Subgroups — C₂×#3164 #60659

Order 256.

927 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544	C₂	1	2	yes	maximal
2	#3164	C₂	1	16	yes	maximal, direct product factor
3	C₂×#501	C₂	1	4	yes	maximal, complemented
4	C₂×#500	C₂	1	2	yes	maximal, complemented
5	C₂×#499	C₂	1	2	yes	maximal, complemented
6	C₂×#428	C₂	1	2	yes	maximal
7	C₂×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×#107	C₂	1	1	yes	maximal, characteristic
9	C₂×#405	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈	C₂²	1	1	yes	characteristic
12	#544	—	2	8	no	—
13	#544	C₂²	1	16	yes	—
14	C₂×C₄⋊Q₈	—	2	2	no	—
15	C₂×C₄⋊Q₈	C₂²	1	4	yes	—
16	C₂×#125	—	2	2	no	—
17	C₂×#125	C₂²	1	4	yes	—
18	C₂×C₂²⋊Q₈	—	2	4	no	—
19	C₂×#117	C₂²	1	2	yes	—
20	C₂×#117	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄	C₂²	1	2	yes	—
22	#501	C₂²	1	32	yes	complemented
23	#500	C₂²	1	16	yes	complemented
24	#499	C₂²	1	16	yes	complemented
25	#428	C₂²	1	16	yes	—
26	#427	C₂²	1	8	yes	—
27	C₂×#107	C₂²	1	1	yes	characteristic
28	C₂×#107	C₂²	1	2	yes	—
29	C₂×#107	C₂²	1	1	yes	characteristic
30	C₂×#107	C₂²	1	4	yes	—
31	C₂×#107	C₂²	1	2	yes	—
32	C₂×#107	C₂²	1	8	yes	—
33	C₂×#106	C₂²	1	4	yes	—
34	C₂×#106	C₂²	1	2	yes	—
35	#405	C₂²	1	8	yes	—
36	C₂×#97	C₂²	1	2	yes	—
37	C₂×C₄×C₈	C₂²	1	2	yes	—
38	C₂²×Q₈	—	2	4	no	—
39	C₂³×C₄	—	2	2	no	—
40	C₂³×C₄	C₂³	1	1	yes	characteristic
41	C₂×M₁₆	Q₈	1	8	yes	complemented
42	C₂×M₁₆	C₂³	1	4	yes	—
43	C₂²×C₈	Q₈	1	8	yes	complemented
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	2	yes	—
47	C₂²×C₈	C₂³	1	4	yes	—
48	C₄⋊Q₈	—	2	16	no	—
49	C₄⋊Q₈	—	2	8	no	—
50	C₄⋊Q₈	C₂³	1	16	yes	—
51	#125	—	2	16	no	—
52	#125	—	2	8	no	—
53	#125	C₂³	1	16	yes	—
54	C₂²⋊Q₈	—	2	32	no	—
55	#117	—	2	4	no	—
56	#117	D₄	1	4	yes	—
57	#117	C₂³	1	8	yes	—
58	#117	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄⋊C₄	—	2	4	no	—
61	C₂×C₄⋊C₄	—	2	4	no	—
62	C₂×C₄⋊C₄	—	2	4	no	—
63	C₂×C₄⋊C₄	—	2	4	no	—
64	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄	C₂³	1	8	yes	—
68	C₂×C₄⋊C₄	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄	C₂³	1	2	yes	—
70	C₂×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₂²⋊C₄	D₄	1	2	yes	—
72	C₂×C₄²	—	2	2	no	—
73	C₂×C₄²	C₂³	1	2	yes	—
74	#107	C₂³	1	8	yes	—
75	#107	C₂³	1	8	yes	—
76	#107	C₂³	1	4	yes	—
77	#107	C₂³	1	4	yes	—
78	#107	C₂³	1	16	yes	—
79	#106	C₂³	1	8	yes	—
80	#106	C₂³	1	16	yes	—
81	#97	C₂³	1	8	yes	—
82	C₄×C₈	C₂³	1	8	yes	—
83	C₂⁴	C₂×D₄	1	1	yes	characteristic
84	C₂×Q₈	—	2	16	no	—
85	C₂×Q₈	—	8	8	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	8	no	—
90	C₂²×C₄	—	8	4	no	—
91	C₂²×C₄	C₂×D₄	1	2	yes	—
92	C₂²×C₄	C₂×D₄	1	2	yes	—
93	C₂²×C₄	C₂×D₄	1	2	yes	—
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	C₂×Q₈	1	4	yes	—
96	C₂²×C₄	C₂×Q₈	1	4	yes	—
97	C₂²×C₄	C₂⁴	1	2	yes	—
98	C₂²×C₄	C₂⁴	1	1	yes	characteristic
99	M₁₆	C₂×Q₈	1	16	yes	complemented
100	C₂×C₈	C₂×Q₈	1	4	yes	—
101	C₂×C₈	C₂×Q₈	1	4	yes	—
102	C₂×C₈	C₂×Q₈	1	8	yes	—
103	C₂×C₈	C₂×Q₈	1	16	yes	complemented
104	C₂×C₈	C₂⁴	1	2	yes	—
105	C₂×C₈	C₂⁴	1	4	yes	—
106	C₂×C₈	C₂⁴	1	2	yes	—
107	C₄⋊C₄	—	2	8	no	—
108	C₄⋊C₄	—	2	4	no	—
109	C₄⋊C₄	—	2	16	no	—
110	C₄⋊C₄	—	2	16	no	—
111	C₄⋊C₄	—	2	16	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	C₂×D₄	1	4	yes	—
115	C₄⋊C₄	C₂×D₄	1	4	yes	—
116	C₄⋊C₄	C₂⁴	1	4	yes	—
117	C₄⋊C₄	C₂⁴	1	8	yes	—
118	C₄⋊C₄	C₂⁴	1	4	yes	—
119	C₂²⋊C₄	—	2	8	no	—
120	C₂²⋊C₄	C₂×D₄	1	8	yes	—
121	C₄²	—	2	8	no	—
122	C₄²	C₂×D₄	1	4	yes	—
123	C₄²	C₂⁴	1	4	yes	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	2	no	—
126	C₂³	C₄⋊Q₈	1	4	yes	—
127	C₂³	C₂²×D₄	1	2	yes	—
128	C₂³	C₂²×D₄	1	1	yes	characteristic
129	Q₈	—	8	16	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	4	8	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	8	8	no	—
139	C₂×C₄	C₄⋊Q₈	1	8	yes	—
140	C₂×C₄	C₄⋊Q₈	1	4	yes	—
141	C₂×C₄	C₂²×D₄	1	4	yes	—
142	C₂×C₄	C₂²×D₄	1	4	yes	—
143	C₂×C₄	C₂²×D₄	1	2	yes	—
144	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄	C₂²×Q₈	1	2	yes	—
146	C₂×C₄	C₂²×Q₈	1	2	yes	—
147	C₂×C₄	C₂²×Q₈	1	4	yes	—
148	C₂×C₄	C₂²×Q₈	1	4	yes	—
149	C₂×C₄	C₂⁵	1	1	yes	characteristic
150	C₈	C₂²×Q₈	1	4	yes	—
151	C₈	C₂²×Q₈	1	4	yes	—
152	C₈	C₂²×Q₈	1	8	yes	—
153	C₂²	—	2	8	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂×C₄⋊Q₈	1	4	yes	—
157	C₂²	C₂×C₄⋊Q₈	1	2	yes	—
158	C₂²	#576	1	2	yes	—
159	C₂²	#578	1	2	yes	—
160	C₂²	C₂³×D₄	1	1	yes	characteristic
161	C₄	—	2	4	no	—
162	C₄	—	2	4	no	—
163	C₄	—	4	8	no	—
164	C₄	—	4	4	no	—
165	C₄	—	4	4	no	—
166	C₄	C₂×C₄⋊Q₈	1	4	yes	—
167	C₄	C₂×C₄⋊Q₈	1	2	yes	—
168	C₄	C₂³×Q₈	1	1	yes	characteristic
169	C₄	C₂³×Q₈	1	1	yes	characteristic
170	C₂	—	2	4	no	—
171	C₂	#3164	1	4	yes	direct product factor
172	C₂	C₂²×C₄⋊Q₈	1	1	yes	characteristic
173	C₂	C₂×#576	1	1	yes	characteristic
174	C₂	C₂×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	C₂⁴	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #3164	C₂	direct (N × H)
3. C₂×#501	C₂	semidirect
4. C₂×#500	C₂	semidirect
5. C₂×#499	C₂	semidirect
22. #501	C₂²	semidirect
23. #500	C₂²	semidirect
24. #499	C₂²	semidirect
41. C₂×M₁₆	Q₈	semidirect
43. C₂²×C₈	Q₈	semidirect
99. M₁₆	C₂×Q₈	semidirect
103. C₂×C₈	C₂×Q₈	semidirect
171. C₂	#3164	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×#544	2
#3164	2
C₂×#501	2
C₂×#500	2
C₂×#499	2
C₂×#428	2
C₂×#427	2
C₂²×#107	2
C₂×#405	2

Subgroups — C2×#3164 #60659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3164 #60659