Subgroups — #134669

Order 512.

376 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#12785	C₂	1	2	yes	maximal
3	#12004	C₂	1	2	yes	maximal, complemented
4	#7152	C₂	1	2	yes	maximal, complemented
5	C₂×#431	C₄	1	1	yes	characteristic
6	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
8	#2150	C₂²	1	4	yes	—
9	C₂×#349	—	2	2	no	—
10	C₈×M₁₆	—	2	4	no	—
11	C₂×#322	C₄	1	1	yes	characteristic
12	C₂×#322	C₂²	1	1	yes	characteristic
13	C₂×C₈²	C₄	1	1	yes	characteristic
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1377	—	2	2	no	—
16	#1370	—	2	4	no	—
17	C₂²×M₁₆	D₄	1	1	yes	characteristic
18	C₂²×M₁₆	Q₈	1	1	yes	characteristic
19	C₂³×C₈	C₈	1	2	yes	—
20	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₂×M₃₂	—	2	4	no	—
22	#431	—	2	2	no	—
23	#431	—	2	2	no	—
24	C₂×C₄⋊C₈	—	2	1	no	—
25	C₂×C₂²⋊C₈	—	2	1	no	—
26	C₄×M₁₆	—	2	2	no	—
27	C₄×M₁₆	—	2	2	no	—
28	C₂×#97	D₄	1	1	yes	characteristic
29	C₂×#97	Q₈	1	1	yes	characteristic
30	C₂×C₄×C₈	—	2	2	no	—
31	C₂×C₄×C₈	—	2	1	no	—
32	C₂×C₄×C₈	—	2	1	no	—
33	C₂×C₄×C₈	C₈	1	2	yes	—
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	C₄⋊C₁₆	—	2	4	no	—
41	C₄⋊C₁₆	—	2	4	no	—
42	#349	—	4	4	no	—
43	#322	—	2	2	no	—
44	#322	—	2	2	no	—
45	C₈²	—	2	2	no	—
46	C₈²	D₄	1	4	yes	—
47	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
48	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
49	C₂³×C₄	M₁₆	1	1	yes	characteristic
50	C₂×M₁₆	—	2	2	no	—
51	C₂×M₁₆	—	2	2	no	—
52	C₂×M₁₆	—	2	2	no	—
53	C₂×M₁₆	—	2	2	no	—
54	C₂×M₁₆	—	2	2	no	—
55	C₂×M₁₆	D₈	1	2	yes	—
56	C₂×M₁₆	Q₁₆	1	2	yes	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	—	2	1	no	—
59	C₂²×C₈	—	2	2	no	—
60	C₂²×C₈	—	2	2	no	—
61	C₂²×C₈	—	4	4	no	—
62	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂×C₈	1	2	yes	—
65	C₂²×C₈	C₂×C₈	1	2	yes	—
66	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₄²	—	2	1	no	—
69	C₂×C₄²	—	2	1	no	—
70	C₂×C₄²	—	2	1	no	—
71	C₂×C₄²	—	2	1	no	—
72	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄²	M₁₆	1	1	yes	characteristic
75	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
76	M₃₂	—	4	4	no	—
77	M₃₂	—	4	4	no	—
78	C₂×C₁₆	—	4	4	no	—
79	C₄⋊C₈	—	2	4	no	—
80	C₂²⋊C₈	—	2	4	no	—
81	#97	—	2	2	no	—
82	#97	—	2	2	no	—
83	C₄×C₈	—	2	2	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂²⋊C₄	1	2	yes	—
92	C₄×C₈	C₂²⋊C₄	1	2	yes	—
93	C₄×C₈	M₁₆	1	2	yes	—
94	C₄×C₈	M₁₆	1	1	yes	characteristic
95	C₄×C₈	M₁₆	1	1	yes	characteristic
96	C₄×C₈	C₂×D₄	1	2	yes	—
97	C₄×C₈	#41	1	2	yes	—
98	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	4	2	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₄	#107	1	2	yes	—
113	C₂²×C₄	#108	1	2	yes	—
114	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
116	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
117	M₁₆	—	2	8	no	—
118	M₁₆	—	2	4	no	—
119	M₁₆	—	2	4	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	C₂²⋊C₈	1	2	yes	—
136	C₂×C₈	C₂²⋊C₈	1	2	yes	—
137	C₂×C₈	#107	1	2	yes	—
138	C₂×C₈	#108	1	2	yes	—
139	C₂×C₈	#121	1	2	yes	—
140	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₈	C₂×D₈	1	1	yes	characteristic
142	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
143	C₂×C₈	#135	1	2	yes	—
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	—	2	1	no	—
150	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₄²	#117	1	1	yes	characteristic
152	C₄²	C₂×M₁₆	1	1	yes	characteristic
153	C₄²	C₂×M₁₆	1	1	yes	characteristic
154	C₁₆	—	8	4	no	—
155	C₂³	—	2	1	no	—
156	C₂³	—	2	1	no	—
157	C₂³	—	4	1	no	—
158	C₂³	—	4	1	no	—
159	C₂³	#335	1	2	yes	—
160	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	#335	1	2	yes	—
178	C₂×C₄	#380	1	1	yes	characteristic
179	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
180	C₂×C₄	#407	1	1	yes	characteristic
181	C₂×C₄	#423	1	1	yes	characteristic
182	C₂×C₄	C₂×#107	1	1	yes	characteristic
183	C₂×C₄	#427	1	1	yes	characteristic
184	C₂×C₄	C₂×#108	1	1	yes	characteristic
185	C₂×C₄	C₂×#108	1	1	yes	characteristic
186	C₂×C₄	#431	1	1	yes	characteristic
187	C₂×C₄	#432	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	2	2	no	—
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	—	4	2	no	—
193	C₈	—	4	2	no	—
194	C₈	—	4	2	no	—
195	C₈	#349	1	4	yes	complemented
196	C₈	#466	1	2	yes	—
197	C₈	#467	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	1	no	—
204	C₂²	—	4	1	no	—
205	C₂²	C₂×#335	1	1	yes	characteristic
206	C₂²	#1564	1	1	yes	characteristic
207	C₂²	#1797	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	4	1	no	—
212	C₄	—	4	1	no	—
213	C₄	#1567	1	1	yes	characteristic
214	C₄	#1568	1	1	yes	characteristic
215	C₄	#1808	1	1	yes	characteristic
216	C₄	#1810	1	1	yes	characteristic
217	C₄	C₂×#349	1	2	yes	—
218	C₄	#2166	1	2	yes	—
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	—	4	2	no	—
222	C₂	#8596	1	1	yes	characteristic
223	C₂	#12785	1	1	yes	characteristic
224	C₂	#12790	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×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
3. #12004	C₂	semidirect
4. #7152	C₂	semidirect
195. C₈	#349	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×M₁₆	2
#12785	2
#12004	2
#7152	2