Finite Group Databasebeta

← #13160226

Subgroups — #13160226

Order 1728.

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

normal subgroups only

This group has 644 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
2C3⋊D3×D24C211yesmaximal, characteristic, complemented
3#11722198C211yesmaximal, characteristic, complemented
4C32×#828C211yesmaximal, characteristic, complemented
5#11722006C211yesmaximal, characteristic, complemented
6#11722004C211yesmaximal, characteristic, complemented
7#11722002C211yesmaximal, characteristic, complemented
8#10592613—31nomaximal
9#10591038—34nomaximal
10C3×#3832—21no—
11C3⋊D3×D12—21no—
12C33×Q16C2211yescharacteristic, complemented
13#88852C2211yescharacteristic
14C3×#3755—21no—
15C3×#3745C2211yescharacteristic, complemented
16C3×C8×C3⋊D3C2211yescharacteristic
17C32×D24C2211yescharacteristic
18#88790—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic, complemented
20C32×C3⋊C16C2211yescharacteristic, complemented
21#64128D311yescharacteristic, complemented
22D8×C3⋊D3—31no—
23C3×#919—34no—
24D3×D24—34no—
25C32×SD32—31no—
26#63655—31no—
27#63655—34no—
28#63655—64no—
29#63653—31no—
30#63626—31no—
31#63625—31no—
32C3×#828D314yescomplemented
33#63546—34no—
34#63545—34no—
35#63543—34no—
36C2×D3×C3⋊D3—41no—
37C33×Q8—21no—
38C4⋊(C3⋊(C3⋊D3))—21no—
39C3×C4⋊(C3⋊D3)—21no—
40C3×C4⋊(C3⋊D3)—41no—
41C3×C4×C3⋊D3—41no—
42C3×C4×C3⋊D3D411yescharacteristic
43C32×D12—21no—
44#6291—41no—
45C33×C8C2311yescharacteristic
46C3×C3⋊(C3⋊C8)D411yescharacteristic
47#5193—91no—
48#4914—94no—
49#3832—21no—
50D4×C3⋊D3—61no—
51C3×#239—64no—
52D3×D12—64no—
53C32×Q16—24no—
54C32×Q16C2×D314yescomplemented
55C32×Q16C2×D311yescharacteristic, complemented
56C32×D8—31no—
57#3755—21no—
58#3753—61no—
59#3745—31no—
60#3745—34no—
61#3745—64no—
62#3745C2×D311yescharacteristic, complemented
63C8×C3⋊D3C2×D311yescharacteristic
64C3×#215—64no—
65C3×D24—34no—
66C3×D24C2×D314yes—
67C3×D3×C8—34no—
68#3714—64no—
69C32×C16—31no—
70C3⋊(C3⋊C16)—31no—
71C3⋊(C3⋊C16)—34no—
72C3⋊(C3⋊C16)—64no—
73C3×C3⋊C16C2×D314yescomplemented
74C2×C3⋊(C3⋊D3)—41no—
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3D811yescharacteristic
77C2×C32×D3—41no—
78D3×C3⋊D3—82no—
79C33×C4—41no—
80C33×C4C2×D411yescharacteristic
81C3×C3⋊Dic3D811yescharacteristic
82C3×#135—91no—
83#919—34no—
84D3×D8—94no—
85C2×D24—91no—
86C3×SD32—34no—
87#828—94no—
88#828—91no—
89#828—184no—
90#828C3⋊D311yescharacteristic, complemented
91#826—94no—
92#826—91no—
93#826—91no—
94#812—91no—
95D48—94no—
96#798—94no—
97C22×C3⋊D3—121no—
98C2×D32—124no—
99C32×Q8—21no—
100C32×Q8—24no—
101C32×Q8—44no—
102C32×D4—61no—
103#638—121no—
104C4⋊(C3⋊D3)—21no—
105C4⋊(C3⋊D3)—41no—
106C4⋊(C3⋊D3)—61no—
107C4⋊(C3⋊D3)—64no—
108C4⋊(C3⋊D3)—124no—
109C4×C3⋊D3—41no—
110C4×C3⋊D3#6711yescharacteristic
111C3×D12—24no—
112C3×D12—64no—
113C3×D12—124no—
114C3×C4×D3—34no—
115C3×C4×D3—124no—
116#626—124no—
117C32×C8—24no—
118C32×C8C22×D314yes—
119C32×C8C22×D311yescharacteristic
120C3⋊(C3⋊C8)#6711yescharacteristic
121C3×C3⋊C8—34no—
122#509—271noSylow
123C2×C33C2×D811yescharacteristic
124C3⋊(C3⋊D3)—81no—
125C3×C3⋊D3—21no—
126C3×C3⋊D3—81no—
127C32×D3—81no—
128C3×#41—181no—
129#239—64no—
130D3×D4—184no—
131C2×D12—181no—
132C3×Q16—24no—
133C3×Q16D3214yescomplemented
134C3×Q16C2×C3⋊D311yescharacteristic, complemented
135C3×SD16—181no—
136C3×D8—34no—
137C3×D8—91no—
138C2×C3×C8—91no—
139#215—64no—
140#213—184no—
141D24—34no—
142D24—94no—
143D24—91no—
144D24—184no—
145D24—181no—
146D24C2×C3⋊D311yescharacteristic
147D3×C8—34no—
148C3×C16—34no—
149C3⋊C16—94no—
150C3⋊C16—91no—
151C3⋊C16—184no—
152C3⋊C16C2×C3⋊D311yescharacteristic, complemented
153C22×C32—121no—
154C2×C3⋊D3—41no—
155C2×C3⋊D3—121no—
156C2×C3⋊D3—124no—
157C2×C3⋊D3—244no—
158C2×C3⋊D3—242no—
159C2×C3⋊D3#21311yescharacteristic
160C2×C3×D3—34no—
161C2×C3×D3—44no—
162C2×C3×D3—124no—
163D32—248no—
164C32×C4—24no—
165C32×C4—41no—
166C32×C4—44no—
167C32×C4—84no—
168C32×C4D3×D414yes—
169C32×C4C2×#6711yescharacteristic
170C3⋊Dic3#21311yescharacteristic
171C3×Dic3—34no—
172#135—91no—
173C2×D8—271no—
174SD32—31no—
175SD32—271no—
176D16—271no—
177D16—271no—
178M32—271no—
179C33#50911yescharacteristic, complemented, Sylow
180C22×D3—364no—
181C22×D3—361no—
182C3×Q8—24no—
183C3×Q8—21no—
184C3×Q8—44no—
185C3×D4—64no—
186C3×D4—181no—
187C3×D4—361no—
188C2×C3×C4—91no—
189C2×C3×C4—361no—
190#67—364no—
191D12—21no—
192D12—64no—
193D12—124no—
194D12—184no—
195D12—181no—
196D12—364no—
197D12—361no—
198C4×D3—34no—
199C4×D3—124no—
200C3×C8—24no—
201C3×C8—91no—
202C3×C8C2×D3214yes—
203C3×C8C22×C3⋊D311yescharacteristic
204C3⋊C8—34no—
205C2×C32—24no—
206C2×C32—241no—
207C2×C32D3×D814yes—
208C2×C32C2×#21311yescharacteristic
209C3⋊D3—21no—
210C3⋊D3—81no—
211C3⋊D3—241no—
212C3⋊D3—244no—
213C3⋊D3—484no—
214C3×D3—64no—
215C3×D3—84no—
216C3×D3—244no—
217#41—181no—
218C2×D4—541no—
219Q16D3×C3⋊D311yescharacteristic, complemented
220SD16—181no—
221D8—31no—
222D8—91no—
223D8—271no—
224D8—541no—
225C2×C8—91no—
226C16—31no—
227C16—271no—
228C22×C3—91no—
229C22×C3—124no—
230C22×C3—361no—
231C2×D3—34no—
232C2×D3—41no—
233C2×D3—124no—
234C2×D3—361no—
235C2×D3—364no—
236C2×D3—724no—
237C2×D3—728no—
238C2×D3—722no—
239C3×C4—24no—
240C3×C4—41no—
241C3×C4—44no—
242C3×C4—84no—
243C3×C4—91no—
244C3×C4D3×#6714yes—
245C3×C4D4×C3⋊D311yescharacteristic
246Dic3—34no—
247C32—24no—
248C32#491414yescomplemented
249C32#519311yescharacteristic, complemented
250C23—1081no—
251Q8—21no—
252D4—61no—
253D4—181no—
254D4—361no—
255D4—541no—
256D4—1081no—
257C2×C4—91no—
258C2×C4—361no—
259C8—91no—
260C8C2×D3×C3⋊D311yescharacteristic
261C2×C3—24no—
262C2×C3—181no—
263C2×C3—244no—
264C2×C3—721no—
265C2×C3D3×#21314yes—
266C2×C3D8×C3⋊D311yescharacteristic
267D3—64no—
268D3—81no—
269D3—244no—
270D3—724no—
271D3—721no—
272D3—1444no—
273C22—91no—
274C22—121no—
275C22—361no—
276C22—1081no—
277C22—2162no—
278C4—41no—
279C4—91no—
280C4C3⋊D3×#6711yescharacteristic
281C3—24no—
282C3#1059103814yescomplemented
283C3#1059261311yescharacteristic, complemented
284C2—181no—
285C2—241no—
286C2—721no—
287C2—2161no—
288C2C3⋊D3×#21311yescharacteristic
subgroupG/H
centerC2C3⋊D3×#213
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C2×D8
hypercenterQ16D3×C3⋊D3
hypocenterC33#509
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. C3⋊D3×D24C2semidirect
3. #11722198C2semidirect
4. C32×#828C2semidirect
5. #11722006C2semidirect
6. #11722004C2semidirect
7. #11722002C2semidirect
12. C33×Q16C22semidirect
15. C3×#3745C22semidirect
19. C3⋊(C3⋊(C3⋊C16))C22semidirect
20. C32×C3⋊C16C22semidirect
21. #64128D3semidirect
32. C3×#828D3semidirect
54. C32×Q16C2×D3semidirect
55. C32×Q16C2×D3semidirect
62. #3745C2×D3semidirect
73. C3×C3⋊C16C2×D3semidirect
90. #828C3⋊D3semidirect
133. C3×Q16D32semidirect
134. C3×Q16C2×C3⋊D3semidirect
152. C3⋊C16C2×C3⋊D3semidirect
179. C33#509semidirect
219. Q16D3×C3⋊D3semidirect
248. C32#4914semidirect
249. C32#5193semidirect
282. C3#10591038semidirect
283. C3#10592613semidirect

Maximal subgroups

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

subgroupindex
C3×#641282
C3⋊D3×D242
#117221982
C32×#8282
#117220062
#117220042
#117220022
#105926133
#105910383