic, complemented, Sylow
176C3×Q8—181no—
177C3×D4—24no—
178C3×D4—21no—
179C3×D4—44no—
180C3×D4—64no—
181C3×D4—361no—
182C2×C3×C4—91no—
183C2×C3×C4—361no—
184#67—124no—
185#67—364no—
186#67—361no—
187C2×Dic3—124no—
188C2×Dic3—364no—
189D12—21no—
190C4×D3—34no—
191C4×D3—361no—
192Dic6—64no—
193Dic6—184no—
194Dic6—181no—
195Dic6—364no—
196C3×C8—24no—
197C3×C8—91no—
198C3×C8C2×D3214yes—
199C3×C8C22×C3⋊D311yescharacteristic
200C3⋊C8—34no—
201C2×C32—24no—
202C2×C32—81no—
203C2×C32—84no—
204C2×C32—164no—
205C2×C32—241no—
206C2×C32D3×D814yes—
207C2×C32C2×#21311yescharacteristic
208C3⋊D3—21no—
209C3×D3—64no—
210C3×D3—84no—
211#41—181no—
212#41—541no—
213Q16—91no—
214Q16—271no—
215SD16—181no—
216SD16—541no—
217D8—31no—
218D8D3×C3⋊D311yescharacteristic
219C2×C8—91no—
220C16—31no—
221C16—271no—
222C22×C3—41no—
223C22×C3—44no—
224C22×C3—84no—
225C22×C3—91no—
226C22×C3—124no—
227C2×D3—34no—
228C2×D3—41no—
229C3×C4—24no—
230C3×C4—91no—
231C3×C4—361no—
232C3×C4D3×#6714yes—
233C3×C4D4×C3⋊D311yescharacteristic
234Dic3—34no—
235Dic3—124no—
236Dic3—361no—
237Dic3—364no—
238Dic3—724no—
239C32—24no—
240C32#491214yescomplemented
241C32#519411yescharacteristic, complemented
242Q8—181no—
243Q8—541no—
244D4—21no—
245D4—61no—
246D4—361no—
247D4—1081no—
248C2×C4—91no—
249C2×C4—361no—
250C2×C4—1081no—
251C8—91no—
252C8C2×D3×C3⋊D311yescharacteristic
253C2×C3—24no—
254C2×C3—84no—
255C2×C3—81no—
256C2×C3—164no—
257C2×C3—181no—
258C2×C3—244no—
259C2×C3D3×#21314yes—
260C2×C3D8×C3⋊D311yescharacteristic
261D3—64no—
262D3—81no—
263C22—41no—
264C22—91no—
265C22—121no—
266C4—91no—
267C4—361no—
268C4—1081no—
269C4C3⋊D3×#6711yescharacteristic
270C3—24no—
271C3#1059103214yescomplemented
272C3#1059261111yescharacteristic, complemented
273C2—81no—
274C2—181no—
275C2—241no—
276C2C3⋊D3×#21311yescharacteristic
subgroupG/H
centerC2C3⋊D3×#213
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×D8C22
socleC2×C33C2×D8
hypercenterD8D3×C3⋊D3
hypocenterC33#508
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
1. C3×#64121C2semidirect
2. #11723339C2semidirect
3. #11722197C2semidirect
4. C32×#826C2semidirect
5. #11722015C2semidirect
6. #11722007C2semidirect
7. C3⋊D3×C3⋊C16C2semidirect
22. #64121D3semidirect
32. C3×#826D3semidirect
92. #826C3⋊D3semidirect
175. C33#508semidirect
240. C32#4912semidirect
241. C32#5194semidirect
271. C3#10591032semidirect
272. C3#10592611semidirect

Maximal subgroups

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

subgroupindex
C3×#641212
#117233392
#117221972
C32×#8262
#117220152
#117220072
C3⋊D3×C3⋊C162
#105926113
#105910323