Subgroups — #498552

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 277 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33713	C₂	1	1	yes	maximal, characteristic
2	C₂×#2178	C₂	1	2	yes	maximal, complemented
3	#23179	C₂	1	2	yes	maximal, complemented
4	#13251	C₂	1	2	yes	maximal, complemented
5	#13241	C₂	1	2	yes	maximal, complemented
6	#13011	C₂	1	4	yes	maximal, complemented
7	C₄×#368	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#508	C₂²	1	2	yes	—
10	C₂×#508	C₂²	1	2	yes	—
11	#3147	C₂²	1	2	yes	—
12	C₂×#443	C₂²	1	4	yes	—
13	#2935	—	2	2	no	—
14	#2249	C₄	1	2	yes	complemented
15	#2249	C₄	1	2	yes	complemented
16	#2248	C₄	1	2	yes	complemented
17	#2247	C₄	1	1	yes	characteristic, complemented
18	#2246	C₄	1	1	yes	characteristic, complemented
19	#2178	—	2	8	no	—
20	C₂×#368	C₂²	1	1	yes	characteristic
21	C₂×#368	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₂×#359	C₂²	1	4	yes	—
24	C₂×#359	C₂²	1	4	yes	—
25	C₂×C₄×C₁₆	C₂²	1	2	yes	—
26	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
27	#1980	—	2	2	no	—
28	#1974	C₂²	1	4	yes	—
29	#1876	—	2	4	no	—
30	C₄×#108	C₂²	1	2	yes	—
31	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	#1381	C₂²	1	2	yes	—
33	#1374	C₂²	1	2	yes	—
34	C₂×#135	—	2	2	no	—
35	C₂×#135	C₂³	1	4	yes	—
36	C₂×#131	—	2	4	no	—
37	#532	—	2	2	no	—
38	#508	—	2	4	no	—
39	#508	—	2	4	no	—
40	#508	—	4	2	no	—
41	#508	—	4	2	no	—
42	C₂×Q₃₂	C₂×C₄	1	2	yes	—
43	C₂×Q₃₂	C₂×C₄	1	2	yes	—
44	C₂×SD₃₂	C₂×C₄	1	4	yes	—
45	C₂×SD₃₂	C₂×C₄	1	4	yes	—
46	C₂×D₁₆	C₂×C₄	1	2	yes	—
47	C₂×D₁₆	C₂×C₄	1	2	yes	—
48	C₂²×C₁₆	C₂³	1	2	yes	—
49	C₂²×C₁₆	C₂³	1	1	yes	characteristic
50	C₂²×C₁₆	C₂³	1	1	yes	characteristic
51	#495	—	2	2	no	—
52	#495	C₂×C₄	1	4	yes	—
53	#494	C₂×C₄	1	2	yes	—
54	#493	C₂×C₄	1	2	yes	—
55	#492	—	2	2	no	—
56	#443	—	2	8	no	—
57	#443	—	4	4	no	—
58	#436	—	4	2	no	—
59	#435	—	4	4	no	—
60	C₈×D₄	—	4	2	no	—
61	C₂×#108	C₂³	1	2	yes	—
62	C₂×#108	C₂³	1	2	yes	—
63	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₂×#104	—	2	4	no	—
65	C₂×#104	—	2	4	no	—
66	#408	—	2	4	no	—
67	C₄×M₁₆	—	2	2	no	—
68	C₂×C₄×C₈	C₂³	1	2	yes	—
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
71	#368	—	2	2	no	—
72	#368	D₄	1	4	yes	complemented
73	C₄⋊C₁₆	—	2	1	no	—
74	C₄⋊C₁₆	—	2	1	no	—
75	#359	—	2	8	no	—
76	#359	—	4	4	no	—
77	C₄×C₁₆	—	2	4	no	—
78	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
79	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
80	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
81	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
82	#341	—	2	2	no	—
83	#336	D₄	1	2	yes	—
84	C₂×#41	—	2	4	no	—
85	#135	—	2	4	no	—
86	#135	—	2	4	no	—
87	#135	—	2	4	no	—
88	#135	—	2	4	no	—
89	#135	—	8	2	no	—
90	#135	—	8	2	no	—
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	C₂²×C₄	1	4	yes	—
93	C₂×SD₁₆	—	2	4	no	—
94	C₂×SD₁₆	—	2	4	no	—
95	C₂×D₈	—	2	2	no	—
96	C₂×D₈	C₂²×C₄	1	4	yes	—
97	#131	—	4	8	no	—
98	#131	—	8	4	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	4	4	no	—
102	C₂²×C₈	—	4	4	no	—
103	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₈	C₂×D₄	1	2	yes	—
105	C₂²×C₈	C₂⁴	1	1	yes	characteristic
106	C₄⋊Q₈	—	2	2	no	—
107	#127	—	2	2	no	—
108	#124	—	2	4	no	—
109	#121	—	4	2	no	—
110	#121	—	4	2	no	—
111	C₄×D₄	—	4	2	no	—
112	C₄×D₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	2	1	no	—
114	C₂×C₄²	C₂×D₄	1	2	yes	—
115	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
116	Q₃₂	—	2	4	no	—
117	Q₃₂	—	4	2	no	—
118	SD₃₂	—	2	8	no	—
119	SD₃₂	—	4	4	no	—
120	D₁₆	—	2	4	no	—
121	D₁₆	—	4	2	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₁₆	C₂×D₄	1	2	yes	—
132	C₂×C₁₆	#41	1	2	yes	—
133	#108	—	2	2	no	—
134	#108	—	2	2	no	—
135	#108	—	2	2	no	—
136	#108	C₂×D₄	1	4	yes	—
137	C₄⋊C₈	D₈	1	1	yes	characteristic
138	C₄⋊C₈	D₈	1	2	yes	—
139	C₄⋊C₈	D₈	1	1	yes	characteristic
140	#104	—	4	8	no	—
141	#104	—	8	4	no	—
142	C₂²⋊C₈	—	4	4	no	—
143	C₂²⋊C₈	—	4	4	no	—
144	#97	—	4	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	4	2	no	—
148	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
149	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
150	C₄×C₈	#41	1	1	yes	characteristic
151	C₄×C₈	#41	1	1	yes	characteristic
152	#95	—	2	1	no	—
153	#41	—	4	4	no	—
154	#41	—	4	4	no	—
155	#41	—	4	4	no	—
156	#41	—	4	4	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	4	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	#135	1	1	yes	characteristic
165	C₂²×C₄	#135	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
167	Q₁₆	—	2	4	no	—
168	Q₁₆	—	2	4	no	—
169	Q₁₆	—	8	2	no	—
170	SD₁₆	—	4	4	no	—
171	SD₁₆	—	4	4	no	—
172	SD₁₆	—	8	4	no	—
173	D₈	—	2	4	no	—
174	D₈	—	2	4	no	—
175	D₈	—	8	2	no	—
176	M₁₆	—	2	4	no	—
177	M₁₆	—	4	2	no	—
178	M₁₆	—	4	2	no	—
179	M₁₆	—	4	2	no	—
180	M₁₆	—	8	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	8	8	no	—
188	C₂×C₈	C₄×D₄	1	1	yes	characteristic
189	C₂×C₈	C₄×D₄	1	1	yes	characteristic
190	C₂×C₈	C₂×D₈	1	2	yes	—
191	C₂×C₈	C₂×D₈	1	2	yes	—
192	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₈	C₂×#41	1	1	yes	characteristic
195	C₂×C₈	C₂×#41	1	1	yes	characteristic
196	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₂²⋊C₄	—	4	4	no	—
202	C₂²⋊C₄	—	4	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	C₄×D₄	1	1	yes	characteristic
206	C₄²	C₄×D₄	1	1	yes	characteristic
207	C₄²	C₂×D₈	1	1	yes	characteristic
208	C₄²	C₂×D₈	1	1	yes	characteristic
209	C₁₆	—	2	1	no	—
210	C₁₆	—	2	1	no	—
211	C₁₆	—	4	2	no	—
212	C₁₆	C₄×D₄	1	2	yes	—
213	C₁₆	C₄×D₄	1	2	yes	—
214	C₂³	—	4	4	no	—
215	C₂³	C₂×#135	1	1	yes	characteristic
216	Q₈	—	4	4	no	—
217	Q₈	—	4	4	no	—
218	D₄	—	4	4	no	—
219	D₄	—	4	4	no	—
220	D₄	—	8	4	no	—
221	D₄	—	8	4	no	—
222	D₄	—	8	4	no	—
223	D₄	—	16	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	C₄×D₈	1	1	yes	characteristic
233	C₂×C₄	C₄×D₈	1	1	yes	characteristic
234	C₂×C₄	C₄×D₈	1	1	yes	characteristic
235	C₂×C₄	C₄×D₈	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
237	C₂×C₄	D₄²	1	1	yes	characteristic
238	C₂×C₄	#550	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
240	C₂×C₄	#576	1	1	yes	characteristic
241	C₂×C₄	#576	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	4	4	no	—
247	C₈	—	8	2	no	—
248	C₈	C₂×C₄×D₄	1	1	yes	characteristic
249	C₈	#518	1	1	yes	characteristic
250	C₈	#550	1	1	yes	characteristic
251	C₈	#559	1	1	yes	characteristic
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	8	4	no	—
255	C₂²	—	8	4	no	—
256	C₂²	—	8	4	no	—
257	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
258	C₂²	#3287	1	1	yes	characteristic
259	C₂²	#3396	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	4	no	—
264	C₄	—	8	4	no	—
265	C₄	#2178	1	2	yes	complemented
266	C₄	#2366	1	1	yes	characteristic
267	C₄	#2948	1	1	yes	characteristic
268	C₄	D₄×D₈	1	1	yes	characteristic
269	C₄	#3391	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	16	4	no	—
273	C₂	#23185	1	1	yes	characteristic
274	C₂	C₂×#2178	1	1	yes	characteristic
275	C₂	#33465	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	#33713	C₂
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
2. C₂×#2178	C₂	semidirect
3. #23179	C₂	semidirect
4. #13251	C₂	semidirect
5. #13241	C₂	semidirect
6. #13011	C₂	semidirect
7. C₄×#368	C₂	semidirect
8. C₁₆×C₄⋊C₄	C₂	semidirect
14. #2249	C₄	semidirect
15. #2249	C₄	semidirect
16. #2248	C₄	semidirect
17. #2247	C₄	semidirect
18. #2246	C₄	semidirect
72. #368	D₄	semidirect
265. C₄	#2178	semidirect

Maximal subgroups

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

subgroup	index
#33713	2
C₂×#2178	2
#23179	2
#13251	2
#13241	2
#13011	2
C₄×#368	2
C₁₆×C₄⋊C₄	2