Subgroups — C₂×#8826 #191464

Order 512.

814 conjugacy classes of subgroups.

The subgroup lattice has 169 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1848	C₂	1	2	yes	maximal
2	C₂×#1567	C₂	1	4	yes	maximal
3	C₂²×#322	C₂	1	1	yes	maximal, characteristic
4	#8826	C₂	1	8	yes	maximal, direct product factor
5	C₂²×#108	—	2	4	no	—
6	C₂×#423	C₂²	1	4	yes	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₈	C₂²	1	2	yes	—
9	#1848	—	2	4	no	—
10	#1848	C₂²	1	8	yes	—
11	#1567	—	2	8	no	—
12	#1567	C₂²	1	16	yes	—
13	C₂×#335	—	2	8	no	—
14	C₂×#322	C₄	1	4	yes	—
15	C₂×#322	C₄	1	4	yes	—
16	C₂×#322	C₂²	1	4	yes	—
17	C₂²×M₁₆	—	2	4	no	—
18	C₂³×C₈	D₄	1	2	yes	—
19	C₂³×C₈	Q₈	1	4	yes	—
20	C₂²×C₄²	C₂³	1	1	yes	characteristic
21	C₂×#108	—	2	16	no	—
22	C₂×#108	—	4	16	no	—
23	#423	—	2	8	no	—
24	#423	—	2	4	no	—
25	#423	C₂³	1	8	yes	—
26	C₂×C₄⋊C₈	—	2	8	no	—
27	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
28	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
29	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₈	D₄	1	2	yes	—
32	C₂×C₄×C₈	D₄	1	4	yes	—
33	C₂×C₄×C₈	D₄	1	4	yes	—
34	C₂×C₄×C₈	Q₈	1	4	yes	—
35	C₂×C₄×C₈	Q₈	1	2	yes	—
36	C₂×C₄×C₈	Q₈	1	2	yes	—
37	C₂×C₄×C₈	C₂³	1	4	yes	—
38	C₂×C₄×C₈	C₂³	1	2	yes	—
39	#335	—	2	32	no	—
40	#322	C₂×C₄	1	16	yes	—
41	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂³×C₄	C₂×Q₈	1	2	yes	—
43	C₂×M₁₆	—	2	8	no	—
44	C₂×M₁₆	—	4	4	no	—
45	C₂×M₁₆	—	4	16	no	—
46	C₂²×C₈	—	4	8	no	—
47	C₂²×C₈	—	4	4	no	—
48	C₂²×C₈	C₄⋊C₄	1	8	yes	—
49	C₂²×C₈	C₄⋊C₄	1	8	yes	—
50	C₂²×C₈	C₄⋊C₄	1	4	yes	—
51	C₂²×C₈	C₄⋊C₄	1	4	yes	—
52	C₂²×C₈	D₈	1	8	yes	—
53	C₂²×C₈	Q₁₆	1	8	yes	—
54	C₂²×C₈	C₂×D₄	1	4	yes	—
55	C₂²×C₈	C₂×Q₈	1	8	yes	—
56	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄²	C₂×D₄	1	2	yes	—
60	C₂×C₄²	C₂×D₄	1	2	yes	—
61	C₂×C₄²	C₂×Q₈	1	2	yes	—
62	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄²	C₂⁴	1	1	yes	characteristic
65	#108	—	4	32	no	—
66	C₄⋊C₈	—	2	16	no	—
67	C₄⋊C₈	—	4	8	no	—
68	C₄×C₈	C₄⋊C₄	1	8	yes	—
69	C₄×C₈	C₄⋊C₄	1	8	yes	—
70	C₄×C₈	C₄⋊C₄	1	4	yes	—
71	C₄×C₈	C₄⋊C₄	1	4	yes	—
72	C₄×C₈	C₂²×C₄	1	8	yes	—
73	C₄×C₈	C₂²×C₄	1	4	yes	—
74	C₄×C₈	C₂×D₄	1	8	yes	—
75	C₄×C₈	C₂×Q₈	1	4	yes	—
76	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
77	C₂²×C₄	—	2	8	no	—
78	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
79	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
82	C₂²×C₄	#127	1	1	yes	characteristic
83	C₂²×C₄	#127	1	1	yes	characteristic
84	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
85	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
86	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
87	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₈	1	4	yes	—
89	C₂²×C₄	C₂×Q₁₆	1	4	yes	—
90	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂²×Q₈	1	2	yes	—
92	M₁₆	—	4	16	no	—
93	M₁₆	—	8	8	no	—
94	C₂×C₈	—	2	16	no	—
95	C₂×C₈	—	4	8	no	—
96	C₂×C₈	—	4	8	no	—
97	C₂×C₈	—	4	8	no	—
98	C₂×C₈	—	8	8	no	—
99	C₂×C₈	#107	1	16	yes	—
100	C₂×C₈	#107	1	16	yes	—
101	C₂×C₈	C₂×C₄⋊C₄	1	16	yes	—
102	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
103	C₂×C₈	C₂×D₈	1	8	yes	—
104	C₂×C₈	C₂×Q₁₆	1	8	yes	—
105	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₄²	C₂×C₄⋊C₄	1	2	yes	—
108	C₄²	C₂×C₄⋊C₄	1	2	yes	—
109	C₄²	#127	1	2	yes	—
110	C₄²	C₄⋊Q₈	1	2	yes	—
111	C₄²	C₄⋊Q₈	1	2	yes	—
112	C₄²	C₂³×C₄	1	1	yes	characteristic
113	C₄²	C₂²×D₄	1	2	yes	—
114	C₄²	C₂²×Q₈	1	1	yes	characteristic
115	C₂³	—	2	4	no	—
116	C₂³	#384	1	1	yes	characteristic
117	C₂³	#384	1	1	yes	characteristic
118	C₂³	#500	1	4	yes	—
119	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	#384	1	2	yes	—
126	C₂×C₄	#384	1	1	yes	characteristic
127	C₂×C₄	#384	1	1	yes	characteristic
128	C₂×C₄	#384	1	2	yes	—
129	C₂×C₄	C₂×#107	1	4	yes	—
130	C₂×C₄	C₂×#107	1	4	yes	—
131	C₂×C₄	#430	1	8	yes	—
132	C₂×C₄	#493	1	2	yes	—
133	C₂×C₄	#493	1	2	yes	—
134	C₂×C₄	#494	1	2	yes	—
135	C₂×C₄	#494	1	2	yes	—
136	C₂×C₄	#500	1	4	yes	—
137	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄	C₂²×C₄⋊C₄	1	2	yes	—
139	C₂×C₄	C₂×#127	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊Q₈	1	2	yes	—
141	C₂×C₄	C₂²×D₈	1	2	yes	—
142	C₂×C₄	C₂²×Q₁₆	1	2	yes	—
143	C₈	—	2	16	no	—
144	C₈	—	8	8	no	—
145	C₈	C₂×#107	1	16	yes	—
146	C₂²	—	2	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	8	no	—
149	C₂²	#1845	1	2	yes	—
150	C₂²	#1845	1	2	yes	—
151	C₂²	#1849	1	4	yes	—
152	C₂²	C₂×#384	1	1	yes	characteristic
153	C₂²	C₂×#500	1	2	yes	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	#1845	1	2	yes	—
157	C₄	#1845	1	2	yes	—
158	C₄	#1849	1	4	yes	—
159	C₄	C₂²×#107	1	2	yes	—
160	C₄	C₂×#430	1	4	yes	—
161	C₄	C₂×#493	1	1	yes	characteristic
162	C₄	C₂×#494	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	2	2	no	—
165	C₂	#8826	1	4	yes	direct product factor
166	C₂	C₂×#1845	1	1	yes	characteristic
167	C₂	C₂×#1849	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	#384
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #8826	C₂	direct (N × H)
165. C₂	#8826	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×#1848	2
C₂×#1567	2
C₂²×#322	2
#8826	2

Subgroups — C2×#8826 #191464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#8826 #191464