Subgroups — #149196

Order 512.

572 conjugacy classes of subgroups.

The subgroup lattice has 281 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#365	C₂	1	2	yes	maximal
2	C₂×#1846	C₂	1	1	yes	maximal, characteristic
3	C₂×#1386	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	C₂²	1	1	yes	characteristic
8	C₂×#427	C₂²	1	1	yes	characteristic
9	C₂²×#107	—	2	1	no	—
10	C₂²×#106	C₂²	1	2	yes	—
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×#382	—	2	1	no	—
13	C₂×#365	—	2	4	no	—
14	C₂×#365	—	2	4	no	—
15	C₂×#365	—	2	4	no	—
16	#1846	—	2	4	no	—
17	C₂×#339	—	2	1	no	—
18	#1729	—	2	1	no	—
19	#1386	—	2	4	no	—
20	#1376	C₄	1	4	yes	complemented
21	C₂²×M₁₆	—	2	1	no	—
22	C₂³×C₈	D₄	1	1	yes	characteristic
23	C₂³×C₈	C₂³	1	1	yes	characteristic
24	C₂×#117	—	2	1	no	—
25	C₂²×C₄⋊C₄	—	2	2	no	—
26	C₂²×C₄²	D₄	1	1	yes	characteristic
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	C₂×M₃₂	D₄	1	4	yes	—
32	C₂×M₃₂	Q₈	1	4	yes	—
33	C₂²×C₁₆	—	2	2	no	—
34	C₂×#108	—	2	2	no	—
35	C₂×#108	—	2	2	no	—
36	C₂×#108	—	2	1	no	—
37	C₂×#108	D₄	1	2	yes	—
38	#427	—	2	2	no	—
39	#427	—	2	2	no	—
40	C₂×#107	—	2	4	no	—
41	C₂×#107	—	4	2	no	—
42	C₂×#107	—	4	2	no	—
43	C₂×#107	Q₈	1	2	yes	—
44	C₂×#106	—	2	2	no	—
45	C₂×#106	—	2	2	no	—
46	C₂×#106	—	2	2	no	—
47	C₂×#106	—	2	2	no	—
48	C₂×#106	—	2	2	no	—
49	C₂×#106	—	2	1	no	—
50	C₂×#106	D₄	1	4	yes	—
51	C₂×C₄×C₈	—	2	2	no	—
52	C₂×C₄×C₈	—	2	2	no	—
53	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄×C₈	Q₈	1	2	yes	—
55	#382	—	2	4	no	—
56	#382	—	4	2	no	—
57	C₂×#95	—	4	2	no	—
58	#365	—	2	16	no	—
59	#339	—	4	2	no	—
60	#336	—	4	2	no	—
61	C₂³×C₄	—	2	1	no	—
62	C₂³×C₄	—	4	2	no	—
63	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×M₁₆	—	2	2	no	—
65	C₂×M₁₆	—	4	1	no	—
66	C₂×M₁₆	—	4	1	no	—
67	C₂×M₁₆	—	4	1	no	—
68	C₂×M₁₆	—	4	1	no	—
69	C₂×M₁₆	—	4	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	—	4	1	no	—
75	C₂²×C₈	C₄⋊C₄	1	2	yes	—
76	C₂²×C₈	SD₁₆	1	2	yes	—
77	C₂²×C₈	SD₁₆	1	2	yes	—
78	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₄	1	2	yes	—
80	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₄	1	2	yes	—
82	C₂²×C₈	C₂×Q₈	1	2	yes	—
83	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₈	#41	1	2	yes	—
86	C₂²×C₈	#41	1	1	yes	characteristic
87	C₂²×C₈	#41	1	1	yes	characteristic
88	#117	—	4	1	no	—
89	#117	—	4	1	no	—
90	#117	—	4	1	no	—
91	#117	—	4	1	no	—
92	C₂×C₄⋊C₄	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	4	2	no	—
95	C₂×C₄⋊C₄	—	4	2	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄⋊C₄	—	4	2	no	—
99	C₂×C₂²⋊C₄	—	4	1	no	—
100	C₂×C₄²	—	2	1	no	—
101	C₂×C₄²	—	2	1	no	—
102	C₂×C₄²	—	2	1	no	—
103	C₂×C₄²	—	4	1	no	—
104	C₂×C₄²	C₄⋊C₄	1	2	yes	—
105	C₂×C₄²	SD₁₆	1	2	yes	—
106	C₂×C₄²	SD₁₆	1	2	yes	—
107	M₃₂	—	2	4	no	—
108	M₃₂	—	2	4	no	—
109	M₃₂	—	2	4	no	—
110	M₃₂	C₄⋊C₄	1	8	yes	—
111	C₂×C₁₆	—	2	4	no	—
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	—	2	4	no	—
114	#108	—	2	4	no	—
115	#108	—	4	2	no	—
116	#107	—	2	4	no	—
117	#107	—	4	8	no	—
118	#106	—	2	4	no	—
119	#106	—	2	4	no	—
120	#106	—	4	2	no	—
121	#106	—	4	2	no	—
122	#106	—	4	2	no	—
123	C₄×C₈	—	2	4	no	—
124	C₄×C₈	—	2	4	no	—
125	#95	—	4	8	no	—
126	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	4	4	no	—
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	8	8	no	—
134	C₂²×C₄	—	8	2	no	—
135	C₂²×C₄	—	8	2	no	—
136	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄	#120	1	1	yes	characteristic
138	C₂²×C₄	#121	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂²×C₄	#123	1	1	yes	characteristic
141	C₂²×C₄	#127	1	1	yes	characteristic
142	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
143	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
144	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
146	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
147	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂²×C₄	#137	1	1	yes	characteristic
149	C₂²×C₄	#137	1	1	yes	characteristic
150	M₁₆	—	4	2	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	8	1	no	—
153	M₁₆	—	8	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	8	1	no	—
162	C₂×C₈	—	8	1	no	—
163	C₂×C₈	#106	1	4	yes	—
164	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×C₈	C₄×D₄	1	1	yes	characteristic
166	C₂×C₈	C₄×D₄	1	2	yes	—
167	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
168	C₂×C₈	#120	1	2	yes	—
169	C₂×C₈	#121	1	2	yes	—
170	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
171	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
172	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
173	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
174	C₂×C₈	#123	1	2	yes	—
175	C₂×C₈	#125	1	2	yes	—
176	C₂×C₈	#125	1	1	yes	characteristic
177	C₂×C₈	C₄⋊Q₈	1	2	yes	—
178	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
179	C₂×C₈	C₄⋊Q₈	1	2	yes	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	8	2	no	—
185	C₄⋊C₄	—	8	2	no	—
186	C₂²⋊C₄	—	8	1	no	—
187	C₂²⋊C₄	—	8	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	8	1	no	—
192	C₄²	—	8	1	no	—
193	C₄²	#106	1	4	yes	—
194	C₁₆	—	2	8	no	—
195	C₂³	—	2	4	no	—
196	C₂³	#428	1	1	yes	characteristic
197	C₂³	#450	1	1	yes	characteristic
198	C₂³	#460	1	1	yes	characteristic
199	C₂³	#476	1	1	yes	characteristic
200	C₂³	#481	1	1	yes	characteristic
201	C₂³	#496	1	1	yes	characteristic
202	C₂³	#501	1	1	yes	characteristic
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	8	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	—	8	4	no	—
210	C₂×C₄	—	8	4	no	—
211	C₂×C₄	—	16	4	no	—
212	C₂×C₄	—	16	4	no	—
213	C₂×C₄	#384	1	1	yes	characteristic
214	C₂×C₄	#385	1	1	yes	characteristic
215	C₂×C₄	#389	1	1	yes	characteristic
216	C₂×C₄	#392	1	1	yes	characteristic
217	C₂×C₄	#395	1	1	yes	characteristic
218	C₂×C₄	#398	1	1	yes	characteristic
219	C₂×C₄	#399	1	1	yes	characteristic
220	C₂×C₄	C₂×#106	1	1	yes	characteristic
221	C₂×C₄	C₂×#106	1	1	yes	characteristic
222	C₂×C₄	#428	1	1	yes	characteristic
223	C₂×C₄	#448	1	1	yes	characteristic
224	C₂×C₄	#448	1	1	yes	characteristic
225	C₂×C₄	#450	1	1	yes	characteristic
226	C₂×C₄	#460	1	1	yes	characteristic
227	C₂×C₄	#461	1	1	yes	characteristic
228	C₂×C₄	#461	1	1	yes	characteristic
229	C₂×C₄	#475	1	1	yes	characteristic
230	C₂×C₄	#475	1	1	yes	characteristic
231	C₂×C₄	#476	1	1	yes	characteristic
232	C₂×C₄	#481	1	1	yes	characteristic
233	C₂×C₄	#483	1	1	yes	characteristic
234	C₂×C₄	#483	1	1	yes	characteristic
235	C₂×C₄	#492	1	1	yes	characteristic
236	C₂×C₄	#492	1	1	yes	characteristic
237	C₂×C₄	#497	1	1	yes	characteristic
238	C₂×C₄	#498	1	1	yes	characteristic
239	C₂×C₄	#498	1	1	yes	characteristic
240	C₂×C₄	#501	1	1	yes	characteristic
241	C₈	—	2	4	no	—
242	C₈	—	8	2	no	—
243	C₈	#385	1	2	yes	—
244	C₈	#389	1	2	yes	—
245	C₈	#395	1	2	yes	—
246	C₈	#398	1	2	yes	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	#1847	1	1	yes	characteristic
254	C₂²	#1853	1	1	yes	characteristic
255	C₂²	#1948	1	1	yes	characteristic
256	C₂²	#2014	1	1	yes	characteristic
257	C₂²	#2058	1	1	yes	characteristic
258	C₂²	#2084	1	1	yes	characteristic
259	C₂²	#2094	1	1	yes	characteristic
260	C₂²	#2231	1	4	yes	—
261	C₄	—	4	4	no	—
262	C₄	—	16	4	no	—
263	C₄	—	16	2	no	—
264	C₄	#1444	1	1	yes	characteristic
265	C₄	#1844	1	1	yes	characteristic
266	C₄	#1853	1	1	yes	characteristic
267	C₄	#1948	1	1	yes	characteristic
268	C₄	#2013	1	1	yes	characteristic
269	C₄	#2058	1	1	yes	characteristic
270	C₄	#2084	1	1	yes	characteristic
271	C₄	#2096	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#11323	1	1	yes	characteristic
277	C₂	#13187	1	2	yes	—
278	C₂	#13458	1	2	yes	—
279	C₂	#13534	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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

Maximal subgroups

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

subgroup	index
C₂²×#365	2
C₂×#1846	2
C₂×#1386	2
C₂×#1376	2
#10859	2