Subgroups — C₈×#137 #17851

Order 256.

357 conjugacy classes of subgroups.

The subgroup lattice has 233 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137	C₂	1	1	yes	maximal, characteristic
4	#1579	C₂	1	2	yes	maximal, complemented
5	SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₈×Q₁₆	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆	C₂	1	2	yes	maximal, complemented
8	#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	#1477	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137	C₄	1	1	yes	characteristic
13	C₄×#41	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
15	C₈×Q₈	—	2	1	no	—
16	C₈×Q₈	—	2	1	no	—
17	C₈×Q₈	C₂²	1	2	yes	—
18	C₈×Q₈	C₂²	1	1	yes	characteristic
19	C₈×Q₈	C₂²	1	2	yes	—
20	C₈×Q₈	C₂²	1	1	yes	characteristic
21	#441	C₄	1	2	yes	—
22	#440	C₄	1	2	yes	—
23	C₄×Q₁₆	C₂²	1	2	yes	—
24	C₄×SD₁₆	C₂²	1	2	yes	—
25	C₈×D₄	—	2	1	no	—
26	C₈×D₄	—	2	1	no	—
27	C₈×D₄	C₂²	1	1	yes	characteristic
28	C₈×D₄	C₂²	1	1	yes	characteristic
29	#431	—	2	1	no	—
30	#431	C₂²	1	1	yes	characteristic
31	#431	C₂²	1	1	yes	characteristic
32	#428	C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₈	—	2	1	no	—
34	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	#419	C₄	1	1	yes	characteristic
36	#417	C₄	1	1	yes	characteristic
37	C₄×M₁₆	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	—	2	1	no	—
40	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
41	#335	C₂²	1	2	yes	—
42	#334	C₂²	1	2	yes	—
43	Q₈⋊C₈	C₂²	1	2	yes	—
44	Q₈⋊C₈	C₂²	1	2	yes	—
45	Q₈⋊C₈	C₂²	1	2	yes	—
46	D₄⋊C₈	C₂²	1	2	yes	—
47	#322	C₂²	1	2	yes	—
48	C₈²	C₂²	1	2	yes	—
49	C₂×#41	C₂×C₄	1	1	yes	characteristic
50	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
51	#137	C₈	1	8	yes	direct product factor
52	C₂×Q₁₆	C₂×C₄	1	2	yes	—
53	C₂×SD₁₆	C₂×C₄	1	2	yes	—
54	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	—	2	1	no	—
59	C₂²×C₈	D₄	1	1	yes	characteristic
60	C₂²×C₈	D₄	1	1	yes	characteristic
61	C₄×Q₈	—	2	1	no	—
62	C₄×Q₈	—	2	1	no	—
63	C₄×Q₈	C₂×C₄	1	2	yes	—
64	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈	C₂³	1	2	yes	—
66	C₄×Q₈	C₂³	1	1	yes	characteristic
67	C₄×D₄	—	2	1	no	—
68	C₄×D₄	—	2	1	no	—
69	C₄×D₄	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄	C₂³	1	1	yes	characteristic
71	#117	—	2	1	no	—
72	#117	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	2	1	no	—
74	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²	—	2	1	no	—
76	C₂×C₄²	—	2	1	no	—
77	C₂×C₄²	C₂³	1	1	yes	characteristic
78	#107	C₂×C₄	1	2	yes	—
79	#106	C₂×C₄	1	2	yes	—
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	2	1	no	—
83	C₄⋊C₈	—	2	1	no	—
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	C₂³	1	2	yes	—
87	C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₄⋊C₈	C₂³	1	1	yes	characteristic
89	#103	C₂×C₄	1	2	yes	—
90	#103	C₂×C₄	1	2	yes	—
91	#103	C₂×C₄	1	2	yes	—
92	#102	C₂×C₄	1	2	yes	—
93	C₂²⋊C₈	—	2	1	no	—
94	C₂²⋊C₈	—	2	1	no	—
95	C₂²⋊C₈	—	2	1	no	—
96	C₂²⋊C₈	—	2	1	no	—
97	#97	C₂×C₄	1	2	yes	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	D₄	1	1	yes	characteristic
104	C₄×C₈	D₄	1	1	yes	characteristic
105	C₄×C₈	C₂³	1	2	yes	—
106	C₄×C₈	C₂³	1	2	yes	—
107	C₄×C₈	C₂³	1	1	yes	characteristic
108	C₄×C₈	C₂³	1	1	yes	characteristic
109	#41	—	2	2	no	—
110	#41	C₂×C₈	1	4	yes	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	2	2	no	—
113	C₂×Q₈	C₂×C₈	1	4	yes	complemented
114	C₂×Q₈	C₂²×C₄	1	2	yes	—
115	C₂×Q₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×D₄	—	2	1	no	—
117	C₂×D₄	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄	#41	1	1	yes	characteristic
126	Q₁₆	C₂×C₈	1	8	yes	complemented
127	SD₁₆	C₂×C₈	1	8	yes	complemented
128	M₁₆	—	2	2	no	—
129	M₁₆	C₂×C₈	1	4	yes	complemented
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	C₂²×C₄	1	2	yes	—
140	C₂×C₈	C₂²×C₄	1	2	yes	—
141	C₂×C₈	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	C₂²×C₄	1	2	yes	—
151	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
153	C₂²⋊C₄	—	2	1	no	—
154	C₂²⋊C₄	—	2	1	no	—
155	C₂²⋊C₄	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	2	no	—
161	C₄²	C₂²×C₄	1	1	yes	characteristic
162	C₄²	C₂×D₄	1	1	yes	characteristic
163	C₄²	#41	1	1	yes	characteristic
164	C₄²	C₂⁴	1	1	yes	characteristic
165	C₂³	—	2	1	no	—
166	C₂³	C₄×D₄	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	Q₈	—	2	1	no	—
169	Q₈	—	2	4	no	—
170	Q₈	C₂²×C₈	1	4	yes	—
171	Q₈	C₂²×C₈	1	2	yes	—
172	D₄	—	2	4	no	—
173	D₄	—	2	1	no	—
174	D₄	C₂²×C₈	1	2	yes	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	C₄×D₄	1	1	yes	characteristic
190	C₂×C₄	C₄×D₄	1	1	yes	characteristic
191	C₂×C₄	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄	C₂²×C₈	1	2	yes	—
193	C₂×C₄	#131	1	2	yes	—
194	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄	C₂×#41	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	2	1	no	—
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	—	4	1	no	—
203	C₈	—	4	1	no	—
204	C₈	—	4	2	no	—
205	C₈	C₂²×C₈	1	4	yes	—
206	C₈	#137	1	2	yes	direct product factor
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	C₈×D₄	1	2	yes	—
213	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	4	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	2	no	—
222	C₄	C₈×D₄	1	2	yes	—
223	C₄	C₂³×C₈	1	1	yes	characteristic
224	C₄	C₂×#131	1	1	yes	characteristic
225	C₄	C₂×#137	1	1	yes	characteristic
226	C₄	#575	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	—	4	2	no	—
229	C₂	C₂×C₈×D₄	1	1	yes	characteristic
230	C₂	C₄×#137	1	1	yes	characteristic
231	C₂	#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈	C₂	semidirect
4. #1579	C₂	semidirect
5. SD₁₆⋊C₈	C₂	semidirect
6. C₈×Q₁₆	C₂	semidirect
7. C₈×SD₁₆	C₂	semidirect
8. #1565	C₂	semidirect
9. #1479	C₂	semidirect
11. C₈×M₁₆	C₂	semidirect
51. #137	C₈	direct (N × H)
113. C₂×Q₈	C₂×C₈	semidirect
126. Q₁₆	C₂×C₈	semidirect
127. SD₁₆	C₂×C₈	semidirect
129. M₁₆	C₂×C₈	semidirect
206. C₈	#137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#41	2
C₂×C₈×Q₈	2
C₄×#137	2
#1579	2
SD₁₆⋊C₈	2
C₈×Q₁₆	2
C₈×SD₁₆	2
#1565	2
#1479	2
#1477	2
C₈×M₁₆	2

Subgroups — C8×#137 #17851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#137 #17851