Finite Group Databasebeta

← #13160216

Subgroups — #13160216

Order 1728.

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

normal subgroups only

This group has 616 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#11723341C211yesmaximal, characteristic, complemented
3#11722196C211yesmaximal, characteristic, complemented
4C32×#827C211yesmaximal, characteristic, complemented
5#11722013C211yesmaximal, characteristic, complemented
6#11722012C211yesmaximal, characteristic, complemented
7C3⋊D3×C3⋊C16C211yesmaximal, characteristic, complemented
8#10592615—31nomaximal
9#10591034—34nomaximal
10C3×#3830—21no—
11#89018—21no—
12C33×D8C2211yescharacteristic
13#88852C2211yescharacteristic
14C3×#3754—21no—
15C3×#3746C2211yescharacteristic
16C3×C8×C3⋊D3C2211yescharacteristic
17C32×Dic12C2211yescharacteristic
18#88795—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
20C32×C3⋊C16C2211yescharacteristic
21#64128—31no—
22#64121D311yescharacteristic, complemented
23C3×#912—34no—
24#63808—34no—
25C32×SD32—31no—
26#63656—31no—
27#63653—31no—
28#63653—34no—
29#63653—64no—
30#63627—31no—
31C16×C3⋊D3—31no—
32C3×#827D314yescomplemented
33#63550—34no—
34#63549—34no—
35D3×C3⋊C16—34no—
36C33×D4—21no—
37C4⋊(C3⋊(C3⋊D3))—21no—
38C3×#638—41no—
39C2×C3×C3⋊Dic3—41no—
40C3×C4×C3⋊D3D411yescharacteristic
41C3×#634—21no—
42C32×Dic6—21no—
43#6292—41no—
44#6289—41no—
45C33×C8C2311yescharacteristic
46C3×C3⋊(C3⋊C8)D411yescharacteristic
47#5194—91no—
48#4916—94no—
49#3832—61no—
50#3830—21no—
51C3×#237—64no—
52#3799—64no—
53C32×Q16—31no—
54C32×D8—24no—
55C32×D8C2×D314yes—
56C32×D8C2×D311yescharacteristic
57#3755—61no—
58#3754—21no—
59#3746C2×D311yescharacteristic
60#3745—31no—
61#3745—34no—
62#3745—64no—
63C8×C3⋊D3C2×D311yescharacteristic
64C3×#214—64no—
65C3×Dic12—34no—
66C3×Dic12C2×D314yes—
67C3×D3×C8—34no—
68#3717—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⋊D3)—41no—
76C2×C3×C3⋊D3D811yescharacteristic
77C33×C4C2×D411yescharacteristic
78C3×C3⋊Dic3—41no—
79C3×C3⋊Dic3D811yescharacteristic
80C32×Dic3—41no—
81C3×#135—91no—
82#919—94no—
83#912—34no—
84#904—91no—
85C3×SD32—34no—
86#829—94no—
87#829—91no—
88#828—91no—
89#827C3⋊D311yescharacteristic, complemented
90#826—94no—
91#826—91no—
92#826—184no—
93C2×C3⋊C16—91no—
94#800—94no—
95D3×C16—94no—
96C32×Q8—61no—
97C32×D4—21no—
98C32×D4—24no—
99C32×D4—44no—
100#638—41no—
101C2×C3⋊Dic3—41no—
102C4⋊(C3⋊D3)—61no—
103C4⋊(C3⋊D3)—64no—
104C4⋊(C3⋊D3)—124no—
105C4⋊(C3⋊D3)—121no—
106C4×C3⋊D3—121no—
107C4×C3⋊D3#6711yescharacteristic
108#634—21no—
109C3×#67—124no—
110C2×C3×Dic3—124no—
111C3×C4×D3—34no—
112C3×Dic6—24no—
113C3×Dic6—64no—
114#626—124no—
115#624—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)—81no—
125C3×C3⋊D3—21no—
126C3×#41—181no—
127#239—184no—
128#237—64no—
129#235—181no—
130C3×Q16—34no—
131C3×Q16—91no—
132C3×SD16—181no—
133C3×D8—24no—
134C3×D8D3214yes—
135C3×D8C2×C3⋊D311yescharacteristic
136C2×C3×C8—91no—
137#215—184no—
138#214—64no—
139Dic12—34no—
140Dic12C2×C3⋊D311yescharacteristic
141D24—94no—
142D24—91no—
143D24—184no—
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—
154C2×C3⋊D3—121no—
155C2×C3⋊D3—124no—
156C2×C3⋊D3—244no—
157C2×C3⋊D3#21311yescharacteristic
158C2×C3×D3—34no—
159C32×C4—24no—
160C32×C4—121no—
161C32×C4D3×D414yes—
162C32×C4C2×#6711yescharacteristic
163C3⋊Dic3—41no—
164C3⋊Dic3#21311yescharacteristic
165C3×Dic3—34no—
166C3×Dic3—44no—
167C3×Dic3—124no—
168#135—91no—
169#135—271no—
170Q32—271no—
171SD32—31no—
172SD32—271no—
173D16—271no—
174C2×C16—271no—
175C33#50811yescharacteristic, complemented, Sylow
176C3×Q8—64no—
177C3×Q8—181no—
178C3×D4—24no—
179C3×D4—21no—
180C3×D4—44no—
181C3×D4—361no—
182C2×C3×C4—91no—
183C2×C3×C4—361no—
184#67—124no—
185#67—361no—
186C2×Dic3—124no—
187D12—184no—
188D12—181no—
189D12—364no—
190D12—364no—
191C4×D3—34no—
192C4×D3—364no—
193C4×D3—361no—
194Dic6—21no—
195Dic6—64no—
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×C32D3×D814yes—
206C2×C32C2×#21311yescharacteristic
207C3⋊D3—21no—
208C3⋊D3—241no—
209C3⋊D3—244no—
210C3⋊D3—484no—
211C3×D3—64no—
212#41—181no—
213#41—541no—
214Q16—31no—
215Q16—91no—
216SD16—181no—
217SD16—541no—
218D8—271no—
219D8D3×C3⋊D311yescharacteristic
220C2×C8—91no—
221C16—31no—
222C16—271no—
223C22×C3—41no—
224C22×C3—44no—
225C22×C3—84no—
226C22×C3—91no—
227C2×D3—34no—
228C2×D3—361no—
229C2×D3—364no—
230C2×D3—724no—
231C3×C4—24no—
232C3×C4—91no—
233C3×C4—124no—
234C3×C4—361no—
235C3×C4D3×#6714yes—
236C3×C4D4×C3⋊D311yescharacteristic
237Dic3—34no—
238Dic3—41no—
239Dic3—124no—
240C32—24no—
241C32#491614yescomplemented
242C32#519411yescharacteristic, complemented
243Q8—61no—
244Q8—181no—
245D4—21no—
246D4—361no—
247D4—541no—
248D4—1081no—
249C2×C4—91no—
250C2×C4—361no—
251C2×C4—1081no—
252C8—91no—
253C8C2×D3×C3⋊D311yescharacteristic
254C2×C3—24no—
255C2×C3—84no—
256C2×C3—81no—
257C2×C3—164no—
258C2×C3—181no—
259C2×C3D3×#21314yes—
260C2×C3D8×C3⋊D311yescharacteristic
261D3—64no—
262D3—724no—
263D3—721no—
264D3—1444no—
265C22—41no—
266C22—91no—
267C22—1081no—
268C4—91no—
269C4—121no—
270C4—361no—
271C4C3⋊D3×#6711yescharacteristic
272C3—24no—
273C3#1059103414yescomplemented
274C3#1059261511yescharacteristic, complemented
275C2—81no—
276C2—181no—
277C2—2161no—
278C2C3⋊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. #11723341C2semidirect
3. #11722196C2semidirect
4. C32×#827C2semidirect
5. #11722013C2semidirect
6. #11722012C2semidirect
7. C3⋊D3×C3⋊C16C2semidirect
22. #64121D3semidirect
32. C3×#827D3semidirect
89. #827C3⋊D3semidirect
175. C33#508semidirect
241. C32#4916semidirect
242. C32#5194semidirect
273. C3#10591034semidirect
274. C3#10592615semidirect

Maximal subgroups

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

subgroupindex
C3×#641212
#117233412
#117221962
C32×#8272
#117220132
#117220122
C3⋊D3×C3⋊C162
#105926153
#105910343