Subgroups — C₂×C₃²×#1004 #13181716

Order 1728.

6332 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×#67	C₂	1	4	yes	maximal, complemented
3	C₃²×#1004	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃²×#237	C₂	1	4	yes	maximal, complemented
5	C₂×C₃²×D₃×D₄	C₂	1	4	yes	maximal, complemented
6	C₂×C₃²×#235	C₂	1	2	yes	maximal, complemented
7	C₂×C₃²×H(5,𝔽₂)	—	3	1	no	maximal
8	C₂×C₃×#1004	C₃	1	4	yes	maximal, direct product factor
9	C₂⁴×C₃³	C₂²	1	2	yes	complemented
10	C₂³×C₃²×D₃	C₂²	1	4	yes	complemented
11	C₂×C₃³×D₄	C₂²	1	4	yes	complemented
12	C₂×C₃³×D₄	C₂²	1	8	yes	complemented
13	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃²×#67	C₂²	1	4	yes	complemented
15	C₂×C₃²×#67	C₂²	1	8	yes	complemented
16	C₂×C₃²×#67	C₂²	1	32	yes	complemented
17	C₂²×C₃²×Dic₃	C₂²	1	4	yes	complemented
18	C₃²×#237	C₂²	1	32	yes	complemented
19	C₃²×D₃×D₄	C₂²	1	32	yes	complemented
20	C₃²×#235	C₂²	1	16	yes	complemented
21	C₂×C₃²×D₁₂	C₂²	1	2	yes	complemented
22	C₂×C₃²×C₄×D₃	C₂²	1	4	yes	complemented
23	C₂×C₃²×Dic₆	C₂²	1	2	yes	complemented
24	C₃²×H(5,𝔽₂)	—	3	16	no	—
25	C₂×C₃²×#41	—	3	4	no	—
26	C₂×C₃²×#41	—	3	2	no	—
27	C₂²×C₃²×D₄	—	2	4	no	—
28	C₂²×C₃²×D₄	—	3	4	no	—
29	C₂²×C₃²×D₄	—	3	4	no	—
30	C₂²×C₃²×D₄	D₃	1	1	yes	characteristic, complemented
31	C₂²×C₃²×D₄	C₂×C₃	1	4	yes	complemented
32	C₂²×C₃×#67	C₂×C₃	1	16	yes	complemented
33	C₃×#1004	C₂×C₃	1	64	yes	direct product factor
34	C₂×C₃×#237	C₂×C₃	1	16	yes	complemented
35	C₂×C₃×D₃×D₄	C₂×C₃	1	16	yes	complemented
36	C₂×C₃×#235	C₂×C₃	1	8	yes	complemented
37	C₂³×C₃³	—	4	4	no	—
38	C₂³×C₃³	C₂³	1	4	yes	—
39	C₂³×C₃³	C₂³	1	1	yes	characteristic
40	C₂³×C₃³	C₂³	1	8	yes	complemented
41	C₂²×C₃²×D₃	—	4	8	no	—
42	C₂²×C₃²×D₃	C₂³	1	4	yes	—
43	C₂²×C₃²×D₃	C₂³	1	16	yes	complemented
44	C₃³×D₄	C₂³	1	16	yes	complemented
45	C₂×C₃³×C₄	C₂³	1	2	yes	—
46	C₂×C₃³×C₄	C₂³	1	4	yes	complemented
47	C₃²×#67	C₂³	1	16	yes	complemented
48	C₃²×#67	C₂³	1	32	yes	complemented
49	C₂×C₃²×Dic₃	C₂³	1	4	yes	—
50	C₂×C₃²×Dic₃	C₂³	1	16	yes	complemented
51	C₃²×D₁₂	C₂³	1	8	yes	complemented
52	C₃²×C₄×D₃	C₂³	1	16	yes	complemented
53	C₃²×Dic₆	C₂³	1	8	yes	complemented
54	C₂×C₃×H(5,𝔽₂)	—	3	4	no	—
55	C₂×#1004	C₃²	1	1	yes	characteristic, direct product factor
56	C₂⁴×C₃²	—	2	8	no	—
57	C₂⁴×C₃²	—	3	4	no	—
58	C₂⁴×C₃²	C₂×D₃	1	2	yes	complemented
59	C₂⁴×C₃²	C₂²×C₃	1	8	yes	complemented
60	C₂³×C₃×D₃	C₂²×C₃	1	16	yes	complemented
61	C₃²×#41	—	3	32	no	—
62	C₃²×#41	—	3	16	no	—
63	C₂×C₃²×Q₈	—	3	2	no	—
64	C₂×C₃²×D₄	—	2	16	no	—
65	C₂×C₃²×D₄	—	2	32	no	—
66	C₂×C₃²×D₄	—	3	8	no	—
67	C₂×C₃²×D₄	—	3	4	no	—
68	C₂×C₃²×D₄	—	3	32	no	—
69	C₂×C₃²×D₄	—	3	32	no	—
70	C₂×C₃²×D₄	—	3	2	no	—
71	C₂×C₃²×D₄	C₂×D₃	1	4	yes	complemented
72	C₂×C₃²×D₄	C₂×D₃	1	8	yes	complemented
73	C₂×C₃²×D₄	C₂²×C₃	1	16	yes	complemented
74	C₂×C₃²×D₄	C₂²×C₃	1	32	yes	complemented
75	C₂²×C₃²×C₄	—	2	4	no	—
76	C₂²×C₃²×C₄	—	3	4	no	—
77	C₂²×C₃²×C₄	—	3	4	no	—
78	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic, complemented
79	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	complemented
80	C₂×C₃×#67	C₂²×C₃	1	16	yes	complemented
81	C₂×C₃×#67	C₂²×C₃	1	32	yes	complemented
82	C₂×C₃×#67	C₂²×C₃	1	128	yes	complemented
83	C₂²×C₃×Dic₃	C₂²×C₃	1	16	yes	complemented
84	C₃×#237	C₂²×C₃	1	128	yes	complemented
85	C₃×D₃×D₄	C₂²×C₃	1	128	yes	complemented
86	C₃×#235	C₂²×C₃	1	64	yes	complemented
87	C₂×C₃×D₁₂	C₂²×C₃	1	8	yes	complemented
88	C₂×C₃×C₄×D₃	C₂²×C₃	1	16	yes	complemented
89	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	complemented
90	C₂²×C₃³	—	2	2	no	—
91	C₂²×C₃³	—	2	8	no	—
92	C₂²×C₃³	—	4	8	no	—
93	C₂²×C₃³	C₂⁴	1	8	yes	—
94	C₂²×C₃³	C₂⁴	1	2	yes	—
95	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
96	C₂×C₃²×D₃	—	2	8	no	—
97	C₂×C₃²×D₃	—	4	16	no	—
98	C₂×C₃²×D₃	C₂⁴	1	8	yes	—
99	C₃³×C₄	C₂⁴	1	4	yes	—
100	C₃²×Dic₃	C₂⁴	1	8	yes	—
101	C₃×H(5,𝔽₂)	—	3	64	no	—
102	C₂×C₃×#41	—	3	16	no	—
103	C₂×C₃×#41	—	3	8	no	—
104	C₂²×C₃×D₄	—	2	4	no	—
105	C₂²×C₃×D₄	—	3	16	no	—
106	C₂²×C₃×D₄	—	3	16	no	—
107	C₂²×C₃×D₄	C₃×D₃	1	4	yes	complemented
108	C₂²×C₃×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
109	C₂²×#67	C₂×C₃²	1	4	yes	complemented
110	#1004	C₂×C₃²	1	16	yes	direct product factor
111	C₂×#237	C₂×C₃²	1	4	yes	complemented
112	C₂×D₃×D₄	C₂×C₃²	1	4	yes	complemented
113	C₂×#235	C₂×C₃²	1	2	yes	complemented
114	C₂³×C₃²	—	2	4	no	—
115	C₂³×C₃²	—	2	16	no	—
116	C₂³×C₃²	—	2	32	no	—
117	C₂³×C₃²	—	3	4	no	—
118	C₂³×C₃²	—	3	16	no	—
119	C₂³×C₃²	—	4	4	no	—
120	C₂³×C₃²	—	4	16	no	—
121	C₂³×C₃²	—	4	32	no	—
122	C₂³×C₃²	—	12	8	no	—
123	C₂³×C₃²	C₂²×D₃	1	4	yes	—
124	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
125	C₂³×C₃²	C₂²×D₃	1	8	yes	complemented
126	C₂³×C₃²	C₂³×C₃	1	16	yes	—
127	C₂³×C₃²	C₂³×C₃	1	4	yes	—
128	C₂³×C₃²	C₂³×C₃	1	32	yes	complemented
129	C₂²×C₃×D₃	—	4	32	no	—
130	C₂²×C₃×D₃	C₂³×C₃	1	16	yes	—
131	C₂²×C₃×D₃	C₂³×C₃	1	64	yes	complemented
132	C₃²×Q₈	—	3	8	no	—
133	C₃²×D₄	—	2	64	no	—
134	C₃²×D₄	—	3	32	no	—
135	C₃²×D₄	—	3	16	no	—
136	C₃²×D₄	—	3	8	no	—
137	C₃²×D₄	C₂²×D₃	1	16	yes	complemented
138	C₃²×D₄	C₂³×C₃	1	64	yes	complemented
139	C₂×C₃²×C₄	—	2	8	no	—
140	C₂×C₃²×C₄	—	2	16	no	—
141	C₂×C₃²×C₄	—	3	16	no	—
142	C₂×C₃²×C₄	—	3	4	no	—
143	C₂×C₃²×C₄	—	3	16	no	—
144	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
145	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	complemented
146	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
147	C₂×C₃²×C₄	C₂³×C₃	1	16	yes	complemented
148	C₃×#67	C₂³×C₃	1	64	yes	complemented
149	C₃×#67	C₂³×C₃	1	128	yes	complemented
150	C₂×C₃×Dic₃	C₂³×C₃	1	16	yes	—
151	C₂×C₃×Dic₃	C₂³×C₃	1	64	yes	complemented
152	C₃×D₁₂	C₂³×C₃	1	32	yes	complemented
153	C₃×C₄×D₃	C₂³×C₃	1	64	yes	complemented
154	C₃×Dic₆	C₂³×C₃	1	32	yes	complemented
155	C₂×H(5,𝔽₂)	—	3	1	no	Sylow
156	C₂×C₃³	—	2	2	no	—
157	C₂×C₃³	—	2	8	no	—
158	C₂×C₃³	H(5,𝔽₂)	1	2	yes	complemented
159	C₂×C₃³	C₂⁵	1	1	yes	characteristic
160	C₃²×D₃	—	2	8	no	—
161	C₂⁴×C₃	—	2	8	no	—
162	C₂⁴×C₃	—	3	16	no	—
163	C₂⁴×C₃	C₂×C₃×D₃	1	8	yes	complemented
164	C₂⁴×C₃	C₂²×C₃²	1	2	yes	complemented
165	C₂³×D₃	C₂²×C₃²	1	4	yes	complemented
166	C₃×#41	—	3	128	no	—
167	C₃×#41	—	3	64	no	—
168	C₂×C₃×Q₈	—	3	8	no	—
169	C₂×C₃×D₄	—	2	16	no	—
170	C₂×C₃×D₄	—	2	32	no	—
171	C₂×C₃×D₄	—	3	32	no	—
172	C₂×C₃×D₄	—	3	16	no	—
173	C₂×C₃×D₄	—	3	128	no	—
174	C₂×C₃×D₄	—	3	128	no	—
175	C₂×C₃×D₄	—	3	8	no	—
176	C₂×C₃×D₄	C₂×C₃×D₃	1	16	yes	complemented
177	C₂×C₃×D₄	C₂×C₃×D₃	1	32	yes	complemented
178	C₂×C₃×D₄	C₂²×C₃²	1	4	yes	complemented
179	C₂×C₃×D₄	C₂²×C₃²	1	8	yes	complemented
180	C₂²×C₃×C₄	—	2	4	no	—
181	C₂²×C₃×C₄	—	3	16	no	—
182	C₂²×C₃×C₄	—	3	16	no	—
183	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	complemented
184	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
185	C₂×#67	C₂²×C₃²	1	4	yes	complemented
186	C₂×#67	C₂²×C₃²	1	8	yes	complemented
187	C₂×#67	C₂²×C₃²	1	32	yes	complemented
188	C₂²×Dic₃	C₂²×C₃²	1	4	yes	complemented
189	#237	C₂²×C₃²	1	32	yes	complemented
190	D₃×D₄	C₂²×C₃²	1	32	yes	complemented
191	#235	C₂²×C₃²	1	16	yes	complemented
192	C₂×D₁₂	C₂²×C₃²	1	2	yes	complemented
193	C₂×C₄×D₃	C₂²×C₃²	1	4	yes	complemented
194	C₂×Dic₆	C₂²×C₃²	1	2	yes	complemented
195	C₂²×C₃²	—	2	16	no	—
196	C₂²×C₃²	—	2	2	no	—
197	C₂²×C₃²	—	2	8	no	—
198	C₂²×C₃²	—	2	8	no	—
199	C₂²×C₃²	—	2	8	no	—
200	C₂²×C₃²	—	2	4	no	—
201	C₂²×C₃²	—	2	32	no	—
202	C₂²×C₃²	—	2	32	no	—
203	C₂²×C₃²	—	3	8	no	—
204	C₂²×C₃²	—	4	32	no	—
205	C₂²×C₃²	—	4	8	no	—
206	C₂²×C₃²	—	4	32	no	—
207	C₂²×C₃²	—	4	64	no	—
208	C₂²×C₃²	—	6	8	no	—
209	C₂²×C₃²	—	12	16	no	—
210	C₂²×C₃²	C₂³×D₃	1	8	yes	—
211	C₂²×C₃²	C₂³×D₃	1	2	yes	—
212	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
213	C₂²×C₃²	C₂⁴×C₃	1	32	yes	—
214	C₂²×C₃²	C₂⁴×C₃	1	8	yes	—
215	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
216	C₂×C₃×D₃	—	2	32	no	—
217	C₂×C₃×D₃	—	4	64	no	—
218	C₂×C₃×D₃	C₂⁴×C₃	1	32	yes	—
219	C₃²×C₄	—	2	16	no	—
220	C₃²×C₄	—	3	8	no	—
221	C₃²×C₄	C₂³×D₃	1	4	yes	—
222	C₃²×C₄	C₂⁴×C₃	1	16	yes	—
223	C₃×Dic₃	C₂⁴×C₃	1	32	yes	—
224	H(5,𝔽₂)	—	3	16	no	—
225	C₂×#41	—	3	4	no	—
226	C₂×#41	—	3	2	no	—
227	C₂²×D₄	—	3	4	no	—
228	C₂²×D₄	—	3	4	no	—
229	C₂²×D₄	C₃²×D₃	1	1	yes	characteristic, complemented
230	C₃³	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
231	C₂³×C₃	—	2	4	no	—
232	C₂³×C₃	—	2	16	no	—
233	C₂³×C₃	—	2	32	no	—
234	C₂³×C₃	—	3	16	no	—
235	C₂³×C₃	—	3	64	no	—
236	C₂³×C₃	—	4	16	no	—
237	C₂³×C₃	—	4	4	no	—
238	C₂³×C₃	—	4	32	no	—
239	C₂³×C₃	—	12	32	no	—
240	C₂³×C₃	C₂²×C₃×D₃	1	16	yes	—
241	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
242	C₂³×C₃	C₂²×C₃×D₃	1	32	yes	complemented
243	C₂³×C₃	C₂³×C₃²	1	4	yes	—
244	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
245	C₂³×C₃	C₂³×C₃²	1	8	yes	complemented
246	C₂²×D₃	—	4	8	no	—
247	C₂²×D₃	C₂³×C₃²	1	4	yes	—
248	C₂²×D₃	C₂³×C₃²	1	16	yes	complemented
249	C₃×Q₈	—	3	32	no	—
250	C₃×D₄	—	2	64	no	—
251	C₃×D₄	—	3	128	no	—
252	C₃×D₄	—	3	64	no	—
253	C₃×D₄	—	3	32	no	—
254	C₃×D₄	C₂²×C₃×D₃	1	64	yes	complemented
255	C₃×D₄	C₂³×C₃²	1	16	yes	complemented
256	C₂×C₃×C₄	—	2	8	no	—
257	C₂×C₃×C₄	—	2	16	no	—
258	C₂×C₃×C₄	—	3	16	no	—
259	C₂×C₃×C₄	—	3	64	no	—
260	C₂×C₃×C₄	—	3	64	no	—
261	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
262	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	complemented
263	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
264	C₂×C₃×C₄	C₂³×C₃²	1	4	yes	complemented
265	#67	C₂³×C₃²	1	16	yes	complemented
266	#67	C₂³×C₃²	1	32	yes	complemented
267	C₂×Dic₃	C₂³×C₃²	1	4	yes	—
268	C₂×Dic₃	C₂³×C₃²	1	16	yes	complemented
269	D₁₂	C₂³×C₃²	1	8	yes	complemented
270	C₄×D₃	C₂³×C₃²	1	16	yes	complemented
271	Dic₆	C₂³×C₃²	1	8	yes	complemented
272	C₂×C₃²	—	2	16	no	—
273	C₂×C₃²	—	2	2	no	—
274	C₂×C₃²	—	2	8	no	—
275	C₂×C₃²	—	2	8	no	—
276	C₂×C₃²	—	2	8	no	—
277	C₂×C₃²	—	2	4	no	—
278	C₂×C₃²	—	2	32	no	—
279	C₂×C₃²	—	4	32	no	—
280	C₂×C₃²	—	6	8	no	—
281	C₂×C₃²	#1004	1	2	yes	direct product factor
282	C₂×C₃²	C₃×H(5,𝔽₂)	1	8	yes	complemented
283	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
284	C₂×C₃²	C₂⁵×C₃	1	4	yes	—
285	C₃×D₃	—	2	32	no	—
286	C₂⁴	—	3	4	no	—
287	C₂⁴	C₂×C₃²×D₃	1	2	yes	complemented
288	#41	—	3	32	no	—
289	#41	—	3	16	no	—
290	C₂×Q₈	—	3	2	no	—
291	C₂×D₄	—	3	8	no	—
292	C₂×D₄	—	3	4	no	—
293	C₂×D₄	—	3	32	no	—
294	C₂×D₄	—	3	32	no	—
295	C₂×D₄	—	3	2	no	—
296	C₂×D₄	C₂×C₃²×D₃	1	4	yes	complemented
297	C₂×D₄	C₂×C₃²×D₃	1	8	yes	complemented
298	C₂²×C₄	—	3	4	no	—
299	C₂²×C₄	—	3	4	no	—
300	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
301	C₂²×C₃	—	2	16	no	—
302	C₂²×C₃	—	2	8	no	—
303	C₂²×C₃	—	2	32	no	—
304	C₂²×C₃	—	2	8	no	—
305	C₂²×C₃	—	2	2	no	—
306	C₂²×C₃	—	2	8	no	—
307	C₂²×C₃	—	2	4	no	—
308	C₂²×C₃	—	2	32	no	—
309	C₂²×C₃	—	3	32	no	—
310	C₂²×C₃	—	4	32	no	—
311	C₂²×C₃	—	4	32	no	—
312	C₂²×C₃	—	4	8	no	—
313	C₂²×C₃	—	4	64	no	—
314	C₂²×C₃	—	6	32	no	—
315	C₂²×C₃	—	12	64	no	—
316	C₂²×C₃	C₂³×C₃×D₃	1	32	yes	—
317	C₂²×C₃	C₂³×C₃×D₃	1	8	yes	—
318	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
319	C₂²×C₃	C₂⁴×C₃²	1	8	yes	—
320	C₂²×C₃	C₂⁴×C₃²	1	2	yes	—
321	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
322	C₂×D₃	—	2	8	no	—
323	C₂×D₃	—	4	16	no	—
324	C₂×D₃	C₂⁴×C₃²	1	8	yes	—
325	C₃×C₄	—	2	16	no	—
326	C₃×C₄	—	3	32	no	—
327	C₃×C₄	C₂³×C₃×D₃	1	16	yes	—
328	C₃×C₄	C₂⁴×C₃²	1	4	yes	—
329	Dic₃	C₂⁴×C₃²	1	8	yes	—
330	C₃²	—	2	4	no	—
331	C₃²	C₂×#1004	1	1	yes	characteristic, direct product factor
332	C₃²	C₂×C₃×H(5,𝔽₂)	1	4	yes	complemented
333	C₂³	—	3	4	no	—
334	C₂³	—	3	16	no	—
335	C₂³	—	4	4	no	—
336	C₂³	—	12	8	no	—
337	C₂³	C₂²×C₃²×D₃	1	4	yes	—
338	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic
339	C₂³	C₂²×C₃²×D₃	1	8	yes	complemented
340	Q₈	—	3	8	no	—
341	D₄	—	3	32	no	—
342	D₄	—	3	16	no	—
343	D₄	—	3	8	no	—
344	D₄	C₂²×C₃²×D₃	1	16	yes	complemented
345	C₂×C₄	—	3	4	no	—
346	C₂×C₄	—	3	16	no	—
347	C₂×C₄	—	3	16	no	—
348	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
349	C₂×C₄	C₂²×C₃²×D₃	1	4	yes	complemented
350	C₂×C₃	—	2	16	no	—
351	C₂×C₃	—	2	8	no	—
352	C₂×C₃	—	2	32	no	—
353	C₂×C₃	—	2	8	no	—
354	C₂×C₃	—	2	2	no	—
355	C₂×C₃	—	2	8	no	—
356	C₂×C₃	—	2	4	no	—
357	C₂×C₃	—	4	32	no	—
358	C₂×C₃	—	6	32	no	—
359	C₂×C₃	C₃×#1004	1	8	yes	direct product factor
360	C₂×C₃	C₃²×H(5,𝔽₂)	1	2	yes	complemented
361	C₂×C₃	C₂⁴×C₃×D₃	1	4	yes	—
362	C₂×C₃	C₂⁵×C₃²	1	1	yes	characteristic
363	D₃	—	2	8	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	8	no	—
366	C₂²	—	3	8	no	—
367	C₂²	—	4	8	no	—
368	C₂²	—	6	8	no	—
369	C₂²	—	12	16	no	—
370	C₂²	C₂³×C₃²×D₃	1	8	yes	—
371	C₂²	C₂³×C₃²×D₃	1	2	yes	—
372	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
373	C₄	—	3	8	no	—
374	C₄	C₂³×C₃²×D₃	1	4	yes	—
375	C₃	—	2	4	no	—
376	C₃	C₂×C₃×#1004	1	4	yes	direct product factor
377	C₃	C₂×C₃²×H(5,𝔽₂)	1	1	yes	characteristic, complemented
378	C₂	—	2	2	no	—
379	C₂	—	2	8	no	—
380	C₂	—	6	8	no	—
381	C₂	C₃²×#1004	1	2	yes	direct product factor
382	C₂	C₂⁴×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂×C₃	C₂⁵×C₃²
Frattini	C₂	C₂⁴×C₃²×D₃
Fitting	C₂²×C₃³×D₄	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₂²×C₃²×D₄	D₃
hypocenter	C₃	C₂×C₃²×H(5,𝔽₂)
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×D₄	C₂	semidirect
2. C₂²×C₃²×#67	C₂	semidirect
3. C₃²×#1004	C₂	direct (N × H)
4. C₂×C₃²×#237	C₂	semidirect
5. C₂×C₃²×D₃×D₄	C₂	semidirect
6. C₂×C₃²×#235	C₂	semidirect
8. C₂×C₃×#1004	C₃	direct (N × H)
9. C₂⁴×C₃³	C₂²	semidirect
10. C₂³×C₃²×D₃	C₂²	semidirect
11. C₂×C₃³×D₄	C₂²	semidirect
12. C₂×C₃³×D₄	C₂²	semidirect
13. C₂²×C₃³×C₄	C₂²	semidirect
14. C₂×C₃²×#67	C₂²	semidirect
15. C₂×C₃²×#67	C₂²	semidirect
16. C₂×C₃²×#67	C₂²	semidirect
17. C₂²×C₃²×Dic₃	C₂²	semidirect
18. C₃²×#237	C₂²	semidirect
19. C₃²×D₃×D₄	C₂²	semidirect
20. C₃²×#235	C₂²	semidirect
21. C₂×C₃²×D₁₂	C₂²	semidirect
22. C₂×C₃²×C₄×D₃	C₂²	semidirect
23. C₂×C₃²×Dic₆	C₂²	semidirect
30. C₂²×C₃²×D₄	D₃	semidirect
31. C₂²×C₃²×D₄	C₂×C₃	semidirect
32. C₂²×C₃×#67	C₂×C₃	semidirect
33. C₃×#1004	C₂×C₃	direct (N × H)
34. C₂×C₃×#237	C₂×C₃	semidirect
35. C₂×C₃×D₃×D₄	C₂×C₃	semidirect
36. C₂×C₃×#235	C₂×C₃	semidirect
40. C₂³×C₃³	C₂³	semidirect
43. C₂²×C₃²×D₃	C₂³	semidirect
44. C₃³×D₄	C₂³	semidirect
46. C₂×C₃³×C₄	C₂³	semidirect
47. C₃²×#67	C₂³	semidirect
48. C₃²×#67	C₂³	semidirect
50. C₂×C₃²×Dic₃	C₂³	semidirect
51. C₃²×D₁₂	C₂³	semidirect
52. C₃²×C₄×D₃	C₂³	semidirect
53. C₃²×Dic₆	C₂³	semidirect
55. C₂×#1004	C₃²	direct (N × H)
58. C₂⁴×C₃²	C₂×D₃	semidirect
59. C₂⁴×C₃²	C₂²×C₃	semidirect
60. C₂³×C₃×D₃	C₂²×C₃	semidirect
71. C₂×C₃²×D₄	C₂×D₃	semidirect
72. C₂×C₃²×D₄	C₂×D₃	semidirect
73. C₂×C₃²×D₄	C₂²×C₃	semidirect
74. C₂×C₃²×D₄	C₂²×C₃	semidirect
78. C₂²×C₃²×C₄	C₂×D₃	semidirect
79. C₂²×C₃²×C₄	C₂²×C₃	semidirect
80. C₂×C₃×#67	C₂²×C₃	semidirect
81. C₂×C₃×#67	C₂²×C₃	semidirect
82. C₂×C₃×#67	C₂²×C₃	semidirect
83. C₂²×C₃×Dic₃	C₂²×C₃	semidirect
84. C₃×#237	C₂²×C₃	semidirect
85. C₃×D₃×D₄	C₂²×C₃	semidirect
86. C₃×#235	C₂²×C₃	semidirect
87. C₂×C₃×D₁₂	C₂²×C₃	semidirect
88. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
89. C₂×C₃×Dic₆	C₂²×C₃	semidirect
107. C₂²×C₃×D₄	C₃×D₃	semidirect
108. C₂²×C₃×D₄	C₂×C₃²	semidirect
109. C₂²×#67	C₂×C₃²	semidirect
110. #1004	C₂×C₃²	direct (N × H)
111. C₂×#237	C₂×C₃²	semidirect
112. C₂×D₃×D₄	C₂×C₃²	semidirect
113. C₂×#235	C₂×C₃²	semidirect
125. C₂³×C₃²	C₂²×D₃	semidirect
128. C₂³×C₃²	C₂³×C₃	semidirect
131. C₂²×C₃×D₃	C₂³×C₃	semidirect
137. C₃²×D₄	C₂²×D₃	semidirect
138. C₃²×D₄	C₂³×C₃	semidirect
145. C₂×C₃²×C₄	C₂²×D₃	semidirect
147. C₂×C₃²×C₄	C₂³×C₃	semidirect
148. C₃×#67	C₂³×C₃	semidirect
149. C₃×#67	C₂³×C₃	semidirect
151. C₂×C₃×Dic₃	C₂³×C₃	semidirect
152. C₃×D₁₂	C₂³×C₃	semidirect
153. C₃×C₄×D₃	C₂³×C₃	semidirect
154. C₃×Dic₆	C₂³×C₃	semidirect
158. C₂×C₃³	H(5,𝔽₂)	semidirect
163. C₂⁴×C₃	C₂×C₃×D₃	semidirect
164. C₂⁴×C₃	C₂²×C₃²	semidirect
165. C₂³×D₃	C₂²×C₃²	semidirect
176. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
177. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
178. C₂×C₃×D₄	C₂²×C₃²	semidirect
179. C₂×C₃×D₄	C₂²×C₃²	semidirect
183. C₂²×C₃×C₄	C₂×C₃×D₃	semidirect
184. C₂²×C₃×C₄	C₂²×C₃²	semidirect
185. C₂×#67	C₂²×C₃²	semidirect
186. C₂×#67	C₂²×C₃²	semidirect
187. C₂×#67	C₂²×C₃²	semidirect
188. C₂²×Dic₃	C₂²×C₃²	semidirect
189. #237	C₂²×C₃²	semidirect
190. D₃×D₄	C₂²×C₃²	semidirect
191. #235	C₂²×C₃²	semidirect
192. C₂×D₁₂	C₂²×C₃²	semidirect
193. C₂×C₄×D₃	C₂²×C₃²	semidirect
194. C₂×Dic₆	C₂²×C₃²	semidirect
229. C₂²×D₄	C₃²×D₃	semidirect
230. C₃³	C₂×H(5,𝔽₂)	semidirect
242. C₂³×C₃	C₂²×C₃×D₃	semidirect
245. C₂³×C₃	C₂³×C₃²	semidirect
248. C₂²×D₃	C₂³×C₃²	semidirect
254. C₃×D₄	C₂²×C₃×D₃	semidirect
255. C₃×D₄	C₂³×C₃²	semidirect
262. C₂×C₃×C₄	C₂²×C₃×D₃	semidirect
264. C₂×C₃×C₄	C₂³×C₃²	semidirect
265. #67	C₂³×C₃²	semidirect
266. #67	C₂³×C₃²	semidirect
268. C₂×Dic₃	C₂³×C₃²	semidirect
269. D₁₂	C₂³×C₃²	semidirect
270. C₄×D₃	C₂³×C₃²	semidirect
271. Dic₆	C₂³×C₃²	semidirect
281. C₂×C₃²	#1004	direct (N × H)
282. C₂×C₃²	C₃×H(5,𝔽₂)	semidirect
287. C₂⁴	C₂×C₃²×D₃	semidirect
296. C₂×D₄	C₂×C₃²×D₃	semidirect
297. C₂×D₄	C₂×C₃²×D₃	semidirect
300. C₂²×C₄	C₂×C₃²×D₃	semidirect
331. C₃²	C₂×#1004	direct (N × H)
332. C₃²	C₂×C₃×H(5,𝔽₂)	semidirect
339. C₂³	C₂²×C₃²×D₃	semidirect
344. D₄	C₂²×C₃²×D₃	semidirect
349. C₂×C₄	C₂²×C₃²×D₃	semidirect
359. C₂×C₃	C₃×#1004	direct (N × H)
360. C₂×C₃	C₃²×H(5,𝔽₂)	semidirect
376. C₃	C₂×C₃×#1004	direct (N × H)
377. C₃	C₂×C₃²×H(5,𝔽₂)	semidirect
381. C₂	C₃²×#1004	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃³×D₄	2
C₂²×C₃²×#67	2
C₃²×#1004	2
C₂×C₃²×#237	2
C₂×C₃²×D₃×D₄	2
C₂×C₃²×#235	2
C₂×C₃²×H(5,𝔽₂)	3
C₂×C₃×#1004	3

Subgroups — C2×C32×#1004 #13181716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#1004 #13181716