Finite Group Databasebeta

← #13161467

Subgroups — #13161467

Order 1728.

450 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 75 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#11723649C211yesmaximal, characteristic, complemented
2#11723647C211yesmaximal, characteristic
3C3×#63627C212yesmaximal, complemented
4C3×C16×C3⋊D3C211yesmaximal, characteristic, complemented
5#11722068C212yesmaximal, complemented
6#10590235—31nomaximal
7D3×#800—32nomaximal
8C4⋊#1113—21no—
9#89040—21no—
10C3×#3746C2212yescomplemented
11C3×#3745C2212yes—
12C3×C8×C3⋊D3C2211yescharacteristic
13#88812—21no—
14#88810—21no—
15C33×C16C2211yescharacteristic, complemented
16C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
17#63802—31no—
18D3×Dic12—32no—
19#63797—31no—
20D3×D24—32no—
21#63627D312yescomplemented
22C16×C3⋊D3D311yescharacteristic, complemented
23C3×#800—32no—
24C3×#800—32no—
25C3×#800—62no—
26C3×D3×C16—32no—
27C3×D3×C16—61no—
28#63549—32no—
29#63548—32no—
30#63547—32no—
31C2×#1113—41no—
32C3×C4⋊(C3⋊D3)—22no—
33C3×C4×C3⋊D3D411yescharacteristic
34C3×#634—22no—
35#6296—41no—
36#6295—41no—
37#6294—41no—
38C33×C8C2311yescharacteristic
39C3×C3⋊(C3⋊C8)D411yescharacteristic
40D3×SD32—92no—
41C2×#800—91no—
42#3802—61no—
43D3×D12—62no—
44#3797—61no—
45D3×Dic6—62no—
46#3746C2×D312yescomplemented
47#3745C2×D312yes—
48C8×C3⋊D3C2×D311yescharacteristic
49C3×Dic12—32no—
50C3×Dic12—32no—
51C3×Dic12—62no—
52C3×D24—32no—
53C3×D24—32no—
54C3×D24—62no—
55C3×D3×C8—32no—
56C3×D3×C8—61no—
57#3719—62no—
58#3718—61no—
59#3714—62no—
60#3713—61no—
61C32×C16—21no—
62C32×C16—22no—
63C32×C16—41no—
64C32×C16C2×D312yescomplemented
65C32×C16C2×D311yescharacteristic, complemented
66C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
67C3×C3⋊C16—32no—
68C3×C3⋊C16—61no—
69C2×C3×C3⋊D3—42no—
70C2×C3×C3⋊D3D811yescharacteristic
71#1113—82no—
72C33×C4C2×D411yescharacteristic
73C3×C3⋊Dic3—42no—
74C3×C3⋊Dic3D811yescharacteristic
75D3×Q16—92no—
76D3×D8—92no—
77C2×Dic12—91no—
78C2×D24—91no—
79C3×SD32—92no—
80C2×C3×C16—91no—
81#828—92no—
82#827—92no—
83#800—32no—
84#800—32no—
85#800—62no—
86#800—92no—
87D3×C16—32no—
88D3×C16—61no—
89C2×D32—122no—
90C2×D32—121no—
91C4⋊(C3⋊D3)—22no—
92C4×C3⋊D3D1211yescharacteristic
93#634—22no—
94C3×D12—62no—
95C3×D12—62no—
96C3×D12—122no—
97C3×C4×D3—32no—
98C3×C4×D3—61no—
99C3×Dic6—62no—
100C3×Dic6—62no—
101C3×Dic6—122no—
102#627—122no—
103#627—121no—
104#626—122no—
105#625—121no—
106#624—121no—
107D3×Dic3—122no—
108C32×C8—21no—
109C32×C8—22no—
110C32×C8—41no—
111C32×C8C22×D311yescharacteristic
112C32×C8C22×D312yes—
113C3⋊(C3⋊C8)D1211yescharacteristic
114C3×C3⋊C8—32no—
115C3×C3⋊C8—61no—
116C2×SD32—271noSylow
117C2×C33C2×D811yescharacteristic
118C3×C3⋊D3—82no—
119C3×C3⋊D3SD3212yescomplemented
120D3×Q8—182no—
121D3×D4—182no—
122C2×D12—181no—
123C2×Dic6—181no—
124C3×Q16—92no—
125C3×D8—92no—
126C2×C3×C8—91no—
127#216—182no—
128#213—182no—
129Dic12—32no—
130Dic12—32no—
131Dic12—62no—
132Dic12—181no—
133D24—32no—
134D24—32no—
135D24—62no—
136D24—181no—
137D3×C8—32no—
138D3×C8—61no—
139C3×C16—21no—
140C3×C16—22no—
141C3×C16—41no—
142C3×C16—91no—
143C3×C16D3212yescomplemented
144C3×C16D3211yescharacteristic, complemented
145C3⋊C16—32no—
146C3⋊C16—61no—
147C2×C3⋊D3—42no—
148C2×C3⋊D3D2411yescharacteristic
149C2×C3×D3—32no—
150C2×C3×D3—61no—
151C2×C3×D3—122no—
152C2×C3×D3—122no—
153C2×C3×D3—242no—
154D32—244no—
155D32—242no—
156C32×C4—22no—
157C32×C4—21no—
158C32×C4—41no—
159C32×C4C2×D1211yescharacteristic
160C32×C4D3×D412yes—
161C3⋊Dic3—42no—
162C3⋊Dic3D2411yescharacteristic
163C3×Dic3—32no—
164C3×Dic3—61no—
165C3×Dic3—122no—
166C3×Dic3—122no—
167C3×Dic3—242no—
168C2×Q16—271no—
169C2×D8—271no—
170SD32—92no—
171SD32—272no—
172C2×C16—91no—
173C33C2×SD3211yescharacteristic, complemented, Sylow
174C22×D3—362no—
175C22×D3—361no—
176C3×Q8—182no—
177C3×D4—182no—
178C2×C3×C4—91no—
179#67—362no—
180C2×Dic3—361no—
181D12—62no—
182D12—62no—
183D12—122no—
184D12—361no—
185C4×D3—32no—
186C4×D3—61no—
187C4×D3—362no—
188Dic6—62no—
189Dic6—62no—
190Dic6—122no—
191Dic6—362no—
192Dic6—361no—
193C3×C8—21no—
194C3×C8—22no—
195C3×C8—41no—
196C3×C8—91no—
197C3×C8C2×D3212yes—
198C3×C8C2×D3211yescharacteristic
199C3⋊C8—32no—
200C3⋊C8—61no—
201C2×C32—22no—
202C2×C32—21no—
203C2×C32—41no—
204C2×C32C2×D2411yescharacteristic
205C2×C32D3×D812yes—
206C3⋊D3—82no—
207C3⋊D3#80012yescomplemented
208C3×D3—34no—
209C3×D3—62no—
210C3×D3—242no—
211C3×D3—242no—
212C3×D3—482no—
213C2×Q8—541no—
214C2×D4—541no—
215Q16—92no—
216Q16—541no—
217D8—92no—
218D8—541no—
219C2×C8—91no—
220C16—91no—
221C16#111311yescharacteristic, complemented
222C22×C3—91no—
223C22×C3—362no—
224C2×D3—32no—
225C2×D3—61no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—242no—
229C2×D3—724no—
230C2×D3—722no—
231C3×C4—21no—
232C3×C4—22no—
233C3×C4—41no—
234C3×C4—91no—
235C3×C4—362no—
236C3×C4D3×D1212yes—
237C3×C4#380211yescharacteristic
238Dic3—32no—
239Dic3—61no—
240Dic3—122no—
241Dic3—122no—
242Dic3—242no—
243C32—22no—
244C32—21no—
245C32—41no—
246C32C2×#80011yescharacteristic, complemented
247C32D3×SD3212yescomplemented
248C23—1081no—
249Q8—182no—
250Q8—1081no—
251D4—182no—
252D4—1081no—
253C2×C4—91no—
254C2×C4—1081no—
255C8—91no—
256C8C2×#111311yescharacteristic
257C2×C3—21no—
258C2×C3—22no—
259C2×C3—41no—
260C2×C3—92no—
261C2×C3—722no—
262C2×C3D3×D2412yes—
263C2×C3#6379711yescharacteristic
264D3—34no—
265D3—62no—
266D3—242no—
267D3—242no—
268D3—482no—
269C22—91no—
270C22—362no—
271C22—2162no—
272C4—91no—
273C4—362no—
274C4C4⋊#111311yescharacteristic
275C3—21no—
276C3—22no—
277C3—41no—
278C3D3×#80012yescomplemented
279C3#1059023511yescharacteristic, complemented
280C2—92no—
281C2—722no—
282C2#1172364711yescharacteristic
subgroupG/H
centerC2#11723647
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×C16C22
socleC2×C33C2×D8
prime radical#11723647C2
hypercenterC16#1113
hypocenterC33C2×SD32
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
1. #11723649C2semidirect
3. C3×#63627C2semidirect
4. C3×C16×C3⋊D3C2semidirect
5. #11722068C2semidirect
10. C3×#3746C22semidirect
15. C33×C16C22semidirect
16. C3×C3⋊(C3⋊C16)C22semidirect
21. #63627D3semidirect
22. C16×C3⋊D3D3semidirect
46. #3746C2×D3semidirect
64. C32×C16C2×D3semidirect
65. C32×C16C2×D3semidirect
66. C3⋊(C3⋊C16)C2×D3semidirect
119. C3×C3⋊D3SD32semidirect
143. C3×C16D32semidirect
144. C3×C16D32semidirect
173. C33C2×SD32semidirect
207. C3⋊D3#800semidirect
221. C16#1113semidirect
246. C32C2×#800semidirect
247. C32D3×SD32semidirect
278. C3D3×#800semidirect
279. C3#10590235semidirect

Maximal subgroups

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

subgroupindex
#117236492
#117236472
C3×#636272
C3×C16×C3⋊D32
#117220682
#105902353
D3×#8003