Subgroups — #134673

Order 512.

398 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1372	C₂	1	2	yes	maximal, complemented
3	#8596	C₂	1	2	yes	maximal
4	#7154	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	—	2	4	no	—
9	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1797	—	2	2	no	—
14	C₂×#335	C₂²	1	4	yes	—
15	#1372	—	2	4	no	—
16	#1277	—	2	2	no	—
17	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₈	Q₈	1	1	yes	characteristic
19	C₂²×C₄²	D₄	1	1	yes	characteristic
20	C₂²×C₁₆	—	2	2	no	—
21	C₂²×C₁₆	—	2	2	no	—
22	C₂²×C₁₆	—	2	2	no	—
23	C₂²×C₁₆	—	2	2	no	—
24	C₂²×C₁₆	—	2	2	no	—
25	C₂²×C₁₆	C₈	1	4	yes	complemented
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	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₈	—	2	4	no	—
31	C₂×C₂²⋊C₈	—	4	2	no	—
32	C₂×C₄×C₈	—	2	2	no	—
33	C₂×C₄×C₈	—	2	2	no	—
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₁₆	—	2	2	no	—
39	C₄×C₁₆	—	2	2	no	—
40	C₄×C₁₆	—	2	2	no	—
41	C₄×C₁₆	D₄	1	4	yes	—
42	#336	—	4	2	no	—
43	#335	—	2	4	no	—
44	#335	—	2	4	no	—
45	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
46	C₂³×C₄	D₈	1	1	yes	characteristic
47	C₂³×C₄	Q₁₆	1	1	yes	characteristic
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	—	2	2	no	—
50	C₂²×C₈	—	2	1	no	—
51	C₂²×C₈	—	2	1	no	—
52	C₂²×C₈	—	2	2	no	—
53	C₂²×C₈	—	2	1	no	—
54	C₂²×C₈	—	2	1	no	—
55	C₂²×C₈	—	4	4	no	—
56	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×C₈	1	2	yes	—
59	C₂²×C₈	M₁₆	1	2	yes	—
60	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄²	—	2	2	no	—
63	C₂×C₄²	—	2	2	no	—
64	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄²	D₈	1	1	yes	characteristic
66	C₂×C₄²	Q₁₆	1	1	yes	characteristic
67	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
68	C₂×C₁₆	—	2	8	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	2	1	no	—
73	C₂×C₁₆	—	2	1	no	—
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	2	8	no	—
76	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
77	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
78	C₂×C₁₆	C₂×C₈	1	2	yes	—
79	C₂×C₁₆	C₂×C₈	1	2	yes	—
80	C₂×C₁₆	M₁₆	1	2	yes	—
81	C₂×C₁₆	M₁₆	1	2	yes	—
82	C₄⋊C₈	—	4	4	no	—
83	C₄⋊C₈	—	4	4	no	—
84	C₂²⋊C₈	—	8	4	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	C₂×D₄	1	2	yes	—
90	C₄×C₈	#41	1	2	yes	—
91	C₂⁴	#107	1	1	yes	characteristic
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	C₄⋊C₈	1	2	yes	—
102	C₂²×C₄	#107	1	1	yes	characteristic
103	C₂²×C₄	#107	1	1	yes	characteristic
104	C₂²×C₄	#107	1	1	yes	characteristic
105	C₂²×C₄	#108	1	2	yes	—
106	C₂²×C₄	SD₃₂	1	4	yes	—
107	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
109	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	8	4	no	—
122	C₂×C₈	—	8	4	no	—
123	C₂×C₈	—	8	4	no	—
124	C₂×C₈	C₄⋊C₈	1	2	yes	—
125	C₂×C₈	#108	1	2	yes	—
126	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₈	#117	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
133	C₄²	—	2	1	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	#121	1	2	yes	—
138	C₄²	#135	1	2	yes	—
139	C₁₆	—	2	2	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	2	2	no	—
142	C₁₆	C₂²⋊C₈	1	4	yes	complemented
143	C₂³	—	2	1	no	—
144	C₂³	—	2	1	no	—
145	C₂³	—	2	1	no	—
146	C₂³	—	2	1	no	—
147	C₂³	#335	1	2	yes	—
148	C₂³	#367	1	2	yes	—
149	C₂³	#368	1	2	yes	—
150	C₂³	C₂×#107	1	1	yes	characteristic
151	C₂×C₄	—	2	8	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	#335	1	2	yes	—
166	C₂×C₄	#367	1	2	yes	—
167	C₂×C₄	#368	1	2	yes	—
168	C₂×C₄	#380	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₂×C₄	#423	1	1	yes	characteristic
171	C₂×C₄	#427	1	1	yes	characteristic
172	C₂×C₄	C₂×#108	1	1	yes	characteristic
173	C₂×C₄	C₂×#108	1	1	yes	characteristic
174	C₂×C₄	#466	1	2	yes	—
175	C₂×C₄	#467	1	2	yes	—
176	C₂×C₄	C₂×SD₃₂	1	2	yes	—
177	C₂×C₄	#508	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	16	4	no	—
182	C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
183	C₈	#407	1	1	yes	characteristic
184	C₈	#431	1	1	yes	characteristic
185	C₈	#432	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	#1372	1	4	yes	complemented
193	C₂²	C₂×#335	1	1	yes	characteristic
194	C₂²	#1567	1	1	yes	characteristic
195	C₂²	#1808	1	1	yes	characteristic
196	C₂²	C₂×#367	1	1	yes	characteristic
197	C₂²	#2158	1	1	yes	characteristic
198	C₂²	C₂×#368	1	1	yes	characteristic
199	C₂²	C₂×#368	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	2	no	—
204	C₄	#1564	1	1	yes	characteristic
205	C₄	#1568	1	1	yes	characteristic
206	C₄	#1797	1	1	yes	characteristic
207	C₄	#1810	1	1	yes	characteristic
208	C₄	#2217	1	4	yes	—
209	C₂	—	2	4	no	—
210	C₂	#8596	1	1	yes	characteristic
211	C₂	C₂×#1372	1	1	yes	characteristic
212	C₂	#12008	1	1	yes	characteristic
213	C₂	#12023	1	1	yes	characteristic
214	C₂	#12024	1	1	yes	characteristic
215	C₂	#12835	1	1	yes	characteristic
216	C₂	#12837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×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
2. C₂×#1372	C₂	semidirect
4. #7154	C₂	semidirect
25. C₂²×C₁₆	C₈	semidirect
142. C₁₆	C₂²⋊C₈	semidirect
192. C₂²	#1372	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₆	2
C₂×#1372	2
#8596	2
#7154	2