Subgroups — #225755

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22746	C₂	1	2	yes	maximal
2	M₁₆×Q₁₆	C₂	1	2	yes	maximal
3	C₂×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	#9624	C₂	1	2	yes	maximal
5	#8909	C₂	1	4	yes	maximal
6	#8824	C₂	1	1	yes	maximal, characteristic
7	#8661	C₂	1	2	yes	maximal
8	C₈×#108	C₂	1	1	yes	maximal, characteristic
9	Q₈×M₁₆	C₂²	1	4	yes	—
10	C₂×C₈×Q₈	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	#1837	—	2	2	no	—
15	#1837	C₂²	1	2	yes	—
16	C₄×#108	—	2	2	no	—
17	C₄×#108	C₂²	1	1	yes	characteristic
18	#1591	—	2	2	no	—
19	#1584	—	2	4	no	—
20	C₈×Q₁₆	—	2	2	no	—
21	C₈×Q₁₆	C₄	1	2	yes	—
22	C₈×Q₁₆	C₄	1	1	yes	characteristic
23	C₈×Q₁₆	C₄	1	2	yes	—
24	C₈×Q₁₆	C₄	1	1	yes	characteristic
25	C₈×Q₁₆	C₄	1	1	yes	characteristic
26	C₈×Q₁₆	C₄	1	1	yes	characteristic
27	#1567	C₂²	1	2	yes	—
28	#1566	C₂²	1	2	yes	—
29	#1564	C₂²	1	1	yes	characteristic
30	C₂×#335	C₂²	1	1	yes	characteristic
31	#1487	C₂²	1	4	yes	—
32	C₂×Q₈⋊C₈	C₂²	1	2	yes	—
33	C₈×M₁₆	C₂²	1	2	yes	—
34	C₂×C₈²	C₂²	1	1	yes	characteristic
35	#1301	C₂²	1	4	yes	—
36	#1278	C₂²	1	2	yes	—
37	C₂²×Q₁₆	Q₈	1	1	yes	characteristic
38	C₂×C₄×Q₈	C₂³	1	2	yes	—
39	#446	—	2	4	no	—
40	#446	—	4	4	no	—
41	C₈×Q₈	—	2	2	no	—
42	C₈×Q₈	—	2	2	no	—
43	C₈×Q₈	—	2	4	no	—
44	C₈×Q₈	C₂×C₄	1	2	yes	—
45	C₈×Q₈	C₂×C₄	1	2	yes	—
46	C₈×Q₈	C₂×C₄	1	2	yes	—
47	C₈×Q₈	C₂×C₄	1	2	yes	—
48	C₄×Q₁₆	C₂×C₄	1	2	yes	—
49	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
50	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₁₆	D₄	1	1	yes	characteristic
52	C₄×Q₁₆	D₄	1	2	yes	—
53	C₄×Q₁₆	D₄	1	1	yes	characteristic
54	#431	C₂³	1	2	yes	—
55	C₂×#108	—	2	4	no	—
56	C₂×#108	D₄	1	2	yes	—
57	C₂×#107	Q₈	1	1	yes	characteristic
58	#423	—	2	4	no	—
59	#423	C₂³	1	4	yes	—
60	C₂×C₄⋊C₈	—	2	2	no	—
61	C₂×C₄⋊C₈	C₂³	1	2	yes	—
62	C₂×#103	Q₈	1	2	yes	—
63	C₄×M₁₆	—	2	4	no	—
64	C₄×M₁₆	C₂³	1	2	yes	—
65	C₂×C₄×C₈	—	2	2	no	—
66	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	#341	—	2	4	no	—
70	#341	—	2	2	no	—
71	#341	—	2	2	no	—
72	#341	D₄	1	1	yes	characteristic
73	#341	D₄	1	1	yes	characteristic
74	#335	—	2	2	no	—
75	#335	—	2	2	no	—
76	#335	—	2	1	no	—
77	#335	C₂×C₄	1	2	yes	—
78	#335	C₂×C₄	1	1	yes	characteristic
79	#335	C₂×C₄	1	1	yes	characteristic
80	#334	—	2	1	no	—
81	#334	—	4	2	no	—
82	Q₈⋊C₈	—	2	4	no	—
83	Q₈⋊C₈	—	2	4	no	—
84	Q₈⋊C₈	C₂×C₄	1	2	yes	—
85	Q₈⋊C₈	C₂×C₄	1	2	yes	—
86	Q₈⋊C₈	C₂×C₄	1	2	yes	—
87	Q₈⋊C₈	C₂×C₄	1	2	yes	—
88	#322	—	2	2	no	—
89	C₈²	—	2	2	no	—
90	C₈²	C₂×C₄	1	1	yes	characteristic
91	C₈²	C₂×C₄	1	1	yes	characteristic
92	C₈²	C₂×C₄	1	1	yes	characteristic
93	C₈²	C₂×C₄	1	1	yes	characteristic
94	C₂²×Q₈	C₂×Q₈	1	2	yes	—
95	C₂×Q₁₆	—	2	4	no	—
96	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
97	C₂×Q₁₆	C₄⋊C₄	1	1	yes	characteristic
98	C₂×Q₁₆	C₄⋊C₄	1	1	yes	characteristic
99	C₂×M₁₆	—	2	4	no	—
100	C₂×M₁₆	C₂×D₄	1	2	yes	—
101	C₂×M₁₆	C₂×D₄	1	2	yes	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
107	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₄×Q₈	C₂²×C₄	1	2	yes	—
111	C₄×Q₈	C₂²×C₄	1	2	yes	—
112	C₄×Q₈	C₂×D₄	1	2	yes	—
113	C₄×Q₈	C₂×D₄	1	2	yes	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂×Q₈	1	2	yes	—
116	C₂×C₄²	—	2	2	no	—
117	C₂×C₄²	C₂⁴	1	1	yes	characteristic
118	#108	—	2	4	no	—
119	#108	—	4	8	no	—
120	#107	C₄⋊C₄	1	2	yes	—
121	#107	C₄⋊C₄	1	1	yes	characteristic
122	#107	C₄⋊C₄	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	4	no	—
131	C₄⋊C₈	—	4	4	no	—
132	C₄⋊C₈	—	4	4	no	—
133	C₄⋊C₈	C₂²×C₄	1	2	yes	—
134	C₄⋊C₈	C₂²×C₄	1	2	yes	—
135	#103	C₄⋊C₄	1	2	yes	—
136	#103	C₄⋊C₄	1	2	yes	—
137	#103	C₄⋊C₄	1	2	yes	—
138	#103	C₄⋊C₄	1	2	yes	—
139	C₂²⋊C₈	—	2	2	no	—
140	#97	—	2	2	no	—
141	#97	—	4	4	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	2	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	—	4	4	no	—
150	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
151	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
152	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
153	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
156	C₄×C₈	C₂×D₄	1	1	yes	characteristic
157	C₄×C₈	C₂×D₄	1	1	yes	characteristic
158	C₄×C₈	#41	1	1	yes	characteristic
159	C₄×C₈	#41	1	1	yes	characteristic
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×Q₈	C₂×C₄⋊C₄	1	2	yes	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
168	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
169	C₂²×C₄	2₋¹⁺⁴	1	1	yes	characteristic
170	Q₁₆	—	2	4	no	—
171	Q₁₆	—	2	2	no	—
172	Q₁₆	#108	1	4	yes	complemented
173	M₁₆	—	2	2	no	—
174	M₁₆	—	2	2	no	—
175	M₁₆	—	2	2	no	—
176	M₁₆	—	4	4	no	—
177	M₁₆	—	4	4	no	—
178	M₁₆	#135	1	4	yes	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	4	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₄×D₄	1	1	yes	characteristic
198	C₂×C₈	C₄×D₄	1	1	yes	characteristic
199	C₂×C₈	C₄×D₄	1	1	yes	characteristic
200	C₂×C₈	C₄×D₄	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	C₂×C₄⋊C₄	1	2	yes	—
207	C₄⋊C₄	C₂×C₄⋊C₄	1	2	yes	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	C₂³×C₄	1	1	yes	characteristic
213	C₄²	C₂²×D₄	1	1	yes	characteristic
214	C₄²	C₂×#41	1	1	yes	characteristic
215	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂³	D₄×Q₈	1	1	yes	characteristic
217	Q₈	—	2	2	no	—
218	Q₈	—	2	2	no	—
219	Q₈	—	2	2	no	—
220	Q₈	—	4	4	no	—
221	Q₈	C₂×#108	1	4	yes	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
233	C₂×C₄	#520	1	1	yes	characteristic
234	C₂×C₄	D₄²	1	1	yes	characteristic
235	C₂×C₄	#547	1	1	yes	characteristic
236	C₂×C₄	#554	1	1	yes	characteristic
237	C₂×C₄	C₂×#135	1	2	yes	—
238	C₂×C₄	#578	1	2	yes	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	4	no	—
248	C₈	—	4	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	4	2	no	—
251	C₈	—	4	2	no	—
252	C₈	—	8	2	no	—
253	C₈	—	8	4	no	—
254	C₈	C₂×#108	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	D₄×C₄⋊C₄	1	1	yes	characteristic
258	C₂²	#3383	1	2	yes	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₄	C₂²×#108	1	1	yes	characteristic
267	C₄	#2916	1	1	yes	characteristic
268	C₄	C₄×#135	1	1	yes	characteristic
269	C₄	#2950	1	1	yes	characteristic
270	C₄	#3285	1	2	yes	—
271	C₂	—	2	2	no	—
272	C₂	D₄×#108	1	1	yes	characteristic
273	C₂	#22685	1	1	yes	characteristic
274	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
172. Q₁₆	#108	semidirect

Maximal subgroups

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

subgroup	index
#22746	2
M₁₆×Q₁₆	2
C₂×C₈×Q₁₆	2
#9624	2
#8909	2
#8824	2
#8661	2
C₈×#108	2