Subgroups — #140069

Order 512.

252 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#367	C₂	1	1	yes	maximal, characteristic
2	#9594	C₂	1	1	yes	maximal, characteristic
3	C₈×#107	C₂	1	1	yes	maximal, characteristic
4	#7171	C₂	1	1	yes	maximal, characteristic
5	#7171	C₂	1	1	yes	maximal, characteristic
6	#7154	C₂	1	1	yes	maximal, characteristic
7	#7136	C₂	1	1	yes	maximal, characteristic
8	C₂×#367	C₄	1	1	yes	characteristic
9	C₂×#367	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1939	—	2	1	no	—
13	C₄×#107	C₂²	1	1	yes	characteristic
14	C₄×#107	C₂²	1	1	yes	characteristic
15	C₈×C₄⋊C₄	—	2	1	no	—
16	C₂×#335	C₂²	1	1	yes	characteristic
17	C₂×#335	C₂²	1	1	yes	characteristic
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1380	C₄	1	1	yes	characteristic
20	#1380	C₄	1	1	yes	characteristic
21	#1279	—	2	1	no	—
22	#1279	—	2	1	no	—
23	C₂²×C₁₆	—	2	1	no	—
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×#107	C₂×C₄	1	1	yes	characteristic
27	C₂×#107	C₂×C₄	1	1	yes	characteristic
28	C₂×#107	C₂×C₄	1	1	yes	characteristic
29	C₂×#107	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₈	—	2	1	no	—
31	C₂×C₄⋊C₈	—	2	1	no	—
32	C₂×C₄⋊C₈	—	4	1	no	—
33	C₂×C₄×C₈	—	4	1	no	—
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₄⋊C₄	—	2	1	no	—
38	C₄×C₄⋊C₄	—	2	1	no	—
39	#367	—	2	2	no	—
40	#367	C₈	1	4	yes	complemented
41	C₄×C₁₆	—	2	1	no	—
42	C₄×C₁₆	—	2	1	no	—
43	C₄×C₁₆	—	2	1	no	—
44	C₄×C₁₆	—	2	1	no	—
45	#340	—	2	1	no	—
46	#340	—	2	1	no	—
47	#336	—	4	1	no	—
48	#336	—	4	1	no	—
49	#335	—	2	1	no	—
50	#335	—	2	1	no	—
51	#335	—	2	1	no	—
52	#335	Q₈	1	2	yes	—
53	C₈²	—	2	1	no	—
54	C₈²	—	2	1	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₂²×C₈	—	4	1	no	—
57	C₂²×C₈	—	4	1	no	—
58	C₂²×C₈	D₈	1	1	yes	characteristic
59	C₂²×C₈	D₈	1	1	yes	characteristic
60	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₈	#41	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	2	1	no	—
63	C₂×C₄⋊C₄	—	2	1	no	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	—	2	1	no	—
66	C₂×C₄²	—	4	1	no	—
67	C₂×C₄²	—	4	1	no	—
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	2	1	no	—
73	C₂×C₁₆	—	2	1	no	—
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	4	1	no	—
76	C₂×C₁₆	—	4	1	no	—
77	C₂×C₁₆	C₂×C₈	1	2	yes	—
78	#107	—	2	1	no	—
79	#107	—	2	1	no	—
80	#107	—	2	1	no	—
81	#107	—	2	1	no	—
82	#107	—	2	1	no	—
83	#107	C₂×C₈	1	2	yes	—
84	#107	C₂×C₈	1	2	yes	—
85	#107	M₁₆	1	2	yes	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	4	1	no	—
88	C₄⋊C₈	—	4	1	no	—
89	C₄⋊C₈	—	4	1	no	—
90	C₄⋊C₈	—	8	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	8	2	no	—
96	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₄×C₈	#41	1	1	yes	characteristic
98	C₄×C₈	#41	1	1	yes	characteristic
99	C₄×C₈	#41	1	1	yes	characteristic
100	#95	—	4	1	no	—
101	#95	—	4	1	no	—
102	C₂²×C₄	—	4	1	no	—
103	C₂²×C₄	—	4	1	no	—
104	C₂²×C₄	—	4	1	no	—
105	C₂²×C₄	—	4	1	no	—
106	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
107	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
108	C₂²×C₄	#135	1	1	yes	characteristic
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	8	1	no	—
121	C₂×C₈	—	8	1	no	—
122	C₂×C₈	—	8	1	no	—
123	C₂×C₈	—	8	1	no	—
124	C₂×C₈	—	8	1	no	—
125	C₂×C₈	SD₃₂	1	2	yes	—
126	C₂×C₈	#117	1	1	yes	characteristic
127	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
128	C₂×C₈	#125	1	1	yes	characteristic
129	C₂×C₈	#126	1	1	yes	characteristic
130	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₈	#131	1	1	yes	characteristic
133	C₂×C₈	#131	1	1	yes	characteristic
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄²	—	8	1	no	—
143	C₄²	—	8	1	no	—
144	C₄²	—	8	1	no	—
145	C₄²	—	8	1	no	—
146	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
147	C₄²	#123	1	1	yes	characteristic
148	C₄²	#137	1	1	yes	characteristic
149	C₄²	#137	1	1	yes	characteristic
150	C₁₆	—	2	2	no	—
151	C₁₆	—	4	2	no	—
152	C₁₆	—	4	1	no	—
153	C₂³	C₄×D₈	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	1	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	1	no	—
166	C₂×C₄	—	8	1	no	—
167	C₂×C₄	—	8	1	no	—
168	C₂×C₄	#387	1	1	yes	characteristic
169	C₂×C₄	C₈×D₄	1	1	yes	characteristic
170	C₂×C₄	#435	1	1	yes	characteristic
171	C₂×C₄	#441	1	1	yes	characteristic
172	C₂×C₄	#443	1	1	yes	characteristic
173	C₂×C₄	#444	1	1	yes	characteristic
174	C₂×C₄	#474	1	1	yes	characteristic
175	C₂×C₄	#479	1	1	yes	characteristic
176	C₂×C₄	#480	1	1	yes	characteristic
177	C₂×C₄	#485	1	1	yes	characteristic
178	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
179	C₂×C₄	#508	1	1	yes	characteristic
180	C₂×C₄	#509	1	1	yes	characteristic
181	C₂×C₄	#510	1	1	yes	characteristic
182	C₈	—	2	1	no	—
183	C₈	—	2	1	no	—
184	C₈	—	2	2	no	—
185	C₈	—	4	2	no	—
186	C₈	—	4	1	no	—
187	C₈	—	8	2	no	—
188	C₈	—	16	1	no	—
189	C₈	#431	1	1	yes	characteristic
190	C₈	#433	1	1	yes	characteristic
191	C₈	C₈×Q₈	1	1	yes	characteristic
192	C₈	#446	1	1	yes	characteristic
193	C₂²	C₈×D₈	1	1	yes	characteristic
194	C₂²	#1581	1	1	yes	characteristic
195	C₂²	#1918	1	1	yes	characteristic
196	C₂²	C₄×SD₃₂	1	1	yes	characteristic
197	C₂²	#2175	1	1	yes	characteristic
198	C₂²	#2178	1	1	yes	characteristic
199	C₂²	#2179	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	2	no	—
203	C₄	—	8	2	no	—
204	C₄	—	8	2	no	—
205	C₄	—	16	1	no	—
206	C₄	#1579	1	1	yes	characteristic
207	C₄	#1587	1	1	yes	characteristic
208	C₄	#1916	1	1	yes	characteristic
209	C₄	#1922	1	1	yes	characteristic
210	C₄	#2225	1	1	yes	characteristic
211	C₄	#2228	1	1	yes	characteristic
212	C₄	#2233	1	1	yes	characteristic
213	C₄	#2237	1	1	yes	characteristic
214	C₂	#9533	1	1	yes	characteristic
215	C₂	C₈×SD₃₂	1	1	yes	characteristic
216	C₂	#12043	1	1	yes	characteristic
217	C₂	#12071	1	1	yes	characteristic
218	C₂	#12074	1	1	yes	characteristic
219	C₂	#13110	1	1	yes	characteristic
220	C₂	#13113	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	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ 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
40. #367	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#367	2
#9594	2
C₈×#107	2
#7171	2
#7154	2
#7136	2