Subgroups — C₂×C₃²×#472 #11770114

Order 1152.

1464 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃²×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#472	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#100	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#472	C₃	1	4	yes	maximal, direct product factor
8	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×D₄	C₂²	1	2	yes	—
10	C₃²×AGL(1,ℤ/8)	—	2	4	no	—
11	C₃²×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
12	C₂×C₃²×SD₁₆	—	2	2	no	—
13	C₂×C₃²×D₈	—	2	2	no	—
14	C₂×C₃²×D₈	—	2	1	no	—
15	C₂×C₃²×D₈	—	2	1	no	—
16	C₂×C₃²×D₈	—	2	2	no	—
17	C₂×C₃²×D₈	C₂²	1	4	yes	complemented
18	C₃²×#131	—	2	2	no	—
19	C₃²×#131	C₂²	1	4	yes	complemented
20	C₂×C₃²×M₁₆	—	2	1	no	—
21	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
23	C₂²×C₃²×C₈	—	2	1	no	—
24	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
25	C₃²×#108	C₂²	1	4	yes	complemented
26	C₃²×#100	C₂²	1	8	yes	complemented
27	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	1	8	yes	complemented
28	C₂²×C₃×D₈	C₂×C₃	1	4	yes	complemented
29	C₂×C₃×#131	C₂×C₃	1	4	yes	complemented
30	C₃×#472	C₂×C₃	1	32	yes	direct product factor
31	C₂×C₃×#108	C₂×C₃	1	4	yes	complemented
32	C₂×C₃×#100	C₂×C₃	1	8	yes	complemented
33	C₂⁴×C₃²	—	2	2	no	—
34	C₃²×#41	—	2	1	no	—
35	C₃²×#41	—	2	1	no	—
36	C₃²×#41	D₄	1	2	yes	—
37	C₃²×#41	C₂³	1	2	yes	—
38	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
39	C₂×C₃²×D₄	—	2	1	no	—
40	C₂×C₃²×D₄	—	2	2	no	—
41	C₂×C₃²×D₄	—	2	2	no	—
42	C₂×C₃²×D₄	—	2	2	no	—
43	C₂×C₃²×D₄	—	2	2	no	—
44	C₂×C₃²×D₄	—	2	2	no	—
45	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
46	C₂×C₃²×D₄	C₂³	1	4	yes	—
47	C₂²×C₃²×C₄	—	2	1	no	—
48	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₃²×SD₁₆	—	2	8	no	—
50	C₃²×D₈	—	2	8	no	—
51	C₃²×D₈	—	2	4	no	—
52	C₃²×D₈	—	2	4	no	—
53	C₃²×M₁₆	—	2	1	no	—
54	C₃²×M₁₆	—	2	2	no	—
55	C₃²×M₁₆	—	2	4	no	—
56	C₃²×M₁₆	C₂³	1	2	yes	—
57	C₃²×M₁₆	C₂³	1	4	yes	—
58	C₂×C₃²×C₈	—	2	1	no	—
59	C₂×C₃²×C₈	—	2	2	no	—
60	C₂×C₃²×C₈	—	2	4	no	—
61	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
62	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
63	C₂×C₃²×C₈	D₄	1	2	yes	—
64	C₂×C₃²×C₈	C₂³	1	2	yes	—
65	C₂×#472	C₃²	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃×#41	C₂²×C₃	1	4	yes	—
67	C₂²×C₃×D₄	C₂²×C₃	1	8	yes	—
68	C₃×AGL(1,ℤ/8)	—	2	16	no	—
69	C₃×AGL(1,ℤ/8)	C₂²×C₃	1	32	yes	complemented
70	C₂×C₃×SD₁₆	—	2	8	no	—
71	C₂×C₃×D₈	—	2	8	no	—
72	C₂×C₃×D₈	—	2	4	no	—
73	C₂×C₃×D₈	—	2	4	no	—
74	C₂×C₃×D₈	—	2	8	no	—
75	C₂×C₃×D₈	C₂²×C₃	1	16	yes	complemented
76	C₃×#131	—	2	8	no	—
77	C₃×#131	C₂²×C₃	1	16	yes	complemented
78	C₂×C₃×M₁₆	—	2	4	no	—
79	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
80	C₂×C₃×M₁₆	C₂²×C₃	1	8	yes	—
81	C₂²×C₃×C₈	—	2	4	no	—
82	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
83	C₃×#108	C₂²×C₃	1	16	yes	complemented
84	C₃×#100	C₂²×C₃	1	32	yes	complemented
85	C₂³×C₃²	—	2	1	no	—
86	C₂³×C₃²	—	2	4	no	—
87	C₂³×C₃²	—	4	2	no	—
88	C₂³×C₃²	—	4	2	no	—
89	C₂³×C₃²	—	8	4	no	—
90	C₂³×C₃²	#41	1	1	yes	characteristic
91	C₃²×Q₈	—	2	1	no	—
92	C₃²×Q₈	C₂×D₄	1	2	yes	—
93	C₃²×D₄	—	2	2	no	—
94	C₃²×D₄	—	2	2	no	—
95	C₃²×D₄	—	2	1	no	—
96	C₃²×D₄	—	2	4	no	—
97	C₃²×D₄	—	2	4	no	—
98	C₃²×D₄	—	4	2	no	—
99	C₃²×D₄	—	4	2	no	—
100	C₃²×D₄	C₂×D₄	1	2	yes	—
101	C₂×C₃²×C₄	—	2	1	no	—
102	C₂×C₃²×C₄	—	2	2	no	—
103	C₂×C₃²×C₄	—	2	2	no	—
104	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃²×C₄	#41	1	1	yes	characteristic
109	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
110	C₃²×C₈	—	2	2	no	—
111	C₃²×C₈	—	2	4	no	—
112	C₃²×C₈	C₂×D₄	1	2	yes	—
113	C₃²×C₈	C₂×D₄	1	2	yes	—
114	C₂×AGL(1,ℤ/8)	C₂×C₃²	1	2	yes	complemented
115	C₂²×D₈	C₂×C₃²	1	1	yes	characteristic, complemented
116	C₂×#131	C₂×C₃²	1	1	yes	characteristic, complemented
117	#472	C₂×C₃²	1	8	yes	direct product factor
118	C₂×#108	C₂×C₃²	1	1	yes	characteristic, complemented
119	C₂×#100	C₂×C₃²	1	2	yes	complemented
120	C₂⁴×C₃	—	2	8	no	—
121	C₃×#41	—	2	4	no	—
122	C₃×#41	—	2	4	no	—
123	C₃×#41	C₃×D₄	1	8	yes	—
124	C₃×#41	C₂³×C₃	1	8	yes	—
125	C₂×C₃×Q₈	C₃×D₄	1	4	yes	—
126	C₂×C₃×D₄	—	2	4	no	—
127	C₂×C₃×D₄	—	2	8	no	—
128	C₂×C₃×D₄	—	2	8	no	—
129	C₂×C₃×D₄	—	2	8	no	—
130	C₂×C₃×D₄	—	2	8	no	—
131	C₂×C₃×D₄	—	2	8	no	—
132	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
133	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
134	C₂²×C₃×C₄	—	2	4	no	—
135	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
136	C₃×SD₁₆	—	2	32	no	—
137	C₃×D₈	—	2	32	no	—
138	C₃×D₈	—	2	16	no	—
139	C₃×D₈	—	2	16	no	—
140	C₃×M₁₆	—	2	4	no	—
141	C₃×M₁₆	—	2	8	no	—
142	C₃×M₁₆	—	2	16	no	—
143	C₃×M₁₆	C₂³×C₃	1	8	yes	—
144	C₃×M₁₆	C₂³×C₃	1	16	yes	—
145	C₂×C₃×C₈	—	2	4	no	—
146	C₂×C₃×C₈	—	2	8	no	—
147	C₂×C₃×C₈	—	2	16	no	—
148	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
149	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
150	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
151	C₂×C₃×C₈	C₂³×C₃	1	8	yes	—
152	C₂²×C₃²	—	2	2	no	—
153	C₂²×C₃²	—	2	2	no	—
154	C₂²×C₃²	—	4	2	no	—
155	C₂²×C₃²	—	4	4	no	—
156	C₂²×C₃²	—	8	4	no	—
157	C₂²×C₃²	—	8	4	no	—
158	C₂²×C₃²	—	8	4	no	—
159	C₂²×C₃²	#121	1	1	yes	characteristic
160	C₂²×C₃²	#121	1	1	yes	characteristic
161	C₂²×C₃²	C₂×#41	1	1	yes	characteristic
162	C₃²×C₄	—	2	2	no	—
163	C₃²×C₄	#121	1	1	yes	characteristic
164	C₃²×C₄	#121	1	1	yes	characteristic
165	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
166	C₃²×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₂×#41	C₂²×C₃²	1	1	yes	characteristic
168	C₂²×D₄	C₂²×C₃²	1	2	yes	—
169	AGL(1,ℤ/8)	—	2	4	no	—
170	AGL(1,ℤ/8)	C₂²×C₃²	1	8	yes	complemented
171	C₂×SD₁₆	—	2	2	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	2	no	—
176	C₂×D₈	C₂²×C₃²	1	4	yes	complemented
177	#131	—	2	2	no	—
178	#131	C₂²×C₃²	1	4	yes	complemented
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	C₂²×C₃²	1	1	yes	characteristic
181	C₂×M₁₆	C₂²×C₃²	1	2	yes	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
184	#108	C₂²×C₃²	1	4	yes	complemented
185	#100	C₂²×C₃²	1	8	yes	complemented
186	C₂³×C₃	—	2	4	no	—
187	C₂³×C₃	—	2	16	no	—
188	C₂³×C₃	—	4	8	no	—
189	C₂³×C₃	—	4	8	no	—
190	C₂³×C₃	—	8	16	no	—
191	C₂³×C₃	C₃×#41	1	4	yes	—
192	C₃×Q₈	—	2	4	no	—
193	C₃×Q₈	C₂×C₃×D₄	1	8	yes	—
194	C₃×D₄	—	2	8	no	—
195	C₃×D₄	—	2	8	no	—
196	C₃×D₄	—	2	4	no	—
197	C₃×D₄	—	2	16	no	—
198	C₃×D₄	—	2	16	no	—
199	C₃×D₄	—	4	8	no	—
200	C₃×D₄	—	4	8	no	—
201	C₃×D₄	C₂×C₃×D₄	1	8	yes	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	8	no	—
204	C₂×C₃×C₄	—	2	8	no	—
205	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
206	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
207	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
208	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
209	C₂×C₃×C₄	C₃×#41	1	4	yes	—
210	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
211	C₃×C₈	—	2	8	no	—
212	C₃×C₈	—	2	16	no	—
213	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
214	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
215	C₂×C₃²	—	2	1	no	—
216	C₂×C₃²	—	2	1	no	—
217	C₂×C₃²	—	4	2	no	—
218	C₂×C₃²	—	8	4	no	—
219	C₂×C₃²	#472	1	2	yes	direct product factor
220	C₂×C₃²	C₂×#121	1	1	yes	characteristic
221	C₂⁴	—	2	2	no	—
222	#41	—	2	1	no	—
223	#41	—	2	1	no	—
224	#41	C₃²×D₄	1	2	yes	—
225	#41	C₂³×C₃²	1	2	yes	—
226	C₂×Q₈	C₃²×D₄	1	1	yes	characteristic
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	C₃²×D₄	1	1	yes	characteristic
234	C₂×D₄	C₂³×C₃²	1	4	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
237	SD₁₆	—	2	8	no	—
238	D₈	—	2	8	no	—
239	D₈	—	2	4	no	—
240	D₈	—	2	4	no	—
241	M₁₆	—	2	1	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	C₂³×C₃²	1	2	yes	—
245	M₁₆	C₂³×C₃²	1	4	yes	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
250	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
251	C₂×C₈	C₃²×D₄	1	2	yes	—
252	C₂×C₈	C₂³×C₃²	1	2	yes	—
253	C₂²×C₃	—	2	8	no	—
254	C₂²×C₃	—	2	8	no	—
255	C₂²×C₃	—	4	8	no	—
256	C₂²×C₃	—	4	16	no	—
257	C₂²×C₃	—	8	16	no	—
258	C₂²×C₃	—	8	16	no	—
259	C₂²×C₃	—	8	16	no	—
260	C₂²×C₃	C₃×#121	1	4	yes	—
261	C₂²×C₃	C₃×#121	1	4	yes	—
262	C₂²×C₃	C₂×C₃×#41	1	4	yes	—
263	C₃×C₄	—	2	8	no	—
264	C₃×C₄	C₃×#121	1	4	yes	—
265	C₃×C₄	C₃×#121	1	4	yes	—
266	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
267	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
268	C₃²	C₂×#472	1	1	yes	characteristic, direct product factor, Sylow
269	C₂³	—	2	1	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	8	4	no	—
274	C₂³	C₃²×#41	1	1	yes	characteristic
275	Q₈	—	2	1	no	—
276	Q₈	C₂×C₃²×D₄	1	2	yes	—
277	D₄	—	2	2	no	—
278	D₄	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	1	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	C₂×C₃²×D₄	1	2	yes	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
289	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
290	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
291	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
292	C₂×C₄	C₃²×#41	1	1	yes	characteristic
293	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	2	4	no	—
296	C₈	C₂×C₃²×D₄	1	2	yes	—
297	C₈	C₂×C₃²×D₄	1	2	yes	—
298	C₂×C₃	—	2	4	no	—
299	C₂×C₃	—	2	4	no	—
300	C₂×C₃	—	4	8	no	—
301	C₂×C₃	—	8	16	no	—
302	C₂×C₃	C₃×#472	1	8	yes	direct product factor
303	C₂×C₃	C₂×C₃×#121	1	4	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	4	no	—
309	C₂²	—	8	4	no	—
310	C₂²	—	8	4	no	—
311	C₂²	C₃²×#121	1	1	yes	characteristic
312	C₂²	C₃²×#121	1	1	yes	characteristic
313	C₂²	C₂×C₃²×#41	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	C₃²×#121	1	1	yes	characteristic
316	C₄	C₃²×#121	1	1	yes	characteristic
317	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
318	C₄	C₂²×C₃²×D₄	1	1	yes	characteristic
319	C₃	C₂×C₃×#472	1	4	yes	direct product factor
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	4	2	no	—
323	C₂	—	8	4	no	—
324	C₂	C₃²×#472	1	2	yes	direct product factor
325	C₂	C₂×C₃²×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃²	#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×AGL(1,ℤ/8)	C₂	semidirect
2. C₂²×C₃²×D₈	C₂	semidirect
3. C₂×C₃²×#131	C₂	semidirect
4. C₃²×#472	C₂	direct (N × H)
5. C₂×C₃²×#108	C₂	semidirect
6. C₂×C₃²×#100	C₂	semidirect
7. C₂×C₃×#472	C₃	direct (N × H)
11. C₃²×AGL(1,ℤ/8)	C₂²	semidirect
17. C₂×C₃²×D₈	C₂²	semidirect
19. C₃²×#131	C₂²	semidirect
25. C₃²×#108	C₂²	semidirect
26. C₃²×#100	C₂²	semidirect
27. C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	semidirect
28. C₂²×C₃×D₈	C₂×C₃	semidirect
29. C₂×C₃×#131	C₂×C₃	semidirect
30. C₃×#472	C₂×C₃	direct (N × H)
31. C₂×C₃×#108	C₂×C₃	semidirect
32. C₂×C₃×#100	C₂×C₃	semidirect
65. C₂×#472	C₃²	direct (N × H)
69. C₃×AGL(1,ℤ/8)	C₂²×C₃	semidirect
75. C₂×C₃×D₈	C₂²×C₃	semidirect
77. C₃×#131	C₂²×C₃	semidirect
83. C₃×#108	C₂²×C₃	semidirect
84. C₃×#100	C₂²×C₃	semidirect
114. C₂×AGL(1,ℤ/8)	C₂×C₃²	semidirect
115. C₂²×D₈	C₂×C₃²	semidirect
116. C₂×#131	C₂×C₃²	semidirect
117. #472	C₂×C₃²	direct (N × H)
118. C₂×#108	C₂×C₃²	semidirect
119. C₂×#100	C₂×C₃²	semidirect
170. AGL(1,ℤ/8)	C₂²×C₃²	semidirect
176. C₂×D₈	C₂²×C₃²	semidirect
178. #131	C₂²×C₃²	semidirect
184. #108	C₂²×C₃²	semidirect
185. #100	C₂²×C₃²	semidirect
219. C₂×C₃²	#472	direct (N × H)
268. C₃²	C₂×#472	direct (N × H)
302. C₂×C₃	C₃×#472	direct (N × H)
319. C₃	C₂×C₃×#472	direct (N × H)
324. C₂	C₃²×#472	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×AGL(1,ℤ/8)	2
C₂²×C₃²×D₈	2
C₂×C₃²×#131	2
C₃²×#472	2
C₂×C₃²×#108	2
C₂×C₃²×#100	2
C₂×C₃×#472	3

Subgroups — C2×C32×#472 #11770114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#472 #11770114