Subgroups — #483894

Order 512.

419 conjugacy classes of subgroups.

The subgroup lattice has 269 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33664	C₂	1	1	yes	maximal, characteristic
2	#17958	C₂	1	1	yes	maximal, characteristic
3	C₂×#1591	C₂	1	1	yes	maximal, characteristic
4	#12083	C₂	1	4	yes	maximal, complemented
5	#12072	C₂	1	4	yes	maximal, complemented
6	#12032	C₂	1	1	yes	maximal, characteristic
7	#11658	C₂	1	1	yes	maximal, characteristic
8	#11646	C₂	1	1	yes	maximal, characteristic
9	#11410	C₂	1	1	yes	maximal, characteristic
10	#3170	C₂²	1	1	yes	characteristic
11	#2960	—	2	1	no	—
12	C₂×#446	—	2	1	no	—
13	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
14	#2236	C₄	1	2	yes	—
15	#2227	C₄	1	2	yes	—
16	#2209	C₄	1	2	yes	—
17	#2188	C₄	1	2	yes	—
18	C₄×Q₃₂	C₂²	1	4	yes	—
19	#2162	C₂²	1	1	yes	characteristic
20	#2162	C₂²	1	1	yes	characteristic
21	#1591	—	2	1	no	—
22	#1591	—	2	1	no	—
23	#1591	—	2	1	no	—
24	#1591	C₂²	1	2	yes	—
25	#1591	C₂²	1	2	yes	—
26	#1591	C₂²	1	2	yes	—
27	C₈×Q₁₆	—	2	1	no	—
28	C₈×Q₁₆	C₂²	1	2	yes	—
29	#1567	C₂²	1	1	yes	characteristic
30	C₂×#335	C₂²	1	1	yes	characteristic
31	#1493	—	2	1	no	—
32	C₂×Q₈⋊C₈	—	2	1	no	—
33	C₂×C₈²	C₂²	1	1	yes	characteristic
34	#1371	C₂²	1	4	yes	—
35	#1332	C₂²	1	4	yes	—
36	#1332	C₂²	1	4	yes	—
37	#1312	C₂²	1	2	yes	—
38	C₈×C₁₆	C₂²	1	2	yes	—
39	C₂²×Q₁₆	C₂×C₄	1	1	yes	characteristic
40	#533	—	2	1	no	—
41	C₂×C₄×Q₈	—	2	1	no	—
42	C₂×Q₃₂	C₂×C₄	1	4	yes	—
43	#500	C₂×C₄	1	1	yes	characteristic
44	#494	C₂×C₄	1	1	yes	characteristic
45	#487	—	2	1	no	—
46	#467	—	2	1	no	—
47	#467	C₂×C₄	1	2	yes	—
48	#446	—	2	2	no	—
49	#446	—	2	2	no	—
50	#446	—	2	2	no	—
51	#446	—	4	1	no	—
52	#446	—	4	1	no	—
53	#446	—	4	1	no	—
54	C₈×Q₈	—	2	2	no	—
55	C₈×Q₈	—	4	1	no	—
56	C₄×Q₁₆	—	2	1	no	—
57	C₄×Q₁₆	—	2	1	no	—
58	C₄×Q₁₆	—	2	1	no	—
59	C₄×Q₁₆	C₂×C₄	1	2	yes	—
60	C₄×Q₁₆	C₂³	1	2	yes	—
61	C₄×Q₁₆	C₂³	1	2	yes	—
62	C₂×#107	C₂×C₄	1	1	yes	characteristic
63	#424	—	4	1	no	—
64	#423	—	2	1	no	—
65	C₂×C₄⋊C₈	—	2	1	no	—
66	C₂×C₄⋊C₈	—	4	1	no	—
67	C₂×#103	—	2	1	no	—
68	#405	—	4	1	no	—
69	C₂×#97	—	4	1	no	—
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	#366	C₂×C₄	1	4	yes	—
74	C₄⋊C₁₆	—	2	1	no	—
75	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
76	#358	C₂×C₄	1	4	yes	—
77	#358	C₂×C₄	1	4	yes	—
78	C₂²⋊C₁₆	—	2	1	no	—
79	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
80	C₄×C₁₆	C₂³	1	2	yes	—
81	C₄×C₁₆	C₂³	1	2	yes	—
82	#335	—	2	1	no	—
83	#335	—	2	1	no	—
84	#335	C₂³	1	2	yes	—
85	#335	C₂³	1	2	yes	—
86	Q₈⋊C₈	—	2	2	no	—
87	Q₈⋊C₈	—	2	2	no	—
88	Q₈⋊C₈	—	2	2	no	—
89	Q₈⋊C₈	—	2	2	no	—
90	C₈²	D₄	1	1	yes	characteristic
91	C₈²	D₄	1	1	yes	characteristic
92	C₈²	C₂³	1	1	yes	characteristic
93	C₈²	C₂³	1	1	yes	characteristic
94	C₂²×Q₈	—	2	1	no	—
95	C₂×Q₁₆	—	2	1	no	—
96	C₂×Q₁₆	—	2	1	no	—
97	C₂×Q₁₆	—	4	2	no	—
98	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
99	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	—	4	1	no	—
103	C₂²×C₈	D₈	1	1	yes	characteristic
104	C₂²×C₈	D₈	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	#41	1	1	yes	characteristic
107	C₄⋊Q₈	—	2	1	no	—
108	C₄⋊Q₈	—	2	1	no	—
109	#125	—	4	1	no	—
110	C₂²⋊Q₈	—	2	2	no	—
111	C₂²⋊Q₈	—	4	1	no	—
112	C₄×Q₈	—	2	2	no	—
113	C₄×Q₈	—	2	2	no	—
114	C₄×Q₈	—	2	2	no	—
115	C₄×Q₈	—	4	1	no	—
116	C₄×Q₈	—	4	1	no	—
117	C₄×Q₈	—	4	1	no	—
118	#117	—	4	1	no	—
119	C₂×C₄⋊C₄	—	2	1	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄²	—	4	1	no	—
122	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
123	Q₃₂	—	4	4	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	C₂²×C₄	1	4	yes	—
126	#107	—	2	1	no	—
127	#107	—	2	1	no	—
128	#107	C₂²×C₄	1	2	yes	—
129	#107	C₂²×C₄	1	2	yes	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	4	2	no	—
133	C₄⋊C₈	—	4	2	no	—
134	C₄⋊C₈	—	4	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	C₄⋊C₈	—	4	1	no	—
137	C₄⋊C₈	—	4	1	no	—
138	#103	—	2	2	no	—
139	#103	—	2	2	no	—
140	#103	—	2	2	no	—
141	#103	—	2	2	no	—
142	#97	—	4	2	no	—
143	#97	—	4	2	no	—
144	#97	—	4	2	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	4	2	no	—
147	C₄×C₈	D₈	1	1	yes	characteristic
148	C₄×C₈	D₈	1	1	yes	characteristic
149	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
150	C₄×C₈	C₂×D₄	1	1	yes	characteristic
151	C₄×C₈	C₂×D₄	1	1	yes	characteristic
152	C₄×C₈	C₂×D₄	1	1	yes	characteristic
153	C₄×C₈	C₂×D₄	1	1	yes	characteristic
154	C₄×C₈	C₂×D₄	1	1	yes	characteristic
155	C₄×C₈	#41	1	1	yes	characteristic
156	C₄×C₈	C₂⁴	1	1	yes	characteristic
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	1	no	—
162	C₂×Q₈	—	4	1	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	—	4	1	no	—
165	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
167	C₂²×C₄	#135	1	1	yes	characteristic
168	Q₁₆	—	2	2	no	—
169	Q₁₆	—	2	4	no	—
170	Q₁₆	—	2	2	no	—
171	Q₁₆	—	4	2	no	—
172	Q₁₆	—	4	2	no	—
173	M₁₆	—	8	2	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	8	2	no	—
183	C₂×C₈	Q₃₂	1	4	yes	complemented
184	C₂×C₈	C₄×D₄	1	1	yes	characteristic
185	C₂×C₈	C₄×D₄	1	1	yes	characteristic
186	C₂×C₈	C₂×D₈	1	1	yes	characteristic
187	C₂×C₈	C₂×D₈	1	1	yes	characteristic
188	C₂×C₈	C₂×D₈	1	1	yes	characteristic
189	C₂×C₈	C₂×D₈	1	1	yes	characteristic
190	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
191	C₂×C₈	C₂×#41	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₄⋊C₄	—	4	1	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₄²	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₄²	—	4	2	no	—
206	C₄²	C₄×D₄	1	1	yes	characteristic
207	C₄²	C₂×D₈	1	1	yes	characteristic
208	C₄²	#135	1	1	yes	characteristic
209	C₄²	C₂²×D₄	1	1	yes	characteristic
210	C₁₆	—	2	4	no	—
211	C₁₆	—	4	2	no	—
212	C₂³	C₄×D₈	1	1	yes	characteristic
213	Q₈	—	4	2	no	—
214	Q₈	—	4	2	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	8	2	no	—
218	Q₈	—	8	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	C₄×D₈	1	1	yes	characteristic
229	C₂×C₄	C₄×D₈	1	1	yes	characteristic
230	C₂×C₄	C₄×D₈	1	1	yes	characteristic
231	C₂×C₄	C₂×Q₃₂	1	2	yes	—
232	C₂×C₄	#508	1	2	yes	—
233	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
235	C₂×C₄	C₂×#135	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	1	no	—
243	C₈	—	8	2	no	—
244	C₈	—	8	2	no	—
245	C₈	C₂×Q₃₂	1	2	yes	—
246	C₈	C₂×Q₃₂	1	2	yes	—
247	C₈	#568	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	C₄×Q₃₂	1	2	yes	—
251	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	8	2	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	—	8	2	no	—
259	C₄	C₄×Q₃₂	1	2	yes	—
260	C₄	#2930	1	1	yes	characteristic
261	C₄	#2955	1	1	yes	characteristic
262	C₄	C₂²×Q₃₂	1	1	yes	characteristic
263	C₄	C₂×#508	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	#17928	1	1	yes	characteristic
266	C₂	C₂×C₄×Q₃₂	1	1	yes	characteristic
267	C₂	#33463	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #12083	C₂	semidirect
5. #12072	C₂	semidirect
183. C₂×C₈	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#33664	2
#17958	2
C₂×#1591	2
#12083	2
#12072	2
#12032	2
#11658	2
#11646	2
#11410	2