Subgroups — #498076

Order 512.

440 conjugacy classes of subgroups.

The subgroup lattice has 257 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#446	C₂	1	1	yes	maximal, characteristic
2	#13268	C₂	1	2	yes	maximal
3	#13107	C₂	1	4	yes	maximal
4	#12900	C₂	1	1	yes	maximal, characteristic
5	#12882	C₂	1	2	yes	maximal
6	#12666	C₂	1	2	yes	maximal
7	#12659	C₂	1	2	yes	maximal
8	C₄×#364	C₂	1	1	yes	maximal, characteristic
9	C₂×#446	C₄	1	1	yes	characteristic
10	C₂×#446	C₂²	1	1	yes	characteristic
11	C₄²×Q₈	C₄	1	1	yes	characteristic
12	C₂×#364	C₂²	1	1	yes	characteristic
13	C₂×#364	C₂²	1	2	yes	—
14	C₂×#364	C₂²	1	2	yes	—
15	C₂×#364	C₂²	1	1	yes	characteristic
16	#2150	C₂²	1	2	yes	—
17	C₂×#347	C₂²	1	4	yes	—
18	C₄×M₃₂	C₂²	1	2	yes	—
19	#1771	C₄	1	2	yes	—
20	#1771	C₂²	1	2	yes	—
21	#1770	C₄	1	1	yes	characteristic
22	#1770	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₈	C₄	1	2	yes	—
24	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
25	C₄×#97	C₂²	1	2	yes	—
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	#1377	C₂²	1	4	yes	—
28	#1377	C₂²	1	4	yes	—
29	#1376	C₂²	1	2	yes	—
30	#1375	C₂²	1	4	yes	—
31	C₂×C₄×Q₈	—	2	1	no	—
32	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×M₃₂	C₂³	1	2	yes	—
34	C₂×M₃₂	C₂³	1	4	yes	—
35	C₂×M₃₂	C₂³	1	2	yes	—
36	#446	—	2	2	no	—
37	#446	—	2	2	no	—
38	#446	—	2	1	no	—
39	#446	—	2	1	no	—
40	#446	—	2	1	no	—
41	#446	—	2	1	no	—
42	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₂×#97	C₂×C₄	1	2	yes	—
47	C₂×#97	C₂×C₄	1	2	yes	—
48	C₂×#97	C₂³	1	2	yes	—
49	C₂×#97	C₂³	1	2	yes	—
50	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
53	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₄⋊C₄	—	2	1	no	—
58	C₄×C₄⋊C₄	—	2	2	no	—
59	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₄³	C₂×C₄	1	1	yes	characteristic
62	C₄³	C₂×C₄	1	2	yes	—
63	#364	—	2	2	no	—
64	#364	—	2	4	no	—
65	#364	—	4	2	no	—
66	#364	Q₈	1	4	yes	complemented
67	C₄⋊C₁₆	—	2	2	no	—
68	C₄⋊C₁₆	—	2	2	no	—
69	#347	—	4	4	no	—
70	#346	—	2	2	no	—
71	C₄×C₁₆	—	2	2	no	—
72	#336	C₂×C₄	1	2	yes	—
73	#336	C₂×C₄	1	2	yes	—
74	#336	C₂×C₄	1	2	yes	—
75	C₂²×Q₈	M₁₆	1	1	yes	characteristic
76	C₂²×C₈	C₂²×C₄	1	2	yes	—
77	C₂²×C₈	C₂²×C₄	1	2	yes	—
78	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂⁴	1	1	yes	characteristic
82	C₄×Q₈	—	2	1	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	—	2	1	no	—
86	C₄×Q₈	—	4	4	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
90	C₂×C₄⋊C₄	M₁₆	1	2	yes	—
91	C₂×C₄²	—	2	1	no	—
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	—	2	1	no	—
94	C₂×C₄²	—	2	2	no	—
95	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄²	C₂²×C₄	1	2	yes	—
97	C₂×C₄²	C₂²×C₄	1	2	yes	—
98	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	2	2	no	—
102	M₃₂	—	2	4	no	—
103	M₃₂	—	2	4	no	—
104	M₃₂	—	2	2	no	—
105	M₃₂	C₂×Q₈	1	4	yes	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	4	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	—	2	1	no	—
116	#97	—	2	4	no	—
117	#97	—	2	4	no	—
118	#97	—	2	2	no	—
119	#97	—	2	2	no	—
120	#97	—	2	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	4	1	no	—
126	C₄×C₈	—	4	2	no	—
127	C₄×C₈	M₁₆	1	2	yes	—
128	C₄×C₈	M₁₆	1	2	yes	—
129	C₄×C₈	C₂×Q₈	1	2	yes	—
130	C₄×C₈	#41	1	1	yes	characteristic
131	C₄×C₈	#41	1	1	yes	characteristic
132	C₄×C₈	#41	1	1	yes	characteristic
133	C₄×C₈	#41	1	1	yes	characteristic
134	C₄×C₈	#41	1	2	yes	—
135	#95	—	2	2	no	—
136	#95	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
144	C₂²×C₄	C₂×M₁₆	1	2	yes	—
145	C₂²×C₄	C₂×M₁₆	1	2	yes	—
146	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	#117	1	1	yes	characteristic
159	C₂×C₈	#117	1	1	yes	characteristic
160	C₂×C₈	C₄×Q₈	1	2	yes	—
161	C₂×C₈	C₂×M₁₆	1	2	yes	—
162	C₂×C₈	C₂×M₁₆	1	2	yes	—
163	C₂×C₈	#131	1	2	yes	—
164	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₈	C₂×#41	1	1	yes	characteristic
166	C₂×C₈	C₂×#41	1	1	yes	characteristic
167	C₂×C₈	C₂×#41	1	1	yes	characteristic
168	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	—	4	4	no	—
186	C₄²	—	4	4	no	—
187	C₄²	#117	1	1	yes	characteristic
188	C₄²	#117	1	1	yes	characteristic
189	C₄²	C₄×Q₈	1	2	yes	—
190	C₄²	C₂×M₁₆	1	1	yes	characteristic
191	C₄²	C₂×M₁₆	1	1	yes	characteristic
192	C₁₆	—	2	4	no	—
193	C₁₆	—	4	2	no	—
194	C₁₆	—	4	4	no	—
195	C₂³	C₂²×M₁₆	1	1	yes	characteristic
196	Q₈	—	4	2	no	—
197	Q₈	—	4	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	#432	1	1	yes	characteristic
214	C₂×C₄	#432	1	1	yes	characteristic
215	C₂×C₄	#432	1	1	yes	characteristic
216	C₂×C₄	#432	1	1	yes	characteristic
217	C₂×C₄	C₂×#117	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
219	C₂×C₄	#520	1	1	yes	characteristic
220	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
221	C₂×C₄	C₂×#131	1	1	yes	characteristic
222	C₂×C₄	#568	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	4	2	no	—
228	C₈	#446	1	4	yes	—
229	C₈	#554	1	1	yes	characteristic
230	C₈	#556	1	1	yes	characteristic
231	C₈	#557	1	1	yes	characteristic
232	C₈	#559	1	1	yes	characteristic
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	C₂×#432	1	1	yes	characteristic
236	C₂²	Q₈×M₁₆	1	1	yes	characteristic
237	C₂²	#2991	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	8	2	no	—
246	C₄	#2165	1	2	yes	—
247	C₄	#2355	1	1	yes	characteristic
248	C₄	#2925	1	1	yes	characteristic
249	C₄	C₂×#446	1	1	yes	characteristic
250	C₄	#2989	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	#22825	1	1	yes	characteristic
254	C₂	C₂×#2165	1	1	yes	characteristic
255	C₂	#33428	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂³	C₂²×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
66. #364	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#446	2
#13268	2
#13107	2
#12900	2
#12882	2
#12666	2
#12659	2
C₄×#364	2