Subgroups — #10826

Order 256.

337 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#344	C₂	1	1	yes	maximal, characteristic
3	#1742	C₂	1	1	yes	maximal, characteristic
4	#1738	C₂	1	1	yes	maximal, characteristic
5	#1733	C₂	1	1	yes	maximal, characteristic
6	C₂×#339	C₂	1	1	yes	maximal, characteristic
7	C₂×#339	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×#117	C₂²	1	1	yes	characteristic
13	C₂×#117	C₂²	1	1	yes	characteristic
14	C₂²×C₄²	C₂²	1	1	yes	characteristic
15	C₄×M₁₆	—	2	2	no	—
16	C₄×M₁₆	C₄	1	4	yes	complemented
17	C₂×#97	—	2	1	no	—
18	C₂×C₄×C₈	—	2	1	no	—
19	#344	C₄	1	4	yes	complemented
20	#343	C₄	1	4	yes	complemented
21	#341	—	2	2	no	—
22	#339	—	2	2	no	—
23	#339	—	2	1	no	—
24	#339	—	2	1	no	—
25	#339	—	2	2	no	—
26	C₂³×C₄	—	2	1	no	—
27	C₂³×C₄	C₂³	1	1	yes	characteristic
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	—	2	1	no	—
33	C₂×M₁₆	—	2	1	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂×M₁₆	—	2	1	no	—
36	C₂×M₁₆	—	2	1	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂×M₁₆	—	2	1	no	—
39	C₂×M₁₆	—	2	1	no	—
40	C₂×M₁₆	—	2	1	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	—	2	1	no	—
43	C₂×M₁₆	—	2	1	no	—
44	C₂×M₁₆	C₂×C₄	1	2	yes	—
45	C₂×M₁₆	C₂×C₄	1	2	yes	—
46	C₂×M₁₆	C₂×C₄	1	2	yes	—
47	C₂×M₁₆	C₂×C₄	1	2	yes	—
48	C₂×M₁₆	C₂×C₄	1	2	yes	—
49	C₂×M₁₆	C₂×C₄	1	2	yes	—
50	C₂×M₁₆	D₄	1	2	yes	—
51	C₂×M₁₆	Q₈	1	2	yes	—
52	C₂²×C₈	—	2	1	no	—
53	C₂²×C₈	—	2	1	no	—
54	C₂²×C₈	—	2	1	no	—
55	C₂²×C₈	—	2	1	no	—
56	#117	—	2	1	no	—
57	#117	—	2	1	no	—
58	#117	—	2	1	no	—
59	#117	—	2	1	no	—
60	#117	—	2	1	no	—
61	#117	—	2	1	no	—
62	#117	—	2	1	no	—
63	#117	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄	—	2	1	no	—
67	C₂×C₂²⋊C₄	—	2	1	no	—
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	—	2	1	no	—
73	C₂×C₄²	C₂×C₄	1	2	yes	—
74	C₂×C₄²	D₄	1	1	yes	characteristic
75	C₂×C₄²	D₄	1	1	yes	characteristic
76	C₂×C₄²	D₄	1	2	yes	—
77	#97	—	2	4	no	—
78	C₄×C₈	—	2	4	no	—
79	C₂⁴	#41	1	1	yes	characteristic
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	4	4	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄	#41	1	1	yes	characteristic
95	C₂²×C₄	#41	1	1	yes	characteristic
96	C₂²×C₄	#41	1	1	yes	characteristic
97	C₂²×C₄	#41	1	1	yes	characteristic
98	C₂²×C₄	#41	1	1	yes	characteristic
99	C₂²×C₄	#41	1	1	yes	characteristic
100	C₂²×C₄	#41	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	1	no	—
113	M₁₆	—	4	1	no	—
114	M₁₆	—	4	1	no	—
115	M₁₆	C₄²	1	4	yes	—
116	M₁₆	C₄⋊C₄	1	4	yes	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	4	1	no	—
135	C₂²⋊C₄	—	4	1	no	—
136	C₂²⋊C₄	—	4	1	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	4	1	no	—
143	C₄²	—	4	1	no	—
144	C₄²	—	4	1	no	—
145	C₄²	C₂²⋊C₄	1	2	yes	—
146	C₄²	C₂²⋊C₄	1	2	yes	—
147	C₂³	—	4	2	no	—
148	C₂³	#117	1	1	yes	characteristic
149	C₂³	#117	1	1	yes	characteristic
150	C₂³	#117	1	1	yes	characteristic
151	C₂³	#123	1	1	yes	characteristic
152	C₂³	#123	1	1	yes	characteristic
153	C₂³	#124	1	1	yes	characteristic
154	C₂³	#125	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
166	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄	#117	1	1	yes	characteristic
169	C₂×C₄	#117	1	1	yes	characteristic
170	C₂×C₄	#117	1	1	yes	characteristic
171	C₂×C₄	#117	1	1	yes	characteristic
172	C₂×C₄	#117	1	1	yes	characteristic
173	C₂×C₄	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄	C₄×D₄	1	1	yes	characteristic
177	C₂×C₄	C₄×D₄	1	1	yes	characteristic
178	C₂×C₄	C₄×D₄	1	1	yes	characteristic
179	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
180	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
181	C₂×C₄	#121	1	1	yes	characteristic
182	C₂×C₄	#121	1	1	yes	characteristic
183	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
184	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
185	C₂×C₄	#123	1	1	yes	characteristic
186	C₂×C₄	#123	1	1	yes	characteristic
187	C₂×C₄	#124	1	1	yes	characteristic
188	C₂×C₄	#125	1	1	yes	characteristic
189	C₂×C₄	#126	1	1	yes	characteristic
190	C₂×C₄	#126	1	1	yes	characteristic
191	C₂×C₄	#126	1	1	yes	characteristic
192	C₂×C₄	#126	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₈	—	2	4	no	—
195	C₈	—	4	2	no	—
196	C₈	—	4	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	1	no	—
202	C₂²	—	4	1	no	—
203	C₂²	—	4	1	no	—
204	C₂²	#376	1	1	yes	characteristic
205	C₂²	#381	1	1	yes	characteristic
206	C₂²	#382	1	1	yes	characteristic
207	C₂²	#387	1	1	yes	characteristic
208	C₂²	#387	1	1	yes	characteristic
209	C₂²	#388	1	1	yes	characteristic
210	C₂²	#388	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	4	2	no	—
213	C₄	—	8	2	no	—
214	C₄	—	8	2	no	—
215	C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
216	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
217	C₄	#380	1	1	yes	characteristic
218	C₄	#383	1	1	yes	characteristic
219	C₄	#387	1	1	yes	characteristic
220	C₄	#387	1	1	yes	characteristic
221	C₄	#388	1	1	yes	characteristic
222	C₄	#388	1	1	yes	characteristic
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	2	1	no	—
229	C₂	#1434	1	1	yes	characteristic
230	C₂	#1932	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₄	C₂³
socle	C₂²	#388
prime radical	C₂⁴	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. C₄×M₁₆	C₄	semidirect
19. #344	C₄	semidirect
20. #343	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆	2
C₂×#344	2
#1742	2
#1738	2
#1733	2
C₂×#339	2