Subgroups — #140384

Order 512.

290 conjugacy classes of subgroups.

The subgroup lattice has 207 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12898	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1367	C₂	1	2	yes	maximal
4	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#7151	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	—	2	1	no	—
7	C₂×C₄⋊C₁₆	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	—	2	1	no	—
12	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈²	C₂²	1	2	yes	—
16	#1374	—	2	1	no	—
17	#1367	—	2	4	no	—
18	#1273	C₄	1	2	yes	—
19	C₂²×C₁₆	—	2	1	no	—
20	C₂²×C₁₆	—	2	1	no	—
21	C₂²×C₁₆	—	2	2	no	—
22	C₂×C₄⋊C₈	—	2	1	no	—
23	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
24	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	C₄³	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₁₆	—	2	2	no	—
35	C₄⋊C₁₆	—	2	1	no	—
36	C₄⋊C₁₆	—	2	1	no	—
37	C₄⋊C₁₆	—	2	2	no	—
38	C₄⋊C₁₆	—	4	2	no	—
39	C₄⋊C₁₆	—	4	2	no	—
40	C₄⋊C₁₆	D₄	1	2	yes	—
41	C₄⋊C₁₆	Q₈	1	2	yes	—
42	C₄×C₁₆	—	4	2	no	—
43	#336	—	2	1	no	—
44	#336	C₈	1	2	yes	—
45	C₈²	—	2	2	no	—
46	C₈²	—	2	2	no	—
47	C₂²×C₈	—	2	2	no	—
48	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂×C₈	1	2	yes	—
51	C₂²×C₈	C₂×C₈	1	2	yes	—
52	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄²	M₁₆	1	1	yes	characteristic
59	C₂×C₄²	M₁₆	1	1	yes	characteristic
60	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	4	1	no	—
64	C₂×C₁₆	—	4	1	no	—
65	C₂×C₁₆	—	4	1	no	—
66	C₂×C₁₆	—	4	1	no	—
67	C₂×C₁₆	—	4	2	no	—
68	C₂×C₁₆	—	4	2	no	—
69	C₂×C₁₆	—	4	2	no	—
70	C₄⋊C₈	—	2	4	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₄⋊C₈	C₁₆	1	4	yes	complemented
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	—	2	2	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	SD₁₆	1	2	yes	—
84	C₄×C₈	SD₁₆	1	2	yes	—
85	C₄×C₈	C₂×D₄	1	1	yes	characteristic
86	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
90	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
91	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	#106	1	2	yes	—
112	C₂×C₈	C₂×C₁₆	1	4	yes	—
113	C₂×C₈	#121	1	1	yes	characteristic
114	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₈	#125	1	1	yes	characteristic
116	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₈	#137	1	1	yes	characteristic
121	C₄²	—	2	2	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	4	1	no	—
127	C₄²	#106	1	2	yes	—
128	C₄²	#108	1	2	yes	—
129	C₄²	C₂×C₁₆	1	2	yes	—
130	C₄²	M₃₂	1	2	yes	—
131	C₄²	C₄×D₄	1	1	yes	characteristic
132	C₄²	C₄×Q₈	1	1	yes	characteristic
133	C₄²	#131	1	1	yes	characteristic
134	C₄²	#131	1	1	yes	characteristic
135	C₁₆	—	4	2	no	—
136	C₁₆	—	4	2	no	—
137	C₁₆	—	8	2	no	—
138	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	#334	1	2	yes	—
149	C₂×C₄	#334	1	2	yes	—
150	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
151	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
152	C₂×C₄	#364	1	2	yes	—
153	C₂×C₄	#389	1	1	yes	characteristic
154	C₂×C₄	#424	1	1	yes	characteristic
155	C₂×C₄	C₂×#106	1	1	yes	characteristic
156	C₂×C₄	#428	1	1	yes	characteristic
157	C₂×C₄	C₂×#108	1	1	yes	characteristic
158	C₂×C₄	#430	1	1	yes	characteristic
159	C₂×C₄	C₈×D₄	1	1	yes	characteristic
160	C₂×C₄	#436	1	1	yes	characteristic
161	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
162	C₂×C₄	#446	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
164	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
165	C₂×C₄	#504	1	1	yes	characteristic
166	C₂×C₄	#504	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	2	4	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	4	4	no	—
172	C₈	—	4	1	no	—
173	C₈	—	4	2	no	—
174	C₈	#460	1	1	yes	characteristic
175	C₈	#461	1	1	yes	characteristic
176	C₈	#475	1	1	yes	characteristic
177	C₈	#476	1	1	yes	characteristic
178	C₂²	C₂×#334	1	1	yes	characteristic
179	C₂²	#1565	1	1	yes	characteristic
180	C₂²	#1939	1	1	yes	characteristic
181	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
182	C₂²	#2151	1	1	yes	characteristic
183	C₂²	C₂×#364	1	1	yes	characteristic
184	C₂²	#2153	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	4	1	no	—
190	C₄	#1367	1	4	yes	complemented
191	C₄	#1570	1	1	yes	characteristic
192	C₄	#1574	1	1	yes	characteristic
193	C₄	#1948	1	1	yes	characteristic
194	C₄	#1950	1	1	yes	characteristic
195	C₄	D₄×C₁₆	1	1	yes	characteristic
196	C₄	#2169	1	1	yes	characteristic
197	C₄	Q₈×C₁₆	1	1	yes	characteristic
198	C₄	#2183	1	1	yes	characteristic
199	C₂	#9602	1	1	yes	characteristic
200	C₂	C₂×#1367	1	1	yes	characteristic
201	C₂	#11997	1	1	yes	characteristic
202	C₂	#12015	1	1	yes	characteristic
203	C₂	#12018	1	1	yes	characteristic
204	C₂	#13159	1	1	yes	characteristic
205	C₂	#13163	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×C₄⋊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
72. C₄⋊C₈	C₁₆	semidirect
190. C₄	#1367	semidirect

Maximal subgroups

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

subgroup	index
#12898	2
C₄×C₄⋊C₁₆	2
C₂×#1367	2
C₈×C₄⋊C₈	2
#7151	2