Subgroups — #498537

Order 512.

440 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33728	C₂	1	1	yes	maximal, characteristic
2	#23170	C₂	1	2	yes	maximal
3	#13274	C₂	1	2	yes	maximal
4	#13170	C₂	1	1	yes	maximal, characteristic
5	#13169	C₂	1	1	yes	maximal, characteristic
6	#13113	C₂	1	2	yes	maximal
7	#13113	C₂	1	2	yes	maximal
8	#12676	C₂	1	1	yes	maximal, characteristic
9	#12671	C₂	1	1	yes	maximal, characteristic
10	#12670	C₂	1	1	yes	maximal, characteristic
11	#12663	C₂	1	1	yes	maximal, characteristic
12	#3160	C₂²	1	2	yes	—
13	#2962	—	2	2	no	—
14	#2255	C₄	1	2	yes	complemented
15	#2255	C₄	1	2	yes	complemented
16	#2255	C₄	1	2	yes	complemented
17	#2255	C₄	1	2	yes	complemented
18	C₂×#368	C₂²	1	1	yes	characteristic
19	C₂×#368	C₂²	1	1	yes	characteristic
20	#2158	C₂²	1	2	yes	—
21	#2158	C₂²	1	1	yes	characteristic
22	#2158	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₂×#346	C₂²	1	1	yes	characteristic
25	C₂×#346	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
27	#1986	—	2	2	no	—
28	#1945	C₂²	1	2	yes	—
29	#1945	C₂²	1	2	yes	—
30	#1922	—	2	2	no	—
31	#1922	—	2	2	no	—
32	C₄×#108	C₂²	1	2	yes	—
33	#1772	C₂²	1	2	yes	—
34	#1772	C₂²	1	2	yes	—
35	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	#1382	C₂²	1	2	yes	—
37	#1382	C₂²	1	2	yes	—
38	#1382	C₂²	1	2	yes	—
39	#1382	C₂²	1	2	yes	—
40	#1381	C₂²	1	2	yes	—
41	#1374	C₂²	1	1	yes	characteristic
42	#1374	C₂²	1	1	yes	characteristic
43	#533	—	2	2	no	—
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	C₂²×C₁₆	C₂³	1	1	yes	characteristic
46	C₂²×C₁₆	C₂³	1	1	yes	characteristic
47	C₂²×C₁₆	C₂³	1	1	yes	characteristic
48	#500	C₂×C₄	1	2	yes	—
49	#500	C₂×C₄	1	2	yes	—
50	#499	—	2	2	no	—
51	#499	C₂×C₄	1	2	yes	—
52	#499	C₂×C₄	1	2	yes	—
53	#498	—	2	2	no	—
54	#446	—	4	2	no	—
55	#446	—	4	2	no	—
56	#446	—	4	2	no	—
57	C₈×Q₈	—	4	2	no	—
58	C₂×#108	C₂³	1	2	yes	—
59	C₂×#108	C₂³	1	2	yes	—
60	#427	—	2	2	no	—
61	#427	C₂³	1	2	yes	—
62	#427	C₂³	1	2	yes	—
63	#424	—	2	2	no	—
64	#424	—	2	2	no	—
65	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
66	#405	—	2	2	no	—
67	#405	—	2	2	no	—
68	C₄×M₁₆	—	2	2	no	—
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
73	#368	—	2	2	no	—
74	#368	Q₈	1	4	yes	complemented
75	#367	C₂×C₄	1	2	yes	—
76	#367	C₂×C₄	1	2	yes	—
77	#367	C₂×C₄	1	2	yes	—
78	#367	C₂×C₄	1	2	yes	—
79	#366	C₂×C₄	1	2	yes	—
80	#366	C₂×C₄	1	2	yes	—
81	#366	C₂×C₄	1	1	yes	characteristic
82	#366	C₂×C₄	1	1	yes	characteristic
83	#366	C₂×C₄	1	1	yes	characteristic
84	#366	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₁₆	—	2	2	no	—
86	#346	—	2	2	no	—
87	#346	C₂×C₄	1	2	yes	—
88	#346	C₂×C₄	1	2	yes	—
89	C₄×C₁₆	—	2	2	no	—
90	#341	—	2	2	no	—
91	#339	—	2	2	no	—
92	#339	—	2	2	no	—
93	#339	—	2	2	no	—
94	#339	—	2	2	no	—
95	#336	D₄	1	1	yes	characteristic
96	#336	D₄	1	1	yes	characteristic
97	C₂×M₁₆	—	2	2	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₈	C₂⁴	1	1	yes	characteristic
103	C₄⋊Q₈	—	2	2	no	—
104	C₄⋊Q₈	—	2	2	no	—
105	#125	—	2	2	no	—
106	#125	—	2	2	no	—
107	C₂²⋊Q₈	—	4	2	no	—
108	C₂²⋊Q₈	—	4	2	no	—
109	C₄×Q₈	—	4	2	no	—
110	C₄×Q₈	—	4	2	no	—
111	#117	—	2	2	no	—
112	#117	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	1	no	—
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	C₂²×C₄	1	2	yes	—
126	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₁₆	C₂×Q₈	1	2	yes	—
129	C₂×C₁₆	#41	1	2	yes	—
130	#108	—	2	2	no	—
131	#108	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	C₂×Q₈	1	4	yes	—
134	#107	—	2	2	no	—
135	#107	—	2	2	no	—
136	#107	C₂²×C₄	1	2	yes	—
137	#107	C₂²×C₄	1	2	yes	—
138	#107	C₂²×C₄	1	2	yes	—
139	#107	C₂²×C₄	1	2	yes	—
140	#106	—	2	2	no	—
141	#106	—	2	2	no	—
142	#106	—	2	2	no	—
143	#106	—	2	2	no	—
144	C₄⋊C₈	—	4	2	no	—
145	C₄⋊C₈	—	4	2	no	—
146	C₄⋊C₈	—	4	2	no	—
147	C₄⋊C₈	—	4	2	no	—
148	C₄⋊C₈	D₈	1	2	yes	—
149	C₄⋊C₈	D₈	1	2	yes	—
150	#97	—	4	2	no	—
151	#97	—	4	2	no	—
152	#97	—	4	2	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	4	2	no	—
158	C₄×C₈	—	4	2	no	—
159	C₄×C₈	—	4	2	no	—
160	C₄×C₈	C₂²×C₄	1	2	yes	—
161	C₄×C₈	#41	1	2	yes	—
162	#95	—	2	1	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	#135	1	1	yes	characteristic
168	C₂²×C₄	#135	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
170	M₁₆	—	2	4	no	—
171	M₁₆	—	4	2	no	—
172	M₁₆	—	4	2	no	—
173	M₁₆	—	4	2	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	C₄×D₄	1	2	yes	—
183	C₂×C₈	C₂×D₈	1	1	yes	characteristic
184	C₂×C₈	C₂×D₈	1	2	yes	—
185	C₂×C₈	C₂×D₈	1	1	yes	characteristic
186	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
188	C₂×C₈	C₂×#41	1	1	yes	characteristic
189	C₂×C₈	C₂×#41	1	1	yes	characteristic
190	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
191	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	4	2	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	4	2	no	—
210	C₄²	—	4	2	no	—
211	C₄²	C₄×D₄	1	2	yes	—
212	C₄²	C₂×D₈	1	2	yes	—
213	C₁₆	—	2	2	no	—
214	C₁₆	—	4	1	no	—
215	C₁₆	—	4	1	no	—
216	C₁₆	C₄×Q₈	1	2	yes	—
217	C₁₆	C₄×Q₈	1	2	yes	—
218	C₂³	C₂×#135	1	1	yes	characteristic
219	Q₈	—	8	4	no	—
220	Q₈	—	16	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₄×D₈	1	2	yes	—
234	C₂×C₄	C₄×D₈	1	2	yes	—
235	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂×C₄	#547	1	1	yes	characteristic
237	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
238	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
239	C₂×C₄	#578	1	1	yes	characteristic
240	C₂×C₄	#578	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	4	4	no	—
246	C₈	—	8	2	no	—
247	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
248	C₈	#519	1	1	yes	characteristic
249	C₈	#556	1	1	yes	characteristic
250	C₈	#557	1	1	yes	characteristic
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
254	C₂²	#3342	1	1	yes	characteristic
255	C₂²	#3383	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	8	1	no	—
260	C₄	—	8	2	no	—
261	C₄	—	8	2	no	—
262	C₄	—	8	2	no	—
263	C₄	—	8	2	no	—
264	C₄	#2365	1	1	yes	characteristic
265	C₄	#2950	1	1	yes	characteristic
266	C₄	#3334	1	1	yes	characteristic
267	C₄	Q₈×D₈	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	#23157	1	1	yes	characteristic
271	C₂	#33463	1	1	yes	characteristic
272	C₂	#33464	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. #2255	C₄	semidirect
15. #2255	C₄	semidirect
16. #2255	C₄	semidirect
17. #2255	C₄	semidirect
74. #368	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33728	2
#23170	2
#13274	2
#13170	2
#13169	2
#13113	2
#12676	2
#12671	2
#12670	2
#12663	2