Subgroups — 3₋¹⁺²×#482 #13145206

Order 1728.

424 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×#121	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×#117	C₂	1	1	yes	maximal, characteristic, complemented
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×#102	C₂	1	1	yes	maximal, characteristic, complemented
6	3₋¹⁺²×#102	C₂	1	1	yes	maximal, characteristic, complemented
7	3₋¹⁺²×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#482	C₃	1	1	yes	maximal, characteristic
9	C₉×#482	C₃	1	3	yes	maximal, complemented
10	C₂×D₄×3₋¹⁺²	—	2	1	no	—
11	C₂×D₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
14	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
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×#121	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×#102	C₂×C₃	1	1	yes	characteristic
26	C₃²×#102	C₂×C₃	1	1	yes	characteristic
27	C₃²×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
28	C₉×#121	C₂×C₃	1	3	yes	—
29	C₉×#117	C₂×C₃	1	3	yes	complemented
30	C₉×#107	C₂×C₃	1	3	yes	complemented
31	C₉×#106	C₂×C₃	1	3	yes	complemented
32	C₉×#102	C₂×C₃	1	3	yes	complemented
33	C₉×#102	C₂×C₃	1	3	yes	complemented
34	C₉×C₂²⋊C₈	C₂×C₃	1	3	yes	complemented
35	C₂³×3₋¹⁺²	—	2	1	no	—
36	C₂³×3₋¹⁺²	D₄	1	1	yes	characteristic
37	D₄×3₋¹⁺²	—	2	1	no	—
38	D₄×3₋¹⁺²	—	2	1	no	—
39	D₄×3₋¹⁺²	—	4	2	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	—	2	1	no	—
45	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
46	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
47	C₈×3₋¹⁺²	—	2	1	no	—
48	C₈×3₋¹⁺²	—	2	1	no	—
49	C₃×#482	—	3	1	no	—
50	C₃×#482	C₃²	1	1	yes	characteristic
51	C₂×C₃²×D₄	—	2	1	no	—
52	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃²×C₈	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×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
59	C₃²×C₂²⋊C₄	—	2	1	no	—
60	C₃²×C₂²⋊C₄	—	2	1	no	—
61	C₃²×C₄²	—	2	1	no	—
62	C₂×D₄×C₉	—	2	3	no	—
63	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
64	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
65	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
66	C₂×C₈×C₉	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×C₄⋊C₄	C₂²×C₃	1	3	yes	—
70	C₉×C₂²⋊C₄	—	2	3	no	—
71	C₉×C₂²⋊C₄	—	2	3	no	—
72	C₄²×C₉	—	2	3	no	—
73	C₂²×3₋¹⁺²	—	2	1	no	—
74	C₂²×3₋¹⁺²	—	2	1	no	—
75	C₂²×3₋¹⁺²	—	2	1	no	—
76	C₂²×3₋¹⁺²	—	4	1	no	—
77	C₂²×3₋¹⁺²	—	4	1	no	—
78	C₂²×3₋¹⁺²	—	4	1	no	—
79	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
80	C₄×3₋¹⁺²	—	2	1	no	—
81	C₄×3₋¹⁺²	—	4	1	no	—
82	C₄×3₋¹⁺²	—	4	1	no	—
83	C₄×3₋¹⁺²	—	4	1	no	—
84	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
85	C₄×3₋¹⁺²	#41	1	1	yes	characteristic
86	C₃×#121	—	3	1	no	—
87	C₃×#121	C₂×C₃²	1	1	yes	characteristic
88	C₃×#117	—	3	1	no	—
89	C₃×#117	C₂×C₃²	1	1	yes	characteristic
90	C₃×#107	—	3	1	no	—
91	C₃×#107	C₂×C₃²	1	1	yes	characteristic
92	C₃×#106	—	3	1	no	—
93	C₃×#106	C₂×C₃²	1	1	yes	characteristic
94	C₃×#102	—	3	1	no	—
95	C₃×#102	—	3	1	no	—
96	C₃×#102	C₂×C₃²	1	1	yes	characteristic
97	C₃×#102	C₂×C₃²	1	1	yes	characteristic
98	C₃×C₂²⋊C₈	—	3	1	no	—
99	C₃×C₂²⋊C₈	C₂×C₃²	1	1	yes	characteristic
100	C₂³×C₃²	—	2	1	no	—
101	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
102	C₃²×D₄	—	2	1	no	—
103	C₃²×D₄	—	2	1	no	—
104	C₃²×D₄	—	4	2	no	—
105	C₂×C₃²×C₄	—	2	1	no	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	—	2	1	no	—
108	C₂×C₃²×C₄	—	2	1	no	—
109	C₂×C₃²×C₄	—	2	1	no	—
110	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂³×C₉	—	2	3	no	—
113	C₂³×C₉	C₃×D₄	1	3	yes	—
114	C₃²×C₈	—	2	1	no	—
115	C₃²×C₈	—	2	1	no	—
116	D₄×C₉	—	2	3	no	—
117	D₄×C₉	—	2	3	no	—
118	D₄×C₉	—	4	6	no	—
119	C₂×C₄×C₉	—	2	3	no	—
120	C₂×C₄×C₉	—	2	3	no	—
121	C₂×C₄×C₉	—	2	3	no	—
122	C₂×C₄×C₉	—	2	3	no	—
123	C₂×C₄×C₉	—	2	3	no	—
124	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
125	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
126	C₈×C₉	—	2	3	no	—
127	C₈×C₉	—	2	3	no	—
128	#482	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×3₋¹⁺²	—	4	1	no	—
130	C₂×3₋¹⁺²	—	8	1	no	—
131	C₂×3₋¹⁺²	#123	1	1	yes	characteristic
132	C₂×3₋¹⁺²	#135	1	1	yes	characteristic
133	C₂×3₋¹⁺²	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₃×D₄	—	2	1	no	—
135	C₂×C₃×D₄	—	3	1	no	—
136	C₂×C₃×D₄	—	6	1	no	—
137	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
138	C₂²×C₃×C₄	—	3	1	no	—
139	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
140	C₂×C₃×C₈	—	3	1	no	—
141	C₂×C₃×C₈	—	3	1	no	—
142	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	3	1	no	—
145	C₃×C₄⋊C₄	—	3	1	no	—
146	C₃×C₄⋊C₄	—	3	1	no	—
147	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
148	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
149	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
150	C₃×C₂²⋊C₄	—	2	1	no	—
151	C₃×C₂²⋊C₄	—	2	1	no	—
152	C₃×C₂²⋊C₄	—	6	1	no	—
153	C₃×C₂²⋊C₄	—	6	1	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	—	6	1	no	—
156	C₂²×C₃²	—	2	1	no	—
157	C₂²×C₃²	—	2	1	no	—
158	C₂²×C₃²	—	2	1	no	—
159	C₂²×C₃²	—	4	1	no	—
160	C₂²×C₃²	—	4	1	no	—
161	C₂²×C₃²	—	4	1	no	—
162	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
163	C₃²×C₄	—	2	1	no	—
164	C₃²×C₄	—	4	1	no	—
165	C₃²×C₄	—	4	1	no	—
166	C₃²×C₄	—	4	1	no	—
167	C₃²×C₄	C₃×#41	1	1	yes	characteristic
168	C₃²×C₄	C₃×#41	1	1	yes	characteristic
169	C₂²×C₉	—	2	3	no	—
170	C₂²×C₉	—	2	3	no	—
171	C₂²×C₉	—	2	3	no	—
172	C₂²×C₉	—	4	3	no	—
173	C₂²×C₉	—	4	3	no	—
174	C₂²×C₉	—	4	3	no	—
175	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
176	C₄×C₉	—	2	3	no	—
177	C₄×C₉	—	4	3	no	—
178	C₄×C₉	—	4	3	no	—
179	C₄×C₉	—	4	3	no	—
180	C₄×C₉	C₃×#41	1	3	yes	—
181	C₄×C₉	C₃×#41	1	3	yes	—
182	#121	C₂×3₋¹⁺²	1	1	yes	characteristic
183	#117	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
184	#107	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
185	#106	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
186	#102	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
187	#102	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
188	C₂²⋊C₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
189	3₋¹⁺²	#482	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³×C₃	—	2	1	no	—
191	C₂³×C₃	—	3	1	no	—
192	C₂³×C₃	—	6	1	no	—
193	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
194	C₃×D₄	—	2	1	no	—
195	C₃×D₄	—	2	1	no	—
196	C₃×D₄	—	4	2	no	—
197	C₃×D₄	—	6	1	no	—
198	C₃×D₄	—	6	1	no	—
199	C₃×D₄	—	12	2	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	3	1	no	—
206	C₂×C₃×C₄	—	3	1	no	—
207	C₂×C₃×C₄	—	6	1	no	—
208	C₂×C₃×C₄	—	6	1	no	—
209	C₂×C₃×C₄	—	6	1	no	—
210	C₂×C₃×C₄	—	6	1	no	—
211	C₂×C₃×C₄	—	6	1	no	—
212	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	6	1	no	—
217	C₃×C₈	—	6	1	no	—
218	C₂×C₃²	—	4	1	no	—
219	C₂×C₃²	—	8	1	no	—
220	C₂×C₃²	C₃×#123	1	1	yes	characteristic
221	C₂×C₃²	C₃×#135	1	1	yes	characteristic
222	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂×C₉	—	4	3	no	—
224	C₂×C₉	—	8	3	no	—
225	C₂×C₉	C₃×#123	1	3	yes	—
226	C₂×C₉	C₃×#135	1	3	yes	—
227	C₂×C₉	C₃×AGL(1,ℤ/8)	1	3	yes	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
230	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
231	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
232	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
233	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
234	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
235	C₄⋊C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	3	1	no	—
243	C₂²×C₃	—	4	1	no	—
244	C₂²×C₃	—	4	1	no	—
245	C₂²×C₃	—	4	1	no	—
246	C₂²×C₃	—	6	1	no	—
247	C₂²×C₃	—	6	1	no	—
248	C₂²×C₃	—	6	1	no	—
249	C₂²×C₃	—	12	1	no	—
250	C₂²×C₃	—	12	1	no	—
251	C₂²×C₃	—	12	1	no	—
252	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	—	3	1	no	—
255	C₃×C₄	—	3	1	no	—
256	C₃×C₄	—	4	1	no	—
257	C₃×C₄	—	4	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	—	6	1	no	—
260	C₃×C₄	—	12	1	no	—
261	C₃×C₄	—	12	1	no	—
262	C₃×C₄	—	12	1	no	—
263	C₃×C₄	C₃²×#41	1	1	yes	characteristic
264	C₃×C₄	C₃²×#41	1	1	yes	characteristic
265	C₃²	C₃×#482	1	1	yes	characteristic
266	C₉	C₃×#482	1	3	yes	complemented
267	C₂³	—	2	1	no	—
268	C₂³	D₄×3₋¹⁺²	1	1	yes	characteristic
269	D₄	—	2	1	no	—
270	D₄	—	2	1	no	—
271	D₄	—	4	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
278	C₂×C₄	C₂³×3₋¹⁺²	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₂×C₃	—	3	1	no	—
282	C₂×C₃	—	3	1	no	—
283	C₂×C₃	—	3	1	no	—
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	—	8	1	no	—
286	C₂×C₃	—	12	1	no	—
287	C₂×C₃	—	24	1	no	—
288	C₂×C₃	C₃²×#123	1	1	yes	characteristic
289	C₂×C₃	C₃²×#135	1	1	yes	characteristic
290	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
303	C₄	#41×3₋¹⁺²	1	1	yes	characteristic
304	C₃	—	3	1	no	—
305	C₃	C₃²×#482	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	8	1	no	—
308	C₂	3₋¹⁺²×#123	1	1	yes	characteristic
309	C₂	3₋¹⁺²×#135	1	1	yes	characteristic
310	C₂	3₋¹⁺²×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₃²×#121	C₂×C₃
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
2. 3₋¹⁺²×#117	C₂	semidirect
3. 3₋¹⁺²×#107	C₂	semidirect
4. 3₋¹⁺²×#106	C₂	semidirect
5. 3₋¹⁺²×#102	C₂	semidirect
6. 3₋¹⁺²×#102	C₂	semidirect
7. 3₋¹⁺²×C₂²⋊C₈	C₂	semidirect
9. C₉×#482	C₃	semidirect
29. C₉×#117	C₂×C₃	semidirect
30. C₉×#107	C₂×C₃	semidirect
31. C₉×#106	C₂×C₃	semidirect
32. C₉×#102	C₂×C₃	semidirect
33. C₉×#102	C₂×C₃	semidirect
34. C₉×C₂²⋊C₈	C₂×C₃	semidirect
128. #482	3₋¹⁺²	direct (N × H)
183. #117	C₂×3₋¹⁺²	semidirect
184. #107	C₂×3₋¹⁺²	semidirect
185. #106	C₂×3₋¹⁺²	semidirect
186. #102	C₂×3₋¹⁺²	semidirect
187. #102	C₂×3₋¹⁺²	semidirect
188. C₂²⋊C₈	C₂×3₋¹⁺²	semidirect
189. 3₋¹⁺²	#482	direct (N × H)
266. C₉	C₃×#482	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×#121	2
3₋¹⁺²×#117	2
3₋¹⁺²×#107	2
3₋¹⁺²×#106	2
3₋¹⁺²×#102	2
3₋¹⁺²×C₂²⋊C₈	2
C₃²×#482	3
C₉×#482	3

Subgroups — 3−1+2×#482 #13145206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#482 #13145206