Finite Group Databasebeta

← #13160238

Subgroups — #13160238

Order 1728.

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

normal subgroups only

This group has 600 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×#64128C211yesmaximal, characteristic, complemented
2#11723339C211yesmaximal, characteristic, complemented
3#11722199C211yesmaximal, characteristic, complemented
4C32×#828C211yesmaximal, characteristic, complemented
5#11722011C211yesmaximal, characteristic, complemented
6#11722004C211yesmaximal, characteristic, complemented
7C3⋊D3×C3⋊C16C211yesmaximal, characteristic, complemented
8#10592615—31nomaximal
9#10591046—34nomaximal
10C3×#3832—21no—
11#89014—21no—
12C33×Q16C2211yescharacteristic
13#88853C2211yescharacteristic
14C3×#3755—21no—
15C3×#3745C2211yescharacteristic
16C3×C8×C3⋊D3C2211yescharacteristic
17C32×D24C2211yescharacteristic
18#88794—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
20C32×C3⋊C16C2211yescharacteristic
21#64128D311yescharacteristic, complemented
22#64121—31no—
23C3×#919—34no—
24#63807—34no—
25C32×SD32—31no—
26#63656—31no—
27#63656—34no—
28#63656—64no—
29#63653—31no—
30#63627—31no—
31C16×C3⋊D3—31no—
32C3×#828D314yescomplemented
33#63548—34no—
34#63545—34no—
35D3×C3⋊C16—34no—
36C33×Q8—21no—
37#6308—21no—
38C3×C4⋊(C3⋊D3)—21no—
39C3×C4⋊(C3⋊D3)—41no—
40C3×C4×C3⋊D3—41no—
41C3×C4×C3⋊D3D411yescharacteristic
42C32×D12—21no—
43#6290—41no—
44D3×C3⋊Dic3—41no—
45C33×C8C2311yescharacteristic
46C3×C3⋊(C3⋊C8)D411yescharacteristic
47#5194—91no—
48#4916—94no—
49#3832—21no—
50#3830—61no—
51C3×#239—64no—
52#3795—64no—
53C32×Q16—24no—
54C32×Q16C2×D314yes—
55C32×Q16C2×D311yescharacteristic
56C32×D8—31no—
57#3755—21no—
58#3754—61no—
59#3746—31no—
60#3746—34no—
61#3746—64no—
62#3745C2×D311yescharacteristic
63C8×C3⋊D3C2×D311yescharacteristic
64C3×#215—64no—
65C3×D24—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—
74C2×C3×C3⋊D3—41no—
75C2×C3×C3⋊D3D811yescharacteristic
76C2×C32×D3—41no—
77C33×C4—41no—
78C33×C4C2×D411yescharacteristic
79C3⋊(C3⋊Dic3)—41no—
80C3×C3⋊Dic3D811yescharacteristic
81C3×#135—91no—
82#919—34no—
83#912—94no—
84#904—91no—
85C3×SD32—34no—
86#829—94no—
87#829—91no—
88#829—184no—
89#828C3⋊D311yescharacteristic, complemented
90#827—91no—
91#826—94no—
92#826—91no—
93C2×C3⋊C16—91no—
94#800—94no—
95D3×C16—94no—
96C32×Q8—21no—
97C32×Q8—24no—
98C32×Q8—44no—
99C32×D4—61no—
100#638—121no—
101C2×C3⋊Dic3—121no—
102C4⋊(C3⋊D3)—21no—
103C4⋊(C3⋊D3)—41no—
104C4×C3⋊D3—41no—
105C4×C3⋊D3#6711yescharacteristic
106#634—61no—
107#634—64no—
108#634—124no—
109C3×D12—24no—
110C3×D12—64no—
111C3×D12—124no—
112C3×C4×D3—34no—
113C3×C4×D3—124no—
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×C33C2×D811yescharacteristic
123C3×C3⋊D3—21no—
124C3×C3⋊D3—81no—
125C32×D3—81no—
126C3×#41—181no—
127#239—64no—
128#237—184no—
129#235—181no—
130C3×Q16—24no—
131C3×Q16D3214yes—
132C3×Q16C2×C3⋊D311yescharacteristic
133C3×SD16—181no—
134C3×D8—34no—
135C3×D8—91no—
136C2×C3×C8—91no—
137#215—64no—
138#214—184no—
139Dic12—94no—
140Dic12—91no—
141Dic12—184no—
142D24—34no—
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—121no—
152C2×C3⋊D3—41no—
153C2×C3⋊D3#21311yescharacteristic
154C2×C3×D3—34no—
155C2×C3×D3—44no—
156C2×C3×D3—124no—
157C32×C4—24no—
158C32×C4—41no—
159C32×C4—44no—
160C32×C4—84no—
161C32×C4D3×D414yes—
162C32×C4C2×#6711yescharacteristic
163C3⋊Dic3—121no—
164C3⋊Dic3—124no—
165C3⋊Dic3—244no—
166C3⋊Dic3#21311yescharacteristic
167C3×Dic3—34no—
168#135—91no—
169#135—271no—
170Q32—271no—
171SD32—31no—
172SD32—271no—
173D16—271no—
174C2×C16—271no—
175C33#50811yescharacteristic, complemented, Sylow
176C3×Q8—24no—
177C3×Q8—21no—
178C3×Q8—44no—
179C3×D4—64no—
180C3×D4—181no—
181C3×D4—361no—
182C2×C3×C4—91no—
183C2×C3×C4—361no—
184#67—364no—
185#67—361no—
186C2×Dic3—364no—
187D12—21no—
188D12—64no—
189D12—124no—
190C4×D3—34no—
191C4×D3—124no—
192C4×D3—361no—
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—241no—
203C2×C32D3×D814yes—
204C2×C32C2×#21311yescharacteristic
205C3⋊D3—21no—
206C3⋊D3—81no—
207C3×D3—64no—
208C3×D3—84no—
209C3×D3—244no—
210#41—181no—
211#41—541no—
212Q16—271no—
213Q16D3×C3⋊D311yescharacteristic
214SD16—181no—
215SD16—541no—
216D8—31no—
217D8—91no—
218C2×C8—91no—
219C16—31no—
220C16—271no—
221C22×C3—91no—
222C22×C3—124no—
223C22×C3—361no—
224C2×D3—34no—
225C2×D3—41no—
226C2×D3—124no—
227C3×C4—24no—
228C3×C4—41no—
229C3×C4—44no—
230C3×C4—84no—
231C3×C4—91no—
232C3×C4D3×#6714yes—
233C3×C4D4×C3⋊D311yescharacteristic
234Dic3—34no—
235Dic3—361no—
236Dic3—364no—
237Dic3—724no—
238C32—24no—
239C32#491614yescomplemented
240C32#519411yescharacteristic, complemented
241Q8—21no—
242Q8—541no—
243D4—61no—
244D4—181no—
245D4—361no—
246D4—1081no—
247C2×C4—91no—
248C2×C4—361no—
249C2×C4—1081no—
250C8—91no—
251C8C2×D3×C3⋊D311yescharacteristic
252C2×C3—24no—
253C2×C3—181no—
254C2×C3—244no—
255C2×C3—721no—
256C2×C3D3×#21314yes—
257C2×C3D8×C3⋊D311yescharacteristic
258D3—64no—
259D3—81no—
260D3—244no—
261C22—91no—
262C22—121no—
263C22—361no—
264C4—41no—
265C4—91no—
266C4—1081no—
267C4C3⋊D3×#6711yescharacteristic
268C3—24no—
269C3#1059104614yescomplemented
270C3#1059261511yescharacteristic, complemented
271C2—181no—
272C2—241no—
273C2—721no—
274C2C3⋊D3×#21311yescharacteristic
subgroupG/H
centerC2C3⋊D3×#213
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C2×D8
hypercenterQ16D3×C3⋊D3
hypocenterC33#508
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. C3×#64128C2semidirect
2. #11723339C2semidirect
3. #11722199C2semidirect
4. C32×#828C2semidirect
5. #11722011C2semidirect
6. #11722004C2semidirect
7. C3⋊D3×C3⋊C16C2semidirect
21. #64128D3semidirect
32. C3×#828D3semidirect
89. #828C3⋊D3semidirect
175. C33#508semidirect
239. C32#4916semidirect
240. C32#5194semidirect
269. C3#10591046semidirect
270. C3#10592615semidirect

Maximal subgroups

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

subgroupindex
C3×#641282
#117233392
#117221992
C32×#8282
#117220112
#117220042
C3⋊D3×C3⋊C162
#105926153
#105910463