Subgroups — #147002

Order 512.

512 conjugacy classes of subgroups.

The subgroup lattice has 233 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2108	C₂	1	1	yes	maximal, characteristic
2	C₂×#1833	C₂	1	2	yes	maximal
3	C₂×#1385	C₂	1	1	yes	maximal, characteristic
4	C₂×#1384	C₂	1	1	yes	maximal, characteristic
5	C₂×#1382	C₂	1	2	yes	maximal
6	C₂²×M₃₂	C₂²	1	1	yes	characteristic
7	C₂³×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂×#427	C₂²	1	2	yes	—
10	C₂×#405	C₂²	1	1	yes	characteristic
11	#2108	—	2	4	no	—
12	C₂×#346	C₄	1	2	yes	complemented
13	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
14	#1833	—	2	4	no	—
15	#1833	C₄	1	8	yes	complemented
16	C₂×#344	—	2	2	no	—
17	C₂×H(3,ℤ/4)	—	2	2	no	—
18	#1385	—	2	2	no	—
19	#1384	—	2	2	no	—
20	#1382	—	2	4	no	—
21	C₂²×M₁₆	—	2	2	no	—
22	C₂²×M₁₆	D₄	1	1	yes	characteristic
23	C₂³×C₈	C₂³	1	1	yes	characteristic
24	C₂×#117	—	2	2	no	—
25	C₂×#117	D₄	1	1	yes	characteristic
26	C₂×M₃₂	—	2	2	no	—
27	C₂×M₃₂	—	2	2	no	—
28	C₂×M₃₂	D₄	1	2	yes	—
29	C₂×M₃₂	Q₈	1	2	yes	—
30	C₂²×C₁₆	—	2	2	no	—
31	C₂²×C₁₆	—	2	2	no	—
32	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₆	C₂×C₄	1	2	yes	—
34	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₁₆	D₄	1	2	yes	—
36	C₂²×C₁₆	Q₈	1	2	yes	—
37	C₂×#108	—	2	2	no	—
38	C₂×#108	—	2	2	no	—
39	C₂×#108	—	2	2	no	—
40	C₂×#108	—	2	2	no	—
41	C₂×#108	—	2	2	no	—
42	C₂×#108	C₂×C₄	1	4	yes	—
43	#427	—	2	2	no	—
44	#427	—	2	2	no	—
45	#427	—	2	2	no	—
46	#427	C₂×C₄	1	4	yes	—
47	C₂×#107	—	2	2	no	—
48	C₂×#106	—	2	2	no	—
49	#405	C₂×C₄	1	4	yes	—
50	#405	D₄	1	4	yes	—
51	C₂×#97	Q₈	1	2	yes	—
52	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
53	#346	—	2	4	no	—
54	C₄×C₁₆	—	2	4	no	—
55	#344	—	2	8	no	—
56	H(3,ℤ/4)	—	2	8	no	—
57	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	—	2	1	no	—
60	C₂×M₁₆	—	2	2	no	—
61	C₂×M₁₆	—	2	4	no	—
62	C₂×M₁₆	—	4	2	no	—
63	C₂×M₁₆	—	4	2	no	—
64	C₂×M₁₆	—	4	2	no	—
65	C₂×M₁₆	—	4	2	no	—
66	C₂×M₁₆	—	4	2	no	—
67	C₂×M₁₆	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₈	—	4	2	no	—
69	C₂²×C₈	C₄⋊C₄	1	2	yes	—
70	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₈	C₂²×C₄	1	2	yes	—
72	C₂²×C₈	C₂×D₄	1	2	yes	—
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₈	#41	1	2	yes	—
79	C₂²×C₈	#41	1	2	yes	—
80	#117	—	2	2	no	—
81	#117	—	2	4	no	—
82	#117	—	4	2	no	—
83	#117	—	4	2	no	—
84	#117	—	4	2	no	—
85	#117	C₂²⋊C₄	1	4	yes	—
86	C₂×C₄⋊C₄	—	2	1	no	—
87	C₂×C₄⋊C₄	—	4	2	no	—
88	C₂×C₂²⋊C₄	—	2	1	no	—
89	C₂×C₂²⋊C₄	—	4	2	no	—
90	C₂×C₄²	—	4	2	no	—
91	C₂×C₄²	C₄⋊C₄	1	2	yes	—
92	M₃₂	—	2	4	no	—
93	M₃₂	—	2	4	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	4	no	—
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
103	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
104	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
105	#108	—	2	4	no	—
106	#108	—	2	4	no	—
107	#108	—	4	2	no	—
108	#108	—	4	2	no	—
109	#107	—	2	4	no	—
110	#107	—	4	2	no	—
111	#106	—	2	4	no	—
112	#106	—	4	2	no	—
113	#97	C₄⋊C₄	1	8	yes	—
114	C₄×C₈	C₄²	1	4	yes	—
115	C₄×C₈	C₂²⋊C₄	1	4	yes	—
116	C₂⁴	#135	1	1	yes	characteristic
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	8	2	no	—
119	C₂²×C₄	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄	#124	1	2	yes	—
122	C₂²×C₄	#127	1	1	yes	characteristic
123	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
124	C₂²×C₄	#135	1	1	yes	characteristic
125	C₂²×C₄	#135	1	1	yes	characteristic
126	C₂²×C₄	#135	1	2	yes	—
127	C₂²×C₄	#135	1	1	yes	characteristic
128	C₂²×C₄	#135	1	2	yes	—
129	M₁₆	—	2	4	no	—
130	M₁₆	—	2	4	no	—
131	M₁₆	—	4	4	no	—
132	M₁₆	—	4	4	no	—
133	M₁₆	—	8	2	no	—
134	M₁₆	—	8	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	8	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	#95	1	4	yes	—
141	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₈	#117	1	1	yes	characteristic
147	C₂×C₈	#117	1	1	yes	characteristic
148	C₂×C₈	C₄×D₄	1	2	yes	—
149	C₂×C₈	C₄×D₄	1	2	yes	—
150	C₂×C₈	C₄×Q₈	1	2	yes	—
151	C₂×C₈	C₄×Q₈	1	2	yes	—
152	C₂×C₈	#121	1	2	yes	—
153	C₂×C₈	#121	1	2	yes	—
154	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
156	C₂×C₈	#125	1	2	yes	—
157	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
158	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	8	2	no	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	4	4	no	—
164	C₂²⋊C₄	—	8	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	4	4	no	—
167	C₄²	—	8	2	no	—
168	C₄²	#95	1	4	yes	—
169	C₁₆	—	2	2	no	—
170	C₁₆	—	2	2	no	—
171	C₁₆	—	2	4	no	—
172	C₂³	—	4	2	no	—
173	C₂³	#416	1	2	yes	—
174	C₂³	#427	1	1	yes	characteristic
175	C₂³	#488	1	2	yes	—
176	C₂³	#495	1	1	yes	characteristic
177	C₂³	#499	1	1	yes	characteristic
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	—	16	4	no	—
182	C₂×C₄	C₂×#95	1	1	yes	characteristic
183	C₂×C₄	#382	1	1	yes	characteristic
184	C₂×C₄	#384	1	1	yes	characteristic
185	C₂×C₄	#390	1	2	yes	—
186	C₂×C₄	#391	1	2	yes	—
187	C₂×C₄	#416	1	2	yes	—
188	C₂×C₄	#416	1	2	yes	—
189	C₂×C₄	#416	1	2	yes	—
190	C₂×C₄	#427	1	1	yes	characteristic
191	C₂×C₄	#427	1	1	yes	characteristic
192	C₂×C₄	#427	1	1	yes	characteristic
193	C₂×C₄	#488	1	2	yes	—
194	C₂×C₄	#488	1	2	yes	—
195	C₂×C₄	#488	1	2	yes	—
196	C₂×C₄	#495	1	2	yes	—
197	C₂×C₄	#495	1	1	yes	characteristic
198	C₂×C₄	#499	1	2	yes	—
199	C₂×C₄	#499	1	1	yes	characteristic
200	C₈	—	2	4	no	—
201	C₈	—	8	4	no	—
202	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
203	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
204	C₈	#380	1	1	yes	characteristic
205	C₈	#380	1	1	yes	characteristic
206	C₈	#389	1	2	yes	—
207	C₈	#389	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	#1735	1	1	yes	characteristic
214	C₂²	#1835	1	1	yes	characteristic
215	C₂²	#1846	1	1	yes	characteristic
216	C₂²	#1958	1	2	yes	—
217	C₂²	#1991	1	2	yes	—
218	C₄	—	4	4	no	—
219	C₄	—	16	4	no	—
220	C₄	C₄⋊#95	1	1	yes	characteristic
221	C₄	#1735	1	1	yes	characteristic
222	C₄	#1835	1	1	yes	characteristic
223	C₄	#1846	1	1	yes	characteristic
224	C₄	#1958	1	2	yes	—
225	C₄	#1991	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	2	no	—
228	C₂	—	2	1	no	—
229	C₂	—	2	2	no	—
230	C₂	#10842	1	1	yes	characteristic
231	C₂	#12893	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1835
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. C₂×#346	C₄	semidirect
13. C₂×C₄×C₁₆	C₄	semidirect
15. #1833	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2108	2
C₂×#1833	2
C₂×#1385	2
C₂×#1384	2
C₂×#1382	2