Subgroups — #149205

Order 512.

548 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#368	C₂	1	2	yes	maximal
2	C₂×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×#1381	C₂	1	2	yes	maximal
4	C₂×#1374	C₂	1	1	yes	maximal, characteristic
5	#10864	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₆	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	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂×#423	—	2	1	no	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×#368	—	2	4	no	—
14	C₂×#368	—	2	4	no	—
15	C₂×#368	—	2	4	no	—
16	#1848	—	2	4	no	—
17	C₂×#341	—	2	1	no	—
18	#1728	—	2	1	no	—
19	#1381	—	2	4	no	—
20	#1374	C₄	1	2	yes	—
21	#1374	C₄	1	2	yes	—
22	C₂²×M₁₆	—	2	2	no	—
23	C₂²×M₁₆	—	2	1	no	—
24	C₂²×M₁₆	—	2	1	no	—
25	C₂³×C₈	D₄	1	1	yes	characteristic
26	C₂³×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₄²	D₄	1	1	yes	characteristic
28	C₂²×C₁₆	—	2	2	no	—
29	C₂²×C₁₆	—	2	2	no	—
30	C₂²×C₁₆	—	2	2	no	—
31	C₂²×C₁₆	—	2	2	no	—
32	C₂²×C₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₁₆	D₄	1	4	yes	—
34	C₂²×C₁₆	Q₈	1	4	yes	—
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	2	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	—	2	4	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	2	1	no	—
46	C₂×#108	—	2	1	no	—
47	C₂×#108	—	4	2	no	—
48	C₂×#108	—	4	2	no	—
49	C₂×#108	D₄	1	4	yes	—
50	C₂×#108	D₄	1	2	yes	—
51	C₂×#108	Q₈	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	#368	—	2	16	no	—
62	#341	—	4	2	no	—
63	#336	—	4	2	no	—
64	C₂³×C₄	Q₁₆	1	2	yes	—
65	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂×M₁₆	—	2	4	no	—
67	C₂×M₁₆	—	2	2	no	—
68	C₂×M₁₆	—	2	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	1	no	—
78	C₂×M₁₆	—	4	1	no	—
79	C₂×M₁₆	—	4	1	no	—
80	C₂×M₁₆	—	4	2	no	—
81	C₂×M₁₆	—	4	1	no	—
82	C₂×M₁₆	—	4	2	no	—
83	C₂×M₁₆	—	4	1	no	—
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₂²×C₈	—	4	1	no	—
89	C₂²×C₈	—	4	1	no	—
90	C₂²×C₈	—	4	2	no	—
91	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	D₈	1	2	yes	—
94	C₂²×C₈	Q₁₆	1	2	yes	—
95	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₄	1	2	yes	—
97	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₄	1	2	yes	—
99	C₂²×C₈	C₂×Q₈	1	2	yes	—
100	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₈	#41	1	2	yes	—
103	C₂²×C₈	#41	1	1	yes	characteristic
104	C₂²×C₈	#41	1	1	yes	characteristic
105	C₂×C₄²	—	2	1	no	—
106	C₂×C₄²	—	2	1	no	—
107	C₂×C₄²	—	2	1	no	—
108	C₂×C₄²	—	2	1	no	—
109	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄²	D₈	1	2	yes	—
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	—	2	4	no	—
115	C₂×C₁₆	—	2	4	no	—
116	C₂×C₁₆	—	2	4	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
119	#108	—	2	4	no	—
120	#108	—	2	4	no	—
121	#108	—	2	4	no	—
122	#108	—	2	4	no	—
123	#108	—	4	8	no	—
124	#108	—	4	2	no	—
125	#108	—	4	2	no	—
126	#108	—	4	2	no	—
127	#108	—	4	2	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	C₂×Q₁₆	1	1	yes	characteristic
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	#107	1	2	yes	—
139	C₂²×C₄	#107	1	2	yes	—
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×D₈	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
150	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂²×C₄	#137	1	1	yes	characteristic
153	C₂²×C₄	#137	1	1	yes	characteristic
154	M₁₆	—	4	2	no	—
155	M₁₆	—	4	4	no	—
156	M₁₆	—	4	2	no	—
157	M₁₆	—	4	4	no	—
158	M₁₆	—	4	2	no	—
159	M₁₆	—	4	2	no	—
160	M₁₆	—	8	1	no	—
161	M₁₆	—	8	2	no	—
162	M₁₆	—	8	1	no	—
163	M₁₆	—	8	1	no	—
164	M₁₆	—	8	2	no	—
165	M₁₆	—	8	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	4	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	8	1	no	—
178	C₂×C₈	—	8	2	no	—
179	C₂×C₈	—	8	1	no	—
180	C₂×C₈	—	8	1	no	—
181	C₂×C₈	—	8	1	no	—
182	C₂×C₈	—	8	2	no	—
183	C₂×C₈	#107	1	2	yes	—
184	C₂×C₈	#107	1	2	yes	—
185	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
186	C₂×C₈	C₄×D₄	1	1	yes	characteristic
187	C₂×C₈	C₄×D₄	1	2	yes	—
188	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
189	C₂×C₈	#120	1	2	yes	—
190	C₂×C₈	#121	1	2	yes	—
191	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
192	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
193	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
194	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
195	C₂×C₈	#123	1	2	yes	—
196	C₂×C₈	#125	1	2	yes	—
197	C₂×C₈	#125	1	1	yes	characteristic
198	C₂×C₈	C₄⋊Q₈	1	2	yes	—
199	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
200	C₂×C₈	C₄⋊Q₈	1	2	yes	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₁₆	—	2	8	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₂×#107	1	1	yes	characteristic
211	C₂³	#451	1	1	yes	characteristic
212	C₂³	#462	1	1	yes	characteristic
213	C₂³	#477	1	1	yes	characteristic
214	C₂³	#484	1	1	yes	characteristic
215	C₂³	#494	1	1	yes	characteristic
216	C₂³	#500	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	#368	1	4	yes	complemented
225	C₂×C₄	#384	1	1	yes	characteristic
226	C₂×C₄	#385	1	1	yes	characteristic
227	C₂×C₄	#389	1	1	yes	characteristic
228	C₂×C₄	#392	1	1	yes	characteristic
229	C₂×C₄	#395	1	1	yes	characteristic
230	C₂×C₄	#398	1	1	yes	characteristic
231	C₂×C₄	#399	1	1	yes	characteristic
232	C₂×C₄	C₂×#107	1	1	yes	characteristic
233	C₂×C₄	#428	1	1	yes	characteristic
234	C₂×C₄	#428	1	1	yes	characteristic
235	C₂×C₄	#447	1	1	yes	characteristic
236	C₂×C₄	#447	1	1	yes	characteristic
237	C₂×C₄	#451	1	1	yes	characteristic
238	C₂×C₄	#459	1	1	yes	characteristic
239	C₂×C₄	#459	1	1	yes	characteristic
240	C₂×C₄	#462	1	1	yes	characteristic
241	C₂×C₄	#474	1	1	yes	characteristic
242	C₂×C₄	#474	1	1	yes	characteristic
243	C₂×C₄	#477	1	1	yes	characteristic
244	C₂×C₄	#480	1	1	yes	characteristic
245	C₂×C₄	#480	1	1	yes	characteristic
246	C₂×C₄	#484	1	1	yes	characteristic
247	C₂×C₄	#493	1	1	yes	characteristic
248	C₂×C₄	#496	1	1	yes	characteristic
249	C₂×C₄	#497	1	1	yes	characteristic
250	C₂×C₄	#500	1	1	yes	characteristic
251	C₂×C₄	#501	1	1	yes	characteristic
252	C₂×C₄	#501	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	2	no	—
255	C₈	—	8	2	no	—
256	C₈	—	8	4	no	—
257	C₈	—	8	2	no	—
258	C₈	#385	1	2	yes	—
259	C₈	#389	1	2	yes	—
260	C₈	#395	1	2	yes	—
261	C₈	#398	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	#1845	1	1	yes	characteristic
271	C₂²	#1852	1	1	yes	characteristic
272	C₂²	#1947	1	1	yes	characteristic
273	C₂²	#2016	1	1	yes	characteristic
274	C₂²	#2057	1	1	yes	characteristic
275	C₂²	#2083	1	1	yes	characteristic
276	C₂²	#2095	1	1	yes	characteristic
277	C₂²	C₂×#368	1	2	yes	—
278	C₂²	#2161	1	2	yes	—
279	C₄	—	2	4	no	—
280	C₄	—	4	2	no	—
281	C₄	#1444	1	1	yes	characteristic
282	C₄	#1847	1	1	yes	characteristic
283	C₄	#1852	1	1	yes	characteristic
284	C₄	#1947	1	1	yes	characteristic
285	C₄	#2011	1	1	yes	characteristic
286	C₄	#2057	1	1	yes	characteristic
287	C₄	#2083	1	1	yes	characteristic
288	C₄	#2093	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	#11324	1	1	yes	characteristic
294	C₂	#12909	1	1	yes	characteristic
295	C₂	#12910	1	1	yes	characteristic
296	C₂	#12922	1	2	yes	—
297	C₂	#13179	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ 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
224. C₂×C₄	#368	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#368	2
C₂×#1848	2
C₂×#1381	2
C₂×#1374	2
#10864	2