Subgroups — #128561

Order 512.

378 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20516	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#12590	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₁₆	C₂	1	2	yes	maximal
5	#7167	C₂	1	2	yes	maximal
6	C₂×#446	C₄	1	1	yes	characteristic
7	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
8	#2295	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₁₆	—	2	1	no	—
11	C₂×C₄×C₁₆	C₂²	1	2	yes	—
12	#1771	—	2	1	no	—
13	#1770	C₄	1	1	yes	characteristic
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#1768	—	2	1	no	—
17	#1767	C₄	1	1	yes	characteristic
18	#1726	—	2	1	no	—
19	C₄²×C₈	C₂²	1	1	yes	characteristic
20	#1374	—	2	1	no	—
21	#1374	—	2	1	no	—
22	#1374	—	2	1	no	—
23	Q₈⋊C₁₆	C₄	1	8	yes	complemented
24	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	—	2	2	no	—
26	C₂²×C₁₆	—	2	2	no	—
27	#446	—	2	2	no	—
28	#446	C₈	1	4	yes	—
29	C₈×Q₈	—	2	2	no	—
30	C₈×Q₈	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₈	—	2	1	no	—
32	C₂×C₄⋊C₈	—	2	1	no	—
33	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×#97	—	2	1	no	—
36	C₂×#97	—	2	1	no	—
37	C₂×#97	—	2	1	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	—	2	1	no	—
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	#391	C₈	1	2	yes	—
47	#389	C₈	1	2	yes	—
48	#387	—	2	2	no	—
49	C₄×C₄⋊C₄	—	2	1	no	—
50	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	#376	—	2	1	no	—
52	C₄³	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₁₆	—	2	2	no	—
54	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
55	C₄×C₁₆	—	2	2	no	—
56	C₄×C₁₆	—	4	2	no	—
57	C₄×C₁₆	C₂×C₄	1	4	yes	—
58	#336	—	2	1	no	—
59	#336	—	2	1	no	—
60	#336	—	2	1	no	—
61	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
68	C₄×Q₈	—	2	1	no	—
69	C₄×Q₈	—	2	1	no	—
70	C₄×Q₈	C₄²	1	2	yes	—
71	C₄×Q₈	C₂×C₈	1	2	yes	—
72	C₂×C₄⋊C₄	—	2	1	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
75	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄²	—	2	1	no	—
77	C₂×C₄²	—	2	1	no	—
78	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄²	C₂×C₈	1	2	yes	—
81	C₂×C₄²	M₁₆	1	2	yes	—
82	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₁₆	—	2	4	no	—
84	C₂×C₁₆	—	2	4	no	—
85	C₂×C₁₆	—	4	2	no	—
86	C₂×C₁₆	—	4	2	no	—
87	C₂×C₁₆	—	4	2	no	—
88	C₂×C₁₆	—	4	2	no	—
89	C₄⋊C₈	—	2	4	no	—
90	C₄⋊C₈	—	2	4	no	—
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	C₄²	1	2	yes	—
94	C₄⋊C₈	C₂×C₈	1	2	yes	—
95	#97	—	2	4	no	—
96	#97	—	4	2	no	—
97	#97	—	4	2	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	—	4	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	C₂×C₈	1	2	yes	—
107	C₄×C₈	M₁₆	1	2	yes	—
108	C₄×C₈	C₂²×C₄	1	2	yes	—
109	C₄×C₈	#41	1	2	yes	—
110	#95	—	2	2	no	—
111	#95	—	2	2	no	—
112	#95	—	2	2	no	—
113	#95	—	2	1	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	C₄×C₈	1	2	yes	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
121	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
124	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	C₂²⋊C₈	1	2	yes	—
138	C₂×C₈	C₂²⋊C₈	1	2	yes	—
139	C₂×C₈	C₄×D₄	1	2	yes	—
140	C₂×C₈	C₄×D₄	1	2	yes	—
141	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₈	#137	1	2	yes	—
143	C₂×C₈	#137	1	1	yes	characteristic
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	C₄×C₈	1	2	yes	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	4	no	—
158	C₄²	—	4	1	no	—
159	C₄²	—	4	1	no	—
160	C₄²	C₂×C₄²	1	1	yes	characteristic
161	C₄²	#117	1	1	yes	characteristic
162	C₄²	C₂²×C₈	1	1	yes	characteristic
163	C₄²	C₂×M₁₆	1	1	yes	characteristic
164	C₁₆	—	4	4	no	—
165	C₁₆	—	4	4	no	—
166	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	4	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	#350	1	4	yes	—
183	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
185	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
187	C₂×C₄	#417	1	1	yes	characteristic
188	C₂×C₄	#419	1	1	yes	characteristic
189	C₂×C₄	#421	1	1	yes	characteristic
190	C₂×C₄	#421	1	1	yes	characteristic
191	C₂×C₄	#431	1	1	yes	characteristic
192	C₂×C₄	#431	1	1	yes	characteristic
193	C₂×C₄	#504	1	4	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	2	2	no	—
197	C₈	—	4	2	no	—
198	C₈	—	4	2	no	—
199	C₈	#440	1	2	yes	—
200	C₈	#441	1	2	yes	—
201	C₂²	#1477	1	1	yes	characteristic
202	C₂²	#1479	1	1	yes	characteristic
203	C₂²	C₄×C₂²⋊C₈	1	1	yes	characteristic
204	C₂²	#2113	1	2	yes	—
205	C₂²	C₂×#350	1	2	yes	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	4	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	8	2	no	—
213	C₄	#1477	1	1	yes	characteristic
214	C₄	#1479	1	1	yes	characteristic
215	C₄	#1759	1	1	yes	characteristic
216	C₄	#1763	1	1	yes	characteristic
217	C₄	#2108	1	2	yes	—
218	C₄	#2164	1	2	yes	—
219	C₂	#8171	1	1	yes	characteristic
220	C₂	#11622	1	2	yes	—
221	C₂	#11638	1	2	yes	—
222	C₂	#12600	1	1	yes	characteristic
223	C₂	C₄×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
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
23. Q₈⋊C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20516	2
C₄×C₄⋊C₁₆	2
#12590	2
C₂×Q₈⋊C₁₆	2
#7167	2