Subgroups — #138592

Order 512.

337 conjugacy classes of subgroups.

The subgroup lattice has 257 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1676	C₂	1	1	yes	maximal, characteristic
3	#13282	C₂	1	1	yes	maximal, characteristic
4	C₂×#1365	C₂	1	1	yes	maximal, characteristic
5	#8645	C₂	1	1	yes	maximal, characteristic
6	#7274	C₂	1	1	yes	maximal, characteristic
7	#7154	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	C₂²	1	1	yes	characteristic
9	C₂×#477	—	2	1	no	—
10	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
11	Q₈×C₁₆	—	2	2	no	—
12	Q₈×C₁₆	—	2	2	no	—
13	C₂×C₄⋊C₁₆	—	2	1	no	—
14	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	—	2	1	no	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#1845	C₂²	1	1	yes	characteristic
18	#1803	—	2	1	no	—
19	#1676	—	2	2	no	—
20	#1676	C₄	1	4	yes	complemented
21	C₂×#335	C₂²	1	1	yes	characteristic
22	C₂×#335	C₂²	1	1	yes	characteristic
23	C₂×#333	—	2	1	no	—
24	#1380	—	2	1	no	—
25	#1380	—	2	1	no	—
26	#1365	—	2	2	no	—
27	#1296	—	2	1	no	—
28	C₂×C₄⋊Q₈	—	2	1	no	—
29	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
30	C₂²×C₁₆	—	2	1	no	—
31	C₂²×C₁₆	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	D₄	1	1	yes	characteristic
33	#500	—	2	1	no	—
34	#500	—	2	1	no	—
35	#500	—	2	1	no	—
36	#500	C₂×C₄	1	2	yes	—
37	#477	—	2	4	no	—
38	#477	—	4	2	no	—
39	C₈×Q₈	—	2	1	no	—
40	C₈×Q₈	—	2	1	no	—
41	C₈×Q₈	C₂×C₄	1	2	yes	—
42	C₈×Q₈	D₄	1	2	yes	—
43	C₂×#107	—	2	1	no	—
44	C₂×#107	—	2	1	no	—
45	C₂×#107	—	2	1	no	—
46	C₂×#107	—	2	1	no	—
47	C₂×#107	—	4	1	no	—
48	C₂×C₄⋊C₈	—	2	1	no	—
49	C₂×C₄⋊C₈	—	2	1	no	—
50	C₂×C₄⋊C₈	—	2	1	no	—
51	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
52	C₂×#103	—	4	1	no	—
53	C₂×#103	—	4	1	no	—
54	C₂×C₄×C₈	—	2	1	no	—
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	#384	—	2	1	no	—
57	C₄⋊C₁₆	—	2	4	no	—
58	C₄⋊C₁₆	—	2	1	no	—
59	C₄⋊C₁₆	D₄	1	2	yes	—
60	C₄×C₁₆	—	2	4	no	—
61	C₄×C₁₆	—	2	1	no	—
62	C₄×C₁₆	—	2	1	no	—
63	#340	—	4	1	no	—
64	#335	—	2	1	no	—
65	#335	—	2	1	no	—
66	#335	—	2	1	no	—
67	#335	C₂×C₄	1	2	yes	—
68	#333	—	2	4	no	—
69	C₂²×Q₈	—	4	1	no	—
70	C₂²×Q₈	D₈	1	1	yes	characteristic
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	—	4	1	no	—
73	C₂²×C₈	—	4	1	no	—
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈	#41	1	1	yes	characteristic
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	—	4	1	no	—
78	C₄⋊Q₈	—	4	1	no	—
79	C₄⋊Q₈	—	4	1	no	—
80	C₄×Q₈	—	2	1	no	—
81	C₄×Q₈	—	2	1	no	—
82	C₄×Q₈	C₂²⋊C₄	1	2	yes	—
83	C₄×Q₈	SD₁₆	1	2	yes	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	4	1	no	—
86	C₂×C₄⋊C₄	—	4	1	no	—
87	C₂×C₄⋊C₄	—	4	1	no	—
88	C₂×C₄⋊C₄	—	4	1	no	—
89	C₂×C₄⋊C₄	—	4	1	no	—
90	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
91	C₂×C₄²	—	2	1	no	—
92	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	4	1	no	—
100	C₂×C₁₆	—	4	1	no	—
101	C₂×C₁₆	Q₁₆	1	2	yes	—
102	#107	—	2	2	no	—
103	#107	—	2	2	no	—
104	#107	—	4	1	no	—
105	#107	—	4	1	no	—
106	#107	—	4	4	no	—
107	#107	—	4	1	no	—
108	#107	—	4	1	no	—
109	#107	—	8	1	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	4	1	no	—
115	C₄⋊C₈	—	4	1	no	—
116	C₄⋊C₈	—	4	1	no	—
117	C₄⋊C₈	C₂²⋊C₄	1	2	yes	—
118	#103	—	4	4	no	—
119	#103	—	8	2	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₄×C₈	C₂×D₄	1	1	yes	characteristic
124	C₄×C₈	C₂×D₄	1	1	yes	characteristic
125	C₄×C₈	#41	1	1	yes	characteristic
126	C₂×Q₈	—	2	1	no	—
127	C₂×Q₈	—	2	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	4	2	no	—
130	C₂×Q₈	—	8	1	no	—
131	C₂×Q₈	—	16	1	no	—
132	C₂×Q₈	—	16	1	no	—
133	C₂×Q₈	#102	1	2	yes	—
134	C₂×Q₈	SD₃₂	1	2	yes	—
135	C₂×Q₈	Q₃₂	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	8	1	no	—
139	C₂²×C₄	—	8	1	no	—
140	C₂²×C₄	#121	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂²×C₄	#135	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	8	1	no	—
150	C₂×C₈	—	8	1	no	—
151	C₂×C₈	—	8	1	no	—
152	C₂×C₈	—	8	1	no	—
153	C₂×C₈	—	8	1	no	—
154	C₂×C₈	C₄×D₄	1	1	yes	characteristic
155	C₂×C₈	C₄×D₄	1	1	yes	characteristic
156	C₂×C₈	#120	1	1	yes	characteristic
157	C₂×C₈	#123	1	1	yes	characteristic
158	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
159	C₂×C₈	#135	1	1	yes	characteristic
160	C₂×C₈	#137	1	1	yes	characteristic
161	C₂×C₈	#137	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	8	1	no	—
168	C₄⋊C₄	—	8	1	no	—
169	C₄⋊C₄	—	8	1	no	—
170	C₄⋊C₄	—	8	1	no	—
171	C₄⋊C₄	—	8	1	no	—
172	C₄⋊C₄	—	8	1	no	—
173	C₄⋊C₄	—	16	1	no	—
174	C₄⋊C₄	#102	1	2	yes	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₄²	#121	1	1	yes	characteristic
179	C₄²	C₂×SD₁₆	1	1	yes	characteristic
180	C₄²	#135	1	1	yes	characteristic
181	C₁₆	—	2	2	no	—
182	C₁₆	—	4	1	no	—
183	C₁₆	—	4	2	no	—
184	C₂³	#466	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	2	2	no	—
188	Q₈	—	16	2	no	—
189	Q₈	—	16	2	no	—
190	Q₈	#358	1	4	yes	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	16	1	no	—
201	C₂×C₄	—	16	1	no	—
202	C₂×C₄	—	16	1	no	—
203	C₂×C₄	—	16	1	no	—
204	C₂×C₄	—	16	1	no	—
205	C₂×C₄	#386	1	1	yes	characteristic
206	C₂×C₄	C₂×#102	1	1	yes	characteristic
207	C₂×C₄	#416	1	1	yes	characteristic
208	C₂×C₄	#462	1	1	yes	characteristic
209	C₂×C₄	#464	1	1	yes	characteristic
210	C₂×C₄	#465	1	1	yes	characteristic
211	C₂×C₄	#465	1	1	yes	characteristic
212	C₂×C₄	#466	1	1	yes	characteristic
213	C₂×C₄	#471	1	1	yes	characteristic
214	C₂×C₄	#473	1	1	yes	characteristic
215	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
216	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
217	C₂×C₄	#508	1	1	yes	characteristic
218	C₂×C₄	#508	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	—	8	2	no	—
223	C₈	—	16	1	no	—
224	C₈	C₄×Q₁₆	1	1	yes	characteristic
225	C₈	#441	1	1	yes	characteristic
226	C₈	#451	1	1	yes	characteristic
227	C₈	#452	1	1	yes	characteristic
228	C₂²	#1674	1	1	yes	characteristic
229	C₂²	#1682	1	1	yes	characteristic
230	C₂²	#1893	1	1	yes	characteristic
231	C₂²	#2216	1	1	yes	characteristic
232	C₂²	#2217	1	1	yes	characteristic
233	C₂²	#2221	1	1	yes	characteristic
234	C₂²	#2223	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	16	2	no	—
240	C₄	—	32	1	no	—
241	C₄	#1672	1	1	yes	characteristic
242	C₄	#1680	1	1	yes	characteristic
243	C₄	#1878	1	1	yes	characteristic
244	C₄	#1908	1	1	yes	characteristic
245	C₄	C₂×#358	1	1	yes	characteristic
246	C₄	#2138	1	1	yes	characteristic
247	C₄	#2216	1	1	yes	characteristic
248	C₄	#2217	1	1	yes	characteristic
249	C₂	#9218	1	1	yes	characteristic
250	C₂	#12458	1	1	yes	characteristic
251	C₂	#12459	1	1	yes	characteristic
252	C₂	#12476	1	1	yes	characteristic
253	C₂	#12477	1	1	yes	characteristic
254	C₂	#13060	1	1	yes	characteristic
255	C₂	#13073	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
20. #1676	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₆	2
C₂×#1676	2
#13282	2
C₂×#1365	2
#8645	2
#7274	2
#7154	2