Subgroups — #140827

Order 512.

392 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	C₈×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₁₆	C₂	1	4	yes	maximal
5	C₂×C₈×Q₈	C₂²	1	2	yes	—
6	C₄×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₁₆	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₁₆	—	2	1	no	—
10	C₂×C₄×C₁₆	C₂²	1	2	yes	—
11	#1984	C₄	1	2	yes	—
12	#1939	—	2	1	no	—
13	#1939	—	2	1	no	—
14	#1842	C₄	1	1	yes	characteristic
15	C₈×C₄⋊C₄	—	2	1	no	—
16	C₈×C₄⋊C₄	—	2	1	no	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	#1374	—	2	1	no	—
19	#1374	—	2	1	no	—
20	Q₈⋊C₁₆	—	2	8	no	—
21	C₂×C₄⋊Q₈	C₈	1	1	yes	characteristic
22	C₂×C₄×Q₈	C₂×C₄	1	2	yes	—
23	C₂²×C₁₆	—	2	2	no	—
24	C₂²×C₁₆	—	2	2	no	—
25	C₈×Q₈	—	2	4	no	—
26	C₈×Q₈	—	2	4	no	—
27	C₂×C₄⋊C₈	—	2	2	no	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₄⋊C₈	—	2	1	no	—
30	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₈	—	2	2	no	—
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	#391	C₈	1	2	yes	—
39	#389	—	2	1	no	—
40	#389	—	2	1	no	—
41	#384	C₈	1	1	yes	characteristic
42	C₄×C₄⋊C₄	—	2	1	no	—
43	C₄×C₄⋊C₄	—	2	1	no	—
44	C₄³	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₁₆	—	2	4	no	—
46	C₄⋊C₁₆	—	2	2	no	—
47	C₄⋊C₁₆	D₄	1	4	yes	—
48	C₄×C₁₆	—	2	4	no	—
49	C₄×C₁₆	—	4	2	no	—
50	#336	—	2	2	no	—
51	#336	—	2	2	no	—
52	C₂²×Q₈	C₂×C₈	1	2	yes	—
53	C₂²×C₈	—	2	2	no	—
54	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₄⋊Q₈	—	2	2	no	—
59	C₄⋊Q₈	C₁₆	1	4	yes	complemented
60	C₄×Q₈	—	2	4	no	—
61	C₄×Q₈	—	2	4	no	—
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×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
68	C₂×C₄²	—	2	2	no	—
69	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄²	M₁₆	1	1	yes	characteristic
74	C₂×C₄²	M₁₆	1	1	yes	characteristic
75	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₁₆	—	2	4	no	—
77	C₂×C₁₆	—	2	4	no	—
78	C₂×C₁₆	—	4	2	no	—
79	C₂×C₁₆	—	4	2	no	—
80	C₂×C₁₆	—	4	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₄⋊C₈	—	2	4	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄⋊C₈	—	4	2	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	4	1	no	—
91	C₄×C₈	—	4	4	no	—
92	C₄×C₈	SD₁₆	1	2	yes	—
93	C₄×C₈	Q₁₆	1	2	yes	—
94	C₄×C₈	C₂×D₄	1	2	yes	—
95	C₄×C₈	#41	1	2	yes	—
96	#95	—	2	2	no	—
97	#95	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	—	2	4	no	—
101	C₂×Q₈	C₂×C₁₆	1	4	yes	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
107	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
110	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
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	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	#103	1	2	yes	—
122	C₂×C₈	#121	1	2	yes	—
123	C₂×C₈	#124	1	1	yes	characteristic
124	C₂×C₈	#127	1	1	yes	characteristic
125	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₈	#137	1	2	yes	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	4	1	no	—
139	C₄²	—	4	2	no	—
140	C₄²	—	4	2	no	—
141	C₄²	#103	1	2	yes	—
142	C₄²	#104	1	2	yes	—
143	C₄²	C₂×C₁₆	1	2	yes	—
144	C₄²	M₃₂	1	2	yes	—
145	C₄²	C₄×D₄	1	2	yes	—
146	C₄²	#131	1	2	yes	—
147	C₁₆	—	4	4	no	—
148	C₁₆	—	4	4	no	—
149	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	Q₈	—	2	8	no	—
151	Q₈	—	4	4	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	4	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	Q₈⋊C₈	1	2	yes	—
168	C₂×C₄	Q₈⋊C₈	1	2	yes	—
169	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
170	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
171	C₂×C₄	#350	1	2	yes	—
172	C₂×C₄	#390	1	1	yes	characteristic
173	C₂×C₄	#408	1	1	yes	characteristic
174	C₂×C₄	C₂×#103	1	1	yes	characteristic
175	C₂×C₄	#419	1	1	yes	characteristic
176	C₂×C₄	C₂×#104	1	1	yes	characteristic
177	C₂×C₄	#421	1	1	yes	characteristic
178	C₂×C₄	C₈×D₄	1	2	yes	—
179	C₂×C₄	#436	1	2	yes	—
180	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
181	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
182	C₂×C₄	#504	1	2	yes	—
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	2	2	no	—
186	C₈	—	4	1	no	—
187	C₈	—	8	2	no	—
188	C₈	#461	1	2	yes	—
189	C₈	#462	1	2	yes	—
190	C₂²	C₂×Q₈⋊C₈	1	1	yes	characteristic
191	C₂²	#1479	1	1	yes	characteristic
192	C₂²	#1954	1	1	yes	characteristic
193	C₂²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
194	C₂²	#2113	1	1	yes	characteristic
195	C₂²	C₂×#350	1	1	yes	characteristic
196	C₂²	#2117	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	1	no	—
201	C₄	—	4	4	no	—
202	C₄	—	8	2	no	—
203	C₄	Q₈⋊C₁₆	1	4	yes	complemented
204	C₄	#1481	1	1	yes	characteristic
205	C₄	#1485	1	1	yes	characteristic
206	C₄	#1980	1	1	yes	characteristic
207	C₄	#1981	1	1	yes	characteristic
208	C₄	D₄×C₁₆	1	2	yes	—
209	C₄	#2169	1	2	yes	—
210	C₂	#9673	1	1	yes	characteristic
211	C₂	C₂×Q₈⋊C₁₆	1	1	yes	characteristic
212	C₂	#11606	1	1	yes	characteristic
213	C₂	#11637	1	1	yes	characteristic
214	C₂	#11638	1	1	yes	characteristic
215	C₂	#13189	1	1	yes	characteristic
216	C₂	#13193	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×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
59. C₄⋊Q₈	C₁₆	semidirect
203. C₄	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Q₈	2
C₄×C₄⋊C₁₆	2
C₂×Q₈⋊C₁₆	2