Subgroups — 3₋¹⁺²×#389 #13144927

Order 1728.

680 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₄×3₋¹⁺²	C₂	1	2	yes	maximal
3	C₂×C₄⋊C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
5	3₋¹⁺²×#95	C₂	1	2	yes	maximal
6	C₃²×#389	C₃	1	1	yes	maximal, characteristic
7	C₉×#389	C₃	1	3	yes	maximal, complemented
8	C₂²×C₄×3₋¹⁺²	C₂²	1	2	yes	—
9	C₂²×C₄×3₋¹⁺²	C₂²	1	2	yes	—
10	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×3₋¹⁺²	—	2	4	no	—
14	C₄⋊C₄×3₋¹⁺²	—	2	2	no	—
15	C₄⋊C₄×3₋¹⁺²	C₄	1	4	yes	complemented
16	C₄²×3₋¹⁺²	—	2	2	no	—
17	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	2	yes	—
19	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
21	C₃²×#95	C₂×C₃	1	2	yes	—
22	C₂×C₉×C₄⋊C₄	C₂×C₃	1	3	yes	—
23	C₂×C₉×C₄⋊C₄	C₂×C₃	1	6	yes	—
24	C₂×C₉×C₄⋊C₄	C₂×C₃	1	3	yes	—
25	C₂×C₄²×C₉	C₂×C₃	1	3	yes	—
26	C₉×#95	C₂×C₃	1	6	yes	—
27	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
28	C₂×C₄×3₋¹⁺²	—	2	1	no	—
29	C₂×C₄×3₋¹⁺²	—	2	1	no	—
30	C₂×C₄×3₋¹⁺²	—	2	2	no	—
31	C₂×C₄×3₋¹⁺²	—	2	2	no	—
32	C₂×C₄×3₋¹⁺²	—	2	1	no	—
33	C₂×C₄×3₋¹⁺²	—	2	1	no	—
34	C₂×C₄×3₋¹⁺²	—	2	2	no	—
35	C₂×C₄×3₋¹⁺²	—	2	2	no	—
36	C₂×C₄×3₋¹⁺²	—	2	2	no	—
37	C₂×C₄×3₋¹⁺²	C₂×C₄	1	4	yes	—
38	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
39	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
40	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
41	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
42	C₂×C₄×3₋¹⁺²	Q₈	1	2	yes	—
43	C₃×#389	—	3	1	no	—
44	C₃×#389	C₃²	1	1	yes	characteristic
45	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
46	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
47	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
50	C₃²×C₄⋊C₄	—	2	4	no	—
51	C₃²×C₄⋊C₄	—	2	2	no	—
52	C₃²×C₄⋊C₄	C₃×C₄	1	4	yes	—
53	C₃²×C₄²	—	2	2	no	—
54	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
55	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
56	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
57	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
58	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
59	C₉×C₄⋊C₄	—	2	12	no	—
60	C₉×C₄⋊C₄	—	2	6	no	—
61	C₉×C₄⋊C₄	C₃×C₄	1	12	yes	complemented
62	C₄²×C₉	—	2	6	no	—
63	C₂²×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
64	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
65	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
66	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
67	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
68	C₂²×3₋¹⁺²	#41	1	1	yes	characteristic
69	C₂²×3₋¹⁺²	#41	1	1	yes	characteristic
70	C₄×3₋¹⁺²	—	2	4	no	—
71	C₄×3₋¹⁺²	—	2	2	no	—
72	C₄×3₋¹⁺²	—	2	2	no	—
73	C₄×3₋¹⁺²	—	4	2	no	—
74	C₄×3₋¹⁺²	C₄⋊C₄	1	4	yes	complemented
75	C₂×C₃×C₄⋊C₄	—	3	1	no	—
76	C₂×C₃×C₄⋊C₄	—	3	1	no	—
77	C₂×C₃×C₄⋊C₄	—	3	2	no	—
78	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic
79	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	2	yes	—
80	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic
81	C₂×C₃×C₄²	—	3	1	no	—
82	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
83	C₃×#95	—	3	2	no	—
84	C₃×#95	C₂×C₃²	1	2	yes	—
85	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	—	2	1	no	—
87	C₂×C₃²×C₄	—	2	1	no	—
88	C₂×C₃²×C₄	—	2	2	no	—
89	C₂×C₃²×C₄	—	2	2	no	—
90	C₂×C₃²×C₄	—	2	1	no	—
91	C₂×C₃²×C₄	—	2	1	no	—
92	C₂×C₃²×C₄	—	2	2	no	—
93	C₂×C₃²×C₄	—	2	2	no	—
94	C₂×C₃²×C₄	—	2	2	no	—
95	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
96	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
97	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
98	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
99	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
100	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
101	C₂³×C₉	C₂³×C₃	1	3	yes	—
102	C₂×C₄×C₉	—	2	3	no	—
103	C₂×C₄×C₉	—	2	3	no	—
104	C₂×C₄×C₉	—	2	6	no	—
105	C₂×C₄×C₉	—	2	6	no	—
106	C₂×C₄×C₉	—	2	3	no	—
107	C₂×C₄×C₉	—	2	3	no	—
108	C₂×C₄×C₉	—	2	6	no	—
109	C₂×C₄×C₉	—	2	6	no	—
110	C₂×C₄×C₉	—	2	6	no	—
111	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
112	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
113	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
114	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
115	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
116	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
117	#389	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×3₋¹⁺²	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×3₋¹⁺²	C₄×D₄	1	1	yes	characteristic
120	C₂×3₋¹⁺²	C₄×Q₈	1	1	yes	characteristic
121	C₂×3₋¹⁺²	#121	1	1	yes	characteristic
122	C₂×3₋¹⁺²	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×3₋¹⁺²	#125	1	1	yes	characteristic
124	C₂×3₋¹⁺²	C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	3	1	no	—
126	C₂²×C₃×C₄	—	3	1	no	—
127	C₂²×C₃×C₄	—	3	2	no	—
128	C₂²×C₃×C₄	—	3	1	no	—
129	C₂²×C₃×C₄	—	3	2	no	—
130	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
131	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
132	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
135	C₃×C₄⋊C₄	—	2	4	no	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	3	4	no	—
138	C₃×C₄⋊C₄	—	6	2	no	—
139	C₃×C₄⋊C₄	—	6	4	no	—
140	C₃×C₄⋊C₄	C₃²×C₄	1	4	yes	—
141	C₃×C₄²	—	2	2	no	—
142	C₃×C₄²	—	6	2	no	—
143	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
147	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
148	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
149	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
150	C₃²×C₄	—	2	4	no	—
151	C₃²×C₄	—	2	2	no	—
152	C₃²×C₄	—	2	2	no	—
153	C₃²×C₄	—	4	2	no	—
154	C₃²×C₄	C₃×C₄⋊C₄	1	4	yes	—
155	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
156	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
157	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
158	C₂²×C₉	C₂×C₃×Q₈	1	3	yes	—
159	C₂²×C₉	C₂×C₃×Q₈	1	3	yes	—
160	C₂²×C₉	C₃×#41	1	3	yes	—
161	C₂²×C₉	C₃×#41	1	3	yes	—
162	C₄×C₉	—	2	12	no	—
163	C₄×C₉	—	2	6	no	—
164	C₄×C₉	—	2	6	no	—
165	C₄×C₉	—	4	6	no	—
166	C₄×C₉	C₃×C₄⋊C₄	1	12	yes	complemented
167	C₂×C₄⋊C₄	C₂×3₋¹⁺²	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×3₋¹⁺²	1	2	yes	—
169	C₂×C₄⋊C₄	C₂×3₋¹⁺²	1	1	yes	characteristic
170	C₂×C₄²	C₂×3₋¹⁺²	1	1	yes	characteristic
171	#95	C₂×3₋¹⁺²	1	2	yes	—
172	3₋¹⁺²	#389	1	1	yes	characteristic, direct product factor, Sylow
173	C₂³×C₃	—	3	1	no	—
174	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	3	2	no	—
185	C₂×C₃×C₄	—	3	4	no	—
186	C₂×C₃×C₄	—	3	2	no	—
187	C₂×C₃×C₄	—	3	2	no	—
188	C₂×C₃×C₄	—	3	2	no	—
189	C₂×C₃×C₄	—	3	2	no	—
190	C₂×C₃×C₄	—	6	1	no	—
191	C₂×C₃×C₄	—	6	1	no	—
192	C₂×C₃×C₄	—	6	2	no	—
193	C₂×C₃×C₄	—	6	1	no	—
194	C₂×C₃×C₄	—	6	2	no	—
195	C₂×C₃×C₄	—	6	1	no	—
196	C₂×C₃×C₄	—	6	2	no	—
197	C₂×C₃×C₄	—	6	2	no	—
198	C₂×C₃×C₄	—	6	2	no	—
199	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
200	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
201	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
202	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
203	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
204	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
205	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
207	C₂×C₃²	C₃×C₄×Q₈	1	1	yes	characteristic
208	C₂×C₃²	C₃×#121	1	1	yes	characteristic
209	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
210	C₂×C₃²	C₃×#125	1	1	yes	characteristic
211	C₂×C₃²	C₃×C₄⋊Q₈	1	1	yes	characteristic
212	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
213	C₂×C₉	C₃×C₄×D₄	1	3	yes	—
214	C₂×C₉	C₃×C₄×Q₈	1	3	yes	—
215	C₂×C₉	C₃×#121	1	3	yes	—
216	C₂×C₉	C₃×C₂²⋊Q₈	1	3	yes	—
217	C₂×C₉	C₃×#125	1	3	yes	—
218	C₂×C₉	C₃×C₄⋊Q₈	1	3	yes	—
219	C₂²×C₄	C₂²×3₋¹⁺²	1	2	yes	—
220	C₂²×C₄	C₂²×3₋¹⁺²	1	2	yes	—
221	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
222	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
223	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	C₄×3₋¹⁺²	1	4	yes	complemented
227	C₄²	—	2	2	no	—
228	C₂²×C₃	—	3	1	no	—
229	C₂²×C₃	—	3	1	no	—
230	C₂²×C₃	—	3	1	no	—
231	C₂²×C₃	—	3	1	no	—
232	C₂²×C₃	—	3	1	no	—
233	C₂²×C₃	—	3	1	no	—
234	C₂²×C₃	—	3	1	no	—
235	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
236	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
237	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
239	C₂²×C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
240	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
241	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
242	C₃×C₄	—	2	4	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	3	4	no	—
246	C₃×C₄	—	4	2	no	—
247	C₃×C₄	—	6	4	no	—
248	C₃×C₄	—	6	2	no	—
249	C₃×C₄	—	6	2	no	—
250	C₃×C₄	—	12	2	no	—
251	C₃×C₄	C₃²×C₄⋊C₄	1	4	yes	—
252	C₃²	C₃×#389	1	1	yes	characteristic
253	C₉	C₃×#389	1	3	yes	complemented
254	C₂³	C₂³×3₋¹⁺²	1	1	yes	characteristic
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	C₂×C₄×3₋¹⁺²	1	4	yes	—
265	C₂×C₄	C₂×C₄×3₋¹⁺²	1	2	yes	—
266	C₂×C₄	D₄×3₋¹⁺²	1	2	yes	—
267	C₂×C₄	D₄×3₋¹⁺²	1	2	yes	—
268	C₂×C₄	Q₈×3₋¹⁺²	1	2	yes	—
269	C₂×C₄	Q₈×3₋¹⁺²	1	2	yes	—
270	C₂×C₃	—	3	1	no	—
271	C₂×C₃	—	3	1	no	—
272	C₂×C₃	—	3	1	no	—
273	C₂×C₃	—	3	1	no	—
274	C₂×C₃	—	3	1	no	—
275	C₂×C₃	—	3	1	no	—
276	C₂×C₃	—	3	1	no	—
277	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
278	C₂×C₃	C₃²×C₄×D₄	1	1	yes	characteristic
279	C₂×C₃	C₃²×C₄×Q₈	1	1	yes	characteristic
280	C₂×C₃	C₃²×#121	1	1	yes	characteristic
281	C₂×C₃	C₃²×C₂²⋊Q₈	1	1	yes	characteristic
282	C₂×C₃	C₃²×#125	1	1	yes	characteristic
283	C₂×C₃	C₃²×C₄⋊Q₈	1	1	yes	characteristic
284	C₂²	C₂²×C₄×3₋¹⁺²	1	1	yes	characteristic
285	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
286	C₂²	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
287	C₂²	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
288	C₂²	C₂×Q₈×3₋¹⁺²	1	1	yes	characteristic
289	C₂²	#41×3₋¹⁺²	1	1	yes	characteristic
290	C₂²	#41×3₋¹⁺²	1	1	yes	characteristic
291	C₄	—	2	4	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	2	no	—
295	C₄	C₄⋊C₄×3₋¹⁺²	1	4	yes	complemented
296	C₃	—	3	1	no	—
297	C₃	C₃²×#389	1	1	yes	characteristic
298	C₂	C₂×C₄⋊C₄×3₋¹⁺²	1	1	yes	characteristic
299	C₂	C₄×D₄×3₋¹⁺²	1	1	yes	characteristic
300	C₂	C₄×Q₈×3₋¹⁺²	1	1	yes	characteristic
301	C₂	3₋¹⁺²×#121	1	1	yes	characteristic
302	C₂	3₋¹⁺²×C₂²⋊Q₈	1	1	yes	characteristic
303	C₂	3₋¹⁺²×#125	1	1	yes	characteristic
304	C₂	3₋¹⁺²×C₄⋊Q₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³×C₃	C₂³×C₃²
commutator	C₂²×C₃	C₂²×C₃²×C₄
prime radical	C₂³×C₃²	C₂³×C₃
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₉×#389	C₃	semidirect
15. C₄⋊C₄×3₋¹⁺²	C₄	semidirect
61. C₉×C₄⋊C₄	C₃×C₄	semidirect
74. C₄×3₋¹⁺²	C₄⋊C₄	semidirect
117. #389	3₋¹⁺²	direct (N × H)
166. C₄×C₉	C₃×C₄⋊C₄	semidirect
172. 3₋¹⁺²	#389	direct (N × H)
226. C₄⋊C₄	C₄×3₋¹⁺²	semidirect
253. C₉	C₃×#389	semidirect
295. C₄	C₄⋊C₄×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₄×3₋¹⁺²	2
C₂×C₄²×3₋¹⁺²	2
3₋¹⁺²×#95	2
C₃²×#389	3
C₉×#389	3

Subgroups — 3−1+2×#389 #13144927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#389 #13144927