Subgroups — C₈×#358 #128665

Order 512.

370 conjugacy classes of subgroups.

The subgroup lattice has 217 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄×#358	C₂	1	1	yes	maximal, characteristic
3	C₂×#1332	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₈×#107	C₂	1	1	yes	maximal, characteristic
6	#7171	C₂	1	1	yes	maximal, characteristic
7	#7154	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	—	2	1	no	—
9	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×#358	C₄	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×#107	C₂²	1	1	yes	characteristic
14	C₈×C₄⋊C₄	—	2	1	no	—
15	C₈×Q₁₆	—	2	2	no	—
16	C₈×Q₁₆	C₄	1	4	yes	complemented
17	C₂×#335	C₂²	1	1	yes	characteristic
18	C₂×#335	C₂²	1	1	yes	characteristic
19	C₂×Q₈⋊C₈	—	2	1	no	—
20	C₂×C₈²	C₂²	1	1	yes	characteristic
21	#1380	C₄	1	2	yes	—
22	#1332	C₄	1	4	yes	complemented
23	C₈×C₁₆	—	2	2	no	—
24	#1279	—	2	1	no	—
25	C₂²×Q₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Q₈	—	2	1	no	—
27	C₂²×C₁₆	—	2	1	no	—
28	C₂²×C₁₆	C₂×C₄	1	2	yes	—
29	C₈×Q₈	—	2	4	no	—
30	C₈×Q₈	—	4	2	no	—
31	C₄×Q₁₆	—	2	1	no	—
32	C₄×Q₁₆	—	2	1	no	—
33	C₄×Q₁₆	C₂×C₄	1	2	yes	—
34	C₄×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×#107	C₂×C₄	1	1	yes	characteristic
36	C₂×#107	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₄⋊C₈	—	4	1	no	—
40	C₂×#103	—	2	1	no	—
41	C₂×C₄×C₈	—	4	1	no	—
42	C₂×C₄×C₈	—	4	1	no	—
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	C₄×C₄⋊C₄	—	2	1	no	—
47	#358	C₈	1	8	yes	direct product factor
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	—	2	1	no	—
50	C₄×C₁₆	—	2	1	no	—
51	C₄×C₁₆	C₂×C₄	1	2	yes	—
52	#340	—	2	1	no	—
53	#336	—	4	1	no	—
54	#335	—	2	1	no	—
55	#335	—	2	1	no	—
56	#335	C₂×C₄	1	2	yes	—
57	#335	C₂×C₄	1	2	yes	—
58	Q₈⋊C₈	—	2	4	no	—
59	C₈²	C₂×C₄	1	2	yes	—
60	C₈²	D₄	1	2	yes	—
61	C₂²×Q₈	—	2	1	no	—
62	C₂×Q₁₆	—	2	1	no	—
63	C₂×Q₁₆	—	2	2	no	—
64	C₂×Q₁₆	C₄²	1	2	yes	—
65	C₂×Q₁₆	C₂×C₈	1	4	yes	—
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	—	4	1	no	—
68	C₂²×C₈	—	4	1	no	—
69	C₂²×C₈	D₈	1	1	yes	characteristic
70	C₂²×C₈	D₈	1	1	yes	characteristic
71	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₈	#41	1	1	yes	characteristic
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	—	2	2	no	—
75	C₄×Q₈	—	4	1	no	—
76	C₄×Q₈	—	4	1	no	—
77	C₂×C₄⋊C₄	—	2	1	no	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄⋊C₄	—	4	1	no	—
80	C₂×C₄²	—	4	1	no	—
81	C₂×C₄²	—	4	1	no	—
82	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
83	C₂×C₁₆	—	2	4	no	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	C₂×C₈	1	4	yes	—
88	#107	—	2	1	no	—
89	#107	—	2	2	no	—
90	#107	C₄²	1	2	yes	—
91	#107	C₂×C₈	1	4	yes	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	4	4	no	—
95	C₄⋊C₈	—	4	1	no	—
96	C₄⋊C₈	—	4	1	no	—
97	#103	—	2	2	no	—
98	#103	—	2	2	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	4	4	no	—
101	C₄×C₈	—	8	2	no	—
102	C₄×C₈	C₂²⋊C₄	1	2	yes	—
103	C₄×C₈	C₂²⋊C₄	1	2	yes	—
104	C₄×C₈	SD₁₆	1	2	yes	—
105	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₄×C₈	C₂×D₄	1	1	yes	characteristic
108	C₄×C₈	#41	1	1	yes	characteristic
109	#95	—	4	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	4	2	no	—
114	C₂²×C₄	—	4	1	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	4	1	no	—
117	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
118	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
119	C₂²×C₄	#135	1	1	yes	characteristic
120	Q₁₆	—	2	2	no	—
121	Q₁₆	—	2	4	no	—
122	Q₁₆	C₄×C₈	1	4	yes	complemented
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	8	1	no	—
131	C₂×C₈	—	8	1	no	—
132	C₂×C₈	—	8	1	no	—
133	C₂×C₈	—	8	1	no	—
134	C₂×C₈	#102	1	2	yes	—
135	C₂×C₈	#102	1	2	yes	—
136	C₂×C₈	SD₃₂	1	2	yes	—
137	C₂×C₈	Q₃₂	1	2	yes	—
138	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₈	#117	1	1	yes	characteristic
140	C₂×C₈	C₄×D₄	1	1	yes	characteristic
141	C₂×C₈	C₄×D₄	1	1	yes	characteristic
142	C₂×C₈	C₂²×C₈	1	2	yes	—
143	C₂×C₈	#131	1	2	yes	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	—	4	2	no	—
151	C₄²	—	4	2	no	—
152	C₄²	—	8	1	no	—
153	C₄²	—	8	1	no	—
154	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₄²	C₄×D₄	1	1	yes	characteristic
156	C₄²	C₂×SD₁₆	1	1	yes	characteristic
157	C₄²	#135	1	1	yes	characteristic
158	C₁₆	—	2	4	no	—
159	C₁₆	—	2	4	no	—
160	C₂³	C₄×D₈	1	1	yes	characteristic
161	Q₈	—	2	4	no	—
162	Q₈	—	4	4	no	—
163	Q₈	—	4	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₂×#102	1	1	yes	characteristic
174	C₂×C₄	#416	1	1	yes	characteristic
175	C₂×C₄	C₈×D₄	1	2	yes	—
176	C₂×C₄	C₄×D₈	1	1	yes	characteristic
177	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
178	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
179	C₂×C₄	#443	1	2	yes	—
180	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
181	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
182	C₂×C₄	#508	1	1	yes	characteristic
183	C₂×C₄	#508	1	1	yes	characteristic
184	C₈	—	2	4	no	—
185	C₈	—	2	2	no	—
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	8	2	no	—
189	C₈	—	8	2	no	—
190	C₈	#358	1	4	yes	direct product factor
191	C₈	C₂×C₄×C₈	1	1	yes	characteristic
192	C₈	#405	1	1	yes	characteristic
193	C₈	C₈×D₄	1	2	yes	—
194	C₂²	C₈×D₈	1	2	yes	—
195	C₂²	C₄×#102	1	1	yes	characteristic
196	C₂²	C₄×SD₃₂	1	1	yes	characteristic
197	C₂²	C₄×Q₃₂	1	1	yes	characteristic
198	C₂²	#2178	1	2	yes	—
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	4	4	no	—
202	C₄	—	8	4	no	—
203	C₄	—	8	2	no	—
204	C₄	C₈×SD₁₆	1	2	yes	—
205	C₄	C₈×C₂²⋊C₄	1	1	yes	characteristic
206	C₄	#1764	1	1	yes	characteristic
207	C₄	C₂×#358	1	1	yes	characteristic
208	C₄	#2138	1	1	yes	characteristic
209	C₄	C₄×SD₃₂	1	1	yes	characteristic
210	C₄	C₄×Q₃₂	1	1	yes	characteristic
211	C₂	C₈×#102	1	1	yes	characteristic
212	C₂	C₈×SD₃₂	1	2	yes	—
213	C₂	C₈×Q₃₂	1	2	yes	—
214	C₂	C₄×#358	1	1	yes	characteristic
215	C₂	#12641	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₄×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. C₈×Q₁₆	C₄	semidirect
22. #1332	C₄	semidirect
47. #358	C₈	direct (N × H)
122. Q₁₆	C₄×C₈	semidirect
190. C₈	#358	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₁₆	2
C₄×#358	2
C₂×#1332	2
C₂×C₈×C₁₆	2
C₈×#107	2
#7171	2
#7154	2

Subgroups — C8×#358 #128665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#358 #128665