Subgroups — C₂×C₃×C₄×#124 #10652639

Order 768.

3790 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×#124	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#390	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃×#388	C₂	1	4	yes	maximal, complemented
8	C₂×C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×C₄×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
10	C₂×C₃×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×#124	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
13	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
14	C₂⁴×C₃×C₄	C₂²	1	2	yes	—
15	C₂×C₃×#124	C₄	1	16	yes	direct product factor
16	C₂×C₃×#124	C₂²	1	8	yes	—
17	C₂×C₃×C₄×Q₈	—	2	8	no	—
18	C₂×C₃×C₄×Q₈	C₂²	1	8	yes	complemented
19	C₂×C₃×C₄×D₄	—	2	8	no	—
20	C₂×C₃×C₄×D₄	C₂²	1	8	yes	—
21	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂²×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂²×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₂²×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂²×C₃×C₄²	C₂²	1	2	yes	—
28	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₂²×C₃×C₄²	C₂²	1	2	yes	—
30	C₂²×C₃×C₄²	C₂²	1	2	yes	—
31	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
32	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
33	C₃×#391	C₂²	1	8	yes	complemented
34	C₃×#390	C₂²	1	8	yes	—
35	C₃×#388	C₂²	1	32	yes	complemented
36	C₃×#382	C₂²	1	8	yes	complemented
37	C₃×C₄×C₂²⋊C₄	C₂²	1	32	yes	complemented
38	C₂×C₃×#95	C₂²	1	4	yes	—
39	C₂×C₃×#95	C₂²	1	4	yes	—
40	C₃×C₄³	C₂²	1	8	yes	complemented
41	C₂²×#124	C₂×C₃	1	1	yes	characteristic
42	C₂²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
44	C₄×#124	C₂×C₃	1	16	yes	direct product factor
45	C₂×#391	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#390	C₂×C₃	1	1	yes	characteristic
47	C₂×#388	C₂×C₃	1	4	yes	complemented
48	C₂×#382	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂×C₄×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
50	C₂×C₄³	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂⁵×C₃	C₂³	1	2	yes	—
52	C₂²×C₃×Q₈	—	2	4	no	—
53	C₂²×C₃×Q₈	—	2	4	no	—
54	C₂²×C₃×Q₈	C₂×C₄	1	8	yes	complemented
55	C₂²×C₃×Q₈	C₂³	1	4	yes	—
56	C₂²×C₃×D₄	—	2	4	no	—
57	C₂²×C₃×D₄	—	2	4	no	—
58	C₂²×C₃×D₄	C₂×C₄	1	8	yes	—
59	C₂²×C₃×D₄	C₂³	1	4	yes	—
60	C₂³×C₃×C₄	—	2	8	no	—
61	C₂³×C₃×C₄	—	2	16	no	—
62	C₂³×C₃×C₄	C₂³	1	2	yes	—
63	C₂³×C₃×C₄	C₂³	1	8	yes	—
64	C₂³×C₃×C₄	C₂³	1	2	yes	—
65	C₂³×C₃×C₄	C₂³	1	2	yes	—
66	C₂³×C₃×C₄	C₂³	1	2	yes	—
67	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₂³×C₃×C₄	C₂³	1	2	yes	—
69	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
71	C₃×#124	C₂×C₄	1	64	yes	direct product factor
72	C₃×C₄×Q₈	—	2	32	no	—
73	C₃×C₄×D₄	—	2	32	no	—
74	C₂×C₃×C₄⋊C₄	—	2	4	no	—
75	C₂×C₃×C₄⋊C₄	—	2	8	no	—
76	C₂×C₃×C₄⋊C₄	—	2	4	no	—
77	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
78	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
79	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
80	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
81	C₂×C₃×C₂²⋊C₄	—	2	16	no	—
82	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	32	yes	complemented
83	C₂×C₃×C₂²⋊C₄	C₂³	1	16	yes	—
84	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
85	C₂×C₃×C₂²⋊C₄	C₂³	1	16	yes	—
86	C₂×C₃×C₄²	—	2	8	no	—
87	C₂×C₃×C₄²	—	2	8	no	—
88	C₂×C₃×C₄²	—	2	4	no	—
89	C₂×C₃×C₄²	—	2	8	no	—
90	C₂×C₃×C₄²	C₂×C₄	1	8	yes	complemented
91	C₂×C₃×C₄²	D₄	1	8	yes	—
92	C₂×C₃×C₄²	C₂³	1	8	yes	—
93	C₂×C₃×C₄²	C₂³	1	8	yes	—
94	C₂×C₃×C₄²	C₂³	1	4	yes	—
95	C₂×C₃×C₄²	C₂³	1	8	yes	—
96	C₂×C₃×C₄²	C₂³	1	4	yes	—
97	C₂×C₃×C₄²	C₂³	1	4	yes	—
98	C₃×#95	C₂³	1	16	yes	—
99	C₃×#95	C₂³	1	16	yes	—
100	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
101	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
102	C₂⁴×C₄	C₂²×C₃	1	2	yes	—
103	C₂×#124	C₃×C₄	1	16	yes	direct product factor
104	C₂×#124	C₂²×C₃	1	8	yes	—
105	C₂×C₄×Q₈	—	2	8	no	—
106	C₂×C₄×Q₈	C₂²×C₃	1	8	yes	complemented
107	C₂×C₄×D₄	—	2	8	no	—
108	C₂×C₄×D₄	C₂²×C₃	1	8	yes	—
109	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
110	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
111	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
112	C₂²×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
113	C₂²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
114	C₂²×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
115	C₂²×C₄²	C₂²×C₃	1	2	yes	—
116	C₂²×C₄²	C₂²×C₃	1	2	yes	—
117	C₂²×C₄²	C₂²×C₃	1	2	yes	—
118	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
119	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
120	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
121	#391	C₂²×C₃	1	8	yes	complemented
122	#390	C₂²×C₃	1	8	yes	—
123	#388	C₂²×C₃	1	32	yes	complemented
124	#382	C₂²×C₃	1	8	yes	complemented
125	C₄×C₂²⋊C₄	C₂²×C₃	1	32	yes	complemented
126	C₂×#95	C₂²×C₃	1	4	yes	—
127	C₂×#95	C₂²×C₃	1	4	yes	—
128	C₄³	C₂²×C₃	1	8	yes	complemented
129	C₂⁴×C₃	—	2	4	no	—
130	C₂⁴×C₃	—	2	8	no	—
131	C₂⁴×C₃	—	2	8	no	—
132	C₂⁴×C₃	—	2	4	no	—
133	C₂⁴×C₃	C₂²×C₄	1	8	yes	—
134	C₂⁴×C₃	C₂⁴	1	4	yes	—
135	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
136	C₂×C₃×Q₈	—	2	32	no	—
137	C₂×C₃×Q₈	—	2	8	no	—
138	C₂×C₃×Q₈	—	2	8	no	—
139	C₂×C₃×Q₈	C₂²×C₄	1	16	yes	complemented
140	C₂×C₃×D₄	—	2	32	no	—
141	C₂×C₃×D₄	—	2	8	no	—
142	C₂×C₃×D₄	—	2	8	no	—
143	C₂×C₃×D₄	C₂²×C₄	1	16	yes	—
144	C₂²×C₃×C₄	—	2	4	no	—
145	C₂²×C₃×C₄	—	2	4	no	—
146	C₂²×C₃×C₄	—	2	4	no	—
147	C₂²×C₃×C₄	—	2	4	no	—
148	C₂²×C₃×C₄	—	2	4	no	—
149	C₂²×C₃×C₄	—	2	4	no	—
150	C₂²×C₃×C₄	—	2	4	no	—
151	C₂²×C₃×C₄	—	2	4	no	—
152	C₂²×C₃×C₄	—	2	4	no	—
153	C₂²×C₃×C₄	—	2	8	no	—
154	C₂²×C₃×C₄	—	2	8	no	—
155	C₂²×C₃×C₄	—	2	16	no	—
156	C₂²×C₃×C₄	—	2	32	no	—
157	C₂²×C₃×C₄	—	4	32	no	—
158	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
159	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
160	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
161	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
162	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
163	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
164	C₂²×C₃×C₄	#41	1	8	yes	—
165	C₂²×C₃×C₄	#41	1	4	yes	—
166	C₂²×C₃×C₄	#41	1	8	yes	—
167	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
168	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
169	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
170	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
171	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
172	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
173	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
174	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
175	C₃×C₄⋊C₄	—	2	8	no	—
176	C₃×C₄⋊C₄	—	2	16	no	—
177	C₃×C₄⋊C₄	—	2	8	no	—
178	C₃×C₂²⋊C₄	—	2	16	no	—
179	C₃×C₂²⋊C₄	—	2	32	no	—
180	C₃×C₂²⋊C₄	C₂²×C₄	1	64	yes	complemented
181	C₃×C₄²	—	2	16	no	—
182	C₃×C₄²	—	2	16	no	—
183	C₃×C₄²	—	2	8	no	—
184	C₃×C₄²	—	2	16	no	—
185	C₃×C₄²	C₂²×C₄	1	16	yes	complemented
186	C₃×C₄²	C₂×D₄	1	16	yes	—
187	C₂⁵	C₂³×C₃	1	2	yes	—
188	C₂²×Q₈	—	2	4	no	—
189	C₂²×Q₈	—	2	4	no	—
190	C₂²×Q₈	C₂×C₃×C₄	1	8	yes	complemented
191	C₂²×Q₈	C₂³×C₃	1	4	yes	—
192	C₂²×D₄	—	2	4	no	—
193	C₂²×D₄	—	2	4	no	—
194	C₂²×D₄	C₂×C₃×C₄	1	8	yes	—
195	C₂²×D₄	C₂³×C₃	1	4	yes	—
196	C₂³×C₄	—	2	8	no	—
197	C₂³×C₄	—	2	16	no	—
198	C₂³×C₄	C₂³×C₃	1	2	yes	—
199	C₂³×C₄	C₂³×C₃	1	2	yes	—
200	C₂³×C₄	C₂³×C₃	1	8	yes	—
201	C₂³×C₄	C₂³×C₃	1	2	yes	—
202	C₂³×C₄	C₂³×C₃	1	2	yes	—
203	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₃	1	2	yes	—
205	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
206	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
207	#124	C₂×C₃×C₄	1	64	yes	direct product factor
208	C₄×Q₈	—	2	32	no	—
209	C₄×D₄	—	2	32	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	8	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
214	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
215	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
216	C₂×C₂²⋊C₄	—	2	8	no	—
217	C₂×C₂²⋊C₄	—	2	16	no	—
218	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	32	yes	complemented
219	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
220	C₂×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
221	C₂×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
222	C₂×C₄²	—	2	8	no	—
223	C₂×C₄²	—	2	8	no	—
224	C₂×C₄²	—	2	4	no	—
225	C₂×C₄²	—	2	8	no	—
226	C₂×C₄²	C₂×C₃×C₄	1	8	yes	complemented
227	C₂×C₄²	C₃×D₄	1	8	yes	—
228	C₂×C₄²	C₂³×C₃	1	8	yes	—
229	C₂×C₄²	C₂³×C₃	1	8	yes	—
230	C₂×C₄²	C₂³×C₃	1	8	yes	—
231	C₂×C₄²	C₂³×C₃	1	4	yes	—
232	C₂×C₄²	C₂³×C₃	1	4	yes	—
233	C₂×C₄²	C₂³×C₃	1	4	yes	—
234	#95	C₂³×C₃	1	16	yes	—
235	#95	C₂³×C₃	1	16	yes	—
236	C₂³×C₃	—	2	16	no	—
237	C₂³×C₃	—	2	32	no	—
238	C₂³×C₃	—	2	4	no	—
239	C₂³×C₃	—	2	8	no	—
240	C₂³×C₃	—	2	8	no	—
241	C₂³×C₃	—	2	4	no	—
242	C₂³×C₃	—	4	8	no	—
243	C₂³×C₃	—	4	16	no	—
244	C₂³×C₃	—	4	8	no	—
245	C₂³×C₃	C₂³×C₄	1	8	yes	—
246	C₂³×C₃	C₂³×C₄	1	2	yes	—
247	C₂³×C₃	C₂²×D₄	1	2	yes	—
248	C₂³×C₃	C₂×#41	1	2	yes	—
249	C₂³×C₃	C₂×#41	1	4	yes	—
250	C₂³×C₃	C₂×#41	1	4	yes	—
251	C₂³×C₃	C₂⁵	1	1	yes	characteristic
252	C₃×Q₈	—	2	32	no	—
253	C₃×D₄	—	2	32	no	—
254	C₂×C₃×C₄	—	2	16	no	—
255	C₂×C₃×C₄	—	2	16	no	—
256	C₂×C₃×C₄	—	2	16	no	—
257	C₂×C₃×C₄	—	2	8	no	—
258	C₂×C₃×C₄	—	2	4	no	—
259	C₂×C₃×C₄	—	2	4	no	—
260	C₂×C₃×C₄	—	2	4	no	—
261	C₂×C₃×C₄	—	2	4	no	—
262	C₂×C₃×C₄	—	2	4	no	—
263	C₂×C₃×C₄	—	2	4	no	—
264	C₂×C₃×C₄	—	2	4	no	—
265	C₂×C₃×C₄	—	2	4	no	—
266	C₂×C₃×C₄	—	2	4	no	—
267	C₂×C₃×C₄	—	2	8	no	—
268	C₂×C₃×C₄	—	2	8	no	—
269	C₂×C₃×C₄	—	4	8	no	—
270	C₂×C₃×C₄	—	4	8	no	—
271	C₂×C₃×C₄	—	4	32	no	—
272	C₂×C₃×C₄	C₄×D₄	1	16	yes	—
273	C₂×C₃×C₄	#124	1	16	yes	direct product factor
274	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
275	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
276	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
277	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
278	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
279	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
280	C₂×C₃×C₄	C₂×#41	1	8	yes	—
281	C₂×C₃×C₄	C₂×#41	1	4	yes	—
282	C₂×C₃×C₄	C₂×#41	1	8	yes	—
283	C₂⁴	—	2	4	no	—
284	C₂⁴	—	2	8	no	—
285	C₂⁴	—	2	8	no	—
286	C₂⁴	—	2	4	no	—
287	C₂⁴	C₂²×C₃×C₄	1	8	yes	—
288	C₂⁴	C₂⁴×C₃	1	4	yes	—
289	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
290	C₂×Q₈	—	2	32	no	—
291	C₂×Q₈	—	2	8	no	—
292	C₂×Q₈	—	2	8	no	—
293	C₂×Q₈	C₂²×C₃×C₄	1	16	yes	complemented
294	C₂×D₄	—	2	32	no	—
295	C₂×D₄	—	2	8	no	—
296	C₂×D₄	—	2	8	no	—
297	C₂×D₄	C₂²×C₃×C₄	1	16	yes	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	—	2	4	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	4	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	4	no	—
307	C₂²×C₄	—	2	8	no	—
308	C₂²×C₄	—	2	8	no	—
309	C₂²×C₄	—	2	16	no	—
310	C₂²×C₄	—	2	32	no	—
311	C₂²×C₄	—	4	32	no	—
312	C₂²×C₄	C₂²×C₃×C₄	1	8	yes	—
313	C₂²×C₄	C₂²×C₃×C₄	1	8	yes	—
314	C₂²×C₄	C₂²×C₃×C₄	1	4	yes	—
315	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
316	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
317	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
318	C₂²×C₄	C₃×#41	1	8	yes	—
319	C₂²×C₄	C₃×#41	1	4	yes	—
320	C₂²×C₄	C₃×#41	1	8	yes	—
321	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
322	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
323	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
324	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
325	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
326	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
327	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
328	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	16	no	—
331	C₄⋊C₄	—	2	8	no	—
332	C₂²⋊C₄	—	2	16	no	—
333	C₂²⋊C₄	—	2	32	no	—
334	C₂²⋊C₄	C₂²×C₃×C₄	1	64	yes	complemented
335	C₄²	—	2	16	no	—
336	C₄²	—	2	16	no	—
337	C₄²	—	2	8	no	—
338	C₄²	—	2	16	no	—
339	C₄²	C₂²×C₃×C₄	1	16	yes	complemented
340	C₄²	C₂×C₃×D₄	1	16	yes	—
341	C₂²×C₃	—	2	8	no	—
342	C₂²×C₃	—	2	16	no	—
343	C₂²×C₃	—	4	4	no	—
344	C₂²×C₃	—	4	8	no	—
345	C₂²×C₃	—	4	4	no	—
346	C₂²×C₃	—	4	32	no	—
347	C₂²×C₃	C₂×C₄×D₄	1	4	yes	—
348	C₂²×C₃	C₄×#41	1	8	yes	—
349	C₂²×C₃	C₂×#124	1	4	yes	—
350	C₂²×C₃	#529	1	8	yes	—
351	C₂²×C₃	#532	1	4	yes	—
352	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
353	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
354	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
355	C₂²×C₃	C₂²×#41	1	2	yes	—
356	C₂²×C₃	C₂²×#41	1	2	yes	—
357	C₃×C₄	—	2	8	no	—
358	C₃×C₄	—	2	8	no	—
359	C₃×C₄	—	2	8	no	—
360	C₃×C₄	—	2	4	no	—
361	C₃×C₄	—	4	4	no	—
362	C₃×C₄	—	4	4	no	—
363	C₃×C₄	C₂×C₄×D₄	1	8	yes	—
364	C₃×C₄	C₂×#124	1	8	yes	direct product factor
365	C₂³	—	2	4	no	—
366	C₂³	—	2	8	no	—
367	C₂³	—	2	8	no	—
368	C₂³	—	2	4	no	—
369	C₂³	—	2	16	no	—
370	C₂³	—	2	32	no	—
371	C₂³	—	4	8	no	—
372	C₂³	—	4	16	no	—
373	C₂³	—	4	8	no	—
374	C₂³	C₂³×C₃×C₄	1	8	yes	—
375	C₂³	C₂³×C₃×C₄	1	2	yes	—
376	C₂³	C₂²×C₃×D₄	1	2	yes	—
377	C₂³	C₂×C₃×#41	1	2	yes	—
378	C₂³	C₂×C₃×#41	1	4	yes	—
379	C₂³	C₂×C₃×#41	1	4	yes	—
380	C₂³	C₂⁵×C₃	1	1	yes	characteristic
381	Q₈	—	2	32	no	—
382	D₄	—	2	32	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	2	4	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	16	no	—
391	C₂×C₄	—	2	16	no	—
392	C₂×C₄	—	2	16	no	—
393	C₂×C₄	—	2	8	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	8	no	—
397	C₂×C₄	—	2	8	no	—
398	C₂×C₄	—	4	8	no	—
399	C₂×C₄	—	4	8	no	—
400	C₂×C₄	—	4	32	no	—
401	C₂×C₄	C₃×C₄×D₄	1	16	yes	—
402	C₂×C₄	C₃×#124	1	16	yes	direct product factor
403	C₂×C₄	C₂³×C₃×C₄	1	8	yes	—
404	C₂×C₄	C₂³×C₃×C₄	1	8	yes	—
405	C₂×C₄	C₂³×C₃×C₄	1	4	yes	—
406	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
407	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
408	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
409	C₂×C₄	C₂×C₃×#41	1	8	yes	—
410	C₂×C₄	C₂×C₃×#41	1	4	yes	—
411	C₂×C₄	C₂×C₃×#41	1	8	yes	—
412	C₂×C₃	—	4	8	no	—
413	C₂×C₃	C₄×#124	1	8	yes	direct product factor
414	C₂×C₃	C₂²×C₄×D₄	1	1	yes	characteristic
415	C₂×C₃	C₂×C₄×#41	1	2	yes	—
416	C₂×C₃	C₂²×#124	1	1	yes	characteristic
417	C₂×C₃	C₂×#529	1	2	yes	—
418	C₂×C₃	C₂×#532	1	1	yes	characteristic
419	C₂²	—	2	8	no	—
420	C₂²	—	2	16	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	4	8	no	—
423	C₂²	—	4	4	no	—
424	C₂²	—	4	32	no	—
425	C₂²	C₂×C₃×C₄×D₄	1	4	yes	—
426	C₂²	C₃×C₄×#41	1	8	yes	—
427	C₂²	C₂×C₃×#124	1	4	yes	—
428	C₂²	C₃×#529	1	8	yes	—
429	C₂²	C₃×#532	1	4	yes	—
430	C₂²	C₂⁴×C₃×C₄	1	1	yes	characteristic
431	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
432	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
433	C₂²	C₂²×C₃×#41	1	2	yes	—
434	C₂²	C₂²×C₃×#41	1	2	yes	—
435	C₄	—	2	8	no	—
436	C₄	—	2	8	no	—
437	C₄	—	2	8	no	—
438	C₄	—	2	4	no	—
439	C₄	—	4	4	no	—
440	C₄	—	4	4	no	—
441	C₄	C₂×C₃×C₄×D₄	1	8	yes	—
442	C₄	C₂×C₃×#124	1	8	yes	direct product factor
443	C₃	C₂×C₄×#124	1	1	yes	characteristic, direct product factor, Sylow
444	C₂	—	4	8	no	—
445	C₂	C₃×C₄×#124	1	8	yes	direct product factor
446	C₂	C₂²×C₃×C₄×D₄	1	1	yes	characteristic
447	C₂	C₂×C₃×C₄×#41	1	2	yes	—
448	C₂	C₂²×C₃×#124	1	1	yes	characteristic
449	C₂	C₂×C₃×#529	1	2	yes	—
450	C₂	C₂×C₃×#532	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₄	C₂³
commutator	C₂²	C₂⁴×C₃×C₄
Frattini, cosocle	C₂³	C₂⁵×C₃
socle	C₂⁴×C₃	C₂⁴
prime radical	C₂³×C₃×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₄×Q₈	C₂	semidirect
4. C₃×C₄×#124	C₂	direct (N × H)
5. C₂×C₃×#391	C₂	semidirect
7. C₂×C₃×#388	C₂	semidirect
8. C₂×C₃×#382	C₂	semidirect
9. C₂×C₃×C₄×C₂²⋊C₄	C₂	semidirect
10. C₂×C₃×C₄³	C₂	semidirect
11. C₂×C₄×#124	C₃	direct (N × H)
15. C₂×C₃×#124	C₄	direct (N × H)
18. C₂×C₃×C₄×Q₈	C₂²	semidirect
33. C₃×#391	C₂²	semidirect
35. C₃×#388	C₂²	semidirect
36. C₃×#382	C₂²	semidirect
37. C₃×C₄×C₂²⋊C₄	C₂²	semidirect
40. C₃×C₄³	C₂²	semidirect
42. C₂²×C₄×Q₈	C₂×C₃	semidirect
44. C₄×#124	C₂×C₃	direct (N × H)
45. C₂×#391	C₂×C₃	semidirect
47. C₂×#388	C₂×C₃	semidirect
48. C₂×#382	C₂×C₃	semidirect
49. C₂×C₄×C₂²⋊C₄	C₂×C₃	semidirect
50. C₂×C₄³	C₂×C₃	semidirect
54. C₂²×C₃×Q₈	C₂×C₄	semidirect
71. C₃×#124	C₂×C₄	direct (N × H)
82. C₂×C₃×C₂²⋊C₄	C₂×C₄	semidirect
90. C₂×C₃×C₄²	C₂×C₄	semidirect
103. C₂×#124	C₃×C₄	direct (N × H)
106. C₂×C₄×Q₈	C₂²×C₃	semidirect
121. #391	C₂²×C₃	semidirect
123. #388	C₂²×C₃	semidirect
124. #382	C₂²×C₃	semidirect
125. C₄×C₂²⋊C₄	C₂²×C₃	semidirect
128. C₄³	C₂²×C₃	semidirect
139. C₂×C₃×Q₈	C₂²×C₄	semidirect
180. C₃×C₂²⋊C₄	C₂²×C₄	semidirect
185. C₃×C₄²	C₂²×C₄	semidirect
190. C₂²×Q₈	C₂×C₃×C₄	semidirect
207. #124	C₂×C₃×C₄	direct (N × H)
218. C₂×C₂²⋊C₄	C₂×C₃×C₄	semidirect
226. C₂×C₄²	C₂×C₃×C₄	semidirect
273. C₂×C₃×C₄	#124	direct (N × H)
293. C₂×Q₈	C₂²×C₃×C₄	semidirect
334. C₂²⋊C₄	C₂²×C₃×C₄	semidirect
339. C₄²	C₂²×C₃×C₄	semidirect
364. C₃×C₄	C₂×#124	direct (N × H)
402. C₂×C₄	C₃×#124	direct (N × H)
413. C₂×C₃	C₄×#124	direct (N × H)
442. C₄	C₂×C₃×#124	direct (N × H)
443. C₃	C₂×C₄×#124	direct (N × H)
445. C₂	C₃×C₄×#124	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×C₃×#124	2
C₂²×C₃×C₄×Q₈	2
C₂²×C₃×C₄×D₄	2
C₃×C₄×#124	2
C₂×C₃×#391	2
C₂×C₃×#390	2
C₂×C₃×#388	2
C₂×C₃×#382	2
C₂×C₃×C₄×C₂²⋊C₄	2
C₂×C₃×C₄³	2
C₂×C₄×#124	3

Subgroups — C2×C3×C4×#124 #10652639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×#124 #10652639