Subgroups — #149197

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2155	C₂	1	2	yes	maximal
2	C₂×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×#1388	C₂	1	2	yes	maximal
4	C₂×#1376	C₂	1	1	yes	maximal, characteristic
5	#10859	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×C₄×C₈	C₂²	1	1	yes	characteristic
13	#2155	—	2	4	no	—
14	#2155	—	2	4	no	—
15	C₂×#365	—	2	4	no	—
16	#1848	—	2	4	no	—
17	C₂×#339	—	2	1	no	—
18	#1729	—	2	1	no	—
19	#1388	—	2	4	no	—
20	#1376	C₄	1	2	yes	complemented
21	#1376	C₄	1	2	yes	complemented
22	C₂²×M₁₆	—	2	2	no	—
23	C₂²×M₁₆	—	2	1	no	—
24	C₂³×C₈	D₄	1	1	yes	characteristic
25	C₂³×C₈	C₂³	1	1	yes	characteristic
26	C₂×#117	—	2	1	no	—
27	C₂²×C₄²	D₄	1	1	yes	characteristic
28	C₂×M₃₂	—	2	2	no	—
29	C₂×M₃₂	—	2	2	no	—
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	C₂×C₄	1	4	yes	—
32	C₂×M₃₂	D₄	1	4	yes	—
33	C₂×M₃₂	Q₈	1	4	yes	—
34	C₂²×C₁₆	—	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	2	no	—
39	C₂×#108	—	2	2	no	—
40	C₂×#108	—	2	2	no	—
41	C₂×#108	—	2	4	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	—	2	1	no	—
44	C₂×#108	—	4	2	no	—
45	C₂×#108	—	4	2	no	—
46	C₂×#108	D₄	1	4	yes	—
47	C₂×#108	Q₈	1	2	yes	—
48	#427	—	2	2	no	—
49	#427	—	2	2	no	—
50	C₂×#107	—	2	1	no	—
51	C₂×#106	D₄	1	2	yes	—
52	#423	—	2	4	no	—
53	#423	—	4	2	no	—
54	C₂×C₄⋊C₈	—	4	2	no	—
55	C₂×C₄×C₈	—	2	2	no	—
56	C₂×C₄×C₈	—	2	2	no	—
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
60	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
61	#365	—	2	16	no	—
62	#339	—	4	2	no	—
63	#336	—	4	2	no	—
64	C₂³×C₄	—	2	1	no	—
65	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×M₁₆	—	2	4	no	—
67	C₂×M₁₆	—	2	2	no	—
68	C₂×M₁₆	—	4	2	no	—
69	C₂×M₁₆	—	4	1	no	—
70	C₂×M₁₆	—	4	2	no	—
71	C₂×M₁₆	—	4	1	no	—
72	C₂×M₁₆	—	4	1	no	—
73	C₂×M₁₆	—	4	2	no	—
74	C₂×M₁₆	—	4	1	no	—
75	C₂×M₁₆	—	4	2	no	—
76	C₂×M₁₆	—	4	1	no	—
77	C₂×M₁₆	—	4	2	no	—
78	C₂²×C₈	—	2	1	no	—
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	—	4	1	no	—
83	C₂²×C₈	—	4	2	no	—
84	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₈	SD₁₆	1	2	yes	—
87	C₂²×C₈	SD₁₆	1	2	yes	—
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	2	yes	—
90	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×D₄	1	2	yes	—
92	C₂²×C₈	C₂×Q₈	1	2	yes	—
93	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₈	#41	1	2	yes	—
96	C₂²×C₈	#41	1	1	yes	characteristic
97	C₂²×C₈	#41	1	1	yes	characteristic
98	#117	—	4	1	no	—
99	#117	—	4	1	no	—
100	#117	—	4	1	no	—
101	#117	—	4	1	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₂²⋊C₄	—	4	1	no	—
104	C₂×C₄²	—	2	1	no	—
105	C₂×C₄²	—	2	1	no	—
106	C₂×C₄²	—	2	1	no	—
107	C₂×C₄²	—	4	1	no	—
108	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄²	SD₁₆	1	2	yes	—
111	C₂×C₄²	SD₁₆	1	2	yes	—
112	M₃₂	—	2	4	no	—
113	M₃₂	—	2	4	no	—
114	M₃₂	—	2	4	no	—
115	M₃₂	C₄⋊C₄	1	8	yes	—
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	2	4	no	—
119	#108	—	2	4	no	—
120	#108	—	2	4	no	—
121	#108	—	2	4	no	—
122	#108	—	4	8	no	—
123	#108	—	4	2	no	—
124	#108	—	4	2	no	—
125	#108	—	4	2	no	—
126	#107	—	4	2	no	—
127	#106	—	2	4	no	—
128	C₄⋊C₈	—	4	8	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂⁴	#137	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	4	4	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₄	#120	1	1	yes	characteristic
142	C₂²×C₄	#121	1	1	yes	characteristic
143	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂²×C₄	#123	1	1	yes	characteristic
145	C₂²×C₄	#127	1	1	yes	characteristic
146	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
147	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
148	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
151	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂²×C₄	#137	1	1	yes	characteristic
154	M₁₆	—	4	4	no	—
155	M₁₆	—	4	2	no	—
156	M₁₆	—	4	4	no	—
157	M₁₆	—	4	2	no	—
158	M₁₆	—	8	2	no	—
159	M₁₆	—	8	1	no	—
160	M₁₆	—	8	2	no	—
161	M₁₆	—	8	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	4	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	8	2	no	—
173	C₂×C₈	—	8	1	no	—
174	C₂×C₈	—	8	1	no	—
175	C₂×C₈	—	8	2	no	—
176	C₂×C₈	#106	1	2	yes	—
177	C₂×C₈	#106	1	2	yes	—
178	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
179	C₂×C₈	C₄×D₄	1	1	yes	characteristic
180	C₂×C₈	C₄×D₄	1	2	yes	—
181	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
182	C₂×C₈	#120	1	2	yes	—
183	C₂×C₈	#121	1	2	yes	—
184	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
185	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
186	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
187	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
188	C₂×C₈	#123	1	2	yes	—
189	C₂×C₈	#125	1	2	yes	—
190	C₂×C₈	#125	1	1	yes	characteristic
191	C₂×C₈	C₄⋊Q₈	1	2	yes	—
192	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
193	C₂×C₈	C₄⋊Q₈	1	2	yes	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	8	1	no	—
197	C₂²⋊C₄	—	8	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	8	1	no	—
203	C₄²	—	8	1	no	—
204	C₄²	#106	1	2	yes	—
205	C₄²	#106	1	2	yes	—
206	C₁₆	—	2	8	no	—
207	C₂³	—	4	2	no	—
208	C₂³	#428	1	1	yes	characteristic
209	C₂³	#448	1	1	yes	characteristic
210	C₂³	#461	1	1	yes	characteristic
211	C₂³	#475	1	1	yes	characteristic
212	C₂³	#483	1	1	yes	characteristic
213	C₂³	#497	1	1	yes	characteristic
214	C₂³	#501	1	1	yes	characteristic
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	#384	1	1	yes	characteristic
224	C₂×C₄	#385	1	1	yes	characteristic
225	C₂×C₄	#389	1	1	yes	characteristic
226	C₂×C₄	#392	1	1	yes	characteristic
227	C₂×C₄	#395	1	1	yes	characteristic
228	C₂×C₄	#398	1	1	yes	characteristic
229	C₂×C₄	#399	1	1	yes	characteristic
230	C₂×C₄	C₂×#106	1	1	yes	characteristic
231	C₂×C₄	C₂×#106	1	1	yes	characteristic
232	C₂×C₄	#428	1	1	yes	characteristic
233	C₂×C₄	#448	1	1	yes	characteristic
234	C₂×C₄	#450	1	1	yes	characteristic
235	C₂×C₄	#450	1	1	yes	characteristic
236	C₂×C₄	#460	1	1	yes	characteristic
237	C₂×C₄	#460	1	1	yes	characteristic
238	C₂×C₄	#461	1	1	yes	characteristic
239	C₂×C₄	#475	1	1	yes	characteristic
240	C₂×C₄	#476	1	1	yes	characteristic
241	C₂×C₄	#476	1	1	yes	characteristic
242	C₂×C₄	#481	1	1	yes	characteristic
243	C₂×C₄	#481	1	1	yes	characteristic
244	C₂×C₄	#483	1	1	yes	characteristic
245	C₂×C₄	#492	1	1	yes	characteristic
246	C₂×C₄	#492	1	1	yes	characteristic
247	C₂×C₄	#496	1	1	yes	characteristic
248	C₂×C₄	#498	1	1	yes	characteristic
249	C₂×C₄	#498	1	1	yes	characteristic
250	C₂×C₄	#501	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₈	—	8	2	no	—
254	C₈	—	8	4	no	—
255	C₈	#385	1	2	yes	—
256	C₈	#389	1	2	yes	—
257	C₈	#395	1	2	yes	—
258	C₈	#398	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	#1847	1	1	yes	characteristic
267	C₂²	#1853	1	1	yes	characteristic
268	C₂²	#1948	1	1	yes	characteristic
269	C₂²	#2013	1	1	yes	characteristic
270	C₂²	#2058	1	1	yes	characteristic
271	C₂²	#2084	1	1	yes	characteristic
272	C₂²	#2096	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	16	2	no	—
276	C₄	#1444	1	1	yes	characteristic
277	C₄	#1844	1	1	yes	characteristic
278	C₄	#1853	1	1	yes	characteristic
279	C₄	#1948	1	1	yes	characteristic
280	C₄	#2014	1	1	yes	characteristic
281	C₄	#2058	1	1	yes	characteristic
282	C₄	#2084	1	1	yes	characteristic
283	C₄	#2094	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	#11323	1	1	yes	characteristic
291	C₂	#13188	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1948
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
20. #1376	C₄	semidirect
21. #1376	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2155	2
C₂×#1848	2
C₂×#1388	2
C₂×#1376	2
#10859	2