Subgroups — C₂×C₃²×#476 #11770120

Order 1152.

1440 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃²×#476	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃²×#106	C₂	1	2	yes	maximal, complemented
5	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#102	C₂	1	2	yes	maximal
7	C₂×C₃×#476	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃²×Q₈	—	2	1	no	—
9	C₂²×C₃²×D₄	C₂²	1	1	yes	characteristic
10	C₂³×C₃²×C₄	—	2	1	no	—
11	C₂²×C₃²×C₈	C₂²	1	2	yes	—
12	C₃²×C₄⋊Q₈	—	2	2	no	—
13	C₃²×C₄⋊Q₈	C₂²	1	4	yes	complemented
14	C₃²×C₄×D₄	—	2	2	no	—
15	C₃²×C₄×D₄	C₂²	1	4	yes	—
16	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
17	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃²×C₂²⋊C₄	—	2	1	no	—
20	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
21	C₃²×#106	C₂²	1	8	yes	complemented
22	C₃²×C₄⋊C₈	C₂²	1	4	yes	complemented
23	C₃²×#102	C₂²	1	8	yes	—
24	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	4	yes	complemented
25	C₂×C₃×C₄×D₄	C₂×C₃	1	4	yes	—
26	C₃×#476	C₂×C₃	1	32	yes	direct product factor
27	C₂×C₃×#106	C₂×C₃	1	8	yes	complemented
28	C₂×C₃×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
29	C₂×C₃×#102	C₂×C₃	1	8	yes	—
30	C₂⁴×C₃²	—	2	1	no	—
31	C₂×C₃²×Q₈	—	2	4	no	—
32	C₂×C₃²×Q₈	—	4	4	no	—
33	C₂×C₃²×D₄	—	2	1	no	—
34	C₂×C₃²×D₄	—	2	2	no	—
35	C₂×C₃²×D₄	Q₈	1	4	yes	complemented
36	C₂×C₃²×D₄	C₂³	1	2	yes	—
37	C₂²×C₃²×C₄	—	2	1	no	—
38	C₂²×C₃²×C₄	—	2	2	no	—
39	C₂²×C₃²×C₄	—	2	4	no	—
40	C₂²×C₃²×C₄	—	4	4	no	—
41	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
42	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
43	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
44	C₂×C₃²×C₈	—	2	4	no	—
45	C₂×C₃²×C₈	C₂³	1	4	yes	—
46	C₃²×C₄⋊C₄	—	2	4	no	—
47	C₃²×C₄⋊C₄	—	2	1	no	—
48	C₃²×C₄⋊C₄	—	2	2	no	—
49	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
50	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
51	C₃²×C₂²⋊C₄	—	2	4	no	—
52	C₃²×C₄²	D₄	1	2	yes	—
53	C₃²×C₄²	C₂³	1	2	yes	—
54	C₂×#476	C₃²	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₃×Q₈	—	2	4	no	—
56	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
57	C₂³×C₃×C₄	—	2	4	no	—
58	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
59	C₃×C₄⋊Q₈	—	2	8	no	—
60	C₃×C₄⋊Q₈	C₂²×C₃	1	16	yes	complemented
61	C₃×C₄×D₄	—	2	8	no	—
62	C₃×C₄×D₄	C₂²×C₃	1	16	yes	—
63	C₂×C₃×C₄⋊C₄	—	2	4	no	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
66	C₂×C₃×C₂²⋊C₄	—	2	4	no	—
67	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
68	C₃×#106	C₂²×C₃	1	32	yes	complemented
69	C₃×C₄⋊C₈	C₂²×C₃	1	16	yes	complemented
70	C₃×#102	C₂²×C₃	1	32	yes	—
71	C₂³×C₃²	—	2	4	no	—
72	C₂³×C₃²	—	2	2	no	—
73	C₂³×C₃²	—	4	2	no	—
74	C₂³×C₃²	—	4	2	no	—
75	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
76	C₃²×Q₈	—	4	8	no	—
77	C₃²×D₄	—	2	4	no	—
78	C₃²×D₄	—	2	2	no	—
79	C₃²×D₄	C₂×Q₈	1	4	yes	complemented
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	4	2	no	—
84	C₂×C₃²×C₄	—	4	4	no	—
85	C₂×C₃²×C₄	—	4	8	no	—
86	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
87	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
88	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
91	C₂×C₃²×C₄	#41	1	2	yes	—
92	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
93	C₃²×C₈	—	2	4	no	—
94	C₂×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
95	C₂×C₄×D₄	C₂×C₃²	1	1	yes	characteristic
96	#476	C₂×C₃²	1	8	yes	direct product factor
97	C₂×#106	C₂×C₃²	1	2	yes	complemented
98	C₂×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic, complemented
99	C₂×#102	C₂×C₃²	1	2	yes	—
100	C₂⁴×C₃	—	2	4	no	—
101	C₂×C₃×Q₈	—	2	16	no	—
102	C₂×C₃×Q₈	—	4	16	no	—
103	C₂×C₃×D₄	—	2	4	no	—
104	C₂×C₃×D₄	—	2	8	no	—
105	C₂×C₃×D₄	C₃×Q₈	1	16	yes	complemented
106	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
107	C₂²×C₃×C₄	—	2	4	no	—
108	C₂²×C₃×C₄	—	2	8	no	—
109	C₂²×C₃×C₄	—	2	16	no	—
110	C₂²×C₃×C₄	—	4	16	no	—
111	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
112	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
113	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
114	C₂×C₃×C₈	—	2	16	no	—
115	C₂×C₃×C₈	C₂³×C₃	1	16	yes	—
116	C₃×C₄⋊C₄	—	2	16	no	—
117	C₃×C₄⋊C₄	—	2	4	no	—
118	C₃×C₄⋊C₄	—	2	8	no	—
119	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
120	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
121	C₃×C₂²⋊C₄	—	2	16	no	—
122	C₃×C₄²	C₃×D₄	1	8	yes	—
123	C₃×C₄²	C₂³×C₃	1	8	yes	—
124	C₂²×C₃²	—	2	4	no	—
125	C₂²×C₃²	—	4	2	no	—
126	C₂²×C₃²	—	4	2	no	—
127	C₂²×C₃²	—	4	8	no	—
128	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
129	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
130	C₂²×C₃²	AGL(1,ℤ/8)	1	2	yes	—
131	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
132	C₃²×C₄	—	2	2	no	—
133	C₃²×C₄	—	4	2	no	—
134	C₃²×C₄	—	4	4	no	—
135	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
136	C₃²×C₄	C₂×SD₁₆	1	4	yes	—
137	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
138	C₃²×C₄	C₂×#41	1	1	yes	characteristic
139	C₂²×Q₈	—	2	1	no	—
140	C₂²×D₄	C₂²×C₃²	1	1	yes	characteristic
141	C₂³×C₄	—	2	1	no	—
142	C₂²×C₈	C₂²×C₃²	1	2	yes	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	C₂²×C₃²	1	4	yes	complemented
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	C₂²×C₃²	1	4	yes	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
150	C₂×C₂²⋊C₄	—	2	1	no	—
151	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
152	#106	C₂²×C₃²	1	8	yes	complemented
153	C₄⋊C₈	C₂²×C₃²	1	4	yes	complemented
154	#102	C₂²×C₃²	1	8	yes	—
155	C₂³×C₃	—	2	16	no	—
156	C₂³×C₃	—	2	8	no	—
157	C₂³×C₃	—	4	8	no	—
158	C₂³×C₃	—	4	8	no	—
159	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
160	C₃×Q₈	—	4	32	no	—
161	C₃×D₄	—	2	16	no	—
162	C₃×D₄	—	2	8	no	—
163	C₃×D₄	C₂×C₃×Q₈	1	16	yes	complemented
164	C₂×C₃×C₄	—	2	8	no	—
165	C₂×C₃×C₄	—	2	8	no	—
166	C₂×C₃×C₄	—	2	16	no	—
167	C₂×C₃×C₄	—	4	8	no	—
168	C₂×C₃×C₄	—	4	16	no	—
169	C₂×C₃×C₄	—	4	32	no	—
170	C₂×C₃×C₄	C₃×SD₁₆	1	16	yes	—
171	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
172	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
173	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
174	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
175	C₂×C₃×C₄	C₃×#41	1	8	yes	—
176	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
177	C₃×C₈	—	2	16	no	—
178	C₂×C₃²	—	4	4	no	—
179	C₂×C₃²	#476	1	4	yes	direct product factor
180	C₂×C₃²	C₂×C₂²⋊Q₈	1	1	yes	characteristic
181	C₂×C₃²	C₂²×SD₁₆	1	1	yes	characteristic
182	C₂×C₃²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂⁴	—	2	1	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₃²×Q₈	1	4	yes	complemented
189	C₂×D₄	C₂³×C₃²	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	C₂³×C₃²	1	4	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
203	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	C₃²×D₄	1	2	yes	—
206	C₄²	C₂³×C₃²	1	2	yes	—
207	C₂²×C₃	—	2	16	no	—
208	C₂²×C₃	—	4	8	no	—
209	C₂²×C₃	—	4	8	no	—
210	C₂²×C₃	—	4	32	no	—
211	C₂²×C₃	C₃×C₂²⋊Q₈	1	8	yes	—
212	C₂²×C₃	C₂×C₃×SD₁₆	1	8	yes	—
213	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	8	yes	—
214	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
215	C₃×C₄	—	2	8	no	—
216	C₃×C₄	—	4	8	no	—
217	C₃×C₄	—	4	16	no	—
218	C₃×C₄	C₃×C₂²⋊Q₈	1	8	yes	—
219	C₃×C₄	C₂×C₃×SD₁₆	1	16	yes	—
220	C₃×C₄	C₂²×C₃×Q₈	1	4	yes	—
221	C₃×C₄	C₂×C₃×#41	1	4	yes	—
222	C₃²	C₂×#476	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³	—	2	2	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
228	Q₈	—	4	8	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	4	no	—
231	D₄	C₂×C₃²×Q₈	1	4	yes	complemented
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	C₃²×SD₁₆	1	4	yes	—
239	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
240	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
242	C₂×C₄	C₂×C₃²×Q₈	1	2	yes	—
243	C₂×C₄	C₃²×#41	1	2	yes	—
244	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
245	C₈	—	2	4	no	—
246	C₂×C₃	—	4	16	no	—
247	C₂×C₃	C₃×#476	1	16	yes	direct product factor
248	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	4	yes	—
249	C₂×C₃	C₂²×C₃×SD₁₆	1	4	yes	—
250	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	4	yes	—
251	C₂²	—	2	4	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	8	no	—
255	C₂²	C₃²×C₂²⋊Q₈	1	2	yes	—
256	C₂²	C₂×C₃²×SD₁₆	1	2	yes	—
257	C₂²	C₃²×AGL(1,ℤ/8)	1	2	yes	—
258	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	4	no	—
262	C₄	C₃²×C₂²⋊Q₈	1	2	yes	—
263	C₄	C₂×C₃²×SD₁₆	1	4	yes	—
264	C₄	C₂²×C₃²×Q₈	1	1	yes	characteristic
265	C₄	C₂×C₃²×#41	1	1	yes	characteristic
266	C₃	C₂×C₃×#476	1	4	yes	direct product factor
267	C₂	—	4	4	no	—
268	C₂	C₃²×#476	1	4	yes	direct product factor
269	C₂	C₂×C₃²×C₂²⋊Q₈	1	1	yes	characteristic
270	C₂	C₂²×C₃²×SD₁₆	1	1	yes	characteristic
271	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×C₄⋊Q₈	C₂	semidirect
3. C₃²×#476	C₂	direct (N × H)
4. C₂×C₃²×#106	C₂	semidirect
5. C₂×C₃²×C₄⋊C₈	C₂	semidirect
7. C₂×C₃×#476	C₃	direct (N × H)
13. C₃²×C₄⋊Q₈	C₂²	semidirect
21. C₃²×#106	C₂²	semidirect
22. C₃²×C₄⋊C₈	C₂²	semidirect
24. C₂×C₃×C₄⋊Q₈	C₂×C₃	semidirect
26. C₃×#476	C₂×C₃	direct (N × H)
27. C₂×C₃×#106	C₂×C₃	semidirect
28. C₂×C₃×C₄⋊C₈	C₂×C₃	semidirect
35. C₂×C₃²×D₄	Q₈	semidirect
54. C₂×#476	C₃²	direct (N × H)
60. C₃×C₄⋊Q₈	C₂²×C₃	semidirect
68. C₃×#106	C₂²×C₃	semidirect
69. C₃×C₄⋊C₈	C₂²×C₃	semidirect
79. C₃²×D₄	C₂×Q₈	semidirect
94. C₂×C₄⋊Q₈	C₂×C₃²	semidirect
96. #476	C₂×C₃²	direct (N × H)
97. C₂×#106	C₂×C₃²	semidirect
98. C₂×C₄⋊C₈	C₂×C₃²	semidirect
105. C₂×C₃×D₄	C₃×Q₈	semidirect
144. C₄⋊Q₈	C₂²×C₃²	semidirect
152. #106	C₂²×C₃²	semidirect
153. C₄⋊C₈	C₂²×C₃²	semidirect
163. C₃×D₄	C₂×C₃×Q₈	semidirect
179. C₂×C₃²	#476	direct (N × H)
188. C₂×D₄	C₃²×Q₈	semidirect
222. C₃²	C₂×#476	direct (N × H)
231. D₄	C₂×C₃²×Q₈	semidirect
247. C₂×C₃	C₃×#476	direct (N × H)
266. C₃	C₂×C₃×#476	direct (N × H)
268. C₂	C₃²×#476	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×C₄⋊Q₈	2
C₂×C₃²×C₄×D₄	2
C₃²×#476	2
C₂×C₃²×#106	2
C₂×C₃²×C₄⋊C₈	2
C₂×C₃²×#102	2
C₂×C₃×#476	3

Subgroups — C2×C32×#476 #11770120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#476 #11770120