Finite Group Databasebeta

← #13160215

Subgroups — #13160215

Order 1728.

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

normal subgroups only

This group has 652 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×D8×C3⋊D3C211yesmaximal, characteristic, complemented
2#11723341C211yesmaximal, characteristic, complemented
3#11722196C211yesmaximal, characteristic, complemented
4C32×#827C211yesmaximal, characteristic, complemented
5#11722008C211yesmaximal, characteristic, complemented
6#11722006C211yesmaximal, characteristic, complemented
7#11722002C211yesmaximal, characteristic, complemented
8#10592613—31nomaximal
9#10591033—34nomaximal
10C3×D4×C3⋊D3—21no—
11#89018—21no—
12C33×D8C2211yescharacteristic
13#88852C2211yescharacteristic, complemented
14C3×#3753—21no—
15C3×#3745C2211yescharacteristic
16C3×C8×C3⋊D3C2211yescharacteristic
17C32×Dic12C2211yescharacteristic, complemented
18#88795—21no—
19C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic, complemented
20C32×C3⋊C16C2211yescharacteristic, complemented
21#64128—31no—
22D8×C3⋊D3D311yescharacteristic, complemented
23C3×D3×D8—34no—
24#63808—34no—
25C32×SD32—31no—
26#63655—31no—
27#63653—31no—
28#63653—34no—
29#63653—64no—
30#63626—31no—
31#63625—31no—
32C3×#827D314yescomplemented
33#63547—34no—
34#63546—34no—
35#63543—34no—
36C22×C3×C3⋊D3—41no—
37C33×D4—21no—
38C4⋊(C3⋊(C3⋊D3))—21no—
39C3×#638—41no—
40C3×C4⋊(C3⋊D3)—21no—
41C3×C4×C3⋊D3D411yescharacteristic
42C32×Dic6—21no—
43#6292—41no—
44#6289—41no—
45C33×C8C2311yescharacteristic
46C3×C3⋊(C3⋊C8)D411yescharacteristic
47#5147—91no—
48#4914—94no—
49#3832—61no—
50D4×C3⋊D3—21no—
51C3×D3×D4—64no—
52#3799—64no—
53C32×Q16—31no—
54C32×D8—24no—
55C32×D8C2×D314yes—
56C32×D8C2×D311yescharacteristic
57#3755—61no—
58#3753—21no—
59#3745—31no—
60#3745—34no—
61#3745—64no—
62#3745C2×D311yescharacteristic
63C8×C3⋊D3C2×D311yescharacteristic
64C3×#213—64no—
65C3×Dic12C2×D314yescomplemented
66C3×D24—34no—
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×D314yescomplemented
74C22×C33—41no—
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3—82no—
78C2×C3×C3⋊D3D811yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3×C3⋊Dic3D811yescharacteristic
81C32×Dic3—41no—
82C2×C3×D8—91no—
83#919—94no—
84D3×D8—34no—
85#904—91no—
86C3×SD32—34no—
87#828—94no—
88#827—91no—
89#827C3⋊D311yescharacteristic, complemented
90#826—94no—
91#826—91no—
92#826—91no—
93#826—184no—
94#812—91no—
95D48—94no—
96#798—94no—
97C22×C3⋊D3—41no—
98C22×C3×D3—124no—
99C32×Q8—61no—
100C32×D4—21no—
101C32×D4—24no—
102C32×D4—44no—
103#638—41no—
104C4⋊(C3⋊D3)—21no—
105C4⋊(C3⋊D3)—61no—
106C4⋊(C3⋊D3)—64no—
107C4⋊(C3⋊D3)—124no—
108C4⋊(C3⋊D3)—121no—
109C4×C3⋊D3—121no—
110C4×C3⋊D3#6711yescharacteristic
111C3×#67—124no—
112C3×D12—64no—
113C3×C4×D3—34no—
114C3×Dic6—24no—
115#626—124no—
116#624—124no—
117C32×C8—24no—
118C32×C8C22×D314yes—
119C32×C8C22×D311yescharacteristic
120C3⋊(C3⋊C8)#6711yescharacteristic
121C3×C3⋊C8—34no—
122#509—271noSylow
123C2×C33—81no—
124C2×C33C2×D811yescharacteristic
125C3⋊(C3⋊D3)—81no—
126C3×C3⋊D3—21no—
127C3×C3⋊D3—81no—
128C2×C3×D4—181no—
129#239—184no—
130D3×D4—64no—
131#235—181no—
132C3×Q16—34no—
133C3×D8—24no—
134C3×D8—91no—
135C3×D8—181no—
136C3×D8D3214yes—
137C3×D8C2×C3⋊D311yescharacteristic
138C2×C3×C8—91no—
139#215—184no—
140#213—64no—
141Dic12C2×C3⋊D311yescharacteristic, complemented
142D24—34no—
143D24—94no—
144D24—91no—
145D24—184no—
146#204—181no—
147D3×C8—34no—
148C3×C16—34no—
149C3⋊C16—94no—
150C3⋊C16—91no—
151C3⋊C16—184no—
152C3⋊C16C2×C3⋊D311yescharacteristic, complemented
153C22×C32—41no—
154C22×C32—44no—
155C22×C32—84no—
156C2×C3⋊D3—41no—
157C2×C3⋊D3—82no—
158C2×C3⋊D3—121no—
159C2×C3⋊D3—124no—
160C2×C3⋊D3—244no—
161C2×C3⋊D3#21311yescharacteristic
162C2×C3×D3—34no—
163C2×C3×D3—124no—
164C2×C3×D3—248no—
165C32×C4—24no—
166C32×C4—121no—
167C32×C4D3×D414yes—
168C32×C4C2×#6711yescharacteristic
169C3⋊Dic3#21311yescharacteristic
170C3×Dic3—34no—
171C3×Dic3—44no—
172#135—271no—
173C2×D8—91no—
174SD32—31no—
175SD32—271no—
176D16—271no—
177D16—271no—
178M32—271no—
179C33#50911yescharacteristic, complemented, Sylow
180C23×C3—361no—
181C22×D3—124no—
182C3×Q8—64no—
183C3×D4—24no—
184C3×D4—21no—
185C3×D4—44no—
186C3×D4—181no—
187C3×D4—361no—
188C2×C3×C4—91no—
189#67—124no—
190#67—361no—
191D12—64no—
192D12—184no—
193D12—181no—
194D12—364no—
195D12—364no—
196C4×D3—34no—
197C4×D3—364no—
198C4×D3—361no—
199Dic6—21no—
200C3×C8—24no—
201C3×C8—91no—
202C3×C8C2×D3214yes—
203C3×C8C22×C3⋊D311yescharacteristic
204C3⋊C8—34no—
205C2×C32—24no—
206C2×C32—81no—
207C2×C32—84no—
208C2×C32—164no—
209C2×C32D3×D814yes—
210C2×C32C2×#21311yescharacteristic
211C3⋊D3—21no—
212C3⋊D3—81no—
213C3⋊D3—241no—
214C3⋊D3—244no—
215C3⋊D3—484no—
216C3×D3—64no—
217C3×D3—244no—
218#41—541no—
219C2×D4—181no—
220Q16—31no—
221SD16—541no—
222D8—91no—
223D8—181no—
224D8—271no—
225D8D3×C3⋊D311yescharacteristic
226C2×C8—91no—
227C16—31no—
228C16—271no—
229C22×C3—41no—
230C22×C3—44no—
231C22×C3—84no—
232C22×C3—91no—
233C22×C3—361no—
234C22×C3—722no—
235C2×D3—34no—
236C2×D3—124no—
237C2×D3—248no—
238C2×D3—361no—
239C2×D3—364no—
240C2×D3—724no—
241C3×C4—24no—
242C3×C4—91no—
243C3×C4—124no—
244C3×C4D3×#6714yes—
245C3×C4D4×C3⋊D311yescharacteristic
246Dic3—34no—
247Dic3—41no—
248C32—24no—
249C32#491414yescomplemented
250C32#514711yescharacteristic, complemented
251C23—361no—
252Q8—61no—
253D4—21no—
254D4—181no—
255D4—361no—
256D4—541no—
257D4—1081no—
258C2×C4—91no—
259C2×C4—1081no—
260C8—91no—
261C8C2×D3×C3⋊D311yescharacteristic
262C2×C3—24no—
263C2×C3—84no—
264C2×C3—81no—
265C2×C3—164no—
266C2×C3—181no—
267C2×C3—721no—
268C2×C3D3×#21314yes—
269C2×C3D8×C3⋊D311yescharacteristic
270D3—64no—
271D3—244no—
272D3—724no—
273D3—721no—
274D3—1444no—
275C22—41no—
276C22—91no—
277C22—361no—
278C22—722no—
279C22—1081no—
280C4—91no—
281C4—121no—
282C4C3⋊D3×#6711yescharacteristic
283C3—24no—
284C3#1059103314yescomplemented
285C3#1059261311yescharacteristic, complemented
286C2—81no—
287C2—181no—
288C2—721no—
289C2—2161no—
290C2C3⋊D3×#21311yescharacteristic
subgroupG/H
centerC2C3⋊D3×#213
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×D8C22
socleC2×C33C2×D8
hypercenterD8D3×C3⋊D3
hypocenterC33#509
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×D8×C3⋊D3C2semidirect
2. #11723341C2semidirect
3. #11722196C2semidirect
4. C32×#827C2semidirect
5. #11722008C2semidirect
6. #11722006C2semidirect
7. #11722002C2semidirect
13. #88852C22semidirect
17. C32×Dic12C22semidirect
19. C3⋊(C3⋊(C3⋊C16))C22semidirect
20. C32×C3⋊C16C22semidirect
22. D8×C3⋊D3D3semidirect
32. C3×#827D3semidirect
65. C3×Dic12C2×D3semidirect
73. C3×C3⋊C16C2×D3semidirect
89. #827C3⋊D3semidirect
141. Dic12C2×C3⋊D3semidirect
152. C3⋊C16C2×C3⋊D3semidirect
179. C33#509semidirect
249. C32#4914semidirect
250. C32#5147semidirect
284. C3#10591033semidirect
285. C3#10592613semidirect

Maximal subgroups

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

subgroupindex
C3×D8×C3⋊D32
#117233412
#117221962
C32×#8272
#117220082
#117220062
#117220022
#105926133
#105910333