Subgroups — C₂×#33726 #7623210

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3160	C₂	1	2	yes	maximal
2	#33726	C₂	1	16	yes	maximal, direct product factor
3	C₂×#2254	C₂	1	2	yes	maximal, complemented
4	C₂×#2253	C₂	1	4	yes	maximal, complemented
5	C₂×#2252	C₂	1	2	yes	maximal, complemented
6	C₂×#2158	C₂	1	4	yes	maximal
7	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×#533	—	2	2	no	—
9	C₂³×C₁₆	C₂²	1	2	yes	—
10	#3160	—	2	8	no	—
11	#3160	C₂²	1	16	yes	—
12	C₂×#500	C₂²	1	4	yes	—
13	C₂×#499	—	2	2	no	—
14	C₂×#499	C₂²	1	4	yes	—
15	C₂×#498	—	2	2	no	—
16	C₂×#427	—	2	2	no	—
17	C₂×#427	C₂²	1	4	yes	—
18	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
19	#2254	C₂²	1	16	yes	complemented
20	#2253	C₂²	1	32	yes	complemented
21	#2252	C₂²	1	16	yes	complemented
22	#2158	C₂²	1	32	yes	—
23	C₂×#367	C₂²	1	8	yes	—
24	C₂×#366	C₂²	1	8	yes	—
25	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₁₆	C₂²	1	2	yes	—
27	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₁₆	C₂²	1	8	yes	—
29	C₂³×C₈	D₄	1	1	yes	characteristic
30	C₂³×C₈	C₂³	1	1	yes	characteristic
31	#533	—	2	16	no	—
32	#533	—	4	8	no	—
33	C₂×C₄⋊Q₈	—	2	4	no	—
34	C₂×#125	—	2	4	no	—
35	C₂×C₂²⋊Q₈	—	4	4	no	—
36	C₂×C₄×Q₈	—	4	4	no	—
37	C₂×#117	—	2	4	no	—
38	C₂²×C₄²	D₄	1	1	yes	characteristic
39	C₂²×C₁₆	Q₈	1	16	yes	complemented
40	C₂²×C₁₆	C₂³	1	2	yes	—
41	C₂²×C₁₆	C₂³	1	2	yes	—
42	C₂²×C₁₆	C₂³	1	8	yes	—
43	#500	—	2	8	no	—
44	#500	C₂³	1	16	yes	—
45	#499	—	2	16	no	—
46	#499	—	2	8	no	—
47	#499	C₂³	1	16	yes	—
48	#498	—	2	16	no	—
49	#427	—	2	8	no	—
50	#427	—	2	16	no	—
51	#427	C₂³	1	16	yes	—
52	C₂×#107	—	2	4	no	—
53	C₂×#107	C₂³	1	8	yes	—
54	C₂×#106	—	2	4	no	—
55	C₂×#106	—	2	4	no	—
56	C₂×C₄×C₈	D₄	1	1	yes	characteristic
57	C₂×C₄×C₈	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	D₄	1	4	yes	—
59	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂³	1	4	yes	—
62	#367	C₂³	1	32	yes	—
63	#366	C₂³	1	32	yes	—
64	C₄×C₁₆	C₂³	1	4	yes	—
65	C₄×C₁₆	C₂³	1	8	yes	—
66	C₄×C₁₆	C₂³	1	4	yes	—
67	C₂²×Q₈	—	4	4	no	—
68	C₂³×C₄	D₈	1	2	yes	—
69	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₈	Q₁₆	1	8	yes	—
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	2	yes	—
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈	C₂×D₄	1	4	yes	—
76	C₂²×C₈	C₂×Q₈	1	8	yes	—
77	C₂²×C₈	C₂⁴	1	1	yes	characteristic
78	C₂²×C₈	C₂⁴	1	2	yes	—
79	C₄⋊Q₈	—	2	16	no	—
80	C₄⋊Q₈	—	4	8	no	—
81	#125	—	2	16	no	—
82	#125	—	4	8	no	—
83	C₂²⋊Q₈	—	4	32	no	—
84	C₄×Q₈	—	4	32	no	—
85	#117	—	2	16	no	—
86	#117	—	4	8	no	—
87	C₂×C₄⋊C₄	—	2	8	no	—
88	C₂×C₄⋊C₄	—	4	8	no	—
89	C₂×C₄⋊C₄	—	4	4	no	—
90	C₂×C₂²⋊C₄	—	4	4	no	—
91	C₂×C₄²	—	4	4	no	—
92	C₂×C₄²	D₈	1	4	yes	—
93	C₂×C₄²	D₈	1	2	yes	—
94	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₄	1	4	yes	—
97	C₂×C₁₆	C₂×Q₈	1	8	yes	—
98	C₂×C₁₆	C₂×Q₈	1	8	yes	—
99	C₂×C₁₆	C₂×Q₈	1	32	yes	complemented
100	C₂×C₁₆	C₂⁴	1	4	yes	—
101	C₂×C₁₆	C₂⁴	1	4	yes	—
102	#107	—	2	16	no	—
103	#107	—	2	8	no	—
104	#107	C₂⁴	1	16	yes	—
105	#106	—	2	16	no	—
106	#106	—	2	16	no	—
107	C₄×C₈	C₂×D₄	1	4	yes	—
108	C₄×C₈	C₂×D₄	1	4	yes	—
109	C₄×C₈	C₂×D₄	1	2	yes	—
110	C₄×C₈	C₂×D₄	1	2	yes	—
111	C₄×C₈	C₂⁴	1	2	yes	—
112	C₄×C₈	C₂⁴	1	2	yes	—
113	C₂⁴	C₂×D₈	1	1	yes	characteristic
114	C₂×Q₈	—	4	16	no	—
115	C₂×Q₈	—	8	16	no	—
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	4	8	no	—
118	C₂²×C₄	C₄⋊Q₈	1	4	yes	—
119	C₂²×C₄	C₂×D₈	1	2	yes	—
120	C₂²×C₄	C₂×D₈	1	8	yes	—
121	C₂²×C₄	C₂×D₈	1	2	yes	—
122	C₂²×C₄	C₂×D₈	1	2	yes	—
123	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
124	C₂²×C₄	C₂×Q₁₆	1	4	yes	—
125	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₄	1	2	yes	—
127	C₂×C₈	C₄⋊Q₈	1	4	yes	—
128	C₂×C₈	C₄⋊Q₈	1	8	yes	—
129	C₂×C₈	C₂×Q₁₆	1	4	yes	—
130	C₂×C₈	C₂×Q₁₆	1	4	yes	—
131	C₂×C₈	C₂×Q₁₆	1	16	yes	—
132	C₂×C₈	C₂²×D₄	1	2	yes	—
133	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₈	C₂²×D₄	1	2	yes	—
135	C₂×C₈	C₂²×D₄	1	2	yes	—
136	C₂×C₈	C₂²×Q₈	1	4	yes	—
137	C₂×C₈	C₂²×Q₈	1	8	yes	—
138	C₂×C₈	C₂⁵	1	1	yes	characteristic
139	C₄⋊C₄	—	2	16	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	32	no	—
142	C₄⋊C₄	—	4	16	no	—
143	C₂²⋊C₄	—	4	16	no	—
144	C₄²	—	4	16	no	—
145	C₄²	C₂×D₈	1	8	yes	—
146	C₄²	C₂×D₈	1	2	yes	—
147	C₄²	C₂×D₈	1	2	yes	—
148	C₄²	C₂²×D₄	1	2	yes	—
149	C₄²	C₂²×D₄	1	2	yes	—
150	C₁₆	C₂²×Q₈	1	8	yes	—
151	C₁₆	C₂²×Q₈	1	8	yes	—
152	C₂³	—	2	4	no	—
153	C₂³	#500	1	4	yes	—
154	C₂³	C₂²×D₈	1	2	yes	—
155	C₂³	C₂²×D₈	1	1	yes	characteristic
156	Q₈	—	8	32	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	16	no	—
159	C₂×C₄	—	4	16	no	—
160	C₂×C₄	—	8	16	no	—
161	C₂×C₄	#500	1	8	yes	—
162	C₂×C₄	#500	1	4	yes	—
163	C₂×C₄	#508	1	8	yes	—
164	C₂×C₄	C₂×C₄⋊Q₈	1	4	yes	—
165	C₂×C₄	C₂×C₄⋊Q₈	1	2	yes	—
166	C₂×C₄	C₂²×D₈	1	4	yes	—
167	C₂×C₄	C₂²×D₈	1	4	yes	—
168	C₂×C₄	C₂²×D₈	1	2	yes	—
169	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
170	C₂×C₄	C₂²×Q₁₆	1	4	yes	—
171	C₂×C₄	C₂²×Q₁₆	1	2	yes	—
172	C₂×C₄	C₂³×D₄	1	1	yes	characteristic
173	C₈	C₂×C₄⋊Q₈	1	4	yes	—
174	C₈	C₂×C₄⋊Q₈	1	2	yes	—
175	C₈	C₂²×Q₁₆	1	4	yes	—
176	C₈	C₂²×Q₁₆	1	4	yes	—
177	C₈	C₂³×Q₈	1	2	yes	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	8	no	—
180	C₂²	C₂×#500	1	4	yes	—
181	C₂²	C₂×#500	1	2	yes	—
182	C₂²	C₂×#508	1	4	yes	—
183	C₂²	C₂³×D₈	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	8	16	no	—
186	C₄	C₂×#500	1	4	yes	—
187	C₄	C₂×#500	1	2	yes	—
188	C₄	C₂×#508	1	8	yes	—
189	C₄	C₂²×C₄⋊Q₈	1	1	yes	characteristic
190	C₄	C₂³×Q₁₆	1	1	yes	characteristic
191	C₂	—	2	4	no	—
192	C₂	#33726	1	4	yes	direct product factor
193	C₂	C₂²×#500	1	1	yes	characteristic
194	C₂	C₂²×#508	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂³	C₂²×D₈
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #33726	C₂	direct (N × H)
3. C₂×#2254	C₂	semidirect
4. C₂×#2253	C₂	semidirect
5. C₂×#2252	C₂	semidirect
19. #2254	C₂²	semidirect
20. #2253	C₂²	semidirect
21. #2252	C₂²	semidirect
39. C₂²×C₁₆	Q₈	semidirect
99. C₂×C₁₆	C₂×Q₈	semidirect
192. C₂	#33726	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×#3160	2
#33726	2
C₂×#2254	2
C₂×#2253	2
C₂×#2252	2
C₂×#2158	2
C₂²×C₄×C₁₆	2

Subgroups — C2×#33726 #7623210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#33726 #7623210