Subgroups — #497888

Order 512.

614 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	C₂×#2178	C₂	1	2	yes	maximal
2	C₄×#508	C₂	1	1	yes	maximal, characteristic
3	#22688	C₂	1	2	yes	maximal, complemented
4	#13169	C₂	1	2	yes	maximal, complemented
5	#12939	C₂	1	4	yes	maximal, complemented
6	#12909	C₂	1	1	yes	maximal, characteristic, complemented
7	#12815	C₂	1	2	yes	maximal, complemented
8	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#508	C₂²	1	1	yes	characteristic
10	C₂×#443	C₂²	1	4	yes	—
11	C₄×#135	C₂²	1	2	yes	—
12	#2906	—	2	2	no	—
13	#2178	—	2	8	no	—
14	C₄×Q₃₂	C₄	1	1	yes	characteristic, complemented
15	C₄×Q₃₂	C₄	1	1	yes	characteristic, complemented
16	C₄×SD₃₂	C₄	1	2	yes	complemented
17	C₄×SD₃₂	C₄	1	2	yes	complemented
18	C₄×D₁₆	C₄	1	1	yes	characteristic, complemented
19	C₄×D₁₆	C₄	1	1	yes	characteristic, complemented
20	C₂×#368	C₂²	1	2	yes	—
21	C₂×#368	C₂²	1	2	yes	—
22	#2158	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₂×#359	C₂²	1	4	yes	—
25	#2138	C₂²	1	2	yes	—
26	C₂×C₄×C₁₆	C₂²	1	2	yes	—
27	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
28	#1945	C₂²	1	4	yes	—
29	#1866	—	2	4	no	—
30	#1848	C₂²	1	2	yes	—
31	#1806	—	2	2	no	—
32	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	#1382	C₂²	1	4	yes	—
34	#1374	C₂²	1	2	yes	—
35	C₂×#135	C₂³	1	2	yes	—
36	C₂×#131	—	2	4	no	—
37	C₄×#41	—	2	2	no	—
38	#508	D₄	1	4	yes	—
39	#508	Q₈	1	4	yes	—
40	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
41	C₂×SD₃₂	C₂×C₄	1	2	yes	—
42	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₁₆	C₂³	1	2	yes	—
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	C₂²×C₁₆	C₂³	1	1	yes	characteristic
46	#443	—	2	8	no	—
47	#443	—	4	4	no	—
48	C₄×Q₁₆	C₂×C₄	1	2	yes	—
49	C₄×Q₁₆	C₂×C₄	1	2	yes	—
50	C₄×SD₁₆	—	2	2	no	—
51	C₄×SD₁₆	—	2	2	no	—
52	C₄×D₈	C₂×C₄	1	2	yes	—
53	C₄×D₈	C₂×C₄	1	2	yes	—
54	C₂×#108	—	2	2	no	—
55	C₂×#108	C₂³	1	4	yes	—
56	#427	C₂³	1	2	yes	—
57	#424	—	2	4	no	—
58	#424	—	4	4	no	—
59	#423	—	2	2	no	—
60	#423	—	4	2	no	—
61	C₂×C₄⋊C₈	—	4	2	no	—
62	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₂×#104	—	2	4	no	—
64	#416	—	2	2	no	—
65	C₂×C₄×C₈	C₂³	1	2	yes	—
66	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
68	#368	—	2	4	no	—
69	#368	—	4	2	no	—
70	#367	C₂×C₄	1	2	yes	—
71	#366	C₂×C₄	1	1	yes	characteristic
72	#366	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₁₆	—	2	1	no	—
74	C₄⋊C₁₆	—	2	1	no	—
75	#359	—	2	8	no	—
76	#358	C₂×C₄	1	2	yes	—
77	#358	C₂×C₄	1	2	yes	—
78	#357	C₂×C₄	1	2	yes	—
79	#357	C₂×C₄	1	2	yes	—
80	C₄×C₁₆	—	2	4	no	—
81	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
82	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
83	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
84	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
85	#339	—	2	4	no	—
86	#336	D₄	1	2	yes	—
87	C₂×#41	—	2	2	no	—
88	#135	—	2	2	no	—
89	#135	—	2	2	no	—
90	#135	C₂×D₄	1	4	yes	—
91	#135	C₂×Q₈	1	4	yes	—
92	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
93	C₂×SD₁₆	—	2	2	no	—
94	C₂×D₈	C₂²×C₄	1	2	yes	—
95	#131	—	4	8	no	—
96	#131	—	8	4	no	—
97	C₂×M₁₆	—	2	4	no	—
98	C₂×M₁₆	—	4	4	no	—
99	C₂²×C₈	—	4	4	no	—
100	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₈	C₂×D₄	1	2	yes	—
102	C₂²×C₈	C₂⁴	1	1	yes	characteristic
103	C₄×Q₈	—	2	2	no	—
104	C₄×Q₈	—	2	2	no	—
105	C₄×D₄	—	2	2	no	—
106	C₄×D₄	—	2	2	no	—
107	C₄×D₄	—	4	2	no	—
108	C₄×D₄	—	4	2	no	—
109	#117	—	2	2	no	—
110	#117	—	4	2	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄²	—	4	2	no	—
113	C₂×C₄²	C₂×D₄	1	2	yes	—
114	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
115	Q₃₂	C₄⋊C₄	1	4	yes	complemented
116	SD₃₂	C₄⋊C₄	1	8	yes	complemented
117	D₁₆	C₄⋊C₄	1	4	yes	complemented
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₁₆	C₂×D₄	1	2	yes	—
128	C₂×C₁₆	C₂×Q₈	1	2	yes	—
129	#108	—	2	4	no	—
130	#108	—	2	4	no	—
131	#108	—	8	2	no	—
132	#107	C₂²×C₄	1	2	yes	—
133	#107	C₂²×C₄	1	2	yes	—
134	#106	—	2	2	no	—
135	C₄⋊C₈	—	4	4	no	—
136	C₄⋊C₈	—	4	4	no	—
137	C₄⋊C₈	—	4	2	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₄⋊C₈	D₈	1	1	yes	characteristic
140	C₄⋊C₈	D₈	1	1	yes	characteristic
141	C₄⋊C₈	D₈	1	2	yes	—
142	#104	—	4	8	no	—
143	#103	—	2	2	no	—
144	#103	—	2	2	no	—
145	#102	—	2	2	no	—
146	#102	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
150	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
151	C₄×C₈	#41	1	1	yes	characteristic
152	C₄×C₈	#41	1	1	yes	characteristic
153	#95	—	2	1	no	—
154	#41	—	2	4	no	—
155	#41	—	2	4	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	#135	1	1	yes	characteristic
166	C₂²×C₄	#135	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
168	Q₁₆	—	2	2	no	—
169	Q₁₆	C₂×C₄⋊C₄	1	4	yes	—
170	SD₁₆	—	2	4	no	—
171	SD₁₆	—	2	4	no	—
172	D₈	—	2	2	no	—
173	D₈	C₂×C₄⋊C₄	1	4	yes	—
174	M₁₆	—	4	4	no	—
175	M₁₆	—	4	4	no	—
176	M₁₆	—	8	8	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	4	4	no	—
182	C₂×C₈	—	8	8	no	—
183	C₂×C₈	C₄×D₄	1	1	yes	characteristic
184	C₂×C₈	C₄×D₄	1	1	yes	characteristic
185	C₂×C₈	C₂×D₈	1	2	yes	—
186	C₂×C₈	C₂×D₈	1	2	yes	—
187	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
190	C₂×C₈	C₂×#41	1	1	yes	characteristic
191	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	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	C₂²⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	4	2	no	—
200	C₂²⋊C₄	—	4	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₄²	—	4	2	no	—
206	C₄²	C₄×D₄	1	1	yes	characteristic
207	C₄²	C₄×D₄	1	1	yes	characteristic
208	C₄²	C₂×D₈	1	1	yes	characteristic
209	C₄²	C₂×D₈	1	1	yes	characteristic
210	C₁₆	—	2	1	no	—
211	C₁₆	—	2	1	no	—
212	C₁₆	—	4	2	no	—
213	C₁₆	C₂×C₄⋊C₄	1	2	yes	—
214	C₁₆	C₂×C₄⋊C₄	1	2	yes	—
215	C₂³	—	4	2	no	—
216	C₂³	C₂×#135	1	1	yes	characteristic
217	Q₈	—	2	4	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	4	no	—
222	D₄	—	4	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	4	no	—
234	C₂×C₄	—	16	2	no	—
235	C₂×C₄	C₄×D₈	1	1	yes	characteristic
236	C₂×C₄	C₄×D₈	1	1	yes	characteristic
237	C₂×C₄	C₄×D₈	1	1	yes	characteristic
238	C₂×C₄	C₄×D₈	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
240	C₂×C₄	D₄²	1	1	yes	characteristic
241	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
242	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
243	C₂×C₄	#576	1	1	yes	characteristic
244	C₂×C₄	#578	1	1	yes	characteristic
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	8	4	no	—
250	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
251	C₈	#520	1	1	yes	characteristic
252	C₈	#547	1	1	yes	characteristic
253	C₈	#554	1	1	yes	characteristic
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
260	C₂²	#3283	1	1	yes	characteristic
261	C₂²	#3380	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	4	4	no	—
266	C₄	—	4	4	no	—
267	C₄	—	8	2	no	—
268	C₄	—	8	2	no	—
269	C₄	#2178	1	2	yes	complemented
270	C₄	D₄×C₄⋊C₄	1	1	yes	characteristic
271	C₄	#2951	1	1	yes	characteristic
272	C₄	D₄×D₈	1	1	yes	characteristic
273	C₄	Q₈×D₈	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	8	4	no	—
277	C₂	C₄⋊C₄×D₈	1	1	yes	characteristic
278	C₂	C₂×#2178	1	1	yes	characteristic
279	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	C₂×#508	C₂²
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
3. #22688	C₂	semidirect
4. #13169	C₂	semidirect
5. #12939	C₂	semidirect
6. #12909	C₂	semidirect
7. #12815	C₂	semidirect
8. C₁₆×C₄⋊C₄	C₂	semidirect
14. C₄×Q₃₂	C₄	semidirect
15. C₄×Q₃₂	C₄	semidirect
16. C₄×SD₃₂	C₄	semidirect
17. C₄×SD₃₂	C₄	semidirect
18. C₄×D₁₆	C₄	semidirect
19. C₄×D₁₆	C₄	semidirect
115. Q₃₂	C₄⋊C₄	semidirect
116. SD₃₂	C₄⋊C₄	semidirect
117. D₁₆	C₄⋊C₄	semidirect
269. C₄	#2178	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2178	2
C₄×#508	2
#22688	2
#13169	2
#12939	2
#12909	2
#12815	2
C₁₆×C₄⋊C₄	2