Subgroups — C₃²×D₃×#120 #13178966

Order 1728.

4492 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×D₃×D₄	C₂	1	3	yes	maximal, complemented
3	C₃³×#120	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#953	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×#937	C₂	1	3	yes	maximal, complemented
6	C₃²×#882	C₂	1	3	yes	maximal, complemented
7	C₃²×D₃×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
8	C₂×C₃²×#120	—	3	1	no	maximal
9	C₃×D₃×#120	C₃	1	4	yes	maximal, direct product factor
10	C₂⁴×C₃³	C₂²	1	1	yes	characteristic, complemented
11	C₂³×C₃²×D₃	—	2	12	no	—
12	C₂³×C₃²×D₃	C₂²	1	3	yes	—
13	C₂³×C₃²×D₃	C₂²	1	3	yes	complemented
14	C₂³×C₃²×D₃	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₃³×D₄	C₂²	1	3	yes	complemented
16	C₂×C₃²×#67	C₂²	1	3	yes	complemented
17	C₂×C₃²×#67	C₂²	1	3	yes	complemented
18	C₃²×D₃×D₄	—	2	12	no	—
19	C₂×C₃²×D₁₂	C₂²	1	3	yes	complemented
20	C₂×C₃²×C₄×D₃	C₂²	1	3	yes	complemented
21	C₃³×C₂²⋊C₄	C₂²	1	3	yes	complemented
22	C₃²×#217	C₂²	1	3	yes	complemented
23	C₃²×(C₂×D₃)⋊C₄	C₂²	1	6	yes	complemented
24	C₂⁵×C₃²	—	3	1	no	—
25	C₂⁴×C₃×D₃	C₂×C₃	1	4	yes	complemented
26	C₂²×C₃²×D₄	—	3	3	no	—
27	C₂×C₃×D₃×D₄	C₂×C₃	1	12	yes	complemented
28	C₃²×#120	—	2	4	no	—
29	C₃²×#120	—	3	3	no	—
30	C₃²×#120	—	3	3	no	—
31	C₃²×#120	—	3	1	no	—
32	C₃²×#120	D₃	1	1	yes	characteristic, direct product factor
33	C₃²×#120	C₂×C₃	1	4	yes	complemented
34	C₂×C₃²×C₂²⋊C₄	—	3	3	no	—
35	C₃×#953	C₂×C₃	1	4	yes	complemented
36	C₃×#937	C₂×C₃	1	12	yes	complemented
37	C₃×#882	C₂×C₃	1	12	yes	complemented
38	C₃×D₃×C₂²⋊C₄	C₂×C₃	1	12	yes	complemented
39	C₂³×C₃³	—	2	6	no	—
40	C₂³×C₃³	C₂³	1	3	yes	—
41	C₂³×C₃³	C₂³	1	1	yes	characteristic, complemented
42	C₂²×C₃²×D₃	—	2	32	no	—
43	C₂²×C₃²×D₃	—	2	6	no	—
44	C₂²×C₃²×D₃	—	2	12	no	—
45	C₂²×C₃²×D₃	—	2	12	no	—
46	C₂²×C₃²×D₃	—	2	6	no	—
47	C₂²×C₃²×D₃	D₄	1	12	yes	complemented
48	C₂²×C₃²×D₃	C₂³	1	3	yes	—
49	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic
50	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic, complemented
51	C₃³×D₄	—	2	6	no	—
52	C₂×C₃³×C₄	C₂³	1	3	yes	complemented
53	C₃²×#67	—	2	6	no	—
54	C₃²×#67	—	2	6	no	—
55	C₂×C₃²×Dic₃	C₂³	1	3	yes	complemented
56	C₃²×D₁₂	—	2	6	no	—
57	C₃²×C₄×D₃	—	2	6	no	—
58	C₂×C₃×#120	—	3	4	no	—
59	D₃×#120	C₃²	1	1	yes	characteristic, direct product factor
60	C₂⁴×C₃²	—	2	4	no	—
61	C₂⁴×C₃²	—	3	3	no	—
62	C₂⁴×C₃²	—	3	3	no	—
63	C₂⁴×C₃²	—	3	1	no	—
64	C₂⁴×C₃²	—	6	12	no	—
65	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic, complemented
66	C₂⁴×C₃²	C₂²×C₃	1	4	yes	complemented
67	C₂³×C₃×D₃	—	2	48	no	—
68	C₂³×C₃×D₃	C₂²×C₃	1	12	yes	—
69	C₂³×C₃×D₃	C₂²×C₃	1	12	yes	complemented
70	C₂³×C₃×D₃	C₂²×C₃	1	4	yes	complemented
71	C₂×C₃²×D₄	—	2	12	no	—
72	C₂×C₃²×D₄	—	3	3	no	—
73	C₂×C₃²×D₄	—	3	3	no	—
74	C₂×C₃²×D₄	—	3	3	no	—
75	C₂×C₃²×D₄	—	6	12	no	—
76	C₂×C₃²×D₄	C₂×D₃	1	3	yes	complemented
77	C₂×C₃²×D₄	C₂²×C₃	1	12	yes	complemented
78	C₂²×C₃²×C₄	—	3	3	no	—
79	C₂×C₃×#67	C₂²×C₃	1	12	yes	complemented
80	C₂×C₃×#67	C₂²×C₃	1	12	yes	complemented
81	C₃×D₃×D₄	—	2	48	no	—
82	C₂×C₃×D₁₂	C₂²×C₃	1	12	yes	complemented
83	C₂×C₃×C₄×D₃	C₂²×C₃	1	12	yes	complemented
84	C₃²×C₂²⋊C₄	—	2	12	no	—
85	C₃²×C₂²⋊C₄	—	3	3	no	—
86	C₃²×C₂²⋊C₄	—	3	6	no	—
87	C₃²×C₂²⋊C₄	C₂×D₃	1	3	yes	complemented
88	C₃²×C₂²⋊C₄	C₂²×C₃	1	12	yes	complemented
89	C₃×#217	C₂²×C₃	1	12	yes	complemented
90	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	24	yes	complemented
91	C₂²×C₃³	—	2	8	no	—
92	C₂²×C₃³	—	2	6	no	—
93	C₂²×C₃³	—	2	3	no	—
94	C₂²×C₃³	C₂×D₄	1	6	yes	complemented
95	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
96	C₂×C₃²×D₃	—	2	24	no	—
97	C₂×C₃²×D₃	—	2	6	no	—
98	C₂×C₃²×D₃	—	2	24	no	—
99	C₂×C₃²×D₃	—	2	3	no	—
100	C₂×C₃²×D₃	—	4	4	no	—
101	C₂×C₃²×D₃	C₂×D₄	1	6	yes	—
102	C₂×C₃²×D₃	C₂×D₄	1	6	yes	complemented
103	C₃³×C₄	—	2	3	no	—
104	C₃²×Dic₃	—	2	3	no	—
105	C₂⁵×C₃	—	3	4	no	—
106	C₂⁴×D₃	C₂×C₃²	1	1	yes	characteristic, complemented
107	C₂²×C₃×D₄	—	3	12	no	—
108	C₂×D₃×D₄	C₂×C₃²	1	3	yes	complemented
109	C₃×#120	—	2	4	no	—
110	C₃×#120	—	3	4	no	—
111	C₃×#120	—	3	12	no	—
112	C₃×#120	—	3	12	no	—
113	C₃×#120	C₃×D₃	1	4	yes	direct product factor
114	C₃×#120	C₂×C₃²	1	1	yes	characteristic, complemented
115	C₂×C₃×C₂²⋊C₄	—	3	12	no	—
116	#953	C₂×C₃²	1	1	yes	characteristic, complemented
117	#937	C₂×C₃²	1	3	yes	complemented
118	#882	C₂×C₃²	1	3	yes	complemented
119	D₃×C₂²⋊C₄	C₂×C₃²	1	3	yes	complemented
120	C₂³×C₃²	—	2	6	no	—
121	C₂³×C₃²	—	2	12	no	—
122	C₂³×C₃²	—	2	4	no	—
123	C₂³×C₃²	—	2	24	no	—
124	C₂³×C₃²	—	3	1	no	—
125	C₂³×C₃²	—	3	12	no	—
126	C₂³×C₃²	—	3	3	no	—
127	C₂³×C₃²	—	3	1	no	—
128	C₂³×C₃²	—	4	24	no	—
129	C₂³×C₃²	—	6	12	no	—
130	C₂³×C₃²	—	6	12	no	—
131	C₂³×C₃²	—	6	6	no	—
132	C₂³×C₃²	—	6	32	no	—
133	C₂³×C₃²	—	6	6	no	—
134	C₂³×C₃²	C₂²×D₃	1	3	yes	—
135	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic, complemented
136	C₂³×C₃²	C₂³×C₃	1	12	yes	—
137	C₂³×C₃²	C₂³×C₃	1	4	yes	complemented
138	C₂²×C₃×D₃	—	2	128	no	—
139	C₂²×C₃×D₃	—	2	24	no	—
140	C₂²×C₃×D₃	—	2	48	no	—
141	C₂²×C₃×D₃	—	2	48	no	—
142	C₂²×C₃×D₃	—	2	24	no	—
143	C₂²×C₃×D₃	C₃×D₄	1	48	yes	complemented
144	C₂²×C₃×D₃	C₂³×C₃	1	12	yes	—
145	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
146	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	complemented
147	C₃²×D₄	—	2	6	no	—
148	C₃²×D₄	—	2	24	no	—
149	C₃²×D₄	—	4	24	no	—
150	C₃²×D₄	—	6	6	no	—
151	C₃²×D₄	—	6	6	no	—
152	C₃²×D₄	—	6	6	no	—
153	C₂×C₃²×C₄	—	2	12	no	—
154	C₂×C₃²×C₄	—	3	3	no	—
155	C₂×C₃²×C₄	—	6	6	no	—
156	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	complemented
157	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	complemented
158	C₃×#67	—	2	24	no	—
159	C₃×#67	—	2	24	no	—
160	C₂×C₃×Dic₃	C₂³×C₃	1	12	yes	complemented
161	C₃×D₁₂	—	2	24	no	—
162	C₃×C₄×D₃	—	2	24	no	—
163	C₂×#120	—	3	1	no	Sylow
164	C₂×C₃³	—	2	6	no	—
165	C₂×C₃³	—	4	1	no	—
166	C₂×C₃³	C₂²×D₄	1	3	yes	—
167	C₃²×D₃	—	2	6	no	—
168	C₃²×D₃	—	4	1	no	—
169	C₃²×D₃	#120	1	4	yes	direct product factor
170	C₂⁴×C₃	—	2	4	no	—
171	C₂⁴×C₃	—	3	12	no	—
172	C₂⁴×C₃	—	3	12	no	—
173	C₂⁴×C₃	—	3	4	no	—
174	C₂⁴×C₃	—	6	48	no	—
175	C₂⁴×C₃	C₂×C₃×D₃	1	4	yes	complemented
176	C₂⁴×C₃	C₂²×C₃²	1	1	yes	characteristic, complemented
177	C₂³×D₃	—	2	12	no	—
178	C₂³×D₃	C₂²×C₃²	1	3	yes	—
179	C₂³×D₃	C₂²×C₃²	1	3	yes	complemented
180	C₂³×D₃	C₂²×C₃²	1	1	yes	characteristic, complemented
181	C₂×C₃×D₄	—	2	12	no	—
182	C₂×C₃×D₄	—	3	12	no	—
183	C₂×C₃×D₄	—	3	12	no	—
184	C₂×C₃×D₄	—	3	12	no	—
185	C₂×C₃×D₄	—	6	48	no	—
186	C₂×C₃×D₄	C₂×C₃×D₃	1	12	yes	complemented
187	C₂×C₃×D₄	C₂²×C₃²	1	3	yes	complemented
188	C₂²×C₃×C₄	—	3	12	no	—
189	C₂×#67	C₂²×C₃²	1	3	yes	complemented
190	C₂×#67	C₂²×C₃²	1	3	yes	complemented
191	D₃×D₄	—	2	12	no	—
192	C₂×D₁₂	C₂²×C₃²	1	3	yes	complemented
193	C₂×C₄×D₃	C₂²×C₃²	1	3	yes	complemented
194	C₃×C₂²⋊C₄	—	2	12	no	—
195	C₃×C₂²⋊C₄	—	3	12	no	—
196	C₃×C₂²⋊C₄	—	3	24	no	—
197	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	12	yes	complemented
198	C₃×C₂²⋊C₄	C₂²×C₃²	1	3	yes	complemented
199	#217	C₂²×C₃²	1	3	yes	complemented
200	(C₂×D₃)⋊C₄	C₂²×C₃²	1	6	yes	complemented
201	C₂²×C₃²	—	2	8	no	—
202	C₂²×C₃²	—	2	6	no	—
203	C₂²×C₃²	—	2	3	no	—
204	C₂²×C₃²	—	2	4	no	—
205	C₂²×C₃²	—	2	24	no	—
206	C₂²×C₃²	—	2	32	no	—
207	C₂²×C₃²	—	2	24	no	—
208	C₂²×C₃²	—	2	12	no	—
209	C₂²×C₃²	—	3	6	no	—
210	C₂²×C₃²	—	3	6	no	—
211	C₂²×C₃²	—	4	24	no	—
212	C₂²×C₃²	—	4	12	no	—
213	C₂²×C₃²	—	4	32	no	—
214	C₂²×C₃²	—	6	24	no	—
215	C₂²×C₃²	—	6	24	no	—
216	C₂²×C₃²	—	6	6	no	—
217	C₂²×C₃²	—	6	3	no	—
218	C₂²×C₃²	—	12	4	no	—
219	C₂²×C₃²	D₃×D₄	1	6	yes	complemented
220	C₂²×C₃²	C₂×C₃×D₄	1	24	yes	complemented
221	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
222	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
223	C₂×C₃×D₃	—	2	96	no	—
224	C₂×C₃×D₃	—	2	24	no	—
225	C₂×C₃×D₃	—	2	96	no	—
226	C₂×C₃×D₃	—	2	12	no	—
227	C₂×C₃×D₃	—	4	16	no	—
228	C₂×C₃×D₃	C₂×C₃×D₄	1	24	yes	—
229	C₂×C₃×D₃	C₂×C₃×D₄	1	24	yes	complemented
230	C₃²×C₄	—	2	3	no	—
231	C₃²×C₄	—	2	12	no	—
232	C₃²×C₄	—	4	12	no	—
233	C₃²×C₄	—	6	3	no	—
234	C₃×Dic₃	—	2	12	no	—
235	C₂⁵	—	3	1	no	—
236	C₂²×D₄	—	3	3	no	—
237	#120	—	3	1	no	—
238	#120	—	3	3	no	—
239	#120	—	3	3	no	—
240	#120	C₃²×D₃	1	1	yes	characteristic, direct product factor
241	C₂×C₂²⋊C₄	—	3	3	no	—
242	C₃³	C₂×#120	1	1	yes	characteristic, complemented, Sylow
243	C₂³×C₃	—	2	24	no	—
244	C₂³×C₃	—	2	6	no	—
245	C₂³×C₃	—	2	12	no	—
246	C₂³×C₃	—	2	4	no	—
247	C₂³×C₃	—	3	4	no	—
248	C₂³×C₃	—	3	12	no	—
249	C₂³×C₃	—	3	48	no	—
250	C₂³×C₃	—	3	4	no	—
251	C₂³×C₃	—	4	24	no	—
252	C₂³×C₃	—	6	48	no	—
253	C₂³×C₃	—	6	48	no	—
254	C₂³×C₃	—	6	24	no	—
255	C₂³×C₃	—	6	128	no	—
256	C₂³×C₃	—	6	24	no	—
257	C₂³×C₃	C₂²×C₃×D₃	1	12	yes	—
258	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	complemented
259	C₂³×C₃	C₂³×C₃²	1	3	yes	—
260	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic, complemented
261	C₂²×D₃	—	2	32	no	—
262	C₂²×D₃	—	2	6	no	—
263	C₂²×D₃	—	2	12	no	—
264	C₂²×D₃	—	2	12	no	—
265	C₂²×D₃	—	2	6	no	—
266	C₂²×D₃	C₃²×D₄	1	12	yes	complemented
267	C₂²×D₃	C₂³×C₃²	1	3	yes	—
268	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic
269	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic, complemented
270	C₃×D₄	—	2	24	no	—
271	C₃×D₄	—	2	6	no	—
272	C₃×D₄	—	4	24	no	—
273	C₃×D₄	—	6	24	no	—
274	C₃×D₄	—	6	24	no	—
275	C₃×D₄	—	6	24	no	—
276	C₂×C₃×C₄	—	2	12	no	—
277	C₂×C₃×C₄	—	3	12	no	—
278	C₂×C₃×C₄	—	6	24	no	—
279	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	complemented
280	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	complemented
281	#67	—	2	6	no	—
282	#67	—	2	6	no	—
283	C₂×Dic₃	C₂³×C₃²	1	3	yes	complemented
284	D₁₂	—	2	6	no	—
285	C₄×D₃	—	2	6	no	—
286	C₂×C₃²	—	2	6	no	—
287	C₂×C₃²	—	2	12	no	—
288	C₂×C₃²	—	2	24	no	—
289	C₂×C₃²	—	3	4	no	—
290	C₂×C₃²	—	4	24	no	—
291	C₂×C₃²	—	4	1	no	—
292	C₂×C₃²	—	4	4	no	—
293	C₂×C₃²	—	6	6	no	—
294	C₂×C₃²	—	8	4	no	—
295	C₂×C₃²	—	12	1	no	—
296	C₂×C₃²	C₂×D₃×D₄	1	3	yes	—
297	C₂×C₃²	C₂²×C₃×D₄	1	12	yes	—
298	C₃×D₃	—	2	24	no	—
299	C₃×D₃	—	4	4	no	—
300	C₃×D₃	C₃×#120	1	16	yes	direct product factor
301	C₂⁴	—	3	3	no	—
302	C₂⁴	—	3	3	no	—
303	C₂⁴	—	3	1	no	—
304	C₂⁴	—	6	12	no	—
305	C₂⁴	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
306	C₂×D₄	—	3	3	no	—
307	C₂×D₄	—	3	3	no	—
308	C₂×D₄	—	3	3	no	—
309	C₂×D₄	—	6	12	no	—
310	C₂×D₄	C₂×C₃²×D₃	1	3	yes	complemented
311	C₂²×C₄	—	3	3	no	—
312	C₂²⋊C₄	—	3	3	no	—
313	C₂²⋊C₄	—	3	6	no	—
314	C₂²⋊C₄	C₂×C₃²×D₃	1	3	yes	complemented
315	C₂²×C₃	—	2	8	no	—
316	C₂²×C₃	—	2	32	no	—
317	C₂²×C₃	—	2	24	no	—
318	C₂²×C₃	—	2	6	no	—
319	C₂²×C₃	—	2	12	no	—
320	C₂²×C₃	—	2	3	no	—
321	C₂²×C₃	—	2	4	no	—
322	C₂²×C₃	—	2	24	no	—
323	C₂²×C₃	—	3	24	no	—
324	C₂²×C₃	—	3	24	no	—
325	C₂²×C₃	—	4	24	no	—
326	C₂²×C₃	—	4	12	no	—
327	C₂²×C₃	—	4	32	no	—
328	C₂²×C₃	—	6	96	no	—
329	C₂²×C₃	—	6	96	no	—
330	C₂²×C₃	—	6	24	no	—
331	C₂²×C₃	—	6	12	no	—
332	C₂²×C₃	—	12	16	no	—
333	C₂²×C₃	C₃×D₃×D₄	1	24	yes	complemented
334	C₂²×C₃	C₂×C₃²×D₄	1	6	yes	complemented
335	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
336	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
337	C₂×D₃	—	2	24	no	—
338	C₂×D₃	—	2	6	no	—
339	C₂×D₃	—	2	24	no	—
340	C₂×D₃	—	2	3	no	—
341	C₂×D₃	—	4	4	no	—
342	C₂×D₃	C₂×C₃²×D₄	1	6	yes	—
343	C₂×D₃	C₂×C₃²×D₄	1	6	yes	complemented
344	C₃×C₄	—	2	12	no	—
345	C₃×C₄	—	2	3	no	—
346	C₃×C₄	—	4	12	no	—
347	C₃×C₄	—	6	12	no	—
348	Dic₃	—	2	3	no	—
349	C₃²	—	2	4	no	—
350	C₃²	D₃×#120	1	1	yes	characteristic, direct product factor
351	C₃²	C₂×C₃×#120	1	4	yes	complemented
352	C₂³	—	2	6	no	—
353	C₂³	—	3	1	no	—
354	C₂³	—	3	3	no	—
355	C₂³	—	3	12	no	—
356	C₂³	—	3	1	no	—
357	C₂³	—	6	12	no	—
358	C₂³	—	6	12	no	—
359	C₂³	—	6	6	no	—
360	C₂³	—	6	32	no	—
361	C₂³	—	6	6	no	—
362	C₂³	C₂²×C₃²×D₃	1	3	yes	—
363	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic, complemented
364	D₄	—	2	6	no	—
365	D₄	—	6	6	no	—
366	D₄	—	6	6	no	—
367	D₄	—	6	6	no	—
368	C₂×C₄	—	3	3	no	—
369	C₂×C₄	—	6	6	no	—
370	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	complemented
371	C₂×C₃	—	2	24	no	—
372	C₂×C₃	—	2	6	no	—
373	C₂×C₃	—	2	12	no	—
374	C₂×C₃	—	3	16	no	—
375	C₂×C₃	—	4	24	no	—
376	C₂×C₃	—	4	4	no	—
377	C₂×C₃	—	4	1	no	—
378	C₂×C₃	—	6	24	no	—
379	C₂×C₃	—	8	4	no	—
380	C₂×C₃	—	12	4	no	—
381	C₂×C₃	C₂×C₃×D₃×D₄	1	12	yes	—
382	C₂×C₃	C₂²×C₃²×D₄	1	3	yes	—
383	D₃	—	2	6	no	—
384	D₃	—	4	1	no	—
385	D₃	C₃²×#120	1	4	yes	direct product factor
386	C₂²	—	2	8	no	—
387	C₂²	—	2	6	no	—
388	C₂²	—	2	3	no	—
389	C₂²	—	3	6	no	—
390	C₂²	—	3	6	no	—
391	C₂²	—	6	24	no	—
392	C₂²	—	6	24	no	—
393	C₂²	—	6	6	no	—
394	C₂²	—	6	3	no	—
395	C₂²	—	12	4	no	—
396	C₂²	C₃²×D₃×D₄	1	6	yes	complemented
397	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
398	C₄	—	2	3	no	—
399	C₄	—	6	3	no	—
400	C₃	—	2	4	no	—
401	C₃	C₃×D₃×#120	1	4	yes	direct product factor
402	C₃	C₂×C₃²×#120	1	1	yes	characteristic, complemented
403	C₂	—	2	6	no	—
404	C₂	—	3	4	no	—
405	C₂	—	4	1	no	—
406	C₂	—	6	6	no	—
407	C₂	—	12	1	no	—
408	C₂	C₂×C₃²×D₃×D₄	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×#120	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃²×#120	D₃
hypocenter	C₃	C₂×C₃²×#120
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#120

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃²×D₃	C₂	semidirect
2. C₂×C₃²×D₃×D₄	C₂	semidirect
3. C₃³×#120	C₂	semidirect
4. C₃²×#953	C₂	semidirect
5. C₃²×#937	C₂	semidirect
6. C₃²×#882	C₂	semidirect
7. C₃²×D₃×C₂²⋊C₄	C₂	semidirect
9. C₃×D₃×#120	C₃	direct (N × H)
10. C₂⁴×C₃³	C₂²	semidirect
13. C₂³×C₃²×D₃	C₂²	semidirect
14. C₂³×C₃²×D₃	C₂²	semidirect
15. C₂×C₃³×D₄	C₂²	semidirect
16. C₂×C₃²×#67	C₂²	semidirect
17. C₂×C₃²×#67	C₂²	semidirect
19. C₂×C₃²×D₁₂	C₂²	semidirect
20. C₂×C₃²×C₄×D₃	C₂²	semidirect
21. C₃³×C₂²⋊C₄	C₂²	semidirect
22. C₃²×#217	C₂²	semidirect
23. C₃²×(C₂×D₃)⋊C₄	C₂²	semidirect
25. C₂⁴×C₃×D₃	C₂×C₃	semidirect
27. C₂×C₃×D₃×D₄	C₂×C₃	semidirect
32. C₃²×#120	D₃	direct (N × H)
33. C₃²×#120	C₂×C₃	semidirect
35. C₃×#953	C₂×C₃	semidirect
36. C₃×#937	C₂×C₃	semidirect
37. C₃×#882	C₂×C₃	semidirect
38. C₃×D₃×C₂²⋊C₄	C₂×C₃	semidirect
41. C₂³×C₃³	C₂³	semidirect
47. C₂²×C₃²×D₃	D₄	semidirect
50. C₂²×C₃²×D₃	C₂³	semidirect
52. C₂×C₃³×C₄	C₂³	semidirect
55. C₂×C₃²×Dic₃	C₂³	semidirect
59. D₃×#120	C₃²	direct (N × H)
65. C₂⁴×C₃²	C₂×D₃	semidirect
66. C₂⁴×C₃²	C₂²×C₃	semidirect
69. C₂³×C₃×D₃	C₂²×C₃	semidirect
70. C₂³×C₃×D₃	C₂²×C₃	semidirect
76. C₂×C₃²×D₄	C₂×D₃	semidirect
77. C₂×C₃²×D₄	C₂²×C₃	semidirect
79. C₂×C₃×#67	C₂²×C₃	semidirect
80. C₂×C₃×#67	C₂²×C₃	semidirect
82. C₂×C₃×D₁₂	C₂²×C₃	semidirect
83. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
87. C₃²×C₂²⋊C₄	C₂×D₃	semidirect
88. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
89. C₃×#217	C₂²×C₃	semidirect
90. C₃×(C₂×D₃)⋊C₄	C₂²×C₃	semidirect
94. C₂²×C₃³	C₂×D₄	semidirect
102. C₂×C₃²×D₃	C₂×D₄	semidirect
106. C₂⁴×D₃	C₂×C₃²	semidirect
108. C₂×D₃×D₄	C₂×C₃²	semidirect
113. C₃×#120	C₃×D₃	direct (N × H)
114. C₃×#120	C₂×C₃²	semidirect
116. #953	C₂×C₃²	semidirect
117. #937	C₂×C₃²	semidirect
118. #882	C₂×C₃²	semidirect
119. D₃×C₂²⋊C₄	C₂×C₃²	semidirect
135. C₂³×C₃²	C₂²×D₃	semidirect
137. C₂³×C₃²	C₂³×C₃	semidirect
143. C₂²×C₃×D₃	C₃×D₄	semidirect
146. C₂²×C₃×D₃	C₂³×C₃	semidirect
156. C₂×C₃²×C₄	C₂²×D₃	semidirect
157. C₂×C₃²×C₄	C₂³×C₃	semidirect
160. C₂×C₃×Dic₃	C₂³×C₃	semidirect
169. C₃²×D₃	#120	direct (N × H)
175. C₂⁴×C₃	C₂×C₃×D₃	semidirect
176. C₂⁴×C₃	C₂²×C₃²	semidirect
179. C₂³×D₃	C₂²×C₃²	semidirect
180. C₂³×D₃	C₂²×C₃²	semidirect
186. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
187. C₂×C₃×D₄	C₂²×C₃²	semidirect
189. C₂×#67	C₂²×C₃²	semidirect
190. C₂×#67	C₂²×C₃²	semidirect
192. C₂×D₁₂	C₂²×C₃²	semidirect
193. C₂×C₄×D₃	C₂²×C₃²	semidirect
197. C₃×C₂²⋊C₄	C₂×C₃×D₃	semidirect
198. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
199. #217	C₂²×C₃²	semidirect
200. (C₂×D₃)⋊C₄	C₂²×C₃²	semidirect
219. C₂²×C₃²	D₃×D₄	semidirect
220. C₂²×C₃²	C₂×C₃×D₄	semidirect
229. C₂×C₃×D₃	C₂×C₃×D₄	semidirect
240. #120	C₃²×D₃	direct (N × H)
242. C₃³	C₂×#120	semidirect
258. C₂³×C₃	C₂²×C₃×D₃	semidirect
260. C₂³×C₃	C₂³×C₃²	semidirect
266. C₂²×D₃	C₃²×D₄	semidirect
269. C₂²×D₃	C₂³×C₃²	semidirect
279. C₂×C₃×C₄	C₂²×C₃×D₃	semidirect
280. C₂×C₃×C₄	C₂³×C₃²	semidirect
283. C₂×Dic₃	C₂³×C₃²	semidirect
300. C₃×D₃	C₃×#120	direct (N × H)
305. C₂⁴	C₂×C₃²×D₃	semidirect
310. C₂×D₄	C₂×C₃²×D₃	semidirect
314. C₂²⋊C₄	C₂×C₃²×D₃	semidirect
333. C₂²×C₃	C₃×D₃×D₄	semidirect
334. C₂²×C₃	C₂×C₃²×D₄	semidirect
343. C₂×D₃	C₂×C₃²×D₄	semidirect
350. C₃²	D₃×#120	direct (N × H)
351. C₃²	C₂×C₃×#120	semidirect
363. C₂³	C₂²×C₃²×D₃	semidirect
370. C₂×C₄	C₂²×C₃²×D₃	semidirect
385. D₃	C₃²×#120	direct (N × H)
396. C₂²	C₃²×D₃×D₄	semidirect
401. C₃	C₃×D₃×#120	direct (N × H)
402. C₃	C₂×C₃²×#120	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₃²×D₃	2
C₂×C₃²×D₃×D₄	2
C₃³×#120	2
C₃²×#953	2
C₃²×#937	2
C₃²×#882	2
C₃²×D₃×C₂²⋊C₄	2
C₂×C₃²×#120	3
C₃×D₃×#120	3

Subgroups — C32×D3×#120 #13178966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₃×#120 #13178966