Subgroups — #23042

Order 256.

412 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3067	C₂	1	1	yes	maximal, characteristic
2	#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	#2956	C₂	1	2	yes	maximal, complemented
4	#2916	C₂	1	1	yes	maximal, characteristic, complemented
5	#1974	C₂	1	1	yes	maximal, characteristic, complemented
6	#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	#1932	C₂	1	2	yes	maximal, complemented
8	#1881	C₂	1	2	yes	maximal, complemented
9	#1858	C₂	1	2	yes	maximal, complemented
10	C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
11	#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	#575	C₂²	1	2	yes	—
13	C₂×#135	C₂²	1	1	yes	characteristic
14	#568	C₂²	1	2	yes	—
15	#568	C₂²	1	2	yes	—
16	C₂×#131	C₂²	1	1	yes	characteristic
17	#473	C₄	1	2	yes	complemented
18	#472	C₄	1	2	yes	complemented
19	#471	C₄	1	4	yes	complemented
20	#444	—	2	2	no	—
21	#444	—	2	4	no	—
22	#443	—	2	2	no	—
23	#443	—	2	4	no	—
24	#430	—	2	2	no	—
25	#430	C₂²	1	2	yes	—
26	#430	C₂²	1	2	yes	—
27	C₂×#108	—	2	2	no	—
28	C₂×#108	C₂²	1	1	yes	characteristic
29	C₂×#108	C₂²	1	1	yes	characteristic
30	C₂×#108	C₂²	1	1	yes	characteristic
31	#421	C₂²	1	2	yes	—
32	C₂×#104	C₂²	1	2	yes	—
33	C₂×#104	C₂²	1	1	yes	characteristic
34	#416	C₂²	1	1	yes	characteristic
35	#413	C₂²	1	2	yes	—
36	#408	C₂²	1	2	yes	—
37	#405	C₂²	1	1	yes	characteristic
38	C₄×M₁₆	C₂²	1	2	yes	—
39	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
40	#344	C₂²	1	2	yes	—
41	#341	C₂²	1	1	yes	characteristic
42	#341	C₂²	1	2	yes	—
43	#339	C₂²	1	1	yes	characteristic
44	C₂×#41	C₂³	1	2	yes	—
45	C₂×#41	C₂³	1	1	yes	characteristic
46	#137	—	2	2	no	—
47	#137	C₂×C₄	1	4	yes	—
48	AGL(1,ℤ/8)	—	2	2	no	—
49	AGL(1,ℤ/8)	C₂×C₄	1	4	yes	—
50	#135	—	2	4	no	—
51	#135	—	2	1	no	—
52	#135	—	2	1	no	—
53	#135	—	2	1	no	—
54	#135	—	2	1	no	—
55	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂×SD₁₆	C₂×C₄	1	2	yes	—
57	C₂×D₈	C₂×C₄	1	1	yes	characteristic
58	#131	—	2	1	no	—
59	#131	—	2	1	no	—
60	#131	—	2	4	no	—
61	#131	—	2	4	no	—
62	#131	—	2	4	no	—
63	#131	—	4	2	no	—
64	#131	C₂×C₄	1	2	yes	—
65	#131	C₂×C₄	1	2	yes	—
66	C₂×M₁₆	—	2	1	no	—
67	C₂×M₁₆	—	2	4	no	—
68	C₂×M₁₆	—	4	2	no	—
69	C₂×M₁₆	C₂³	1	2	yes	—
70	C₂×M₁₆	C₂³	1	1	yes	characteristic
71	C₂×M₁₆	C₂³	1	2	yes	—
72	C₂×M₁₆	C₂³	1	2	yes	—
73	C₂×M₁₆	C₂³	1	1	yes	characteristic
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₈	C₂³	1	1	yes	characteristic
77	#117	C₂³	1	1	yes	characteristic
78	C₂×C₄²	C₂³	1	1	yes	characteristic
79	#108	—	2	4	no	—
80	#108	—	2	4	no	—
81	#108	—	2	1	no	—
82	#108	—	2	1	no	—
83	#108	C₂×C₄	1	2	yes	—
84	#108	C₂×C₄	1	2	yes	—
85	#108	D₄	1	2	yes	—
86	#108	D₄	1	2	yes	—
87	#104	—	2	4	no	—
88	#104	—	2	2	no	—
89	#104	—	2	2	no	—
90	#104	D₄	1	2	yes	—
91	#104	D₄	1	2	yes	—
92	#103	—	2	1	no	—
93	#102	—	2	1	no	—
94	#101	C₂×C₄	1	4	yes	—
95	#100	C₂×C₄	1	4	yes	—
96	C₂²⋊C₈	—	4	2	no	—
97	#97	—	2	1	no	—
98	#97	—	2	2	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	2	no	—
103	#41	—	2	1	no	—
104	#41	—	2	1	no	—
105	#41	—	2	2	no	—
106	#41	—	2	2	no	—
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	#41	C₂²×C₄	1	2	yes	—
110	#41	C₂×D₄	1	2	yes	—
111	C₂×Q₈	C₂²×C₄	1	2	yes	—
112	C₂×Q₈	#41	1	1	yes	characteristic
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	C₂²×C₄	1	2	yes	—
116	C₂×D₄	#41	1	1	yes	characteristic
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₂⁴	1	1	yes	characteristic
121	Q₁₆	—	2	4	no	—
122	Q₁₆	—	2	1	no	—
123	Q₁₆	—	2	1	no	—
124	SD₁₆	—	2	4	no	—
125	SD₁₆	—	2	4	no	—
126	SD₁₆	—	2	2	no	—
127	SD₁₆	—	2	2	no	—
128	D₈	—	2	4	no	—
129	D₈	—	2	1	no	—
130	D₈	—	2	1	no	—
131	M₁₆	—	2	1	no	—
132	M₁₆	—	2	2	no	—
133	M₁₆	—	2	1	no	—
134	M₁₆	—	2	2	no	—
135	M₁₆	—	2	2	no	—
136	M₁₆	—	2	2	no	—
137	M₁₆	—	2	2	no	—
138	M₁₆	—	2	2	no	—
139	M₁₆	—	4	4	no	—
140	M₁₆	—	4	4	no	—
141	M₁₆	C₂²×C₄	1	2	yes	—
142	M₁₆	C₂²×C₄	1	4	yes	—
143	M₁₆	C₂×D₄	1	4	yes	—
144	M₁₆	C₂×D₄	1	2	yes	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	4	4	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈	#41	1	1	yes	characteristic
162	C₂×C₈	#41	1	1	yes	characteristic
163	C₄⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	C₂×D₄	1	1	yes	characteristic
168	C₄²	C₂×D₄	1	1	yes	characteristic
169	C₂³	—	2	1	no	—
170	C₂³	—	2	2	no	—
171	C₂³	2₋¹⁺⁴	1	1	yes	characteristic
172	Q₈	—	2	1	no	—
173	Q₈	—	2	2	no	—
174	Q₈	—	2	2	no	—
175	Q₈	C₄×D₄	1	2	yes	—
176	D₄	—	2	2	no	—
177	D₄	—	2	2	no	—
178	D₄	—	2	2	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	1	no	—
181	D₄	—	4	2	no	—
182	D₄	—	4	2	no	—
183	D₄	C₄×D₄	1	2	yes	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₄	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	C₈	—	2	2	no	—
201	C₈	—	2	4	no	—
202	C₈	—	2	2	no	—
203	C₈	—	2	4	no	—
204	C₈	—	4	2	no	—
205	C₈	—	4	1	no	—
206	C₈	C₄×D₄	1	1	yes	characteristic
207	C₈	C₄×D₄	1	1	yes	characteristic
208	C₈	C₄×D₄	1	1	yes	characteristic
209	C₈	C₄×D₄	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	8	1	no	—
214	C₂²	—	8	2	no	—
215	C₂²	#520	1	1	yes	characteristic
216	C₂²	#547	1	1	yes	characteristic
217	C₂²	#548	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	8	1	no	—
223	C₄	C₂×C₄×D₄	1	1	yes	characteristic
224	C₄	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄	D₄²	1	1	yes	characteristic
226	C₄	#553	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	—	4	2	no	—
231	C₂	—	8	2	no	—
232	C₂	#2361	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄²
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2361
prime radical	#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2957	C₂	semidirect
3. #2956	C₂	semidirect
4. #2916	C₂	semidirect
5. #1974	C₂	semidirect
6. #1953	C₂	semidirect
7. #1932	C₂	semidirect
8. #1881	C₂	semidirect
9. #1858	C₂	semidirect
10. C₄×#108	C₂	semidirect
11. #1764	C₂	semidirect
17. #473	C₄	semidirect
18. #472	C₄	semidirect
19. #471	C₄	semidirect

Maximal subgroups

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

subgroup	index
#3067	2
#2957	2
#2956	2
#2916	2
#1974	2
#1953	2
#1932	2
#1881	2
#1858	2
C₄×#108	2
#1764	2