Subgroups — #134684

Order 512.

396 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#12783	C₂	1	2	yes	maximal
3	C₂×#1370	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	#7151	C₂	1	2	yes	maximal, complemented
6	C₂×#431	C₄	1	1	yes	characteristic
7	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	C₂×C₂²⋊C₁₆	—	2	2	no	—
12	C₈×M₁₆	—	2	2	no	—
13	C₈×M₁₆	—	2	2	no	—
14	C₂×#322	C₄	1	1	yes	characteristic
15	C₂×#322	C₄	1	1	yes	characteristic
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1374	—	2	2	no	—
19	#1370	—	2	2	no	—
20	#1367	—	2	2	no	—
21	C₂²×M₁₆	C₈	1	1	yes	characteristic
22	C₂²×M₁₆	C₈	1	1	yes	characteristic
23	C₂³×C₈	D₄	1	1	yes	characteristic
24	C₂³×C₈	Q₈	1	1	yes	characteristic
25	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	—	2	2	no	—
27	C₂²×C₁₆	—	2	2	no	—
28	#431	—	2	2	no	—
29	#431	—	2	2	no	—
30	C₂×C₄⋊C₈	—	2	1	no	—
31	C₂×C₂²⋊C₈	—	2	1	no	—
32	C₄×M₁₆	—	2	2	no	—
33	C₄×M₁₆	—	2	2	no	—
34	C₂×#97	C₈	1	1	yes	characteristic
35	C₂×#97	C₈	1	1	yes	characteristic
36	C₂×C₄×C₈	—	2	1	no	—
37	C₂×C₄×C₈	—	2	1	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	—	2	1	no	—
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊C₁₆	—	2	2	no	—
49	C₄⋊C₁₆	—	2	2	no	—
50	C₄⋊C₁₆	—	2	2	no	—
51	C₄⋊C₁₆	—	2	2	no	—
52	C₂²⋊C₁₆	—	4	4	no	—
53	#322	—	2	2	no	—
54	#322	—	2	2	no	—
55	C₈²	—	2	1	no	—
56	C₈²	—	2	1	no	—
57	C₈²	D₄	1	2	yes	—
58	C₈²	D₄	1	2	yes	—
59	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
61	C₂³×C₄	M₁₆	1	1	yes	characteristic
62	C₂×M₁₆	—	2	2	no	—
63	C₂×M₁₆	—	2	2	no	—
64	C₂×M₁₆	—	2	2	no	—
65	C₂×M₁₆	—	2	2	no	—
66	C₂×M₁₆	—	2	2	no	—
67	C₂×M₁₆	C₁₆	1	4	yes	complemented
68	C₂²×C₈	—	2	1	no	—
69	C₂²×C₈	—	2	1	no	—
70	C₂²×C₈	—	2	1	no	—
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₂²×C₈	—	2	1	no	—
74	C₂²×C₈	—	2	2	no	—
75	C₂²×C₈	—	2	2	no	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	—	2	1	no	—
87	C₂×C₄²	—	2	1	no	—
88	C₂×C₄²	—	2	1	no	—
89	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
91	C₂×C₄²	M₁₆	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₁₆	—	4	2	no	—
94	C₂×C₁₆	—	4	2	no	—
95	C₂×C₁₆	—	4	2	no	—
96	C₂×C₁₆	—	4	2	no	—
97	C₂×C₁₆	—	4	2	no	—
98	C₂×C₁₆	—	4	2	no	—
99	C₄⋊C₈	—	2	4	no	—
100	C₂²⋊C₈	—	2	4	no	—
101	#97	—	2	2	no	—
102	#97	—	2	2	no	—
103	C₄×C₈	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	C₂²⋊C₄	1	2	yes	—
114	C₄×C₈	C₂²⋊C₄	1	2	yes	—
115	C₄×C₈	M₁₆	1	2	yes	—
116	C₄×C₈	M₁₆	1	2	yes	—
117	C₄×C₈	C₂×D₄	1	1	yes	characteristic
118	C₄×C₈	C₂×D₄	1	1	yes	characteristic
119	C₄×C₈	#41	1	1	yes	characteristic
120	C₄×C₈	#41	1	1	yes	characteristic
121	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	2	1	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
134	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₁₆	1	2	yes	—
137	C₂²×C₄	M₃₂	1	2	yes	—
138	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
141	M₁₆	—	2	8	no	—
142	M₁₆	—	2	8	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	C₂²⋊C₈	1	2	yes	—
164	C₂×C₈	C₂²⋊C₈	1	2	yes	—
165	C₂×C₈	C₂×C₁₆	1	2	yes	—
166	C₂×C₈	C₂×C₁₆	1	2	yes	—
167	C₂×C₈	M₃₂	1	2	yes	—
168	C₂×C₈	M₃₂	1	2	yes	—
169	C₂×C₈	#121	1	1	yes	characteristic
170	C₂×C₈	#121	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
173	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₈	#137	1	2	yes	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
182	C₄²	#117	1	1	yes	characteristic
183	C₄²	C₂×M₁₆	1	1	yes	characteristic
184	C₄²	C₂×M₁₆	1	1	yes	characteristic
185	C₁₆	—	8	2	no	—
186	C₁₆	—	8	2	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	2	no	—
191	C₂³	C₄⋊C₁₆	1	2	yes	—
192	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
210	C₂×C₄	#380	1	1	yes	characteristic
211	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
212	C₂×C₄	#407	1	1	yes	characteristic
213	C₂×C₄	#423	1	1	yes	characteristic
214	C₂×C₄	#428	1	2	yes	—
215	C₂×C₄	#430	1	2	yes	—
216	C₂×C₄	#431	1	1	yes	characteristic
217	C₂×C₄	#432	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
219	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
220	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
221	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	—	4	2	no	—
228	C₈	—	4	2	no	—
229	C₈	—	4	1	no	—
230	C₈	—	4	1	no	—
231	C₈	C₂²⋊C₁₆	1	4	yes	complemented
232	C₈	#468	1	2	yes	—
233	C₈	#469	1	1	yes	characteristic
234	C₈	#470	1	1	yes	characteristic
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	1	no	—
240	C₂²	#1565	1	2	yes	—
241	C₂²	#1797	1	1	yes	characteristic
242	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
243	C₂²	#2150	1	1	yes	characteristic
244	C₂²	#2153	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	1	no	—
250	C₄	#1569	1	2	yes	—
251	C₄	#1809	1	1	yes	characteristic
252	C₄	#1811	1	1	yes	characteristic
253	C₄	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
254	C₄	#2112	1	1	yes	characteristic
255	C₄	#2162	1	1	yes	characteristic
256	C₄	#2163	1	1	yes	characteristic
257	C₂	—	4	2	no	—
258	C₂	#8597	1	1	yes	characteristic
259	C₂	#12015	1	2	yes	—
260	C₂	#12017	1	2	yes	—
261	C₂	#12783	1	1	yes	characteristic
262	C₂	#12789	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₄⋊C₈
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#1370	C₂	semidirect
4. C₂×#1367	C₂	semidirect
5. #7151	C₂	semidirect
67. C₂×M₁₆	C₁₆	semidirect
231. C₈	C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×M₁₆	2
#12783	2
C₂×#1370	2
C₂×#1367	2
#7151	2