Finite Group Databasebeta

← #13160230

Subgroups — #13160230

Order 1728.

788 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 96 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
1#11723960C211yesmaximal, characteristic, complemented
2#11723322C211yesmaximal, characteristic, complemented
3C3×#63655C211yesmaximal, characteristic, complemented
4C32×#828C211yesmaximal, characteristic, complemented
5#11722006C211yesmaximal, characteristic, complemented
6#11722005C211yesmaximal, characteristic, complemented
7#11722003C211yesmaximal, characteristic, complemented
8#10592613—31nomaximal
9#10591040—34nomaximal
10#89080—21no—
11#89026—21no—
12C33×Q16C2211yescharacteristic, complemented
13#88862—21no—
14#88852C2211yescharacteristic, complemented
15C8×C3⋊(C3⋊D3)C2211yescharacteristic
16C3×#3745C2211yescharacteristic
17C32×D24C2211yescharacteristic
18#88789—21no—
19C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
20C32×C3⋊C16C2211yescharacteristic, complemented
21#64128—31no—
22#64128—34no—
23#64128—64no—
24D8×C3⋊D3—31no—
25#63797—34no—
26C32×SD32—31no—
27#63655D311yescharacteristic, complemented
28#63653—31no—
29#63626—31no—
30#63625—31no—
31C3×#828—34no—
32C3×#828D314yescomplemented
33#63546—34no—
34#63546—34no—
35#63544—34no—
36C2×D3×C3⋊D3—41no—
37C33×Q8—21no—
38C4⋊(C3⋊(C3⋊D3))—21no—
39C4⋊(C3⋊(C3⋊D3))—41no—
40C4×C3⋊(C3⋊D3)—41no—
41C4×C3⋊(C3⋊D3)D411yescharacteristic
42C3×C4⋊(C3⋊D3)—21no—
43C32×D12—21no—
44#6290—41no—
45C33×C8C2311yescharacteristic
46C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
47#4914—94no—
48#4914—91no—
49#3832—61no—
50#3832—64no—
51#3832—68no—
52D4×C3⋊D3—61no—
53#3802—64no—
54C32×Q16—24no—
55C32×Q16C2×D314yescomplemented
56C32×Q16C2×D311yescharacteristic, complemented
57C32×D8—31no—
58#3755—61no—
59#3755—64no—
60#3755—68no—
61#3753—61no—
62#3745—31no—
63#3745—34no—
64#3745—64no—
65#3745C2×D311yescharacteristic
66C8×C3⋊D3—31no—
67C8×C3⋊D3—34no—
68C8×C3⋊D3—64no—
69C3×D24—34no—
70C3×D24C2×D314yes—
71#3713—64no—
72C32×C16—31no—
73C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
74C3×C3⋊C16—34no—
75C3×C3⋊C16C2×D314yescomplemented
76C2×C3⋊(C3⋊D3)—41no—
77C2×C3⋊(C3⋊D3)D811yescharacteristic
78C2×C3×C3⋊D3—41no—
79C2×C32×D3—41no—
80D3×C3⋊D3—82no—
81C33×C4—41no—
82C33×C4C2×D411yescharacteristic
83C3⋊(C3⋊Dic3)D811yescharacteristic
84#919—94no—
85#919—91no—
86#919—184no—
87D3×D8—94no—
88D3×D8—91no—
89C3×SD32—34no—
90C3×SD32—91no—
91#828—34no—
92#828C3⋊D311yescharacteristic, complemented
93#826—94no—
94#826—91no—
95D48—94no—
96D48—91no—
97#798—94no—
98#798—91no—
99C22×C3⋊D3—121no—
100C2×D32—124no—
101C32×Q8—28no—
102C32×Q8—24no—
103C32×Q8—21no—
104C32×D4—61no—
105#638—121no—
106C4⋊(C3⋊D3)—21no—
107C4⋊(C3⋊D3)—68no—
108C4⋊(C3⋊D3)—64no—
109C4⋊(C3⋊D3)—61no—
110C4⋊(C3⋊D3)—121no—
111C4⋊(C3⋊D3)—124no—
112C4⋊(C3⋊D3)—128no—
113C4×C3⋊D3—31no—
114C4×C3⋊D3—34no—
115C4×C3⋊D3—64no—
116C4×C3⋊D3—121no—
117C4×C3⋊D3—124no—
118C4×C3⋊D3—128no—
119C3×D12—24no—
120C3×D12—64no—
121#625—124no—
122C32×C8—24no—
123C32×C8C22×D314yes—
124C32×C8C22×D311yescharacteristic
125C3⋊(C3⋊C8)—31no—
126C3⋊(C3⋊C8)—34no—
127C3⋊(C3⋊C8)—64no—
128#509—271noSylow
129C2×C33C2×D811yescharacteristic
130C3⋊(C3⋊D3)—21no—
131C3⋊(C3⋊D3)—81no—
132C3×C3⋊D3—81no—
133C32×D3—81no—
134#239—184no—
135#239—181no—
136#239—188no—
137D3×D4—181no—
138D3×D4—184no—
139C3×Q16—24no—
140C3×Q16D3214yescomplemented
141C3×Q16C2×C3⋊D311yescharacteristic, complemented
142C3×D8—34no—
143C3×D8—91no—
144#215—184no—
145#215—181no—
146#215—188no—
147#213—181no—
148#213—184no—
149D24—34no—
150D24—94no—
151D24—91no—
152D24—184no—
153D24C2×C3⋊D311yescharacteristic
154D3×C8—94no—
155D3×C8—91no—
156D3×C8—184no—
157C3×C16—34no—
158C3×C16—91no—
159C3⋊C16—34no—
160C3⋊C16C2×C3⋊D311yescharacteristic, complemented
161C22×C32—121no—
162C2×C3⋊D3—31no—
163C2×C3⋊D3—34no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—64no—
166C2×C3⋊D3—128no—
167C2×C3⋊D3—124no—
168C2×C3⋊D3—121no—
169C2×C3⋊D3—242no—
170C2×C3×D3—44no—
171C2×C3×D3—124no—
172D32—248no—
173C32×C4—24no—
174C32×C4—48no—
175C32×C4—44no—
176C32×C4—41no—
177C32×C4D3×D414yes—
178C32×C4D3×D411yescharacteristic
179C3⋊Dic3—31no—
180C3⋊Dic3—34no—
181C3⋊Dic3—64no—
182#135—271no—
183C2×D8—271no—
184SD32—31no—
185SD32—91no—
186D16—271no—
187D16—271no—
188M32—271no—
189C33#50911yescharacteristic, complemented, Sylow
190C22×D3—361no—
191C22×D3—364no—
192C3×Q8—28no—
193C3×Q8—24no—
194C3×Q8—21no—
195C3×D4—64no—
196C3×D4—181no—
197#67—361no—
198#67—364no—
199D12—21no—
200D12—64no—
201D12—188no—
202D12—184no—
203D12—181no—
204D12—361no—
205D12—364no—
206D12—368no—
207C4×D3—94no—
208C4×D3—91no—
209C4×D3—184no—
210C4×D3—361no—
211C4×D3—364no—
212C4×D3—368no—
213C3×C8—24no—
214C3×C8C2×D3214yes—
215C3×C8C22×C3⋊D311yescharacteristic
216C3⋊C8—94no—
217C3⋊C8—91no—
218C3⋊C8—184no—
219C2×C32—24no—
220C2×C32—241no—
221C2×C32D3×D814yes—
222C2×C32D3×D811yescharacteristic
223C3⋊D3—61no—
224C3⋊D3—64no—
225C3⋊D3—81no—
226C3⋊D3—124no—
227C3⋊D3—248no—
228C3⋊D3—244no—
229C3⋊D3—241no—
230C3×D3—84no—
231C3×D3—244no—
232#41—541no—
233C2×D4—541no—
234Q16D3×C3⋊D311yescharacteristic, complemented
235SD16—541no—
236D8—31no—
237D8—91no—
238D8—271no—
239D8—541no—
240C2×C8—271no—
241C16—31no—
242C16—91no—
243C22×C3—124no—
244C22×C3—361no—
245C2×D3—41no—
246C2×D3—91no—
247C2×D3—94no—
248C2×D3—124no—
249C2×D3—184no—
250C2×D3—368no—
251C2×D3—361no—
252C2×D3—364no—
253C2×D3—722no—
254C2×D3—728no—
255C3×C4—24no—
256C3×C4—48no—
257C3×C4—41no—
258C3×C4—44no—
259C3×C4#381114yes—
260C3×C4D4×C3⋊D311yescharacteristic
261Dic3—91no—
262Dic3—94no—
263Dic3—184no—
264C32—24no—
265C32#491414yescomplemented
266C32#491411yescharacteristic, complemented
267C23—1081no—
268Q8—21no—
269D4—61no—
270D4—181no—
271D4—541no—
272D4—1081no—
273D4—1081no—
274C2×C4—271no—
275C2×C4—1081no—
276C8—271no—
277C8C2×D3×C3⋊D311yescharacteristic
278C2×C3—24no—
279C2×C3—244no—
280C2×C3—721no—
281C2×C3#6392614yes—
282C2×C3D8×C3⋊D311yescharacteristic
283D3—81no—
284D3—184no—
285D3—181no—
286D3—244no—
287D3—364no—
288D3—728no—
289D3—724no—
290D3—721no—
291C22—121no—
292C22—271no—
293C22—361no—
294C22—1081no—
295C22—2162no—
296C4—41no—
297C4—271no—
298C4#8903911yescharacteristic
299C3—24no—
300C3#1059104014yescomplemented
301C3#1059261311yescharacteristic, complemented
302C2—241no—
303C2—541no—
304C2—721no—
305C2—2161no—
306C2#1172354711yescharacteristic
subgroupG/H
centerC2#11723547
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. #11723960C2semidirect
2. #11723322C2semidirect
3. C3×#63655C2semidirect
4. C32×#828C2semidirect
5. #11722006C2semidirect
6. #11722005C2semidirect
7. #11722003C2semidirect
12. C33×Q16C22semidirect
14. #88852C22semidirect
19. C3×C3⋊(C3⋊C16)C22semidirect
20. C32×C3⋊C16C22semidirect
27. #63655D3semidirect
32. C3×#828D3semidirect
55. C32×Q16C2×D3semidirect
56. C32×Q16C2×D3semidirect
73. C3⋊(C3⋊C16)C2×D3semidirect
75. C3×C3⋊C16C2×D3semidirect
92. #828C3⋊D3semidirect
140. C3×Q16D32semidirect
141. C3×Q16C2×C3⋊D3semidirect
160. C3⋊C16C2×C3⋊D3semidirect
189. C33#509semidirect
234. Q16D3×C3⋊D3semidirect
265. C32#4914semidirect
266. C32#4914semidirect
300. C3#10591040semidirect
301. C3#10592613semidirect

Maximal subgroups

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

subgroupindex
#117239602
#117233222
C3×#636552
C32×#8282
#117220062
#117220052
#117220032
#105926133
#105910403