Subgroups — #140379

Order 512.

264 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1372	C₂	1	1	yes	maximal, characteristic
3	C₂×#1371	C₂	1	1	yes	maximal, characteristic
4	#8807	C₂	1	1	yes	maximal, characteristic
5	C₄×#335	C₂	1	1	yes	maximal, characteristic
6	#7154	C₂	1	1	yes	maximal, characteristic
7	#7154	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1831	—	2	1	no	—
13	C₄×C₄⋊C₈	—	2	1	no	—
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×#335	C₂²	1	1	yes	characteristic
16	C₂×#335	C₂²	1	1	yes	characteristic
17	C₂×#335	C₂²	1	1	yes	characteristic
18	C₂×#335	C₂²	1	1	yes	characteristic
19	C₂×#334	—	2	1	no	—
20	#1374	C₄	1	1	yes	characteristic
21	#1374	C₄	1	1	yes	characteristic
22	#1372	—	2	2	no	—
23	#1371	—	2	2	no	—
24	#1277	—	2	1	no	—
25	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊C₈	—	2	1	no	—
30	C₂×C₄⋊C₈	—	2	1	no	—
31	C₂×C₄⋊C₈	—	2	1	no	—
32	C₂×C₄⋊C₈	—	2	1	no	—
33	C₂×C₄×C₈	—	4	1	no	—
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	C₄³	D₄	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄⋊C₁₆	C₈	1	4	yes	complemented
43	C₄×C₁₆	—	2	1	no	—
44	C₄×C₁₆	—	2	1	no	—
45	C₄×C₁₆	—	2	1	no	—
46	C₄×C₁₆	—	2	1	no	—
47	#336	—	4	1	no	—
48	#336	—	4	1	no	—
49	#336	—	4	1	no	—
50	#335	—	2	1	no	—
51	#335	—	2	1	no	—
52	#335	—	2	1	no	—
53	#335	—	2	1	no	—
54	#335	—	2	1	no	—
55	#335	—	2	1	no	—
56	#335	D₄	1	2	yes	—
57	#335	Q₈	1	2	yes	—
58	#334	—	4	2	no	—
59	C₂²×C₈	—	4	1	no	—
60	C₂²×C₈	—	4	1	no	—
61	C₂²×C₈	—	4	1	no	—
62	C₂²×C₈	—	4	1	no	—
63	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄²	D₈	1	1	yes	characteristic
70	C₂×C₄²	D₈	1	1	yes	characteristic
71	C₂×C₄²	Q₁₆	1	1	yes	characteristic
72	C₂×C₄²	Q₁₆	1	1	yes	characteristic
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	C₂×C₈	1	2	yes	—
85	C₂×C₁₆	C₂×C₈	1	2	yes	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	4	1	no	—
89	C₄⋊C₈	—	4	1	no	—
90	C₄⋊C₈	—	4	1	no	—
91	C₄⋊C₈	—	4	1	no	—
92	C₄⋊C₈	—	4	1	no	—
93	C₄⋊C₈	—	4	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	4	1	no	—
100	C₄×C₈	—	8	2	no	—
101	C₄×C₈	C₂×C₈	1	2	yes	—
102	C₄×C₈	M₁₆	1	2	yes	—
103	C₄×C₈	C₂×D₄	1	1	yes	characteristic
104	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₄×C₈	#41	1	1	yes	characteristic
106	C₄×C₈	#41	1	1	yes	characteristic
107	C₂²×C₄	#107	1	1	yes	characteristic
108	C₂²×C₄	#107	1	1	yes	characteristic
109	C₂²×C₄	#107	1	1	yes	characteristic
110	C₂²×C₄	#107	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
113	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	8	1	no	—
124	C₂×C₈	—	8	1	no	—
125	C₂×C₈	—	8	1	no	—
126	C₂×C₈	—	8	1	no	—
127	C₂×C₈	—	8	1	no	—
128	C₂×C₈	—	8	1	no	—
129	C₂×C₈	—	8	1	no	—
130	C₂×C₈	—	8	1	no	—
131	C₂×C₈	—	8	1	no	—
132	C₂×C₈	—	8	1	no	—
133	C₂×C₈	C₄⋊C₈	1	2	yes	—
134	C₂×C₈	C₄×D₄	1	1	yes	characteristic
135	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
136	C₂×C₈	#125	1	1	yes	characteristic
137	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
138	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₈	#131	1	1	yes	characteristic
141	C₂×C₈	#131	1	1	yes	characteristic
142	C₄²	—	2	2	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	—	2	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	4	1	no	—
150	C₄²	C₄⋊C₈	1	2	yes	—
151	C₄²	#108	1	2	yes	—
152	C₄²	#121	1	1	yes	characteristic
153	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
154	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₄²	#137	1	1	yes	characteristic
156	C₁₆	—	2	2	no	—
157	C₁₆	—	2	2	no	—
158	C₁₆	—	4	1	no	—
159	C₁₆	—	4	1	no	—
160	C₂³	C₂×#107	1	1	yes	characteristic
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	#335	1	2	yes	—
173	C₂×C₄	#335	1	2	yes	—
174	C₂×C₄	#368	1	2	yes	—
175	C₂×C₄	#389	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
177	C₂×C₄	#424	1	1	yes	characteristic
178	C₂×C₄	#428	1	1	yes	characteristic
179	C₂×C₄	C₂×#108	1	1	yes	characteristic
180	C₂×C₄	#430	1	1	yes	characteristic
181	C₂×C₄	#459	1	1	yes	characteristic
182	C₂×C₄	#462	1	1	yes	characteristic
183	C₂×C₄	#474	1	1	yes	characteristic
184	C₂×C₄	#477	1	1	yes	characteristic
185	C₂×C₄	#508	1	1	yes	characteristic
186	C₂×C₄	#508	1	1	yes	characteristic
187	C₂×C₄	#509	1	1	yes	characteristic
188	C₂×C₄	#510	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	4	1	no	—
193	C₈	—	8	2	no	—
194	C₈	—	8	2	no	—
195	C₈	—	16	1	no	—
196	C₈	—	16	1	no	—
197	C₈	C₈×D₄	1	1	yes	characteristic
198	C₈	#436	1	1	yes	characteristic
199	C₈	C₈×Q₈	1	1	yes	characteristic
200	C₈	#446	1	1	yes	characteristic
201	C₂²	C₂×#335	1	1	yes	characteristic
202	C₂²	#1571	1	1	yes	characteristic
203	C₂²	#1947	1	1	yes	characteristic
204	C₂²	#2158	1	1	yes	characteristic
205	C₂²	#2159	1	1	yes	characteristic
206	C₂²	C₂×#368	1	1	yes	characteristic
207	C₂²	#2161	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	4	1	no	—
214	C₄	#1565	1	1	yes	characteristic
215	C₄	#1574	1	1	yes	characteristic
216	C₄	#1939	1	1	yes	characteristic
217	C₄	#1950	1	1	yes	characteristic
218	C₄	#2210	1	1	yes	characteristic
219	C₄	#2211	1	1	yes	characteristic
220	C₄	#2228	1	1	yes	characteristic
221	C₄	#2229	1	1	yes	characteristic
222	C₂	#9601	1	1	yes	characteristic
223	C₂	#12008	1	1	yes	characteristic
224	C₂	#12009	1	1	yes	characteristic
225	C₂	#12034	1	1	yes	characteristic
226	C₂	#12035	1	1	yes	characteristic
227	C₂	#13178	1	1	yes	characteristic
228	C₂	#13179	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ 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
42. C₄⋊C₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₁₆	2
C₂×#1372	2
C₂×#1371	2
#8807	2
C₄×#335	2
#7154	2