Subgroups — 3₋¹⁺²×#485 #13145215

Order 1728.

384 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×#117	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×#103	C₂	1	1	yes	maximal, characteristic, complemented
6	3₋¹⁺²×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	3₋¹⁺²×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₃²×#485	C₃	1	1	yes	maximal, characteristic
9	C₉×#485	C₃	1	3	yes	maximal, complemented
10	C₂×Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
15	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×3₋¹⁺²	—	2	1	no	—
19	C₂²⋊C₄×3₋¹⁺²	—	2	1	no	—
20	C₄²×3₋¹⁺²	—	2	1	no	—
21	C₃²×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₃²×#117	C₂×C₃	1	1	yes	characteristic
23	C₃²×#107	C₂×C₃	1	1	yes	characteristic
24	C₃²×#106	C₂×C₃	1	1	yes	characteristic
25	C₃²×#103	C₂×C₃	1	1	yes	characteristic
26	C₃²×#103	C₂×C₃	1	1	yes	characteristic
27	C₃²×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
28	C₉×C₂²⋊Q₈	C₂×C₃	1	3	yes	—
29	C₉×#117	C₂×C₃	1	3	yes	—
30	C₉×#107	C₂×C₃	1	3	yes	complemented
31	C₉×#106	C₂×C₃	1	3	yes	complemented
32	C₉×#103	C₂×C₃	1	3	yes	complemented
33	C₉×#103	C₂×C₃	1	3	yes	complemented
34	C₉×C₂²⋊C₈	C₂×C₃	1	3	yes	—
35	C₂³×3₋¹⁺²	D₄	1	1	yes	characteristic
36	Q₈×3₋¹⁺²	—	2	1	no	—
37	Q₈×3₋¹⁺²	—	2	1	no	—
38	C₂×C₄×3₋¹⁺²	—	2	1	no	—
39	C₂×C₄×3₋¹⁺²	—	2	1	no	—
40	C₂×C₄×3₋¹⁺²	—	2	1	no	—
41	C₂×C₄×3₋¹⁺²	—	2	1	no	—
42	C₂×C₄×3₋¹⁺²	—	2	1	no	—
43	C₂×C₄×3₋¹⁺²	—	2	1	no	—
44	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
45	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
46	C₈×3₋¹⁺²	—	2	1	no	—
47	C₈×3₋¹⁺²	—	2	1	no	—
48	C₃×#485	—	3	1	no	—
49	C₃×#485	C₃²	1	1	yes	characteristic
50	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
51	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₃²×C₄⋊C₄	—	2	1	no	—
55	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
56	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
57	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
58	C₃²×C₂²⋊C₄	—	2	1	no	—
59	C₃²×C₂²⋊C₄	—	2	1	no	—
60	C₃²×C₄²	—	2	1	no	—
61	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
62	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
63	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
64	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
65	C₉×C₄⋊C₄	—	2	3	no	—
66	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
67	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
68	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
69	C₉×C₂²⋊C₄	—	2	3	no	—
70	C₉×C₂²⋊C₄	—	2	3	no	—
71	C₄²×C₉	—	2	3	no	—
72	C₂²×3₋¹⁺²	—	2	1	no	—
73	C₂²×3₋¹⁺²	—	2	1	no	—
74	C₂²×3₋¹⁺²	—	2	1	no	—
75	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
76	C₄×3₋¹⁺²	—	2	1	no	—
77	C₄×3₋¹⁺²	—	4	1	no	—
78	C₄×3₋¹⁺²	—	4	1	no	—
79	C₄×3₋¹⁺²	—	4	1	no	—
80	C₄×3₋¹⁺²	—	4	1	no	—
81	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
82	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
83	C₃×C₂²⋊Q₈	—	3	1	no	—
84	C₃×C₂²⋊Q₈	C₂×C₃²	1	1	yes	characteristic
85	C₃×#117	—	3	1	no	—
86	C₃×#117	C₂×C₃²	1	1	yes	characteristic
87	C₃×#107	—	3	1	no	—
88	C₃×#107	C₂×C₃²	1	1	yes	characteristic
89	C₃×#106	—	3	1	no	—
90	C₃×#106	C₂×C₃²	1	1	yes	characteristic
91	C₃×#103	—	3	1	no	—
92	C₃×#103	—	3	1	no	—
93	C₃×#103	C₂×C₃²	1	1	yes	characteristic
94	C₃×#103	C₂×C₃²	1	1	yes	characteristic
95	C₃×C₂²⋊C₈	—	3	1	no	—
96	C₃×C₂²⋊C₈	C₂×C₃²	1	1	yes	characteristic
97	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
98	C₃²×Q₈	—	2	1	no	—
99	C₃²×Q₈	—	2	1	no	—
100	C₂×C₃²×C₄	—	2	1	no	—
101	C₂×C₃²×C₄	—	2	1	no	—
102	C₂×C₃²×C₄	—	2	1	no	—
103	C₂×C₃²×C₄	—	2	1	no	—
104	C₂×C₃²×C₄	—	2	1	no	—
105	C₂×C₃²×C₄	—	2	1	no	—
106	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
107	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
108	C₂³×C₉	C₃×D₄	1	3	yes	—
109	C₃²×C₈	—	2	1	no	—
110	C₃²×C₈	—	2	1	no	—
111	Q₈×C₉	—	2	3	no	—
112	Q₈×C₉	—	2	3	no	—
113	C₂×C₄×C₉	—	2	3	no	—
114	C₂×C₄×C₉	—	2	3	no	—
115	C₂×C₄×C₉	—	2	3	no	—
116	C₂×C₄×C₉	—	2	3	no	—
117	C₂×C₄×C₉	—	2	3	no	—
118	C₂×C₄×C₉	—	2	3	no	—
119	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
120	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
121	C₈×C₉	—	2	3	no	—
122	C₈×C₉	—	2	3	no	—
123	#485	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×3₋¹⁺²	—	4	1	no	—
125	C₂×3₋¹⁺²	#123	1	1	yes	characteristic
126	C₂×3₋¹⁺²	#135	1	1	yes	characteristic
127	C₂×3₋¹⁺²	#137	1	1	yes	characteristic
128	C₂×C₃×Q₈	—	3	1	no	—
129	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
130	C₂²×C₃×C₄	—	3	1	no	—
131	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
132	C₂×C₃×C₈	—	3	1	no	—
133	C₂×C₃×C₈	—	3	1	no	—
134	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	—	3	1	no	—
138	C₃×C₄⋊C₄	—	3	1	no	—
139	C₃×C₄⋊C₄	—	3	1	no	—
140	C₃×C₄⋊C₄	—	6	1	no	—
141	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
142	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
143	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
144	C₃×C₂²⋊C₄	—	2	1	no	—
145	C₃×C₂²⋊C₄	—	2	1	no	—
146	C₃×C₂²⋊C₄	—	6	1	no	—
147	C₃×C₂²⋊C₄	—	6	1	no	—
148	C₃×C₄²	—	2	1	no	—
149	C₃×C₄²	—	6	1	no	—
150	C₂²×C₃²	—	2	1	no	—
151	C₂²×C₃²	—	2	1	no	—
152	C₂²×C₃²	—	2	1	no	—
153	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
154	C₃²×C₄	—	2	1	no	—
155	C₃²×C₄	—	4	1	no	—
156	C₃²×C₄	—	4	1	no	—
157	C₃²×C₄	—	4	1	no	—
158	C₃²×C₄	—	4	1	no	—
159	C₃²×C₄	C₃×#41	1	1	yes	characteristic
160	C₃²×C₄	C₃×#41	1	1	yes	characteristic
161	C₂²×C₉	—	2	3	no	—
162	C₂²×C₉	—	2	3	no	—
163	C₂²×C₉	—	2	3	no	—
164	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
165	C₄×C₉	—	2	3	no	—
166	C₄×C₉	—	4	3	no	—
167	C₄×C₉	—	4	3	no	—
168	C₄×C₉	—	4	3	no	—
169	C₄×C₉	—	4	3	no	—
170	C₄×C₉	C₃×#41	1	3	yes	—
171	C₄×C₉	C₃×#41	1	3	yes	—
172	C₂²⋊Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic
173	#117	C₂×3₋¹⁺²	1	1	yes	characteristic
174	#107	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
175	#106	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
176	#103	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
177	#103	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
178	C₂²⋊C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
179	3₋¹⁺²	#485	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×C₃	—	3	1	no	—
181	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
182	C₃×Q₈	—	2	1	no	—
183	C₃×Q₈	—	2	1	no	—
184	C₃×Q₈	—	6	1	no	—
185	C₃×Q₈	—	6	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	3	1	no	—
193	C₂×C₃×C₄	—	3	1	no	—
194	C₂×C₃×C₄	—	6	1	no	—
195	C₂×C₃×C₄	—	6	1	no	—
196	C₂×C₃×C₄	—	6	1	no	—
197	C₂×C₃×C₄	—	6	1	no	—
198	C₂×C₃×C₄	—	6	1	no	—
199	C₂×C₃×C₄	—	6	1	no	—
200	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
202	C₃×C₈	—	2	1	no	—
203	C₃×C₈	—	2	1	no	—
204	C₃×C₈	—	6	1	no	—
205	C₃×C₈	—	6	1	no	—
206	C₂×C₃²	—	4	1	no	—
207	C₂×C₃²	C₃×#123	1	1	yes	characteristic
208	C₂×C₃²	C₃×#135	1	1	yes	characteristic
209	C₂×C₃²	C₃×#137	1	1	yes	characteristic
210	C₂×C₉	—	4	3	no	—
211	C₂×C₉	C₃×#123	1	3	yes	—
212	C₂×C₉	C₃×#135	1	3	yes	—
213	C₂×C₉	C₃×#137	1	3	yes	—
214	C₂×Q₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
215	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
216	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
217	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
220	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
221	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	1	no	—
228	C₂²×C₃	—	3	1	no	—
229	C₂²×C₃	—	6	1	no	—
230	C₂²×C₃	—	6	1	no	—
231	C₂²×C₃	—	6	1	no	—
232	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	3	1	no	—
235	C₃×C₄	—	3	1	no	—
236	C₃×C₄	—	4	1	no	—
237	C₃×C₄	—	4	1	no	—
238	C₃×C₄	—	4	1	no	—
239	C₃×C₄	—	4	1	no	—
240	C₃×C₄	—	6	1	no	—
241	C₃×C₄	—	12	1	no	—
242	C₃×C₄	—	12	1	no	—
243	C₃×C₄	—	12	1	no	—
244	C₃×C₄	—	12	1	no	—
245	C₃×C₄	C₃²×#41	1	1	yes	characteristic
246	C₃×C₄	C₃²×#41	1	1	yes	characteristic
247	C₃²	C₃×#485	1	1	yes	characteristic
248	C₉	C₃×#485	1	3	yes	complemented
249	C₂³	D₄×3₋¹⁺²	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
259	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₂×C₃	—	3	1	no	—
263	C₂×C₃	—	3	1	no	—
264	C₂×C₃	—	3	1	no	—
265	C₂×C₃	—	4	1	no	—
266	C₂×C₃	—	12	1	no	—
267	C₂×C₃	C₃²×#123	1	1	yes	characteristic
268	C₂×C₃	C₃²×#135	1	1	yes	characteristic
269	C₂×C₃	C₃²×#137	1	1	yes	characteristic
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
280	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
281	C₃	—	3	1	no	—
282	C₃	C₃²×#485	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	3₋¹⁺²×#123	1	1	yes	characteristic
285	C₂	3₋¹⁺²×#135	1	1	yes	characteristic
286	C₂	3₋¹⁺²×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂³×C₃²	C₃×D₄
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₄×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²×#107	C₂	semidirect
4. 3₋¹⁺²×#106	C₂	semidirect
5. 3₋¹⁺²×#103	C₂	semidirect
6. 3₋¹⁺²×#103	C₂	semidirect
9. C₉×#485	C₃	semidirect
30. C₉×#107	C₂×C₃	semidirect
31. C₉×#106	C₂×C₃	semidirect
32. C₉×#103	C₂×C₃	semidirect
33. C₉×#103	C₂×C₃	semidirect
123. #485	3₋¹⁺²	direct (N × H)
174. #107	C₂×3₋¹⁺²	semidirect
175. #106	C₂×3₋¹⁺²	semidirect
176. #103	C₂×3₋¹⁺²	semidirect
177. #103	C₂×3₋¹⁺²	semidirect
179. 3₋¹⁺²	#485	direct (N × H)
248. C₉	C₃×#485	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×C₂²⋊Q₈	2
3₋¹⁺²×#117	2
3₋¹⁺²×#107	2
3₋¹⁺²×#106	2
3₋¹⁺²×#103	2
3₋¹⁺²×C₂²⋊C₈	2
C₃²×#485	3
C₉×#485	3

Subgroups — 3−1+2×#485 #13145215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#485 #13145215