Subgroups — C₂×C₃²×#478 #11770121

Order 1152.

1344 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	C₂×C₃²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃²×#478	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃²×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃²×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃²×#102	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃²×#102	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃×#478	C₃	1	4	yes	maximal, direct product factor
10	C₂²×C₃²×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃²×C₄	—	2	1	no	—
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃²×#125	—	2	2	no	—
15	C₃²×#125	C₂²	1	4	yes	complemented
16	C₃²×C₄×D₄	—	2	2	no	—
17	C₃²×C₄×D₄	C₂²	1	4	yes	—
18	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
19	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
20	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃²×C₂²⋊C₄	—	2	1	no	—
24	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
25	C₃²×#107	C₂²	1	4	yes	complemented
26	C₃²×#106	C₂²	1	4	yes	complemented
27	C₃²×C₄⋊C₈	C₂²	1	4	yes	complemented
28	C₃²×#102	C₂²	1	4	yes	—
29	C₃²×#102	C₂²	1	4	yes	—
30	C₂×C₃×#125	C₂×C₃	1	4	yes	complemented
31	C₂×C₃×C₄×D₄	C₂×C₃	1	4	yes	—
32	C₃×#478	C₂×C₃	1	32	yes	direct product factor
33	C₂×C₃×#107	C₂×C₃	1	4	yes	complemented
34	C₂×C₃×#106	C₂×C₃	1	4	yes	complemented
35	C₂×C₃×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
36	C₂×C₃×#102	C₂×C₃	1	4	yes	—
37	C₂×C₃×#102	C₂×C₃	1	4	yes	—
38	C₂⁴×C₃²	—	2	1	no	—
39	C₂×C₃²×D₄	—	2	1	no	—
40	C₂×C₃²×D₄	—	2	2	no	—
41	C₂×C₃²×D₄	Q₈	1	4	yes	—
42	C₂×C₃²×D₄	C₂³	1	2	yes	—
43	C₂²×C₃²×C₄	—	2	1	no	—
44	C₂²×C₃²×C₄	—	2	1	no	—
45	C₂²×C₃²×C₄	—	2	1	no	—
46	C₂²×C₃²×C₄	—	2	4	no	—
47	C₂²×C₃²×C₄	—	4	4	no	—
48	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
49	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
50	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃²×C₈	—	2	2	no	—
52	C₂×C₃²×C₈	—	2	2	no	—
53	C₂×C₃²×C₈	C₂³	1	2	yes	—
54	C₂×C₃²×C₈	C₂³	1	2	yes	—
55	C₃²×C₄⋊C₄	—	2	4	no	—
56	C₃²×C₄⋊C₄	—	2	4	no	—
57	C₃²×C₄⋊C₄	—	2	1	no	—
58	C₃²×C₄⋊C₄	—	2	1	no	—
59	C₃²×C₄⋊C₄	—	2	1	no	—
60	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
61	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
62	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
63	C₃²×C₂²⋊C₄	—	2	4	no	—
64	C₃²×C₄²	D₄	1	2	yes	—
65	C₃²×C₄²	C₂³	1	2	yes	—
66	C₂×#478	C₃²	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
68	C₂³×C₃×C₄	—	2	4	no	—
69	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
70	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
71	C₃×#125	—	2	8	no	—
72	C₃×#125	C₂²×C₃	1	16	yes	complemented
73	C₃×C₄×D₄	—	2	8	no	—
74	C₃×C₄×D₄	C₂²×C₃	1	16	yes	—
75	C₂×C₃×C₄⋊C₄	—	2	4	no	—
76	C₂×C₃×C₄⋊C₄	—	2	4	no	—
77	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
80	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
81	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
82	C₃×#107	C₂²×C₃	1	16	yes	complemented
83	C₃×#106	C₂²×C₃	1	16	yes	complemented
84	C₃×C₄⋊C₈	C₂²×C₃	1	16	yes	complemented
85	C₃×#102	C₂²×C₃	1	16	yes	—
86	C₃×#102	C₂²×C₃	1	16	yes	—
87	C₂³×C₃²	—	2	4	no	—
88	C₂³×C₃²	—	2	2	no	—
89	C₂³×C₃²	—	4	2	no	—
90	C₂³×C₃²	—	4	2	no	—
91	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
92	C₃²×D₄	—	2	4	no	—
93	C₃²×D₄	—	2	2	no	—
94	C₃²×D₄	C₂×Q₈	1	4	yes	—
95	C₂×C₃²×C₄	—	2	2	no	—
96	C₂×C₃²×C₄	—	2	2	no	—
97	C₂×C₃²×C₄	—	2	2	no	—
98	C₂×C₃²×C₄	—	2	2	no	—
99	C₂×C₃²×C₄	—	2	2	no	—
100	C₂×C₃²×C₄	—	4	8	no	—
101	C₂×C₃²×C₄	—	4	2	no	—
102	C₂×C₃²×C₄	—	4	2	no	—
103	C₂×C₃²×C₄	—	4	2	no	—
104	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
105	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
106	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
108	C₂×C₃²×C₄	#41	1	2	yes	—
109	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
110	C₃²×C₈	—	2	2	no	—
111	C₃²×C₈	—	2	2	no	—
112	C₂×#125	C₂×C₃²	1	1	yes	characteristic, complemented
113	C₂×C₄×D₄	C₂×C₃²	1	1	yes	characteristic
114	#478	C₂×C₃²	1	8	yes	direct product factor
115	C₂×#107	C₂×C₃²	1	1	yes	characteristic, complemented
116	C₂×#106	C₂×C₃²	1	1	yes	characteristic, complemented
117	C₂×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
118	C₂×#102	C₂×C₃²	1	1	yes	characteristic
119	C₂×#102	C₂×C₃²	1	1	yes	characteristic
120	C₂⁴×C₃	—	2	4	no	—
121	C₂×C₃×D₄	—	2	4	no	—
122	C₂×C₃×D₄	—	2	8	no	—
123	C₂×C₃×D₄	C₃×Q₈	1	16	yes	—
124	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
125	C₂²×C₃×C₄	—	2	4	no	—
126	C₂²×C₃×C₄	—	2	4	no	—
127	C₂²×C₃×C₄	—	2	4	no	—
128	C₂²×C₃×C₄	—	2	16	no	—
129	C₂²×C₃×C₄	—	4	16	no	—
130	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
131	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
132	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
133	C₂×C₃×C₈	—	2	8	no	—
134	C₂×C₃×C₈	—	2	8	no	—
135	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
136	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
137	C₃×C₄⋊C₄	—	2	16	no	—
138	C₃×C₄⋊C₄	—	2	16	no	—
139	C₃×C₄⋊C₄	—	2	4	no	—
140	C₃×C₄⋊C₄	—	2	4	no	—
141	C₃×C₄⋊C₄	—	2	4	no	—
142	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
143	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
144	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
145	C₃×C₂²⋊C₄	—	2	16	no	—
146	C₃×C₄²	C₃×D₄	1	8	yes	—
147	C₃×C₄²	C₂³×C₃	1	8	yes	—
148	C₂²×C₃²	—	2	4	no	—
149	C₂²×C₃²	—	4	2	no	—
150	C₂²×C₃²	—	4	2	no	—
151	C₂²×C₃²	—	4	8	no	—
152	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
153	C₂²×C₃²	#135	1	2	yes	—
154	C₂²×C₃²	AGL(1,ℤ/8)	1	2	yes	—
155	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
156	C₃²×C₄	—	2	2	no	—
157	C₃²×C₄	—	2	2	no	—
158	C₃²×C₄	—	4	2	no	—
159	C₃²×C₄	—	4	2	no	—
160	C₃²×C₄	—	4	2	no	—
161	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
162	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
163	C₃²×C₄	C₂×#41	1	1	yes	characteristic
164	C₂²×D₄	C₂²×C₃²	1	1	yes	characteristic
165	C₂³×C₄	—	2	1	no	—
166	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
167	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
168	#125	—	2	2	no	—
169	#125	C₂²×C₃²	1	4	yes	complemented
170	C₄×D₄	—	2	2	no	—
171	C₄×D₄	C₂²×C₃²	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
177	C₂×C₂²⋊C₄	—	2	1	no	—
178	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
179	#107	C₂²×C₃²	1	4	yes	complemented
180	#106	C₂²×C₃²	1	4	yes	complemented
181	C₄⋊C₈	C₂²×C₃²	1	4	yes	complemented
182	#102	C₂²×C₃²	1	4	yes	—
183	#102	C₂²×C₃²	1	4	yes	—
184	C₂³×C₃	—	2	16	no	—
185	C₂³×C₃	—	2	8	no	—
186	C₂³×C₃	—	4	8	no	—
187	C₂³×C₃	—	4	8	no	—
188	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
189	C₃×D₄	—	2	16	no	—
190	C₃×D₄	—	2	8	no	—
191	C₃×D₄	C₂×C₃×Q₈	1	16	yes	—
192	C₂×C₃×C₄	—	2	8	no	—
193	C₂×C₃×C₄	—	2	8	no	—
194	C₂×C₃×C₄	—	2	8	no	—
195	C₂×C₃×C₄	—	2	8	no	—
196	C₂×C₃×C₄	—	2	8	no	—
197	C₂×C₃×C₄	—	4	32	no	—
198	C₂×C₃×C₄	—	4	8	no	—
199	C₂×C₃×C₄	—	4	8	no	—
200	C₂×C₃×C₄	—	4	8	no	—
201	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
202	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
203	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
204	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
205	C₂×C₃×C₄	C₃×#41	1	8	yes	—
206	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
207	C₃×C₈	—	2	8	no	—
208	C₃×C₈	—	2	8	no	—
209	C₂×C₃²	—	4	4	no	—
210	C₂×C₃²	#478	1	4	yes	direct product factor
211	C₂×C₃²	C₂×C₂²⋊Q₈	1	1	yes	characteristic
212	C₂×C₃²	C₂×#135	1	1	yes	characteristic
213	C₂×C₃²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₂⁴	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	C₃²×Q₈	1	4	yes	—
218	C₂×D₄	C₂³×C₃²	1	2	yes	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
226	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	C₂³×C₃²	1	2	yes	—
230	C₂×C₈	C₂³×C₃²	1	2	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
237	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
238	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₄²	C₃²×D₄	1	2	yes	—
241	C₄²	C₂³×C₃²	1	2	yes	—
242	C₂²×C₃	—	2	16	no	—
243	C₂²×C₃	—	4	8	no	—
244	C₂²×C₃	—	4	8	no	—
245	C₂²×C₃	—	4	32	no	—
246	C₂²×C₃	C₃×C₂²⋊Q₈	1	8	yes	—
247	C₂²×C₃	C₃×#135	1	8	yes	—
248	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	8	yes	—
249	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
250	C₃×C₄	—	2	8	no	—
251	C₃×C₄	—	2	8	no	—
252	C₃×C₄	—	4	8	no	—
253	C₃×C₄	—	4	8	no	—
254	C₃×C₄	—	4	8	no	—
255	C₃×C₄	C₃×C₂²⋊Q₈	1	8	yes	—
256	C₃×C₄	C₂²×C₃×Q₈	1	4	yes	—
257	C₃×C₄	C₂×C₃×#41	1	4	yes	—
258	C₃²	C₂×#478	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³	—	2	2	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
264	D₄	—	2	2	no	—
265	D₄	—	2	4	no	—
266	D₄	C₂×C₃²×Q₈	1	4	yes	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	8	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
277	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
278	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₃²×Q₈	1	2	yes	—
280	C₂×C₄	C₃²×#41	1	2	yes	—
281	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	2	2	no	—
284	C₂×C₃	—	4	16	no	—
285	C₂×C₃	C₃×#478	1	16	yes	direct product factor
286	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	4	yes	—
287	C₂×C₃	C₂×C₃×#135	1	4	yes	—
288	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	4	yes	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	8	no	—
293	C₂²	C₃²×C₂²⋊Q₈	1	2	yes	—
294	C₂²	C₃²×#135	1	2	yes	—
295	C₂²	C₃²×AGL(1,ℤ/8)	1	2	yes	—
296	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	C₃²×C₂²⋊Q₈	1	2	yes	—
303	C₄	C₂²×C₃²×Q₈	1	1	yes	characteristic
304	C₄	C₂×C₃²×#41	1	1	yes	characteristic
305	C₃	C₂×C₃×#478	1	4	yes	direct product factor
306	C₂	—	4	4	no	—
307	C₂	C₃²×#478	1	4	yes	direct product factor
308	C₂	C₂×C₃²×C₂²⋊Q₈	1	1	yes	characteristic
309	C₂	C₂×C₃²×#135	1	1	yes	characteristic
310	C₂	C₂×C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
prime radical	C₂²×C₃²×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ C₂×C₃²×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×#125	C₂	semidirect
3. C₃²×#478	C₂	direct (N × H)
4. C₂×C₃²×#107	C₂	semidirect
5. C₂×C₃²×#106	C₂	semidirect
6. C₂×C₃²×C₄⋊C₈	C₂	semidirect
9. C₂×C₃×#478	C₃	direct (N × H)
15. C₃²×#125	C₂²	semidirect
25. C₃²×#107	C₂²	semidirect
26. C₃²×#106	C₂²	semidirect
27. C₃²×C₄⋊C₈	C₂²	semidirect
30. C₂×C₃×#125	C₂×C₃	semidirect
32. C₃×#478	C₂×C₃	direct (N × H)
33. C₂×C₃×#107	C₂×C₃	semidirect
34. C₂×C₃×#106	C₂×C₃	semidirect
35. C₂×C₃×C₄⋊C₈	C₂×C₃	semidirect
66. C₂×#478	C₃²	direct (N × H)
72. C₃×#125	C₂²×C₃	semidirect
82. C₃×#107	C₂²×C₃	semidirect
83. C₃×#106	C₂²×C₃	semidirect
84. C₃×C₄⋊C₈	C₂²×C₃	semidirect
112. C₂×#125	C₂×C₃²	semidirect
114. #478	C₂×C₃²	direct (N × H)
115. C₂×#107	C₂×C₃²	semidirect
116. C₂×#106	C₂×C₃²	semidirect
117. C₂×C₄⋊C₈	C₂×C₃²	semidirect
169. #125	C₂²×C₃²	semidirect
179. #107	C₂²×C₃²	semidirect
180. #106	C₂²×C₃²	semidirect
181. C₄⋊C₈	C₂²×C₃²	semidirect
210. C₂×C₃²	#478	direct (N × H)
258. C₃²	C₂×#478	direct (N × H)
285. C₂×C₃	C₃×#478	direct (N × H)
305. C₃	C₂×C₃×#478	direct (N × H)
307. C₂	C₃²×#478	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#125	2
C₂×C₃²×C₄×D₄	2
C₃²×#478	2
C₂×C₃²×#107	2
C₂×C₃²×#106	2
C₂×C₃²×C₄⋊C₈	2
C₂×C₃²×#102	2
C₂×C₃×#478	3

Subgroups — C2×C32×#478 #11770121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#478 #11770121