Finite Group Databasebeta

← #13161464

Subgroups — #13161464

Order 1728.

450 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C8×#1113C211yesmaximal, characteristic
2C3×#63625C212yesmaximal, complemented
3C3×C16×C3⋊D3C211yesmaximal, characteristic, complemented
4#11722066C212yesmaximal, complemented
5#11722065C211yesmaximal, characteristic, complemented
6#10590226—31nomaximal
7D3×#798—32nomaximal
8C4×#1113C411yescharacteristic
9C3×C8×C3⋊D3C2211yescharacteristic
10C3×C8×C3⋊D3C2212yes—
11#88808C412yes—
12#88808C411yescharacteristic
13C33×C16C2211yescharacteristic, complemented
14C3×C3⋊(C3⋊C16)C2212yescomplemented
15C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
16D32×C8—32no—
17D32×C8—31no—
18#63625D312yescomplemented
19C16×C3⋊D3D311yescharacteristic, complemented
20C3×#798—32no—
21C3×#798—32no—
22C3×#798—62no—
23C3×D3×C16—32no—
24C3×D3×C16—61no—
25#63544—32no—
26#63543—32no—
27#63543—32no—
28#63542—31no—
29D3×C3⋊C16—32no—
30C2×#1113C811yescharacteristic
31C3×C4×C3⋊D3C2×C412yes—
32C3×C4×C3⋊D3C2×C411yescharacteristic
33#6294C812yes—
34#6294C811yescharacteristic
35C33×C8C2311yescharacteristic
36C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
37C3×C3⋊(C3⋊C8)C2×C412yes—
38D3×M32—92no—
39C2×#798—91no—
40C4×D32—32no—
41C4×D32—31no—
42C8×C3⋊D3C2×D311yescharacteristic
43C8×C3⋊D3C2×D312yes—
44C3×D3×C8—32no—
45C3×D3×C8—32no—
46C3×D3×C8—32no—
47C3×D3×C8—61no—
48C3×D3×C8—62no—
49#3710—32no—
50#3710—31no—
51D3×C3⋊C8—32no—
52D3×C3⋊C8—32no—
53D3×C3⋊C8—32no—
54C32×C16—21no—
55C32×C16—22no—
56C32×C16—41no—
57C32×C16C2×D312yescomplemented
58C32×C16C2×D311yescharacteristic, complemented
59C3⋊(C3⋊C16)C2×D312yescomplemented
60C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
61C3×C3⋊C16—32no—
62C3×C3⋊C16—32no—
63C3×C3⋊C16—32no—
64C3×C3⋊C16—61no—
65C3×C3⋊C16—62no—
66C2×C3×C3⋊D3C2×C812yes—
67C2×C3×C3⋊D3C2×C811yescharacteristic
68#1113—22no—
69C33×C4C22×C411yescharacteristic
70C3×C3⋊Dic3C2×C812yes—
71C3×C3⋊Dic3C2×C811yescharacteristic
72C2×D3×C8—91no—
73C2×D3×C8—92no—
74C3×M32—92no—
75C2×C3×C16—91no—
76#812—92no—
77C2×C3⋊C16—91no—
78#798—32no—
79#798—32no—
80#798—62no—
81#798—92no—
82#798—92no—
83D3×C16—32no—
84D3×C16—61no—
85D3×C16—92no—
86C2×D32—32no—
87C2×D32—31no—
88C4×C3⋊D3C4×D312yes—
89C4×C3⋊D3C4×D311yescharacteristic
90C3×C4×D3—32no—
91C3×C4×D3—32no—
92C3×C4×D3—32no—
93C3×C4×D3—61no—
94C3×C4×D3—62no—
95#624—32no—
96#624—31no—
97D3×Dic3—32no—
98D3×Dic3—32no—
99D3×Dic3—32no—
100C32×C8—21no—
101C32×C8—22no—
102C32×C8—41no—
103C32×C8C22×D312yes—
104C32×C8C22×D311yescharacteristic
105C3⋊(C3⋊C8)C4×D311yescharacteristic
106C3⋊(C3⋊C8)C4×D312yes—
107C3×C3⋊C8—32no—
108C3×C3⋊C8—32no—
109C3×C3⋊C8—32no—
110C3×C3⋊C8—61no—
111C3×C3⋊C8—62no—
112C2×M32—271noSylow
113C2×C33C22×C811yescharacteristic
114C3×C3⋊D3—22no—
115C3×C3⋊D3M3212yescomplemented
116C2×C4×D3—91no—
117C2×C4×D3—92no—
118C2×C3×C8—91no—
119C2×C3×C8—92no—
120C2×C3⋊C8—91no—
121C2×C3⋊C8—92no—
122D3×C8—32no—
123D3×C8—32no—
124D3×C8—32no—
125D3×C8—61no—
126D3×C8—62no—
127D3×C8—92no—
128D3×C8—92no—
129D3×C8—92no—
130C3×C16—21no—
131C3×C16—22no—
132C3×C16—41no—
133C3×C16—92no—
134C3×C16—91no—
135C3×C16D3212yescomplemented
136C3×C16D3211yescharacteristic, complemented
137C3⋊C16—32no—
138C3⋊C16—32no—
139C3⋊C16—32no—
140C3⋊C16—61no—
141C3⋊C16—62no—
142C2×C3⋊D3D3×C812yes—
143C2×C3⋊D3D3×C811yescharacteristic
144C2×C3×D3—32no—
145C2×C3×D3—32no—
146C2×C3×D3—32no—
147C2×C3×D3—61no—
148C2×C3×D3—62no—
149D32—64no—
150D32—62no—
151C32×C4—22no—
152C32×C4—21no—
153C32×C4—41no—
154C32×C4C2×C4×D311yescharacteristic
155C32×C4C2×C4×D312yes—
156C3⋊Dic3D3×C811yescharacteristic
157C3⋊Dic3D3×C812yes—
158C3×Dic3—32no—
159C3×Dic3—32no—
160C3×Dic3—32no—
161C3×Dic3—62no—
162C3×Dic3—61no—
163C22×C8—271no—
164M32—92no—
165M32—272no—
166C2×C16—91no—
167C2×C16—271no—
168C33C2×M3211yescharacteristic, complemented, Sylow
169C22×D3—92no—
170C22×D3—91no—
171C2×C3×C4—91no—
172C2×C3×C4—92no—
173C2×Dic3—91no—
174C2×Dic3—92no—
175C4×D3—32no—
176C4×D3—32no—
177C4×D3—32no—
178C4×D3—61no—
179C4×D3—62no—
180C4×D3—92no—
181C4×D3—92no—
182C4×D3—92no—
183C3×C8—21no—
184C3×C8—22no—
185C3×C8—41no—
186C3×C8—91no—
187C3×C8—92no—
188C3×C8C2×D3212yes—
189C3×C8C2×D3211yescharacteristic
190C3⋊C8—32no—
191C3⋊C8—32no—
192C3⋊C8—32no—
193C3⋊C8—61no—
194C3⋊C8—62no—
195C2×C32—22no—
196C2×C32—21no—
197C2×C32—41no—
198C2×C32C2×D3×C811yescharacteristic
199C2×C32C2×D3×C812yes—
200C3⋊D3—22no—
201C3⋊D3#79812yescomplemented
202C3×D3—34no—
203C3×D3—62no—
204C3×D3—62no—
205C3×D3—62no—
206C3×D3—122no—
207C22×C4—271no—
208C2×C8—91no—
209C2×C8—92no—
210C2×C8—272no—
211C2×C8—271no—
212C16—91no—
213C16—92no—
214C16#111311yescharacteristic, complemented
215C22×C3—91no—
216C22×C3—92no—
217C2×D3—32no—
218C2×D3—32no—
219C2×D3—32no—
220C2×D3—61no—
221C2×D3—62no—
222C2×D3—184no—
223C2×D3—182no—
224C3×C4—21no—
225C3×C4—22no—
226C3×C4—41no—
227C3×C4—92no—
228C3×C4—91no—
229C3×C4C4×D3212yes—
230C3×C4C4×D3211yescharacteristic
231Dic3—32no—
232Dic3—32no—
233Dic3—32no—
234Dic3—61no—
235Dic3—62no—
236C32—22no—
237C32—21no—
238C32—41no—
239C32C2×#79811yescharacteristic, complemented
240C32D3×M3212yescomplemented
241C23—271no—
242C2×C4—91no—
243C2×C4—92no—
244C2×C4—271no—
245C2×C4—272no—
246C8—91no—
247C8—92no—
248C8C2×#111311yescharacteristic
249C2×C3—21no—
250C2×C3—22no—
251C2×C3—41no—
252C2×C3—92no—
253C2×C3—182no—
254C2×C3D32×C812yes—
255C2×C3D32×C811yescharacteristic
256D3—34no—
257D3—62no—
258D3—62no—
259D3—62no—
260D3—122no—
261C22—91no—
262C22—92no—
263C22—542no—
264C4—91no—
265C4—92no—
266C4C4×#111311yescharacteristic
267C3—21no—
268C3—22no—
269C3—41no—
270C3D3×#79812yescomplemented
271C3#1059022611yescharacteristic, complemented
272C2—92no—
273C2—182no—
274C2C8×#111311yescharacteristic
subgroupG/H
center, FrattiniC8C2×#1113
commutator, socleC2×C33C22×C8
FittingC33×C16C22
prime radicalC2×#1113C8
cosocleC33×C8C23
hypercenterC16#1113
hypocenterC33C2×M32
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
2. C3×#63625C2semidirect
3. C3×C16×C3⋊D3C2semidirect
4. #11722066C2semidirect
5. #11722065C2semidirect
13. C33×C16C22semidirect
14. C3×C3⋊(C3⋊C16)C22semidirect
15. C3×C3⋊(C3⋊C16)C22semidirect
18. #63625D3semidirect
19. C16×C3⋊D3D3semidirect
57. C32×C16C2×D3semidirect
58. C32×C16C2×D3semidirect
59. C3⋊(C3⋊C16)C2×D3semidirect
60. C3⋊(C3⋊C16)C2×D3semidirect
115. C3×C3⋊D3M32semidirect
135. C3×C16D32semidirect
136. C3×C16D32semidirect
168. C33C2×M32semidirect
201. C3⋊D3#798semidirect
214. C16#1113semidirect
239. C32C2×#798semidirect
240. C32D3×M32semidirect
270. C3D3×#798semidirect
271. C3#10590226semidirect

Maximal subgroups

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

subgroupindex
C8×#11132
C3×#636252
C3×C16×C3⋊D32
#117220662
#117220652
#105902263
D3×#7983