Subgroups — C₂×C₃²×#475 #11770122

Order 1152.

1248 conjugacy classes of subgroups.

The subgroup lattice has 249 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
2	C₂×C₃²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×#475	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃²×#106	C₂	1	2	yes	maximal
5	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃²×#103	C₂	1	2	yes	maximal
7	C₂×C₃×#475	C₃	1	4	yes	maximal, direct product factor
8	C₂²×C₃²×Q₈	—	2	1	no	—
9	C₂²×C₃²×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	2	yes	—
11	C₃²×C₄⋊Q₈	—	2	2	no	—
12	C₃²×C₄⋊Q₈	C₂²	1	4	yes	—
13	C₃²×C₄×Q₈	—	2	2	no	—
14	C₃²×C₄×Q₈	C₂²	1	4	yes	—
15	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
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×C₄²	—	2	1	no	—
20	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
21	C₃²×#106	C₂²	1	8	yes	—
22	C₃²×C₄⋊C₈	C₂²	1	4	yes	—
23	C₃²×#103	C₂²	1	8	yes	—
24	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	4	yes	—
25	C₂×C₃×C₄×Q₈	C₂×C₃	1	4	yes	—
26	C₃×#475	C₂×C₃	1	32	yes	direct product factor
27	C₂×C₃×#106	C₂×C₃	1	8	yes	—
28	C₂×C₃×C₄⋊C₈	C₂×C₃	1	4	yes	—
29	C₂×C₃×#103	C₂×C₃	1	8	yes	—
30	C₂×C₃²×Q₈	—	2	1	no	—
31	C₂×C₃²×Q₈	—	2	4	no	—
32	C₂×C₃²×Q₈	—	2	2	no	—
33	C₂×C₃²×Q₈	—	4	4	no	—
34	C₂×C₃²×Q₈	Q₈	1	4	yes	complemented
35	C₂×C₃²×Q₈	C₂³	1	2	yes	—
36	C₂²×C₃²×C₄	—	2	1	no	—
37	C₂²×C₃²×C₄	—	2	2	no	—
38	C₂²×C₃²×C₄	—	2	1	no	—
39	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
40	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
41	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃²×C₈	—	2	4	no	—
43	C₂×C₃²×C₈	C₂³	1	4	yes	—
44	C₃²×C₄⋊C₄	—	2	4	no	—
45	C₃²×C₄⋊C₄	—	2	4	no	—
46	C₃²×C₄⋊C₄	—	2	1	no	—
47	C₃²×C₄⋊C₄	—	2	2	no	—
48	C₃²×C₄⋊C₄	C₂³	1	2	yes	—
49	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
50	C₃²×C₄²	—	2	4	no	—
51	C₃²×C₄²	D₄	1	2	yes	—
52	C₃²×C₄²	C₂³	1	2	yes	—
53	C₂×#475	C₃²	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₃×Q₈	—	2	4	no	—
55	C₂²×C₃×Q₈	C₂²×C₃	1	4	yes	—
56	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
57	C₃×C₄⋊Q₈	—	2	8	no	—
58	C₃×C₄⋊Q₈	C₂²×C₃	1	16	yes	—
59	C₃×C₄×Q₈	—	2	8	no	—
60	C₃×C₄×Q₈	C₂²×C₃	1	16	yes	—
61	C₂×C₃×C₄⋊C₄	—	2	4	no	—
62	C₂×C₃×C₄⋊C₄	—	2	4	no	—
63	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
65	C₂×C₃×C₄²	—	2	4	no	—
66	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
67	C₃×#106	C₂²×C₃	1	32	yes	—
68	C₃×C₄⋊C₈	C₂²×C₃	1	16	yes	—
69	C₃×#103	C₂²×C₃	1	32	yes	—
70	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
71	C₃²×Q₈	—	2	4	no	—
72	C₃²×Q₈	—	2	2	no	—
73	C₃²×Q₈	—	4	8	no	—
74	C₃²×Q₈	C₂×Q₈	1	4	yes	complemented
75	C₂×C₃²×C₄	—	2	4	no	—
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	—	2	2	no	—
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	4	no	—
80	C₂×C₃²×C₄	—	4	4	no	—
81	C₂×C₃²×C₄	—	4	2	no	—
82	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
83	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
84	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
87	C₂×C₃²×C₄	#41	1	2	yes	—
88	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
89	C₃²×C₈	—	2	4	no	—
90	C₂×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
91	C₂×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
92	#475	C₂×C₃²	1	8	yes	direct product factor
93	C₂×#106	C₂×C₃²	1	2	yes	—
94	C₂×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
95	C₂×#103	C₂×C₃²	1	2	yes	—
96	C₂×C₃×Q₈	—	2	4	no	—
97	C₂×C₃×Q₈	—	2	16	no	—
98	C₂×C₃×Q₈	—	2	8	no	—
99	C₂×C₃×Q₈	—	4	16	no	—
100	C₂×C₃×Q₈	C₃×Q₈	1	16	yes	complemented
101	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
102	C₂²×C₃×C₄	—	2	4	no	—
103	C₂²×C₃×C₄	—	2	4	no	—
104	C₂²×C₃×C₄	—	2	8	no	—
105	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
106	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
107	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
108	C₂×C₃×C₈	—	2	16	no	—
109	C₂×C₃×C₈	C₂³×C₃	1	16	yes	—
110	C₃×C₄⋊C₄	—	2	16	no	—
111	C₃×C₄⋊C₄	—	2	16	no	—
112	C₃×C₄⋊C₄	—	2	4	no	—
113	C₃×C₄⋊C₄	—	2	8	no	—
114	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
115	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
116	C₃×C₄²	—	2	16	no	—
117	C₃×C₄²	C₃×D₄	1	8	yes	—
118	C₃×C₄²	C₂³×C₃	1	8	yes	—
119	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
120	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
121	C₂²×C₃²	#137	1	2	yes	—
122	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
123	C₃²×C₄	—	2	4	no	—
124	C₃²×C₄	—	2	2	no	—
125	C₃²×C₄	—	4	4	no	—
126	C₃²×C₄	—	4	2	no	—
127	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
128	C₃²×C₄	C₂×SD₁₆	1	4	yes	—
129	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
130	C₃²×C₄	C₂×#41	1	1	yes	characteristic
131	C₂²×Q₈	—	2	1	no	—
132	C₂²×Q₈	C₂²×C₃²	1	1	yes	characteristic
133	C₂²×C₈	C₂²×C₃²	1	2	yes	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	C₂²×C₃²	1	4	yes	—
136	C₄×Q₈	—	2	2	no	—
137	C₄×Q₈	C₂²×C₃²	1	4	yes	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
144	#106	C₂²×C₃²	1	8	yes	—
145	C₄⋊C₈	C₂²×C₃²	1	4	yes	—
146	#103	C₂²×C₃²	1	8	yes	—
147	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
148	C₃×Q₈	—	2	16	no	—
149	C₃×Q₈	—	2	8	no	—
150	C₃×Q₈	—	4	32	no	—
151	C₃×Q₈	C₂×C₃×Q₈	1	16	yes	complemented
152	C₂×C₃×C₄	—	2	16	no	—
153	C₂×C₃×C₄	—	2	8	no	—
154	C₂×C₃×C₄	—	2	8	no	—
155	C₂×C₃×C₄	—	2	8	no	—
156	C₂×C₃×C₄	—	2	16	no	—
157	C₂×C₃×C₄	—	4	16	no	—
158	C₂×C₃×C₄	—	4	8	no	—
159	C₂×C₃×C₄	C₃×SD₁₆	1	16	yes	—
160	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
161	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
162	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
163	C₂×C₃×C₄	C₂×C₃×Q₈	1	8	yes	—
164	C₂×C₃×C₄	C₃×#41	1	8	yes	—
165	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
166	C₃×C₈	—	2	16	no	—
167	C₂×C₃²	#475	1	4	yes	direct product factor
168	C₂×C₃²	C₂×C₂²⋊Q₈	1	1	yes	characteristic
169	C₂×C₃²	C₂²×SD₁₆	1	1	yes	characteristic
170	C₂×C₃²	C₂×#137	1	1	yes	characteristic
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	C₃²×Q₈	1	4	yes	complemented
176	C₂×Q₈	C₂³×C₃²	1	2	yes	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂³×C₃²	1	4	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	C₂³×C₃²	1	4	yes	—
190	C₄⋊C₄	C₂³×C₃²	1	2	yes	—
191	C₄²	—	2	4	no	—
192	C₄²	C₃²×D₄	1	2	yes	—
193	C₄²	C₂³×C₃²	1	2	yes	—
194	C₂²×C₃	C₃×C₂²⋊Q₈	1	8	yes	—
195	C₂²×C₃	C₂×C₃×SD₁₆	1	8	yes	—
196	C₂²×C₃	C₃×#137	1	8	yes	—
197	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
198	C₃×C₄	—	2	8	no	—
199	C₃×C₄	—	2	16	no	—
200	C₃×C₄	—	4	16	no	—
201	C₃×C₄	—	4	8	no	—
202	C₃×C₄	C₃×C₂²⋊Q₈	1	8	yes	—
203	C₃×C₄	C₂×C₃×SD₁₆	1	16	yes	—
204	C₃×C₄	C₂²×C₃×Q₈	1	4	yes	—
205	C₃×C₄	C₂×C₃×#41	1	4	yes	—
206	C₃²	C₂×#475	1	1	yes	characteristic, direct product factor, Sylow
207	C₂³	C₂×C₃²×D₄	1	1	yes	characteristic
208	Q₈	—	2	2	no	—
209	Q₈	—	2	4	no	—
210	Q₈	—	4	8	no	—
211	Q₈	C₂×C₃²×Q₈	1	4	yes	complemented
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₃²×SD₁₆	1	4	yes	—
220	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
221	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
223	C₂×C₄	C₂×C₃²×Q₈	1	2	yes	—
224	C₂×C₄	C₃²×#41	1	2	yes	—
225	C₂×C₄	C₂⁴×C₃²	1	1	yes	characteristic
226	C₈	—	2	4	no	—
227	C₂×C₃	C₃×#475	1	16	yes	direct product factor
228	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	4	yes	—
229	C₂×C₃	C₂²×C₃×SD₁₆	1	4	yes	—
230	C₂×C₃	C₂×C₃×#137	1	4	yes	—
231	C₂²	C₃²×C₂²⋊Q₈	1	2	yes	—
232	C₂²	C₂×C₃²×SD₁₆	1	2	yes	—
233	C₂²	C₃²×#137	1	2	yes	—
234	C₂²	C₂²×C₃²×D₄	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	4	no	—
238	C₄	—	4	2	no	—
239	C₄	C₃²×C₂²⋊Q₈	1	2	yes	—
240	C₄	C₂×C₃²×SD₁₆	1	4	yes	—
241	C₄	C₂²×C₃²×Q₈	1	1	yes	characteristic
242	C₄	C₂×C₃²×#41	1	1	yes	characteristic
243	C₃	C₂×C₃×#475	1	4	yes	direct product factor
244	C₂	C₃²×#475	1	4	yes	direct product factor
245	C₂	C₂×C₃²×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂	C₂²×C₃²×SD₁₆	1	1	yes	characteristic
247	C₂	C₂×C₃²×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
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
3. C₃²×#475	C₂	direct (N × H)
7. C₂×C₃×#475	C₃	direct (N × H)
26. C₃×#475	C₂×C₃	direct (N × H)
34. C₂×C₃²×Q₈	Q₈	semidirect
53. C₂×#475	C₃²	direct (N × H)
74. C₃²×Q₈	C₂×Q₈	semidirect
92. #475	C₂×C₃²	direct (N × H)
100. C₂×C₃×Q₈	C₃×Q₈	semidirect
151. C₃×Q₈	C₂×C₃×Q₈	semidirect
167. C₂×C₃²	#475	direct (N × H)
175. C₂×Q₈	C₃²×Q₈	semidirect
206. C₃²	C₂×#475	direct (N × H)
211. Q₈	C₂×C₃²×Q₈	semidirect
227. C₂×C₃	C₃×#475	direct (N × H)
243. C₃	C₂×C₃×#475	direct (N × H)
244. C₂	C₃²×#475	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×Q₈	2
C₃²×#475	2
C₂×C₃²×#106	2
C₂×C₃²×C₄⋊C₈	2
C₂×C₃²×#103	2
C₂×C₃×#475	3

Subgroups — C2×C32×#475 #11770122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#475 #11770122