Subgroups — #148982

Order 512.

504 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1848	C₂	1	1	yes	maximal, characteristic
2	C₂×#1381	C₂	1	2	yes	maximal
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	#1848	—	2	4	no	—
14	C₂×#341	—	2	1	no	—
15	#1728	—	2	1	no	—
16	#1381	—	2	4	no	—
17	#1381	—	2	4	no	—
18	#1374	C₄	1	2	yes	—
19	#1374	C₄	1	2	yes	—
20	C₂²×M₁₆	—	2	2	no	—
21	C₂²×M₁₆	—	2	1	no	—
22	C₂²×M₁₆	—	2	1	no	—
23	C₂³×C₈	D₄	1	1	yes	characteristic
24	C₂³×C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₄²	D₄	1	1	yes	characteristic
26	C₂²×C₁₆	—	2	2	no	—
27	C₂²×C₁₆	—	2	2	no	—
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	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	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	—	2	1	no	—
45	C₂×#108	—	4	2	no	—
46	C₂×#108	—	4	2	no	—
47	C₂×#108	D₄	1	4	yes	—
48	C₂×#108	D₄	1	2	yes	—
49	C₂×#108	Q₈	1	2	yes	—
50	#423	—	2	4	no	—
51	#423	—	4	2	no	—
52	C₂×C₄⋊C₈	—	4	2	no	—
53	C₂×C₄×C₈	—	2	2	no	—
54	C₂×C₄×C₈	—	2	2	no	—
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
59	#341	—	4	2	no	—
60	#336	—	4	2	no	—
61	C₂³×C₄	Q₁₆	1	2	yes	—
62	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×M₁₆	—	2	4	no	—
64	C₂×M₁₆	—	2	2	no	—
65	C₂×M₁₆	—	2	2	no	—
66	C₂×M₁₆	—	4	1	no	—
67	C₂×M₁₆	—	4	2	no	—
68	C₂×M₁₆	—	4	1	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	1	no	—
76	C₂×M₁₆	—	4	1	no	—
77	C₂×M₁₆	—	4	2	no	—
78	C₂×M₁₆	—	4	1	no	—
79	C₂×M₁₆	—	4	2	no	—
80	C₂×M₁₆	—	4	1	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	—	4	1	no	—
86	C₂²×C₈	—	4	1	no	—
87	C₂²×C₈	—	4	2	no	—
88	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	SD₁₆	1	2	yes	—
91	C₂²×C₈	SD₁₆	1	2	yes	—
92	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	2	yes	—
95	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₈	#41	1	2	yes	—
98	C₂²×C₈	#41	1	2	yes	—
99	C₂²×C₈	#41	1	2	yes	—
100	C₂²×C₈	#41	1	1	yes	characteristic
101	C₂²×C₈	#41	1	1	yes	characteristic
102	C₂×C₄²	—	2	1	no	—
103	C₂×C₄²	—	2	1	no	—
104	C₂×C₄²	—	2	1	no	—
105	C₂×C₄²	—	2	1	no	—
106	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	D₈	1	2	yes	—
109	C₂×C₁₆	—	2	4	no	—
110	C₂×C₁₆	—	2	4	no	—
111	C₂×C₁₆	—	2	4	no	—
112	C₂×C₁₆	—	4	2	no	—
113	C₂×C₁₆	—	4	2	no	—
114	C₂×C₁₆	—	4	2	no	—
115	C₂×C₁₆	—	4	2	no	—
116	#108	—	2	4	no	—
117	#108	—	2	4	no	—
118	#108	—	2	4	no	—
119	#108	—	2	4	no	—
120	#108	—	4	8	no	—
121	#108	—	4	2	no	—
122	#108	—	4	2	no	—
123	#108	—	4	2	no	—
124	#108	—	4	2	no	—
125	C₄⋊C₈	—	4	8	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	#107	1	2	yes	—
136	C₂²×C₄	#107	1	2	yes	—
137	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₄	#120	1	1	yes	characteristic
139	C₂²×C₄	#121	1	1	yes	characteristic
140	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₄	#123	1	1	yes	characteristic
142	C₂²×C₄	#124	1	1	yes	characteristic
143	C₂²×C₄	#124	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₈	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	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂²×C₄	#137	1	1	yes	characteristic
150	C₂²×C₄	#137	1	1	yes	characteristic
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	4	no	—
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	4	no	—
155	M₁₆	—	4	2	no	—
156	M₁₆	—	4	2	no	—
157	M₁₆	—	8	1	no	—
158	M₁₆	—	8	2	no	—
159	M₁₆	—	8	1	no	—
160	M₁₆	—	8	1	no	—
161	M₁₆	—	8	1	no	—
162	M₁₆	—	8	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	4	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	8	1	no	—
175	C₂×C₈	—	8	2	no	—
176	C₂×C₈	—	8	1	no	—
177	C₂×C₈	—	8	1	no	—
178	C₂×C₈	—	8	1	no	—
179	C₂×C₈	—	8	2	no	—
180	C₂×C₈	#106	1	2	yes	—
181	C₂×C₈	#106	1	2	yes	—
182	C₂×C₈	#117	1	2	yes	—
183	C₂×C₈	C₄×D₄	1	2	yes	—
184	C₂×C₈	C₄×D₄	1	1	yes	characteristic
185	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
186	C₂×C₈	#121	1	2	yes	—
187	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
188	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
189	C₂×C₈	#123	1	2	yes	—
190	C₂×C₈	#123	1	2	yes	—
191	C₂×C₈	#124	1	2	yes	—
192	C₂×C₈	#125	1	2	yes	—
193	C₂×C₈	#125	1	2	yes	—
194	C₂×C₈	#125	1	1	yes	characteristic
195	C₂×C₈	#126	1	2	yes	—
196	C₂×C₈	#126	1	2	yes	—
197	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₁₆	—	4	4	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	C₂×#107	1	1	yes	characteristic
208	C₂³	#451	1	1	yes	characteristic
209	C₂³	#462	1	1	yes	characteristic
210	C₂³	#477	1	1	yes	characteristic
211	C₂³	#484	1	1	yes	characteristic
212	C₂³	#487	1	1	yes	characteristic
213	C₂³	#487	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	#368	1	4	yes	—
222	C₂×C₄	#382	1	1	yes	characteristic
223	C₂×C₄	#385	1	1	yes	characteristic
224	C₂×C₄	#389	1	1	yes	characteristic
225	C₂×C₄	#393	1	1	yes	characteristic
226	C₂×C₄	#394	1	1	yes	characteristic
227	C₂×C₄	#397	1	1	yes	characteristic
228	C₂×C₄	#400	1	1	yes	characteristic
229	C₂×C₄	C₂×#106	1	1	yes	characteristic
230	C₂×C₄	#428	1	1	yes	characteristic
231	C₂×C₄	#428	1	1	yes	characteristic
232	C₂×C₄	#447	1	1	yes	characteristic
233	C₂×C₄	#448	1	1	yes	characteristic
234	C₂×C₄	#450	1	1	yes	characteristic
235	C₂×C₄	#459	1	1	yes	characteristic
236	C₂×C₄	#460	1	1	yes	characteristic
237	C₂×C₄	#461	1	1	yes	characteristic
238	C₂×C₄	#474	1	1	yes	characteristic
239	C₂×C₄	#475	1	1	yes	characteristic
240	C₂×C₄	#476	1	1	yes	characteristic
241	C₂×C₄	#480	1	1	yes	characteristic
242	C₂×C₄	#481	1	1	yes	characteristic
243	C₂×C₄	#483	1	1	yes	characteristic
244	C₂×C₄	#486	1	1	yes	characteristic
245	C₂×C₄	#486	1	1	yes	characteristic
246	C₂×C₄	#489	1	1	yes	characteristic
247	C₂×C₄	#489	1	1	yes	characteristic
248	C₂×C₄	#490	1	1	yes	characteristic
249	C₂×C₄	#491	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	8	2	no	—
253	C₈	—	8	2	no	—
254	C₈	—	8	4	no	—
255	C₈	#387	1	2	yes	—
256	C₈	#388	1	2	yes	—
257	C₈	#398	1	2	yes	—
258	C₈	#400	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	#1834	1	1	yes	characteristic
268	C₂²	#1852	1	1	yes	characteristic
269	C₂²	#1947	1	1	yes	characteristic
270	C₂²	#2024	1	1	yes	characteristic
271	C₂²	#2043	1	1	yes	characteristic
272	C₂²	#2072	1	1	yes	characteristic
273	C₂²	#2100	1	1	yes	characteristic
274	C₂²	C₂×#368	1	2	yes	—
275	C₂²	#2161	1	2	yes	—
276	C₄	—	2	4	no	—
277	C₄	—	4	2	no	—
278	C₄	#1442	1	1	yes	characteristic
279	C₄	#1836	1	1	yes	characteristic
280	C₄	#1853	1	1	yes	characteristic
281	C₄	#1948	1	1	yes	characteristic
282	C₄	#2023	1	1	yes	characteristic
283	C₄	#2041	1	1	yes	characteristic
284	C₄	#2071	1	1	yes	characteristic
285	C₄	#2099	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	#11267	1	1	yes	characteristic
291	C₂	#12891	1	1	yes	characteristic
292	C₂	#12892	1	1	yes	characteristic
293	C₂	#12922	1	2	yes	—
294	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

Maximal subgroups

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

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