Subgroups — #109260

Order 512.

581 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1848	C₂	1	1	yes	maximal, characteristic
2	#17793	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#8065	C₂	1	2	yes	maximal
5	#7938	C₂	1	2	yes	maximal
6	#2920	C₂²	1	2	yes	—
7	C₂×#431	C₂²	1	1	yes	characteristic
8	C₂²×#108	C₄	1	4	yes	—
9	C₂×#423	C₂²	1	2	yes	—
10	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#1848	—	2	2	no	—
13	#1848	—	2	1	no	—
14	#1848	—	2	1	no	—
15	#1568	C₄	1	4	yes	—
16	#1567	—	2	1	no	—
17	#1566	—	2	1	no	—
18	#1564	—	2	2	no	—
19	C₈×M₁₆	—	2	1	no	—
20	C₈×M₁₆	—	2	1	no	—
21	C₈×M₁₆	C₄	1	2	yes	—
22	C₈×M₁₆	C₄	1	2	yes	—
23	C₂×#322	—	2	1	no	—
24	C₂×C₈²	—	2	1	no	—
25	#1299	—	2	4	no	—
26	#1283	—	2	4	no	—
27	C₂²×M₁₆	C₂×C₄	1	4	yes	—
28	C₂²×M₁₆	Q₈	1	2	yes	—
29	C₂³×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₄²	C₂³	1	1	yes	characteristic
31	#432	—	2	2	no	—
32	#432	C₂×C₄	1	4	yes	—
33	#431	—	2	1	no	—
34	#431	—	2	1	no	—
35	#431	—	2	2	no	—
36	#431	—	2	2	no	—
37	#431	C₂×C₄	1	2	yes	—
38	#431	D₄	1	2	yes	—
39	C₂×#108	—	2	4	no	—
40	C₂×#108	—	2	4	no	—
41	C₂×#108	—	4	8	no	—
42	#423	—	2	2	no	—
43	#423	—	2	2	no	—
44	#423	—	2	2	no	—
45	#423	—	2	2	no	—
46	#423	—	2	2	no	—
47	C₂×C₄⋊C₈	—	2	2	no	—
48	C₂×C₄⋊C₈	—	2	2	no	—
49	C₂×C₄⋊C₈	Q₈	1	2	yes	—
50	#407	—	2	2	no	—
51	C₂×C₂²⋊C₈	—	2	1	no	—
52	C₄×M₁₆	—	2	1	no	—
53	C₄×M₁₆	—	2	1	no	—
54	C₄×M₁₆	—	2	2	no	—
55	C₄×M₁₆	C₂×C₄	1	2	yes	—
56	C₄×M₁₆	D₄	1	2	yes	—
57	C₂×#97	—	2	1	no	—
58	C₂×C₄×C₈	—	2	1	no	—
59	C₂×C₄×C₈	—	2	1	no	—
60	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	D₄	1	1	yes	characteristic
64	C₂×C₄×C₈	D₄	1	1	yes	characteristic
65	C₂×C₄×C₈	D₄	1	2	yes	—
66	C₂×C₄×C₈	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	D₄	1	1	yes	characteristic
68	C₂×C₄×C₈	Q₈	1	2	yes	—
69	#335	—	2	4	no	—
70	#334	—	2	4	no	—
71	#322	—	2	2	no	—
72	#322	—	2	2	no	—
73	C₈²	—	2	2	no	—
74	C₈²	—	2	2	no	—
75	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂³×C₄	#41	1	1	yes	characteristic
78	C₂×M₁₆	—	2	2	no	—
79	C₂×M₁₆	—	2	8	no	—
80	C₂×M₁₆	—	2	4	no	—
81	C₂×M₁₆	—	2	4	no	—
82	C₂×M₁₆	—	4	2	no	—
83	C₂×M₁₆	—	4	8	no	—
84	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	2	4	no	—
87	C₂²×C₈	—	2	4	no	—
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	—	2	4	no	—
90	C₂²×C₈	C₄²	1	2	yes	—
91	C₂²×C₈	C₄²	1	2	yes	—
92	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
93	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
94	C₂²×C₈	C₄⋊C₄	1	2	yes	—
95	C₂²×C₈	C₄⋊C₄	1	2	yes	—
96	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
105	C₂×C₄²	#41	1	1	yes	characteristic
106	C₂×C₄²	#41	1	1	yes	characteristic
107	C₂×C₄²	#41	1	1	yes	characteristic
108	#108	—	4	8	no	—
109	#108	—	4	8	no	—
110	C₄⋊C₈	—	2	4	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄⋊C₈	—	4	2	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄⋊C₈	—	4	2	no	—
117	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
118	C₂²⋊C₈	—	2	2	no	—
119	C₂²⋊C₈	—	2	2	no	—
120	C₂²⋊C₈	—	2	4	no	—
121	C₂²⋊C₈	—	4	2	no	—
122	#97	—	2	2	no	—
123	#97	—	2	4	no	—
124	#97	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	4	2	no	—
130	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊C₄	1	2	yes	—
135	C₄×C₈	C₂²⋊C₄	1	2	yes	—
136	C₄×C₈	C₄⋊C₄	1	2	yes	—
137	C₄×C₈	C₄⋊C₄	1	2	yes	—
138	C₄×C₈	C₄⋊C₄	1	2	yes	—
139	C₄×C₈	C₄⋊C₄	1	2	yes	—
140	C₂⁴	C₄×Q₈	1	1	yes	characteristic
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	#117	1	1	yes	characteristic
150	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
151	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
153	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
154	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
155	C₂²×C₄	#121	1	1	yes	characteristic
156	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂²×C₄	#124	1	1	yes	characteristic
160	C₂²×C₄	#125	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
163	M₁₆	—	2	8	no	—
164	M₁₆	—	2	4	no	—
165	M₁₆	—	4	8	no	—
166	M₁₆	—	8	4	no	—
167	M₁₆	—	8	4	no	—
168	M₁₆	#108	1	8	yes	complemented
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	—	8	4	no	—
182	C₂×C₈	#95	1	2	yes	—
183	C₂×C₈	#95	1	2	yes	—
184	C₂×C₈	#95	1	2	yes	—
185	C₂×C₈	#95	1	2	yes	—
186	C₂×C₈	#100	1	4	yes	—
187	C₂×C₈	#101	1	4	yes	—
188	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₄²	#117	1	1	yes	characteristic
194	C₄²	C₄×D₄	1	1	yes	characteristic
195	C₄²	C₄×Q₈	1	1	yes	characteristic
196	C₄²	#121	1	1	yes	characteristic
197	C₄²	#121	1	1	yes	characteristic
198	C₄²	#121	1	1	yes	characteristic
199	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
200	C₄²	#124	1	1	yes	characteristic
201	C₄²	#125	1	1	yes	characteristic
202	C₄²	#127	1	1	yes	characteristic
203	C₄²	C₄⋊Q₈	1	1	yes	characteristic
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	4	1	no	—
207	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
208	C₂³	#389	1	1	yes	characteristic
209	C₂³	#391	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	C₂×#95	1	1	yes	characteristic
229	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
230	C₂×C₄	#380	1	1	yes	characteristic
231	C₂×C₄	#380	1	1	yes	characteristic
232	C₂×C₄	#382	1	1	yes	characteristic
233	C₂×C₄	#384	1	1	yes	characteristic
234	C₂×C₄	#389	1	1	yes	characteristic
235	C₂×C₄	#389	1	1	yes	characteristic
236	C₂×C₄	#389	1	1	yes	characteristic
237	C₂×C₄	#390	1	1	yes	characteristic
238	C₂×C₄	#390	1	1	yes	characteristic
239	C₂×C₄	#391	1	1	yes	characteristic
240	C₂×C₄	C₂×#100	1	2	yes	—
241	C₂×C₄	C₂×#101	1	2	yes	—
242	C₂×C₄	C₂×#108	1	4	yes	—
243	C₈	—	2	8	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	4	4	no	—
247	C₈	—	8	4	no	—
248	C₈	—	8	4	no	—
249	C₈	#343	1	4	yes	—
250	C₈	#343	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	4	no	—
259	C₂²	C₄⋊#95	1	1	yes	characteristic
260	C₂²	C₄×#108	1	2	yes	—
261	C₂²	#1997	1	2	yes	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	C₂×#343	1	2	yes	—
270	C₄	#1810	1	2	yes	—
271	C₄	#1814	1	2	yes	—
272	C₄	#1848	1	2	yes	—
273	C₄	#1965	1	1	yes	characteristic
274	C₄	#1966	1	1	yes	characteristic
275	C₂	—	2	2	no	—
276	C₂	—	2	2	no	—
277	C₂	—	4	1	no	—
278	C₂	#10857	1	1	yes	characteristic
279	C₂	#10864	1	1	yes	characteristic
280	C₂	#10872	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
168. M₁₆	#108	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1848	2
#17793	2
C₂×C₈×M₁₆	2
#8065	2
#7938	2