Subgroups — #512794

Order 512.

331 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33480	C₂	1	1	yes	maximal, characteristic
2	#13660	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#13646	C₂	1	1	yes	maximal, characteristic, complemented
5	#13646	C₂	1	2	yes	maximal, complemented
6	#13645	C₂	1	2	yes	maximal, complemented
7	D₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#13641	C₂	1	2	yes	maximal, complemented
9	#13639	C₂	1	1	yes	maximal, characteristic, complemented
10	#13549	C₂	1	2	yes	maximal, complemented
11	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#2966	C₄	1	1	yes	characteristic
13	C₂×M₆₄	C₂²	1	1	yes	characteristic
14	C₂×M₆₄	C₂²	1	1	yes	characteristic
15	C₂×M₆₄	C₂²	1	2	yes	—
16	C₂²×C₃₂	C₂²	1	2	yes	—
17	#2183	C₄	1	1	yes	characteristic
18	Q₈×C₁₆	C₂²	1	1	yes	characteristic
19	#2169	C₄	1	1	yes	characteristic
20	#2168	C₄	1	2	yes	—
21	D₄×C₁₆	C₂²	1	1	yes	characteristic
22	D₄×C₁₆	C₂²	1	2	yes	—
23	#2164	C₄	1	2	yes	—
24	#2163	C₄	1	1	yes	characteristic
25	#2108	C₂²	1	2	yes	—
26	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
27	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
28	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
29	C₄⋊C₃₂	C₂²	1	2	yes	—
30	C₄⋊C₃₂	C₂²	1	2	yes	—
31	C₂²⋊C₃₂	C₂²	1	2	yes	—
32	C₂²⋊C₃₂	C₂²	1	2	yes	—
33	C₂²⋊C₃₂	C₂²	1	2	yes	—
34	#1397	C₂²	1	1	yes	characteristic
35	#1397	C₂²	1	2	yes	—
36	#1397	C₂²	1	1	yes	characteristic
37	C₄×C₃₂	C₂²	1	2	yes	—
38	C₄×C₃₂	C₂²	1	1	yes	characteristic
39	C₄×C₃₂	C₂²	1	1	yes	characteristic
40	C₄×C₃₂	C₂²	1	2	yes	—
41	#529	C₈	1	1	yes	characteristic
42	C₂×M₃₂	C₂×C₄	1	2	yes	—
43	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	C₂²×C₁₆	C₂³	1	2	yes	—
46	#446	C₈	1	1	yes	characteristic
47	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
48	#436	C₈	1	1	yes	characteristic
49	#435	C₈	1	2	yes	—
50	C₈×D₄	C₂×C₄	1	2	yes	—
51	C₈×D₄	C₂×C₄	1	1	yes	characteristic
52	#433	C₈	1	2	yes	—
53	#432	C₈	1	1	yes	characteristic
54	#405	C₂×C₄	1	2	yes	—
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	M₆₄	—	2	2	no	—
57	M₆₄	—	2	2	no	—
58	M₆₄	—	2	4	no	—
59	C₂×C₃₂	—	2	4	no	—
60	C₂×C₃₂	C₂³	1	2	yes	—
61	C₂×C₃₂	C₂³	1	2	yes	—
62	C₂×C₃₂	C₂³	1	1	yes	characteristic
63	C₂×C₃₂	C₂³	1	1	yes	characteristic
64	C₂×C₃₂	C₂³	1	1	yes	characteristic
65	C₂×C₃₂	C₂³	1	1	yes	characteristic
66	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
69	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
70	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
71	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
72	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
73	#346	C₂×C₄	1	2	yes	—
74	#346	C₂×C₄	1	2	yes	—
75	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
76	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
77	C₄×C₁₆	C₂³	1	1	yes	characteristic
78	C₄×C₁₆	C₂³	1	1	yes	characteristic
79	C₄×C₁₆	C₂³	1	2	yes	—
80	C₂×M₁₆	C₂×C₈	1	2	yes	—
81	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₄	1	2	yes	—
84	#126	C₁₆	1	2	yes	—
85	#125	C₁₆	1	1	yes	characteristic
86	#124	C₁₆	1	1	yes	characteristic
87	#123	C₁₆	1	2	yes	—
88	C₂²⋊Q₈	C₁₆	1	1	yes	characteristic
89	#121	C₁₆	1	1	yes	characteristic
90	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
91	C₄×D₄	C₂×C₈	1	2	yes	—
92	C₄×D₄	C₂×C₈	1	1	yes	characteristic
93	#117	C₂×C₈	1	2	yes	—
94	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₁₆	C₂²×C₄	1	2	yes	—
104	C₂×C₁₆	C₂²×C₄	1	2	yes	—
105	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
108	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
109	C₄⋊C₈	C₂×C₈	1	2	yes	—
110	C₄⋊C₈	C₂×C₈	1	2	yes	—
111	C₂²⋊C₈	C₂×C₈	1	2	yes	—
112	C₂²⋊C₈	C₂×C₈	1	2	yes	—
113	C₂²⋊C₈	C₂×C₈	1	2	yes	—
114	#97	C₂×C₈	1	2	yes	—
115	#97	C₂×C₈	1	2	yes	—
116	C₄×C₈	C₂×C₈	1	1	yes	characteristic
117	C₄×C₈	C₂×C₈	1	1	yes	characteristic
118	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₄×C₈	C₂²×C₄	1	2	yes	—
121	C₃₂	—	2	2	no	—
122	C₃₂	—	2	1	no	—
123	C₃₂	—	2	2	no	—
124	C₃₂	—	2	1	no	—
125	C₃₂	#41	1	2	yes	—
126	C₃₂	#41	1	2	yes	—
127	C₂×Q₈	C₂×C₁₆	1	1	yes	characteristic
128	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
129	C₂×D₄	C₂×C₁₆	1	2	yes	—
130	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
132	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂²×C₄	C₂²×C₈	1	2	yes	—
134	M₁₆	—	2	4	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₈	C₂²×C₈	1	2	yes	—
143	C₂×C₈	C₂²×C₈	1	2	yes	—
144	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
145	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
146	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
147	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
148	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
149	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
150	C₄⋊C₄	C₂×C₁₆	1	1	yes	characteristic
151	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
152	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
153	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
154	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
155	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
156	C₂²⋊C₄	C₂×C₁₆	1	2	yes	—
157	C₄²	C₂×C₁₆	1	1	yes	characteristic
158	C₄²	C₂×C₁₆	1	1	yes	characteristic
159	C₄²	C₂²×C₈	1	1	yes	characteristic
160	C₄²	C₂²×C₈	1	1	yes	characteristic
161	C₄²	C₂²×C₈	1	2	yes	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	2	1	no	—
164	C₁₆	—	2	1	no	—
165	C₁₆	—	2	2	no	—
166	C₁₆	#131	1	2	yes	—
167	C₁₆	C₂×#41	1	1	yes	characteristic
168	C₁₆	C₂×#41	1	1	yes	characteristic
169	C₂³	C₂²×C₁₆	1	2	yes	—
170	C₂³	C₂²×C₁₆	1	1	yes	characteristic
171	Q₈	—	2	2	no	—
172	D₄	—	2	2	no	—
173	D₄	—	4	2	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
181	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
183	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₁₆	1	2	yes	—
185	C₂×C₄	C₂²×C₁₆	1	2	yes	—
186	C₂×C₄	C₂²×C₁₆	1	2	yes	—
187	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
188	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	2	no	—
193	C₈	#504	1	2	yes	—
194	C₈	C₄×#41	1	1	yes	characteristic
195	C₈	C₂×#131	1	1	yes	characteristic
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	2	no	—
202	C₂²	C₂³×C₁₆	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	2	no	—
210	C₄	#2258	1	2	yes	—
211	C₄	C₈×#41	1	1	yes	characteristic
212	C₄	C₂×#504	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	—	4	2	no	—
215	C₂	C₁₆×#41	1	1	yes	characteristic
216	C₂	C₂×#2258	1	1	yes	characteristic
217	C₂	#33974	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	#121	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #13660	C₂	semidirect
3. Q₈×C₃₂	C₂	semidirect
4. #13646	C₂	semidirect
5. #13646	C₂	semidirect
6. #13645	C₂	semidirect
7. D₄×C₃₂	C₂	semidirect
8. #13641	C₂	semidirect
9. #13639	C₂	semidirect
10. #13549	C₂	semidirect
11. C₄×M₆₄	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33480	2
#13660	2
Q₈×C₃₂	2
#13646	2
#13645	2
D₄×C₃₂	2
#13641	2
#13639	2
#13549	2
C₄×M₆₄	2