Subgroups — #225770

Order 512.

527 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22733	C₂	1	2	yes	maximal
2	M₁₆×Q₁₆	C₂	1	2	yes	maximal
3	C₂×#1591	C₂	1	1	yes	maximal, characteristic
4	#9624	C₂	1	2	yes	maximal
5	#8907	C₂	1	4	yes	maximal
6	#8823	C₂	1	1	yes	maximal, characteristic
7	#8663	C₂	1	2	yes	maximal
8	C₈×#108	C₂	1	1	yes	maximal, characteristic
9	Q₈×M₁₆	C₂²	1	4	yes	—
10	C₂×#446	C₂²	1	2	yes	—
11	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
12	#1950	—	2	4	no	—
13	#1950	C₂²	1	4	yes	—
14	#1849	—	2	2	no	—
15	#1848	C₂²	1	2	yes	—
16	#1778	—	2	2	no	—
17	C₄×#108	C₂²	1	1	yes	characteristic
18	#1591	—	2	2	no	—
19	#1591	C₄	1	2	yes	—
20	#1591	C₄	1	2	yes	—
21	#1591	C₄	1	2	yes	—
22	#1591	C₄	1	2	yes	—
23	#1584	—	2	4	no	—
24	C₈×Q₁₆	—	2	2	no	—
25	#1567	C₂²	1	2	yes	—
26	#1567	C₂²	1	2	yes	—
27	#1564	C₂²	1	1	yes	characteristic
28	C₂×#335	C₂²	1	1	yes	characteristic
29	#1487	C₂²	1	4	yes	—
30	C₂×Q₈⋊C₈	C₂²	1	2	yes	—
31	C₈×M₁₆	C₂²	1	2	yes	—
32	C₂×C₈²	C₂²	1	1	yes	characteristic
33	#1301	C₂²	1	4	yes	—
34	#1278	C₂²	1	2	yes	—
35	C₂²×Q₁₆	Q₈	1	1	yes	characteristic
36	C₂×C₄×Q₈	C₂³	1	2	yes	—
37	#446	—	2	4	no	—
38	#446	—	2	2	no	—
39	#446	—	2	2	no	—
40	#446	—	4	4	no	—
41	#446	C₂×C₄	1	4	yes	—
42	#446	C₂×C₄	1	4	yes	—
43	C₈×Q₈	—	2	4	no	—
44	C₄×Q₁₆	C₂×C₄	1	2	yes	—
45	C₄×Q₁₆	C₂×C₄	1	2	yes	—
46	C₄×Q₁₆	D₄	1	2	yes	—
47	C₄×Q₁₆	D₄	1	2	yes	—
48	#431	C₂³	1	2	yes	—
49	C₂×#108	—	2	2	no	—
50	C₂×#108	—	2	4	no	—
51	C₂×#108	—	2	2	no	—
52	C₂×#108	D₄	1	1	yes	characteristic
53	C₂×#108	D₄	1	1	yes	characteristic
54	C₂×#107	Q₈	1	1	yes	characteristic
55	#423	—	2	4	no	—
56	#423	C₂³	1	4	yes	—
57	C₂×C₄⋊C₈	—	2	2	no	—
58	C₂×C₄⋊C₈	C₂³	1	2	yes	—
59	C₂×#103	Q₈	1	2	yes	—
60	C₄×M₁₆	—	2	4	no	—
61	C₄×M₁₆	C₂³	1	2	yes	—
62	C₂×#97	—	2	2	no	—
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
66	#341	—	2	4	no	—
67	#341	D₄	1	2	yes	—
68	#335	—	2	2	no	—
69	#335	—	2	2	no	—
70	#335	—	2	1	no	—
71	#335	—	4	2	no	—
72	#335	C₂×C₄	1	2	yes	—
73	#335	C₂×C₄	1	2	yes	—
74	#334	—	2	1	no	—
75	Q₈⋊C₈	—	2	4	no	—
76	Q₈⋊C₈	—	2	4	no	—
77	Q₈⋊C₈	C₂×C₄	1	4	yes	—
78	Q₈⋊C₈	C₂×C₄	1	4	yes	—
79	#322	—	2	2	no	—
80	C₈²	—	2	2	no	—
81	C₈²	C₂×C₄	1	1	yes	characteristic
82	C₈²	C₂×C₄	1	1	yes	characteristic
83	C₈²	C₂×C₄	1	1	yes	characteristic
84	C₈²	C₂×C₄	1	1	yes	characteristic
85	C₂²×Q₈	C₂×Q₈	1	2	yes	—
86	C₂×Q₁₆	—	4	2	no	—
87	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
88	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
89	C₂×M₁₆	—	2	4	no	—
90	C₂×M₁₆	C₂×D₄	1	2	yes	—
91	C₂×M₁₆	C₂×D₄	1	2	yes	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₄×Q₈	—	2	2	no	—
99	C₄×Q₈	—	2	2	no	—
100	C₄×Q₈	C₂²×C₄	1	4	yes	—
101	C₄×Q₈	C₂×D₄	1	4	yes	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	C₂×Q₈	1	2	yes	—
104	C₂×C₄²	—	2	2	no	—
105	C₂×C₄²	C₂⁴	1	1	yes	characteristic
106	#108	—	2	2	no	—
107	#108	—	4	4	no	—
108	#108	—	8	2	no	—
109	#108	—	8	4	no	—
110	#108	Q₁₆	1	4	yes	complemented
111	#107	C₄⋊C₄	1	2	yes	—
112	#107	C₄⋊C₄	1	2	yes	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	2	4	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	4	no	—
119	C₄⋊C₈	—	4	4	no	—
120	C₄⋊C₈	—	4	4	no	—
121	C₄⋊C₈	C₂²×C₄	1	4	yes	—
122	#103	C₄⋊C₄	1	4	yes	—
123	#103	C₄⋊C₄	1	4	yes	—
124	C₂²⋊C₈	—	2	2	no	—
125	#97	—	2	4	no	—
126	#97	—	2	4	no	—
127	#97	—	2	2	no	—
128	#97	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	4	4	no	—
133	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
135	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
136	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
138	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
139	C₄×C₈	C₂×D₄	1	1	yes	characteristic
140	C₄×C₈	C₂×D₄	1	1	yes	characteristic
141	C₄×C₈	#41	1	1	yes	characteristic
142	C₄×C₈	#41	1	1	yes	characteristic
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	C₂×C₄⋊C₄	1	4	yes	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
151	C₂²×C₄	2₋¹⁺⁴	1	1	yes	characteristic
152	Q₁₆	—	2	4	no	—
153	Q₁₆	—	4	2	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	M₁₆	—	2	2	no	—
157	M₁₆	—	4	4	no	—
158	M₁₆	—	4	4	no	—
159	M₁₆	C₂×Q₁₆	1	4	yes	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₈	C₄×D₄	1	1	yes	characteristic
179	C₂×C₈	C₄×D₄	1	1	yes	characteristic
180	C₂×C₈	C₄×D₄	1	1	yes	characteristic
181	C₂×C₈	C₄×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂×Q₁₆	1	2	yes	—
183	C₂×C₈	#135	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	C₂×C₄⋊C₄	1	4	yes	—
188	C₄²	—	2	4	no	—
189	C₄²	—	2	4	no	—
190	C₄²	C₂³×C₄	1	1	yes	characteristic
191	C₄²	C₂²×D₄	1	1	yes	characteristic
192	C₄²	C₂×#41	1	1	yes	characteristic
193	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂³	D₄×Q₈	1	1	yes	characteristic
195	Q₈	—	2	4	no	—
196	Q₈	—	2	4	no	—
197	Q₈	—	8	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
207	C₂×C₄	#520	1	1	yes	characteristic
208	C₂×C₄	D₄²	1	1	yes	characteristic
209	C₂×C₄	#547	1	1	yes	characteristic
210	C₂×C₄	#554	1	1	yes	characteristic
211	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
212	C₂×C₄	C₂×#135	1	1	yes	characteristic
213	C₂×C₄	#576	1	1	yes	characteristic
214	C₂×C₄	#578	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	4	no	—
222	C₈	—	4	1	no	—
223	C₈	—	4	4	no	—
224	C₈	—	4	2	no	—
225	C₈	—	8	2	no	—
226	C₈	—	8	4	no	—
227	C₈	#430	1	2	yes	—
228	C₈	C₄×Q₁₆	1	2	yes	—
229	C₈	C₄×Q₁₆	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	D₄×C₄⋊C₄	1	1	yes	characteristic
233	C₂²	Q₈×Q₁₆	1	1	yes	characteristic
234	C₂²	#3383	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	4	no	—
239	C₄	—	4	4	no	—
240	C₄	C₂×#430	1	1	yes	characteristic
241	C₄	#2916	1	1	yes	characteristic
242	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
243	C₄	#2951	1	1	yes	characteristic
244	C₄	D₄×Q₁₆	1	1	yes	characteristic
245	C₄	#3287	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	#22669	1	1	yes	characteristic
249	C₂	C₄⋊C₄×Q₁₆	1	1	yes	characteristic
250	C₂	#22694	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂³	D₄×Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
110. #108	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22733	2
M₁₆×Q₁₆	2
C₂×#1591	2
#9624	2
#8907	2
#8823	2
#8663	2
C₈×#108	2