Subgroups — #326501

Order 512.

420 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33257	C₂	1	1	yes	maximal, characteristic
2	#21732	C₂	1	1	yes	maximal, characteristic
3	#10369	C₂	1	2	yes	maximal
4	#10367	C₂	1	2	yes	maximal
5	#9755	C₂	1	1	yes	maximal, characteristic
6	#9755	C₂	1	1	yes	maximal, characteristic
7	#8793	C₂	1	1	yes	maximal, characteristic
8	#8790	C₂	1	1	yes	maximal, characteristic
9	#8681	C₂	1	2	yes	maximal
10	#8679	C₂	1	2	yes	maximal
11	#8266	C₂	1	1	yes	maximal, characteristic
12	#3160	C₂²	1	1	yes	characteristic
13	#2960	C₂²	1	2	yes	—
14	#2864	C₂²	1	2	yes	—
15	#1994	C₂²	1	2	yes	—
16	#1945	—	2	1	no	—
17	#1945	—	2	1	no	—
18	#1942	—	2	1	no	—
19	#1942	—	2	1	no	—
20	#1941	—	2	1	no	—
21	#1941	—	2	1	no	—
22	#1848	C₂²	1	1	yes	characteristic
23	#1846	C₂²	1	1	yes	characteristic
24	#1846	C₂²	1	1	yes	characteristic
25	#1835	C₂²	1	2	yes	—
26	#1772	—	2	1	no	—
27	#1772	—	2	1	no	—
28	C₄²×C₈	C₂²	1	1	yes	characteristic
29	#1564	C₂²	1	1	yes	characteristic
30	#1564	C₂²	1	1	yes	characteristic
31	#1518	C₂²	1	2	yes	—
32	#1518	C₂²	1	2	yes	—
33	#1518	C₂²	1	2	yes	—
34	#1518	C₂²	1	2	yes	—
35	#1450	C₂²	1	1	yes	characteristic
36	#1450	C₂²	1	1	yes	characteristic
37	#1286	C₂²	1	2	yes	—
38	#1286	C₂²	1	2	yes	—
39	#1286	C₂²	1	2	yes	—
40	#1286	C₂²	1	2	yes	—
41	#1276	C₂²	1	1	yes	characteristic
42	#1276	C₂²	1	1	yes	characteristic
43	#533	C₂³	1	2	yes	—
44	#500	Q₈	1	1	yes	characteristic
45	#500	Q₈	1	1	yes	characteristic
46	#499	—	2	1	no	—
47	#499	—	2	1	no	—
48	#498	Q₈	1	2	yes	—
49	#446	—	2	2	no	—
50	#446	—	2	2	no	—
51	C₈×Q₈	—	2	2	no	—
52	C₈×Q₈	—	2	2	no	—
53	#431	C₂³	1	1	yes	characteristic
54	#431	C₂³	1	1	yes	characteristic
55	#431	C₂³	1	1	yes	characteristic
56	#431	C₂³	1	1	yes	characteristic
57	C₂×#108	—	2	1	no	—
58	C₂×#108	—	2	1	no	—
59	#427	—	2	1	no	—
60	#427	—	2	1	no	—
61	C₂×#107	—	2	1	no	—
62	C₂×#107	—	2	1	no	—
63	C₂×#106	—	2	1	no	—
64	C₂×#106	—	2	1	no	—
65	#424	—	2	2	no	—
66	#423	C₂³	1	2	yes	—
67	#405	—	2	2	no	—
68	C₂×C₄×C₈	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	D₄	1	1	yes	characteristic
70	C₂×C₄×C₈	D₄	1	1	yes	characteristic
71	C₂×C₄×C₈	D₄	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	#389	—	2	2	no	—
75	#389	—	2	2	no	—
76	#387	—	4	2	no	—
77	#387	—	4	2	no	—
78	#382	C₂³	1	2	yes	—
79	#382	C₂³	1	2	yes	—
80	C₄³	C₂³	1	1	yes	characteristic
81	#340	—	2	2	no	—
82	#340	—	2	2	no	—
83	#339	—	2	2	no	—
84	#339	—	2	2	no	—
85	#335	—	2	1	no	—
86	#335	—	2	1	no	—
87	#334	—	2	1	no	—
88	#334	—	2	1	no	—
89	#328	—	4	2	no	—
90	#328	—	4	2	no	—
91	#328	—	4	2	no	—
92	#328	—	4	2	no	—
93	#322	—	2	1	no	—
94	#322	—	2	1	no	—
95	C₈²	—	2	1	no	—
96	C₈²	—	2	1	no	—
97	C₂×M₁₆	—	2	2	no	—
98	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
102	C₄⋊Q₈	C₂×Q₈	1	2	yes	—
103	C₄⋊Q₈	C₂×Q₈	1	2	yes	—
104	#125	—	2	2	no	—
105	C₂²⋊Q₈	—	2	2	no	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₄×Q₈	—	2	2	no	—
108	C₄×Q₈	—	2	2	no	—
109	#117	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	4	2	no	—
114	C₂×C₄²	—	2	1	no	—
115	C₂×C₄²	—	2	1	no	—
116	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄²	C₂⁴	1	1	yes	characteristic
119	#108	—	4	2	no	—
120	#108	—	4	2	no	—
121	#107	—	2	1	no	—
122	#107	—	2	1	no	—
123	#107	—	2	1	no	—
124	#107	—	2	1	no	—
125	#107	—	4	2	no	—
126	#107	—	4	2	no	—
127	#106	—	2	2	no	—
128	#106	—	2	2	no	—
129	#106	—	4	2	no	—
130	#106	—	4	2	no	—
131	C₄⋊C₈	—	2	1	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	—	2	1	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	4	2	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₂²⋊C₈	—	2	1	no	—
140	C₂²⋊C₈	—	2	1	no	—
141	C₂²⋊C₈	—	2	1	no	—
142	C₂²⋊C₈	—	2	1	no	—
143	#97	—	4	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	—	2	1	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	4	2	no	—
153	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
154	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
155	C₄×C₈	#41	1	1	yes	characteristic
156	C₄×C₈	#41	1	1	yes	characteristic
157	C₄×C₈	#41	1	1	yes	characteristic
158	C₄×C₈	#41	1	1	yes	characteristic
159	C₄×C₈	#41	1	1	yes	characteristic
160	C₄×C₈	#41	1	1	yes	characteristic
161	#95	—	2	2	no	—
162	#95	—	2	2	no	—
163	#95	—	2	2	no	—
164	#95	—	2	2	no	—
165	#95	—	4	2	no	—
166	#95	—	4	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
174	C₂²×C₄	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂²×C₄	H(5,𝔽₂)	1	1	yes	characteristic
176	M₁₆	—	4	2	no	—
177	M₁₆	—	4	2	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
191	C₂×C₈	#124	1	1	yes	characteristic
192	C₂×C₈	#124	1	1	yes	characteristic
193	C₂×C₈	#124	1	1	yes	characteristic
194	C₂×C₈	#124	1	1	yes	characteristic
195	C₂×C₈	#135	1	2	yes	—
196	C₂×C₈	#135	1	2	yes	—
197	C₂×C₈	#137	1	2	yes	—
198	C₂×C₈	#137	1	2	yes	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	8	4	no	—
210	C₄⋊C₄	—	8	4	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	4	2	no	—
217	C₄²	—	4	2	no	—
218	C₄²	—	4	2	no	—
219	C₄²	C₂²×Q₈	1	1	yes	characteristic
220	C₄²	C₂×#41	1	1	yes	characteristic
221	C₄²	C₂×#41	1	1	yes	characteristic
222	C₄²	C₂×#41	1	1	yes	characteristic
223	C₂³	#535	1	1	yes	characteristic
224	Q₈	—	8	2	no	—
225	Q₈	—	8	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₄	C₂×#124	1	1	yes	characteristic
244	C₂×C₄	#554	1	1	yes	characteristic
245	C₂×C₄	#554	1	1	yes	characteristic
246	C₂×C₄	#555	1	1	yes	characteristic
247	C₂×C₄	#555	1	1	yes	characteristic
248	C₂×C₄	C₂×#135	1	1	yes	characteristic
249	C₂×C₄	C₂×#135	1	1	yes	characteristic
250	C₂×C₄	C₂×#137	1	1	yes	characteristic
251	C₂×C₄	C₂×#137	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	8	1	no	—
257	C₈	—	8	1	no	—
258	C₈	—	8	1	no	—
259	C₈	—	8	1	no	—
260	C₈	—	8	2	no	—
261	C₈	#479	1	2	yes	—
262	C₈	#479	1	2	yes	—
263	C₈	#479	1	2	yes	—
264	C₈	#479	1	2	yes	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	#2563	1	1	yes	characteristic
268	C₂²	#3210	1	1	yes	characteristic
269	C₂²	#3210	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	8	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	16	2	no	—
279	C₄	—	16	2	no	—
280	C₄	C₂×#479	1	1	yes	characteristic
281	C₄	C₂×#479	1	1	yes	characteristic
282	C₄	#3131	1	1	yes	characteristic
283	C₄	#3139	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#29144	1	1	yes	characteristic
287	C₂	#29154	1	1	yes	characteristic
288	C₂	#29154	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#33257	2
#21732	2
#10369	2
#10367	2
#9755	2
#8793	2
#8790	2
#8681	2
#8679	2
#8266	2