Subgroups — #24987

Order 256.

319 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#471	C₂	1	1	yes	maximal, characteristic
2	#2949	C₂	1	1	yes	maximal, characteristic, complemented
3	#2082	C₂	1	1	yes	maximal, characteristic, complemented
4	#2081	C₂	1	1	yes	maximal, characteristic, complemented
5	#2049	C₂	1	1	yes	maximal, characteristic, complemented
6	#2048	C₂	1	1	yes	maximal, characteristic
7	#1977	C₂	1	1	yes	maximal, characteristic, complemented
8	#1953	C₂	1	1	yes	maximal, characteristic, complemented
9	#1905	C₂	1	1	yes	maximal, characteristic, complemented
10	#1904	C₂	1	1	yes	maximal, characteristic
11	#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	#1830	C₂	1	1	yes	maximal, characteristic, complemented
13	#1829	C₂	1	1	yes	maximal, characteristic, complemented
14	#1814	C₂	1	1	yes	maximal, characteristic, complemented
15	#1757	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137	C₂²	1	1	yes	characteristic
17	C₂×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆	C₂²	1	1	yes	characteristic
19	C₂×#131	C₂²	1	1	yes	characteristic
20	#519	C₂²	1	1	yes	characteristic
21	#518	C₂²	1	1	yes	characteristic
22	#471	—	2	2	no	—
23	#471	—	2	2	no	—
24	#440	—	2	2	no	—
25	C₄×SD₁₆	—	2	2	no	—
26	#430	C₂²	1	1	yes	characteristic
27	#430	C₂²	1	1	yes	characteristic
28	C₂×#108	C₂²	1	1	yes	characteristic
29	C₂×#108	C₂²	1	1	yes	characteristic
30	#427	C₂²	1	1	yes	characteristic
31	#421	C₂²	1	1	yes	characteristic
32	#421	C₂²	1	1	yes	characteristic
33	#419	C₂²	1	1	yes	characteristic
34	#419	C₂²	1	1	yes	characteristic
35	#418	C₂²	1	1	yes	characteristic
36	#418	C₂²	1	1	yes	characteristic
37	#413	C₂²	1	1	yes	characteristic
38	C₂×#101	C₂²	1	1	yes	characteristic
39	C₂×#101	C₂²	1	1	yes	characteristic
40	C₂×#100	C₂²	1	1	yes	characteristic
41	C₂×#100	C₂²	1	1	yes	characteristic
42	#410	C₂²	1	1	yes	characteristic
43	#405	C₂²	1	1	yes	characteristic
44	#405	C₂²	1	1	yes	characteristic
45	#344	C₂²	1	1	yes	characteristic
46	#344	C₂²	1	1	yes	characteristic
47	#344	C₂²	1	1	yes	characteristic
48	#344	C₂²	1	1	yes	characteristic
49	#344	C₂²	1	1	yes	characteristic
50	#344	C₂²	1	1	yes	characteristic
51	#344	C₂²	1	1	yes	characteristic
52	#344	C₂²	1	1	yes	characteristic
53	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
54	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
55	C₂×#41	C₂³	1	1	yes	characteristic
56	C₂²×Q₈	C₂³	1	1	yes	characteristic
57	C₂²×D₄	C₂³	1	1	yes	characteristic
58	#137	—	2	2	no	—
59	#137	—	4	1	no	—
60	AGL(1,ℤ/8)	—	2	2	no	—
61	AGL(1,ℤ/8)	—	4	1	no	—
62	C₂×Q₁₆	—	2	1	no	—
63	C₂×SD₁₆	—	2	1	no	—
64	C₂×SD₁₆	—	2	1	no	—
65	C₂×SD₁₆	—	2	1	no	—
66	C₂×SD₁₆	—	2	1	no	—
67	C₂×SD₁₆	—	2	1	no	—
68	C₂×SD₁₆	—	2	1	no	—
69	C₂×SD₁₆	D₄	1	2	yes	—
70	C₂×SD₁₆	D₄	1	2	yes	—
71	C₂×D₈	—	2	1	no	—
72	#131	—	2	1	no	—
73	#131	—	2	1	no	—
74	#131	—	2	1	no	—
75	#131	—	2	1	no	—
76	C₂×M₁₆	—	2	1	no	—
77	C₂×M₁₆	C₂³	1	1	yes	characteristic
78	C₂×M₁₆	C₂³	1	1	yes	characteristic
79	C₂×M₁₆	C₂³	1	1	yes	characteristic
80	C₂×M₁₆	C₂³	1	1	yes	characteristic
81	C₂×M₁₆	C₂³	1	1	yes	characteristic
82	C₂×M₁₆	C₂³	1	1	yes	characteristic
83	C₂×M₁₆	C₂³	1	1	yes	characteristic
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	C₂³	1	1	yes	characteristic
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	4	1	no	—
88	C₄×D₄	—	2	2	no	—
89	C₄×D₄	—	4	1	no	—
90	#117	—	2	2	no	—
91	#117	C₂³	1	1	yes	characteristic
92	#117	C₂³	1	1	yes	characteristic
93	#117	C₂³	1	1	yes	characteristic
94	#117	C₂³	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	—	2	2	no	—
96	#108	—	2	1	no	—
97	#108	—	2	1	no	—
98	#108	—	2	1	no	—
99	#108	—	2	1	no	—
100	#108	—	4	1	no	—
101	#108	—	4	1	no	—
102	#107	—	2	1	no	—
103	#106	—	2	1	no	—
104	#104	—	4	1	no	—
105	#104	—	4	1	no	—
106	#103	—	2	2	no	—
107	#103	—	4	1	no	—
108	#102	—	2	2	no	—
109	#102	—	4	1	no	—
110	#101	—	2	2	no	—
111	#101	—	2	1	no	—
112	#101	—	4	1	no	—
113	#100	—	2	2	no	—
114	#100	—	2	1	no	—
115	#100	—	4	1	no	—
116	#99	—	2	2	no	—
117	#97	—	2	1	no	—
118	#97	—	2	1	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	2	1	no	—
121	C₂⁴	—	2	1	no	—
122	#41	—	2	1	no	—
123	#41	—	2	1	no	—
124	#41	—	2	1	no	—
125	#41	—	2	1	no	—
126	C₂×Q₈	—	2	1	no	—
127	C₂×Q₈	—	2	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	C₂×D₄	1	2	yes	—
132	C₂×Q₈	#41	1	1	yes	characteristic
133	C₂×D₄	—	2	1	no	—
134	C₂×D₄	—	2	1	no	—
135	C₂×D₄	—	2	1	no	—
136	C₂×D₄	—	2	1	no	—
137	C₂×D₄	—	2	1	no	—
138	C₂×D₄	—	2	1	no	—
139	C₂×D₄	C₂×D₄	1	2	yes	—
140	C₂×D₄	#41	1	1	yes	characteristic
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂⁴	1	1	yes	characteristic
145	Q₁₆	—	4	2	no	—
146	SD₁₆	—	2	2	no	—
147	SD₁₆	—	2	2	no	—
148	SD₁₆	—	2	2	no	—
149	SD₁₆	—	2	2	no	—
150	SD₁₆	—	4	2	no	—
151	SD₁₆	—	4	2	no	—
152	D₈	—	4	2	no	—
153	M₁₆	—	2	1	no	—
154	M₁₆	—	2	1	no	—
155	M₁₆	—	2	1	no	—
156	M₁₆	—	2	1	no	—
157	M₁₆	—	2	1	no	—
158	M₁₆	—	2	1	no	—
159	M₁₆	—	2	1	no	—
160	M₁₆	—	2	1	no	—
161	M₁₆	—	2	1	no	—
162	M₁₆	—	2	1	no	—
163	M₁₆	—	2	1	no	—
164	M₁₆	—	2	1	no	—
165	M₁₆	—	2	1	no	—
166	M₁₆	—	2	1	no	—
167	M₁₆	—	4	1	no	—
168	M₁₆	—	4	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈	#41	1	1	yes	characteristic
181	C₂×C₈	#41	1	1	yes	characteristic
182	C₂×C₈	#41	1	1	yes	characteristic
183	C₂×C₈	#41	1	1	yes	characteristic
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	#41	1	2	yes	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	4	2	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	4	1	no	—
202	C₂³	—	4	1	no	—
203	C₂³	—	16	1	no	—
204	C₂³	C₂×#41	1	1	yes	characteristic
205	Q₈	—	2	2	no	—
206	Q₈	—	2	1	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	2	1	no	—
209	Q₈	—	4	1	no	—
210	Q₈	—	4	1	no	—
211	D₄	—	2	2	no	—
212	D₄	—	2	1	no	—
213	D₄	—	2	2	no	—
214	D₄	—	2	1	no	—
215	D₄	—	4	1	no	—
216	D₄	—	4	1	no	—
217	D₄	—	4	1	no	—
218	D₄	—	4	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	16	2	no	—
230	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄	C₂×#41	1	1	yes	characteristic
232	C₂×C₄	C₂×#41	1	1	yes	characteristic
233	C₂×C₄	C₂×#41	1	1	yes	characteristic
234	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
235	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
236	C₈	—	4	1	no	—
237	C₈	—	4	1	no	—
238	C₈	—	4	1	no	—
239	C₈	—	4	1	no	—
240	C₈	—	4	1	no	—
241	C₈	—	4	1	no	—
242	C₈	—	4	1	no	—
243	C₈	#123	1	1	yes	characteristic
244	C₈	#123	1	1	yes	characteristic
245	C₈	#123	1	1	yes	characteristic
246	C₈	#123	1	1	yes	characteristic
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	1	no	—
252	C₂²	—	16	1	no	—
253	C₂²	—	16	1	no	—
254	C₂²	#529	1	1	yes	characteristic
255	C₂²	#546	1	1	yes	characteristic
256	C₂²	#555	1	1	yes	characteristic
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	8	1	no	—
261	C₄	—	8	1	no	—
262	C₄	—	8	1	no	—
263	C₄	C₂×#123	1	1	yes	characteristic
264	C₄	#540	1	1	yes	characteristic
265	C₄	#546	1	1	yes	characteristic
266	C₄	#553	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	8	1	no	—
271	C₂	—	8	2	no	—
272	C₂	#2420	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2420
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ 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. #2949	C₂	semidirect
3. #2082	C₂	semidirect
4. #2081	C₂	semidirect
5. #2049	C₂	semidirect
7. #1977	C₂	semidirect
8. #1953	C₂	semidirect
9. #1905	C₂	semidirect
11. #1833	C₂	semidirect
12. #1830	C₂	semidirect
13. #1829	C₂	semidirect
14. #1814	C₂	semidirect
15. #1757	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#471	2
#2949	2
#2082	2
#2081	2
#2049	2
#2048	2
#1977	2
#1953	2
#1905	2
#1904	2
#1833	2
#1830	2
#1829	2
#1814	2
#1757	2