Subgroups — #23235

Order 256.

355 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3131	C₂	1	1	yes	maximal, characteristic
2	#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	#2935	C₂	1	1	yes	maximal, characteristic
4	#1986	C₂	1	1	yes	maximal, characteristic, complemented
5	#1980	C₂	1	1	yes	maximal, characteristic
6	#1938	C₂	1	2	yes	maximal, complemented
7	#1929	C₂	1	2	yes	maximal, complemented
8	#1866	C₂	1	2	yes	maximal, complemented
9	C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
10	#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	#1765	C₂	1	2	yes	maximal, complemented
12	C₂×#131	C₂²	1	2	yes	—
13	#533	C₂²	1	1	yes	characteristic
14	#532	C₂²	1	1	yes	characteristic
15	#488	C₄	1	2	yes	complemented
16	#488	C₄	1	2	yes	complemented
17	#487	C₄	1	1	yes	characteristic, complemented
18	#487	C₄	1	1	yes	characteristic, complemented
19	#486	C₄	1	1	yes	characteristic, complemented
20	#486	C₄	1	1	yes	characteristic, complemented
21	#446	—	2	2	no	—
22	#446	—	2	1	no	—
23	C₈×Q₈	—	2	1	no	—
24	#436	—	2	1	no	—
25	#435	—	2	2	no	—
26	C₈×D₄	—	2	1	no	—
27	C₂×#108	C₂²	1	1	yes	characteristic
28	C₂×#108	C₂²	1	1	yes	characteristic
29	#424	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×#104	C₂²	1	2	yes	—
32	C₂×#104	C₂²	1	2	yes	—
33	#416	C₂²	1	2	yes	—
34	#416	C₂²	1	2	yes	—
35	#408	C₂²	1	2	yes	—
36	#405	C₂²	1	2	yes	—
37	C₄×M₁₆	C₂²	1	1	yes	characteristic
38	C₄×M₁₆	C₂²	1	1	yes	characteristic
39	C₂×#97	C₂²	1	2	yes	—
40	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
42	#341	C₂²	1	1	yes	characteristic
43	#341	C₂²	1	1	yes	characteristic
44	#339	C₂²	1	2	yes	—
45	#339	C₂²	1	2	yes	—
46	#336	C₂²	1	2	yes	—
47	C₂×#41	C₂³	1	2	yes	—
48	#131	—	2	4	no	—
49	#131	—	4	2	no	—
50	C₂×M₁₆	—	2	2	no	—
51	C₂×M₁₆	C₂³	1	1	yes	characteristic
52	C₂×M₁₆	C₂³	1	1	yes	characteristic
53	C₂×M₁₆	C₂³	1	1	yes	characteristic
54	C₂×M₁₆	C₂³	1	1	yes	characteristic
55	C₂²×C₈	—	2	2	no	—
56	C₂²×C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₈	C₂³	1	1	yes	characteristic
58	C₂²×C₈	C₂³	1	2	yes	—
59	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
62	#127	C₂×C₄	1	1	yes	characteristic
63	#125	C₂×C₄	1	2	yes	—
64	#124	C₂×C₄	1	2	yes	—
65	C₂²⋊Q₈	—	2	1	no	—
66	C₂²⋊Q₈	—	2	1	no	—
67	#121	—	2	1	no	—
68	#121	—	2	1	no	—
69	C₄×Q₈	—	2	1	no	—
70	C₄×Q₈	—	2	1	no	—
71	C₄×D₄	—	2	1	no	—
72	C₄×D₄	—	2	1	no	—
73	#117	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄²	C₂³	1	2	yes	—
76	C₂×C₄²	C₂³	1	1	yes	characteristic
77	#108	—	2	2	no	—
78	#108	—	2	2	no	—
79	C₄⋊C₈	—	2	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	D₄	1	1	yes	characteristic
82	C₄⋊C₈	D₄	1	2	yes	—
83	C₄⋊C₈	D₄	1	1	yes	characteristic
84	#104	—	2	4	no	—
85	#104	—	4	2	no	—
86	#103	C₂×C₄	1	2	yes	—
87	#103	C₂×C₄	1	2	yes	—
88	#103	C₂×C₄	1	2	yes	—
89	#103	C₂×C₄	1	2	yes	—
90	#102	C₂×C₄	1	2	yes	—
91	#102	C₂×C₄	1	2	yes	—
92	#102	C₂×C₄	1	2	yes	—
93	#102	C₂×C₄	1	2	yes	—
94	C₂²⋊C₈	—	2	2	no	—
95	C₂²⋊C₈	—	2	2	no	—
96	#97	—	2	2	no	—
97	#97	—	2	1	no	—
98	#97	—	2	1	no	—
99	#97	—	4	2	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	C₂×C₄	1	1	yes	characteristic
104	C₄×C₈	C₂×C₄	1	1	yes	characteristic
105	C₄×C₈	C₂×C₄	1	1	yes	characteristic
106	C₄×C₈	C₂×C₄	1	1	yes	characteristic
107	#95	—	2	1	no	—
108	#41	—	2	2	no	—
109	#41	—	2	2	no	—
110	#41	—	2	2	no	—
111	#41	—	2	2	no	—
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	2	1	no	—
114	C₂×Q₈	C₂²×C₄	1	2	yes	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	—	2	1	no	—
118	C₂×D₄	C₂²×C₄	1	2	yes	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	#41	1	1	yes	characteristic
122	C₂²×C₄	#41	1	1	yes	characteristic
123	C₂²×C₄	C₂⁴	1	1	yes	characteristic
124	M₁₆	—	2	1	no	—
125	M₁₆	—	2	1	no	—
126	M₁₆	—	2	1	no	—
127	M₁₆	—	2	1	no	—
128	M₁₆	—	2	1	no	—
129	M₁₆	—	2	1	no	—
130	M₁₆	—	4	4	no	—
131	M₁₆	#41	1	2	yes	—
132	M₁₆	#41	1	2	yes	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	4	4	no	—
144	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₈	C₂×D₄	1	2	yes	—
149	C₂×C₈	C₂×D₄	1	2	yes	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	C₂²×C₄	1	2	yes	—
158	C₄⋊C₄	C₂²×C₄	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	C₂²×C₄	1	1	yes	characteristic
166	C₄²	C₂²×C₄	1	1	yes	characteristic
167	C₄²	C₂×D₄	1	1	yes	characteristic
168	C₄²	C₂×D₄	1	1	yes	characteristic
169	C₂³	—	2	2	no	—
170	C₂³	C₂×#41	1	1	yes	characteristic
171	Q₈	—	2	2	no	—
172	Q₈	—	2	2	no	—
173	Q₈	—	4	2	no	—
174	Q₈	—	8	1	no	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	4	2	no	—
178	D₄	—	4	2	no	—
179	D₄	—	4	2	no	—
180	D₄	—	8	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄	C₂×#41	1	1	yes	characteristic
198	C₂×C₄	C₂×#41	1	1	yes	characteristic
199	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	4	1	no	—
210	C₈	—	4	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
217	C₂²	#556	1	1	yes	characteristic
218	C₂²	#559	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	8	2	no	—
226	C₄	#444	1	2	yes	—
227	C₄	C₄×#41	1	1	yes	characteristic
228	C₄	#520	1	1	yes	characteristic
229	C₄	#546	1	1	yes	characteristic
230	C₄	#548	1	1	yes	characteristic
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	—	8	2	no	—
234	C₂	#2367	1	1	yes	characteristic
235	C₂	C₂×#444	1	1	yes	characteristic
236	C₂	#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2962	C₂	semidirect
4. #1986	C₂	semidirect
6. #1938	C₂	semidirect
7. #1929	C₂	semidirect
8. #1866	C₂	semidirect
9. C₄×#108	C₂	semidirect
10. #1770	C₂	semidirect
11. #1765	C₂	semidirect
15. #488	C₄	semidirect
16. #488	C₄	semidirect
17. #487	C₄	semidirect
18. #487	C₄	semidirect
19. #486	C₄	semidirect
20. #486	C₄	semidirect

Maximal subgroups

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

subgroup	index
#3131	2
#2962	2
#2935	2
#1986	2
#1980	2
#1938	2
#1929	2
#1866	2
C₄×#108	2
#1770	2
#1765	2