Subgroups — #128727

Order 512.

286 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#12666	C₂	1	1	yes	maximal, characteristic
3	#11607	C₂	1	1	yes	maximal, characteristic
4	#11404	C₂	1	1	yes	maximal, characteristic
5	#8203	C₂	1	1	yes	maximal, characteristic
6	#7168	C₂	1	1	yes	maximal, characteristic
7	#7152	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
9	#2183	—	2	2	no	—
10	#2183	C₄	1	2	yes	complemented
11	Q₈×C₁₆	C₄	1	2	yes	complemented
12	#2150	—	2	1	no	—
13	#2150	C₂²	1	1	yes	characteristic
14	#2150	C₂²	1	1	yes	characteristic
15	C₂×#347	—	2	1	no	—
16	C₄×M₃₂	—	2	1	no	—
17	C₄×M₃₂	C₂²	1	1	yes	characteristic
18	C₄×M₃₂	C₂²	1	1	yes	characteristic
19	#1770	C₂²	1	1	yes	characteristic
20	C₄×#103	C₄	1	1	yes	characteristic
21	C₂×Q₈⋊C₈	C₄	1	1	yes	characteristic
22	C₂×#322	C₂²	1	1	yes	characteristic
23	#1375	—	2	1	no	—
24	Q₈⋊C₁₆	C₄	1	2	yes	complemented
25	Q₈⋊C₁₆	C₄	1	2	yes	complemented
26	#1313	—	2	2	no	—
27	#1279	C₄	1	1	yes	characteristic
28	#1277	C₄	1	1	yes	characteristic
29	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×M₃₂	—	2	1	no	—
31	C₂×M₃₂	—	2	1	no	—
32	C₂×M₃₂	—	2	1	no	—
33	C₂×M₃₂	D₄	1	1	yes	characteristic
34	C₂×M₃₂	D₄	1	1	yes	characteristic
35	C₈×Q₈	—	2	1	no	—
36	C₈×Q₈	—	2	1	no	—
37	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
38	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
39	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
40	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₈	—	2	1	no	—
42	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×#103	—	2	1	no	—
45	C₂×#97	—	2	1	no	—
46	C₂×C₄×C₈	—	2	1	no	—
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₁₆	—	2	2	no	—
52	C₄⋊C₁₆	—	2	2	no	—
53	C₄⋊C₁₆	—	2	1	no	—
54	C₄⋊C₁₆	—	2	1	no	—
55	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
56	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
57	#347	—	4	2	no	—
58	#346	—	2	2	no	—
59	#346	—	2	1	no	—
60	#346	D₄	1	2	yes	—
61	C₄×C₁₆	—	2	2	no	—
62	C₄×C₁₆	C₂×C₄	1	2	yes	—
63	C₄×C₁₆	C₂×C₄	1	2	yes	—
64	#340	—	2	1	no	—
65	#336	—	2	1	no	—
66	Q₈⋊C₈	—	2	1	no	—
67	Q₈⋊C₈	—	2	1	no	—
68	#322	—	2	1	no	—
69	#322	—	2	1	no	—
70	C₂²×Q₈	M₁₆	1	1	yes	characteristic
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₂²×C₈	—	2	1	no	—
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₈	#41	1	1	yes	characteristic
77	C₄×Q₈	—	2	1	no	—
78	C₄×Q₈	—	2	1	no	—
79	C₄×Q₈	C₄²	1	1	yes	characteristic
80	C₄×Q₈	C₄²	1	1	yes	characteristic
81	C₄×Q₈	M₁₆	1	1	yes	characteristic
82	C₄×Q₈	M₁₆	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
86	C₂×C₄²	—	2	1	no	—
87	C₂×C₄²	—	2	1	no	—
88	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
89	M₃₂	—	2	2	no	—
90	M₃₂	—	2	2	no	—
91	M₃₂	—	2	1	no	—
92	M₃₂	—	2	1	no	—
93	M₃₂	—	2	1	no	—
94	M₃₂	—	2	1	no	—
95	M₃₂	—	4	1	no	—
96	M₃₂	—	4	1	no	—
97	M₃₂	—	4	1	no	—
98	M₃₂	—	4	1	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
103	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	—	2	1	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	C₄²	1	1	yes	characteristic
112	C₄⋊C₈	C₄²	1	1	yes	characteristic
113	#103	—	4	2	no	—
114	#97	—	4	1	no	—
115	#97	—	4	1	no	—
116	C₄×C₈	—	2	1	no	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	—	2	1	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	2	1	no	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₄×C₈	C₂×D₄	1	1	yes	characteristic
127	C₄×C₈	#41	1	1	yes	characteristic
128	#95	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂×Q₈	#97	1	1	yes	characteristic
133	C₂×Q₈	#97	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
139	C₂²×C₄	#131	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	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	8	1	no	—
154	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₈	#117	1	1	yes	characteristic
156	C₂×C₈	C₄×D₄	1	1	yes	characteristic
157	C₂×C₈	C₄×D₄	1	1	yes	characteristic
158	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×C₈	#137	1	1	yes	characteristic
160	C₂×C₈	#137	1	1	yes	characteristic
161	C₂×C₈	#137	1	1	yes	characteristic
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	#97	1	1	yes	characteristic
170	C₄⋊C₄	#97	1	1	yes	characteristic
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	1	no	—
177	C₄²	C₂×C₄²	1	1	yes	characteristic
178	C₄²	C₄×D₄	1	1	yes	characteristic
179	C₄²	C₂×M₁₆	1	1	yes	characteristic
180	C₄²	#131	1	1	yes	characteristic
181	C₁₆	—	2	4	no	—
182	C₁₆	—	2	2	no	—
183	C₁₆	—	2	2	no	—
184	C₁₆	—	4	2	no	—
185	C₁₆	—	4	2	no	—
186	C₂³	#435	1	1	yes	characteristic
187	Q₈	—	2	1	no	—
188	Q₈	—	2	1	no	—
189	Q₈	—	2	1	no	—
190	Q₈	—	4	2	no	—
191	Q₈	#347	1	2	yes	complemented
192	C₂×C₄	—	2	1	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	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	8	1	no	—
201	C₂×C₄	—	8	1	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×#97	1	1	yes	characteristic
206	C₂×C₄	#405	1	1	yes	characteristic
207	C₂×C₄	#435	1	1	yes	characteristic
208	C₂×C₄	#435	1	1	yes	characteristic
209	C₂×C₄	#435	1	1	yes	characteristic
210	C₂×C₄	#440	1	1	yes	characteristic
211	C₂×C₄	#441	1	1	yes	characteristic
212	C₂×C₄	#443	1	1	yes	characteristic
213	C₂×C₄	#443	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	4	1	no	—
217	C₈	—	4	1	no	—
218	C₈	—	8	1	no	—
219	C₈	—	8	1	no	—
220	C₈	—	8	1	no	—
221	C₈	#417	1	1	yes	characteristic
222	C₈	#419	1	1	yes	characteristic
223	C₈	#440	1	1	yes	characteristic
224	C₈	#441	1	1	yes	characteristic
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#1587	1	1	yes	characteristic
228	C₂²	#1588	1	1	yes	characteristic
229	C₂²	#1752	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	—	16	1	no	—
238	C₄	#1587	1	1	yes	characteristic
239	C₄	#1588	1	1	yes	characteristic
240	C₄	#1763	1	1	yes	characteristic
241	C₄	#1764	1	1	yes	characteristic
242	C₄	C₂×#347	1	1	yes	characteristic
243	C₄	#2110	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	#8205	1	1	yes	characteristic
247	C₂	#12616	1	1	yes	characteristic
248	C₂	#12618	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂³	#435
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂ ◁ 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
10. #2183	C₄	semidirect
11. Q₈×C₁₆	C₄	semidirect
24. Q₈⋊C₁₆	C₄	semidirect
25. Q₈⋊C₁₆	C₄	semidirect
191. Q₈	#347	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₃₂	2
#12666	2
#11607	2
#11404	2
#8203	2
#7168	2
#7152	2