Subgroups — C₂×#546 #3463

Order 128.

950 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#546	C₂	1	16	yes	maximal, direct product factor
4	C₂×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#123	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#121	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#121	C₂	1	2	yes	maximal, complemented
9	C₂×#120	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
13	C₂⁵	C₂²	1	2	yes	—
14	C₂×#41	—	2	8	no	—
15	C₂×#41	C₂²	1	8	yes	complemented
16	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₄	—	2	8	no	—
18	C₂²×D₄	C₂²	1	1	yes	characteristic
19	C₂²×D₄	C₂²	1	2	yes	—
20	C₂²×D₄	C₂²	1	1	yes	characteristic
21	C₂²×D₄	C₂²	1	1	yes	characteristic
22	C₂²×D₄	C₂²	1	8	yes	complemented
23	C₂²×D₄	C₂²	1	2	yes	complemented
24	C₂²×D₄	C₂²	1	2	yes	complemented
25	C₂³×C₄	C₂²	1	1	yes	characteristic
26	C₂³×C₄	C₂²	1	2	yes	complemented
27	C₂³×C₄	C₂²	1	1	yes	characteristic, complemented
28	C₂³×C₄	C₂²	1	2	yes	complemented
29	#124	C₂²	1	8	yes	complemented
30	#123	C₂²	1	16	yes	complemented
31	C₂²⋊Q₈	C₂²	1	8	yes	complemented
32	#121	C₂²	1	8	yes	complemented
33	#121	C₂²	1	16	yes	complemented
34	#120	C₂²	1	16	yes	complemented
35	C₄×D₄	C₂²	1	8	yes	complemented
36	C₄×D₄	C₂²	1	8	yes	complemented
37	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
40	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂×C₂²⋊C₄	C₂²	1	16	yes	complemented
43	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
44	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
45	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
46	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
48	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
49	C₂⁴	—	2	8	no	—
50	C₂⁴	—	2	8	no	—
51	C₂⁴	—	4	4	no	—
52	C₂⁴	C₂³	1	2	yes	—
53	C₂⁴	C₂³	1	1	yes	characteristic
54	C₂⁴	C₂³	1	8	yes	—
55	C₂⁴	C₂³	1	2	yes	—
56	C₂⁴	C₂³	1	1	yes	characteristic
57	#41	—	2	32	no	—
58	C₂×Q₈	—	2	4	no	—
59	C₂×Q₈	C₂³	1	4	yes	complemented
60	C₂×D₄	—	2	8	no	—
61	C₂×D₄	—	2	8	no	—
62	C₂×D₄	—	2	4	no	—
63	C₂×D₄	—	2	32	no	—
64	C₂×D₄	—	2	4	no	—
65	C₂×D₄	—	4	4	no	—
66	C₂×D₄	D₄	1	8	yes	complemented
67	C₂×D₄	C₂³	1	4	yes	—
68	C₂×D₄	C₂³	1	8	yes	—
69	C₂×D₄	C₂³	1	4	yes	—
70	C₂×D₄	C₂³	1	4	yes	—
71	C₂×D₄	C₂³	1	8	yes	complemented
72	C₂×D₄	C₂³	1	8	yes	complemented
73	C₂²×C₄	—	2	8	no	—
74	C₂²×C₄	—	2	4	no	—
75	C₂²×C₄	—	2	4	no	—
76	C₂²×C₄	—	4	4	no	—
77	C₂²×C₄	C₂³	1	4	yes	—
78	C₂²×C₄	C₂³	1	1	yes	characteristic
79	C₂²×C₄	C₂³	1	1	yes	characteristic
80	C₂²×C₄	C₂³	1	2	yes	—
81	C₂²×C₄	C₂³	1	1	yes	characteristic
82	C₂²×C₄	C₂³	1	2	yes	—
83	C₂²×C₄	C₂³	1	1	yes	characteristic
84	C₂²×C₄	C₂³	1	1	yes	characteristic
85	C₂²×C₄	C₂³	1	8	yes	complemented
86	C₂²×C₄	C₂³	1	4	yes	complemented
87	C₂²×C₄	C₂³	1	8	yes	complemented
88	C₄⋊C₄	C₂³	1	4	yes	complemented
89	C₄⋊C₄	C₂³	1	4	yes	complemented
90	C₄⋊C₄	C₂³	1	8	yes	complemented
91	C₂²⋊C₄	C₂³	1	4	yes	—
92	C₂²⋊C₄	C₂³	1	8	yes	—
93	C₂²⋊C₄	C₂³	1	8	yes	complemented
94	C₂²⋊C₄	C₂³	1	8	yes	complemented
95	C₂²⋊C₄	C₂³	1	8	yes	complemented
96	C₂²⋊C₄	C₂³	1	4	yes	complemented
97	C₂²⋊C₄	C₂³	1	8	yes	complemented
98	C₄²	C₂³	1	4	yes	complemented
99	C₂³	—	2	2	no	—
100	C₂³	—	2	4	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	2	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	—	2	16	no	—
105	C₂³	—	2	4	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	2	4	no	—
108	C₂³	—	2	16	no	—
109	C₂³	—	2	2	no	—
110	C₂³	—	2	4	no	—
111	C₂³	—	4	8	no	—
112	C₂³	—	4	32	no	—
113	C₂³	C₂×D₄	1	8	yes	—
114	C₂³	#41	1	4	yes	complemented
115	C₂³	C₂⁴	1	4	yes	—
116	C₂³	C₂⁴	1	2	yes	—
117	C₂³	C₂⁴	1	4	yes	—
118	C₂³	C₂⁴	1	2	yes	—
119	C₂³	C₂⁴	1	1	yes	characteristic
120	Q₈	—	2	8	no	—
121	D₄	—	2	16	no	—
122	D₄	—	2	16	no	—
123	D₄	—	2	8	no	—
124	D₄	—	2	8	no	—
125	D₄	—	4	8	no	—
126	D₄	C₂×D₄	1	16	yes	complemented
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	16	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	C₂×D₄	1	4	yes	—
138	C₂×C₄	C₂⁴	1	2	yes	—
139	C₂×C₄	C₂⁴	1	2	yes	—
140	C₂×C₄	C₂⁴	1	4	yes	—
141	C₂×C₄	C₂⁴	1	2	yes	—
142	C₂×C₄	C₂⁴	1	4	yes	—
143	C₂×C₄	C₂⁴	1	2	yes	—
144	C₂×C₄	C₂⁴	1	2	yes	—
145	C₂²	—	2	8	no	—
146	C₂²	—	2	16	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	4	no	—
149	C₂²	—	2	4	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	32	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	8	no	—
160	C₂²	C₂²×D₄	1	8	yes	—
161	C₂²	C₂²×D₄	1	2	yes	—
162	C₂²	C₂×#41	1	2	yes	—
163	C₂²	C₂×#41	1	4	yes	complemented
164	C₂²	H(5,𝔽₂)	1	2	yes	—
165	C₂²	C₂⁵	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	4	no	—
172	C₄	C₂²×D₄	1	4	yes	—
173	C₂	—	2	4	no	—
174	C₂	—	2	8	no	—
175	C₂	—	4	2	no	—
176	C₂	—	4	4	no	—
177	C₂	#546	1	4	yes	direct product factor
178	C₂	C₂³×D₄	1	1	yes	characteristic
179	C₂	C₂²×#41	1	1	yes	characteristic
180	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#41	C₂	semidirect
2. C₂³×D₄	C₂	semidirect
3. #546	C₂	direct (N × H)
4. C₂×#124	C₂	semidirect
5. C₂×#123	C₂	semidirect
6. C₂×C₂²⋊Q₈	C₂	semidirect
7. C₂×#121	C₂	semidirect
8. C₂×#121	C₂	semidirect
9. C₂×#120	C₂	semidirect
10. C₂×C₄×D₄	C₂	semidirect
11. C₂×C₄×D₄	C₂	semidirect
12. C₂²×C₂²⋊C₄	C₂	semidirect
15. C₂×#41	C₂²	semidirect
16. C₂²×Q₈	C₂²	semidirect
22. C₂²×D₄	C₂²	semidirect
23. C₂²×D₄	C₂²	semidirect
24. C₂²×D₄	C₂²	semidirect
26. C₂³×C₄	C₂²	semidirect
27. C₂³×C₄	C₂²	semidirect
28. C₂³×C₄	C₂²	semidirect
29. #124	C₂²	semidirect
30. #123	C₂²	semidirect
31. C₂²⋊Q₈	C₂²	semidirect
32. #121	C₂²	semidirect
33. #121	C₂²	semidirect
34. #120	C₂²	semidirect
35. C₄×D₄	C₂²	semidirect
36. C₄×D₄	C₂²	semidirect
37. C₂×C₄⋊C₄	C₂²	semidirect
38. C₂×C₄⋊C₄	C₂²	semidirect
39. C₂×C₄⋊C₄	C₂²	semidirect
42. C₂×C₂²⋊C₄	C₂²	semidirect
43. C₂×C₂²⋊C₄	C₂²	semidirect
44. C₂×C₂²⋊C₄	C₂²	semidirect
45. C₂×C₂²⋊C₄	C₂²	semidirect
46. C₂×C₂²⋊C₄	C₂²	semidirect
47. C₂×C₂²⋊C₄	C₂²	semidirect
48. C₂×C₄²	C₂²	semidirect
59. C₂×Q₈	C₂³	semidirect
66. C₂×D₄	D₄	semidirect
71. C₂×D₄	C₂³	semidirect
72. C₂×D₄	C₂³	semidirect
85. C₂²×C₄	C₂³	semidirect
86. C₂²×C₄	C₂³	semidirect
87. C₂²×C₄	C₂³	semidirect
88. C₄⋊C₄	C₂³	semidirect
89. C₄⋊C₄	C₂³	semidirect
90. C₄⋊C₄	C₂³	semidirect
93. C₂²⋊C₄	C₂³	semidirect
94. C₂²⋊C₄	C₂³	semidirect
95. C₂²⋊C₄	C₂³	semidirect
96. C₂²⋊C₄	C₂³	semidirect
97. C₂²⋊C₄	C₂³	semidirect
98. C₄²	C₂³	semidirect
114. C₂³	#41	semidirect
126. D₄	C₂×D₄	semidirect
163. C₂²	C₂×#41	semidirect
177. C₂	#546	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41	2
C₂³×D₄	2
#546	2
C₂×#124	2
C₂×#123	2
C₂×C₂²⋊Q₈	2
C₂×#121	2
C₂×#120	2
C₂×C₄×D₄	2
C₂²×C₂²⋊C₄	2

Subgroups — C2×#546 #3463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#546 #3463