Subgroups — #148986

Order 512.

484 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×#1849	C₂	1	1	yes	maximal, characteristic
2	C₂×#1388	C₂	1	2	yes	maximal
3	#12569	C₂	1	2	yes	maximal
4	C₂×#1375	C₂	1	1	yes	maximal, characteristic
5	#10860	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₃₂	C₂²	1	2	yes	—
7	C₂²×#108	—	2	1	no	—
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂²×#108	C₂²	1	1	yes	characteristic
10	C₂×#427	C₂²	1	1	yes	characteristic
11	C₂×#423	—	2	1	no	—
12	C₂²×#97	C₂²	1	1	yes	characteristic
13	#1849	—	2	4	no	—
14	C₂×#339	—	2	1	no	—
15	#1729	—	2	1	no	—
16	#1388	—	2	4	no	—
17	#1387	—	2	4	no	—
18	#1375	C₄	1	4	yes	complemented
19	C₂²×M₁₆	—	2	2	no	—
20	C₂²×M₁₆	—	2	1	no	—
21	C₂³×C₈	D₄	1	1	yes	characteristic
22	C₂³×C₈	C₂³	1	1	yes	characteristic
23	C₂×#117	—	2	1	no	—
24	C₂²×C₄²	D₄	1	1	yes	characteristic
25	C₂×M₃₂	—	2	2	no	—
26	C₂×M₃₂	—	2	2	no	—
27	C₂×M₃₂	—	2	2	no	—
28	C₂×M₃₂	—	2	2	no	—
29	C₂×M₃₂	—	2	2	no	—
30	C₂×M₃₂	C₂×C₄	1	4	yes	—
31	C₂²×C₁₆	—	2	2	no	—
32	C₂×#108	—	2	2	no	—
33	C₂×#108	—	2	2	no	—
34	C₂×#108	—	2	2	no	—
35	C₂×#108	—	2	2	no	—
36	C₂×#108	—	2	2	no	—
37	C₂×#108	—	2	2	no	—
38	C₂×#108	—	2	4	no	—
39	C₂×#108	—	2	2	no	—
40	C₂×#108	—	2	1	no	—
41	C₂×#108	—	4	2	no	—
42	C₂×#108	—	4	2	no	—
43	C₂×#108	D₄	1	4	yes	—
44	C₂×#108	Q₈	1	2	yes	—
45	#427	—	2	2	no	—
46	#427	—	2	2	no	—
47	C₂×#107	D₄	1	2	yes	—
48	C₂×#106	—	2	1	no	—
49	#423	—	2	4	no	—
50	#423	—	4	2	no	—
51	C₂×C₄⋊C₈	—	4	2	no	—
52	C₂×#97	—	2	2	no	—
53	C₂×#97	—	2	2	no	—
54	C₂×#97	C₂×C₄	1	2	yes	—
55	C₂×#97	Q₈	1	2	yes	—
56	#339	—	4	2	no	—
57	#336	—	4	2	no	—
58	C₂³×C₄	—	2	1	no	—
59	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×M₁₆	—	2	4	no	—
61	C₂×M₁₆	—	2	2	no	—
62	C₂×M₁₆	—	4	1	no	—
63	C₂×M₁₆	—	4	2	no	—
64	C₂×M₁₆	—	4	1	no	—
65	C₂×M₁₆	—	4	2	no	—
66	C₂×M₁₆	—	4	2	no	—
67	C₂×M₁₆	—	4	1	no	—
68	C₂×M₁₆	—	4	1	no	—
69	C₂×M₁₆	—	4	2	no	—
70	C₂×M₁₆	—	4	2	no	—
71	C₂×M₁₆	—	4	1	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₂²×C₈	—	2	1	no	—
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	4	2	no	—
77	C₂²×C₈	—	4	1	no	—
78	C₂²×C₈	C₄⋊C₄	1	2	yes	—
79	C₂²×C₈	SD₁₆	1	2	yes	—
80	C₂²×C₈	SD₁₆	1	2	yes	—
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₄	1	2	yes	—
84	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₈	#41	1	2	yes	—
87	C₂²×C₈	#41	1	2	yes	—
88	C₂²×C₈	#41	1	2	yes	—
89	C₂²×C₈	#41	1	1	yes	characteristic
90	C₂²×C₈	#41	1	1	yes	characteristic
91	#117	—	4	1	no	—
92	#117	—	4	1	no	—
93	#117	—	4	1	no	—
94	#117	—	4	1	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₂²⋊C₄	—	4	1	no	—
97	C₂×C₄²	—	2	1	no	—
98	C₂×C₄²	—	2	1	no	—
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	—	4	1	no	—
101	C₂×C₄²	C₄⋊C₄	1	2	yes	—
102	C₂×C₄²	D₈	1	2	yes	—
103	C₂×C₄²	Q₁₆	1	2	yes	—
104	M₃₂	—	2	4	no	—
105	M₃₂	—	2	4	no	—
106	M₃₂	—	4	2	no	—
107	M₃₂	—	4	2	no	—
108	C₂×C₁₆	—	2	4	no	—
109	C₂×C₁₆	—	4	2	no	—
110	C₂×C₁₆	—	4	2	no	—
111	#108	—	2	4	no	—
112	#108	—	2	4	no	—
113	#108	—	2	4	no	—
114	#108	—	4	8	no	—
115	#108	—	4	2	no	—
116	#108	—	4	2	no	—
117	#108	—	4	2	no	—
118	#107	—	2	4	no	—
119	#106	—	4	2	no	—
120	C₄⋊C₈	—	4	8	no	—
121	#97	—	2	4	no	—
122	#97	—	2	4	no	—
123	C₂⁴	#137	1	1	yes	characteristic
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	8	2	no	—
129	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₄	#120	1	1	yes	characteristic
131	C₂²×C₄	#121	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂²×C₄	#123	1	1	yes	characteristic
134	C₂²×C₄	#124	1	1	yes	characteristic
135	C₂²×C₄	#124	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
138	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
139	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
140	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂²×C₄	#137	1	1	yes	characteristic
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	4	no	—
145	M₁₆	—	4	4	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	8	1	no	—
148	M₁₆	—	8	2	no	—
149	M₁₆	—	8	1	no	—
150	M₁₆	—	8	2	no	—
151	C₂×C₈	—	2	2	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	—	2	2	no	—
157	C₂×C₈	—	4	4	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	8	1	no	—
160	C₂×C₈	—	8	2	no	—
161	C₂×C₈	—	8	2	no	—
162	C₂×C₈	—	8	1	no	—
163	C₂×C₈	#106	1	4	yes	—
164	C₂×C₈	#117	1	2	yes	—
165	C₂×C₈	C₄×D₄	1	2	yes	—
166	C₂×C₈	C₄×D₄	1	1	yes	characteristic
167	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
168	C₂×C₈	#121	1	2	yes	—
169	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
170	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
171	C₂×C₈	#123	1	2	yes	—
172	C₂×C₈	#123	1	2	yes	—
173	C₂×C₈	#124	1	2	yes	—
174	C₂×C₈	#125	1	2	yes	—
175	C₂×C₈	#125	1	2	yes	—
176	C₂×C₈	#125	1	1	yes	characteristic
177	C₂×C₈	#126	1	2	yes	—
178	C₂×C₈	#126	1	2	yes	—
179	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	8	1	no	—
183	C₂²⋊C₄	—	8	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	8	1	no	—
188	C₄²	—	8	1	no	—
189	C₄²	#107	1	4	yes	—
190	C₁₆	—	4	4	no	—
191	C₂³	—	4	2	no	—
192	C₂³	#428	1	1	yes	characteristic
193	C₂³	#451	1	1	yes	characteristic
194	C₂³	#461	1	1	yes	characteristic
195	C₂³	#475	1	1	yes	characteristic
196	C₂³	#480	1	1	yes	characteristic
197	C₂³	#489	1	1	yes	characteristic
198	C₂³	#491	1	1	yes	characteristic
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	4	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	#382	1	1	yes	characteristic
206	C₂×C₄	#385	1	1	yes	characteristic
207	C₂×C₄	#389	1	1	yes	characteristic
208	C₂×C₄	#393	1	1	yes	characteristic
209	C₂×C₄	#394	1	1	yes	characteristic
210	C₂×C₄	#397	1	1	yes	characteristic
211	C₂×C₄	#400	1	1	yes	characteristic
212	C₂×C₄	C₂×#106	1	1	yes	characteristic
213	C₂×C₄	C₂×#107	1	1	yes	characteristic
214	C₂×C₄	#428	1	1	yes	characteristic
215	C₂×C₄	#447	1	1	yes	characteristic
216	C₂×C₄	#448	1	1	yes	characteristic
217	C₂×C₄	#450	1	1	yes	characteristic
218	C₂×C₄	#459	1	1	yes	characteristic
219	C₂×C₄	#460	1	1	yes	characteristic
220	C₂×C₄	#462	1	1	yes	characteristic
221	C₂×C₄	#474	1	1	yes	characteristic
222	C₂×C₄	#476	1	1	yes	characteristic
223	C₂×C₄	#477	1	1	yes	characteristic
224	C₂×C₄	#481	1	1	yes	characteristic
225	C₂×C₄	#483	1	1	yes	characteristic
226	C₂×C₄	#484	1	1	yes	characteristic
227	C₂×C₄	#486	1	1	yes	characteristic
228	C₂×C₄	#486	1	1	yes	characteristic
229	C₂×C₄	#487	1	1	yes	characteristic
230	C₂×C₄	#487	1	1	yes	characteristic
231	C₂×C₄	#489	1	1	yes	characteristic
232	C₂×C₄	#490	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	—	8	2	no	—
235	C₈	—	8	4	no	—
236	C₈	#387	1	2	yes	—
237	C₈	#388	1	2	yes	—
238	C₈	#398	1	2	yes	—
239	C₈	#400	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	#1836	1	1	yes	characteristic
248	C₂²	#1852	1	1	yes	characteristic
249	C₂²	#1948	1	1	yes	characteristic
250	C₂²	#2024	1	1	yes	characteristic
251	C₂²	#2043	1	1	yes	characteristic
252	C₂²	#2071	1	1	yes	characteristic
253	C₂²	#2099	1	1	yes	characteristic
254	C₄	—	4	4	no	—
255	C₄	—	16	2	no	—
256	C₄	#1442	1	1	yes	characteristic
257	C₄	#1834	1	1	yes	characteristic
258	C₄	#1853	1	1	yes	characteristic
259	C₄	#1947	1	1	yes	characteristic
260	C₄	#2023	1	1	yes	characteristic
261	C₄	#2041	1	1	yes	characteristic
262	C₄	#2072	1	1	yes	characteristic
263	C₄	#2100	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	#11267	1	1	yes	characteristic
271	C₂	#13486	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#486
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2071
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
18. #1375	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1849	2
C₂×#1388	2
#12569	2
C₂×#1375	2
#10860	2