Subgroups — #96857

Order 512.

428 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	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#8261	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄×#333	C₂	1	1	yes	maximal, characteristic
5	#7571	C₂	1	1	yes	maximal, characteristic
6	#7031	C₂	1	1	yes	maximal, characteristic
7	#7031	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	C₄	1	2	yes	—
9	C₄×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#1984	C₄	1	2	yes	—
11	#1939	—	2	1	no	—
12	#1939	—	2	1	no	—
13	#1842	C₂²	1	1	yes	characteristic
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₈×C₄⋊C₄	—	2	1	no	—
16	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₄²×C₈	C₂²	1	1	yes	characteristic
21	C₂×#333	C₄	1	2	yes	—
22	C₂×#322	—	2	1	no	—
23	C₂×C₈²	—	2	1	no	—
24	#1296	C₄	1	2	yes	—
25	#1273	—	2	1	no	—
26	#1273	—	2	1	no	—
27	#1272	C₄	1	4	yes	—
28	C₂×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
30	C₈×Q₈	—	4	4	no	—
31	C₂×C₄⋊C₈	—	2	1	no	—
32	C₂×C₄⋊C₈	—	2	2	no	—
33	C₂×C₄⋊C₈	—	2	1	no	—
34	C₂×C₄⋊C₈	—	2	1	no	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄×C₈	—	2	2	no	—
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	#391	C₂×C₄	1	2	yes	—
51	#389	—	2	1	no	—
52	#389	—	2	1	no	—
53	#384	C₂×C₄	1	1	yes	characteristic
54	C₄×C₄⋊C₄	—	2	1	no	—
55	C₄×C₄⋊C₄	—	2	1	no	—
56	C₄³	C₂³	1	1	yes	characteristic
57	#336	—	2	2	no	—
58	#336	—	2	2	no	—
59	#336	—	2	1	no	—
60	#336	—	2	1	no	—
61	#336	C₂×C₄	1	2	yes	—
62	#336	C₂×C₄	1	2	yes	—
63	#333	C₈	1	8	yes	complemented
64	#322	—	4	2	no	—
65	C₈²	—	4	2	no	—
66	C₂²×Q₈	C₄²	1	2	yes	—
67	C₂²×C₈	—	2	2	no	—
68	C₂²×C₈	—	2	2	no	—
69	C₂²×C₈	—	2	2	no	—
70	C₂²×C₈	—	2	2	no	—
71	C₂²×C₈	—	2	2	no	—
72	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	C₂×C₈	1	4	yes	—
78	C₄×Q₈	—	4	2	no	—
79	C₄×Q₈	—	4	2	no	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	C₄²	1	2	yes	—
86	C₂×C₄²	—	2	2	no	—
87	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄²	#41	1	1	yes	characteristic
93	C₂×C₄²	#41	1	1	yes	characteristic
94	C₄⋊C₈	—	2	4	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	4	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	C₄⋊C₈	—	4	4	no	—
100	C₄⋊C₈	—	4	1	no	—
101	C₄⋊C₈	—	4	1	no	—
102	C₄⋊C₈	—	4	1	no	—
103	C₄⋊C₈	—	4	1	no	—
104	C₄⋊C₈	C₂×C₈	1	4	yes	—
105	C₄⋊C₈	C₂×C₈	1	4	yes	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	—	4	4	no	—
115	C₄×C₈	—	4	1	no	—
116	C₄×C₈	—	4	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	—	4	1	no	—
119	C₄×C₈	—	4	1	no	—
120	C₄×C₈	—	4	1	no	—
121	C₄×C₈	—	4	1	no	—
122	C₄×C₈	—	4	1	no	—
123	C₄×C₈	SD₁₆	1	2	yes	—
124	C₄×C₈	Q₁₆	1	2	yes	—
125	#95	—	2	2	no	—
126	#95	—	2	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	4	2	no	—
129	C₂×Q₈	—	4	2	no	—
130	C₂×Q₈	C₄×C₈	1	4	yes	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄	#117	1	1	yes	characteristic
136	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
139	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	#100	1	2	yes	—
165	C₂×C₈	#103	1	2	yes	—
166	C₂×C₈	#103	1	2	yes	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	4	1	no	—
178	C₄²	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	C₂²×C₈	1	2	yes	—
181	C₄²	#131	1	2	yes	—
182	C₄²	C₂×SD₁₆	1	1	yes	characteristic
183	C₄²	C₂×Q₁₆	1	1	yes	characteristic
184	C₄²	#135	1	1	yes	characteristic
185	C₄²	#135	1	1	yes	characteristic
186	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₄²	#137	1	1	yes	characteristic
188	C₄²	#137	1	1	yes	characteristic
189	C₄²	#137	1	1	yes	characteristic
190	C₂³	C₄×C₂²⋊C₄	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	Q₈	—	4	4	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
209	C₂×C₄	#405	1	1	yes	characteristic
210	C₂×C₄	C₂×#100	1	1	yes	characteristic
211	C₂×C₄	#413	1	1	yes	characteristic
212	C₂×C₄	C₂×#103	1	1	yes	characteristic
213	C₂×C₄	#416	1	1	yes	characteristic
214	C₂×C₄	#417	1	1	yes	characteristic
215	C₂×C₄	#419	1	1	yes	characteristic
216	C₂×C₄	C₈×D₄	1	2	yes	—
217	C₂×C₄	C₈×D₄	1	2	yes	—
218	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
219	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
220	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
221	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
222	C₂×C₄	#440	1	1	yes	characteristic
223	C₂×C₄	#440	1	1	yes	characteristic
224	C₂×C₄	#441	1	1	yes	characteristic
225	C₂×C₄	#441	1	1	yes	characteristic
226	C₂×C₄	#443	1	2	yes	—
227	C₂×C₄	#444	1	2	yes	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₈	—	4	1	no	—
233	C₈	—	4	4	no	—
234	C₈	—	4	4	no	—
235	C₈	—	8	2	no	—
236	C₈	—	8	2	no	—
237	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
238	C₂²	C₄×#100	1	1	yes	characteristic
239	C₂²	#1757	1	1	yes	characteristic
240	C₂²	C₄×#103	1	1	yes	characteristic
241	C₂²	#1763	1	1	yes	characteristic
242	C₂²	#1764	1	1	yes	characteristic
243	C₂²	#1765	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	4	no	—
252	C₄	—	8	2	no	—
253	C₄	#1548	1	1	yes	characteristic
254	C₄	#1550	1	1	yes	characteristic
255	C₄	#1551	1	1	yes	characteristic
256	C₄	#1554	1	1	yes	characteristic
257	C₄	C₈×SD₁₆	1	2	yes	—
258	C₄	C₈×Q₁₆	1	2	yes	—
259	C₄	SD₁₆⋊C₈	1	2	yes	—
260	C₄	#1579	1	2	yes	—
261	C₂	C₈×#100	1	1	yes	characteristic
262	C₂	C₈×#103	1	1	yes	characteristic
263	C₂	#8201	1	1	yes	characteristic
264	C₂	#8235	1	1	yes	characteristic
265	C₂	#8245	1	1	yes	characteristic
266	C₂	#8256	1	1	yes	characteristic
267	C₂	#8258	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
63. #333	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Q₈	2
#8261	2
C₈×C₄⋊C₈	2
C₄×#333	2
#7571	2
#7031	2