Subgroups — C₂×Q₈×#41 #62713

Order 256.

3761 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41	C₂	1	32	yes	maximal, direct product factor
2	C₂×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#557	C₂	1	3	yes	maximal, complemented
4	C₂×#554	C₂	1	9	yes	maximal, complemented
5	C₂×D₄×Q₈	C₂	1	3	yes	maximal, complemented
6	C₂×#533	C₂	1	9	yes	maximal, complemented
7	C₂×C₄×#41	C₂	1	3	yes	maximal
8	C₂²×C₄×Q₈	C₂	1	3	yes	maximal, complemented
9	C₂²×#41	C₂²	1	1	yes	characteristic
10	C₂³×Q₈	C₂²	1	3	yes	—
11	Q₈²	C₂²	1	16	yes	complemented
12	#557	C₂²	1	48	yes	complemented
13	#554	C₂²	1	144	yes	complemented
14	D₄×Q₈	C₂²	1	48	yes	complemented
15	#533	C₂²	1	144	yes	complemented
16	C₂×C₄⋊Q₈	C₂²	1	9	yes	—
17	C₂×C₄⋊Q₈	C₂²	1	9	yes	—
18	C₂×#125	C₂²	1	18	yes	—
19	C₂×C₂²⋊Q₈	C₂²	1	18	yes	—
20	C₂×C₂²⋊Q₈	C₂²	1	18	yes	—
21	C₄×#41	C₂²	1	48	yes	—
22	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈	C₂²	1	9	yes	—
24	C₂×C₄×Q₈	C₂²	1	3	yes	—
25	C₂×C₄×Q₈	C₂²	1	9	yes	—
26	C₂×C₄×Q₈	C₂²	1	3	yes	—
27	C₂×C₄×Q₈	C₂²	1	3	yes	—
28	C₂×C₄×Q₈	C₂²	1	6	yes	—
29	C₂×C₄×Q₈	C₂²	1	48	yes	complemented
30	C₂×C₄×D₄	C₂²	1	9	yes	—
31	C₂×C₄×D₄	C₂²	1	9	yes	—
32	C₂×#117	C₂²	1	9	yes	—
33	C₂²×C₄⋊C₄	C₂²	1	9	yes	—
34	C₂²×C₄²	C₂²	1	9	yes	—
35	C₂×#41	Q₈	1	16	yes	direct product factor
36	C₂×#41	C₂³	1	8	yes	—
37	C₂²×Q₈	—	2	24	no	—
38	C₂²×Q₈	C₂³	1	9	yes	—
39	C₂²×Q₈	C₂³	1	24	yes	—
40	C₂²×Q₈	C₂³	1	6	yes	—
41	C₂²×Q₈	C₂³	1	1	yes	characteristic
42	C₂²×Q₈	C₂³	1	1	yes	characteristic
43	C₂²×D₄	C₂³	1	3	yes	—
44	C₂³×C₄	C₂³	1	9	yes	—
45	C₂³×C₄	C₂³	1	9	yes	—
46	C₂³×C₄	C₂³	1	3	yes	—
47	C₄⋊Q₈	C₂³	1	72	yes	—
48	C₄⋊Q₈	C₂³	1	72	yes	—
49	#125	C₂³	1	144	yes	—
50	C₂²⋊Q₈	C₂³	1	144	yes	—
51	C₂²⋊Q₈	C₂³	1	144	yes	—
52	C₄×Q₈	C₂³	1	8	yes	—
53	C₄×Q₈	C₂³	1	72	yes	—
54	C₄×Q₈	C₂³	1	24	yes	—
55	C₄×Q₈	C₂³	1	72	yes	—
56	C₄×Q₈	C₂³	1	24	yes	—
57	C₄×Q₈	C₂³	1	24	yes	—
58	C₄×Q₈	C₂³	1	48	yes	—
59	C₄×D₄	C₂³	1	72	yes	—
60	C₄×D₄	C₂³	1	72	yes	—
61	#117	C₂³	1	72	yes	—
62	C₂×C₄⋊C₄	C₂³	1	3	yes	—
63	C₂×C₄⋊C₄	C₂³	1	18	yes	—
64	C₂×C₄⋊C₄	C₂³	1	9	yes	—
65	C₂×C₄⋊C₄	C₂³	1	72	yes	—
66	C₂×C₄⋊C₄	C₂³	1	9	yes	—
67	C₂×C₄⋊C₄	C₂³	1	9	yes	—
68	C₂×C₄⋊C₄	C₂³	1	18	yes	—
69	C₂×C₂²⋊C₄	C₂³	1	18	yes	—
70	C₂×C₄²	C₂³	1	9	yes	—
71	C₂×C₄²	C₂³	1	9	yes	—
72	C₂×C₄²	C₂³	1	72	yes	—
73	C₂×C₄²	C₂³	1	9	yes	—
74	C₂×C₄²	C₂³	1	3	yes	—
75	C₂⁴	C₂⁴	1	3	yes	—
76	#41	C₂×Q₈	1	64	yes	direct product factor
77	C₂×Q₈	—	2	96	no	—
78	C₂×Q₈	—	2	36	no	—
79	C₂×Q₈	—	4	12	no	—
80	C₂×Q₈	C₂×Q₈	1	8	yes	complemented
81	C₂×Q₈	#41	1	8	yes	direct product factor
82	C₂×Q₈	C₂⁴	1	36	yes	—
83	C₂×Q₈	C₂⁴	1	24	yes	—
84	C₂×Q₈	C₂⁴	1	4	yes	—
85	C₂×Q₈	C₂⁴	1	4	yes	—
86	C₂×D₄	C₂×Q₈	1	24	yes	complemented
87	C₂×D₄	C₂⁴	1	12	yes	—
88	C₂²×C₄	—	2	36	no	—
89	C₂²×C₄	—	4	18	no	—
90	C₂²×C₄	C₂×Q₈	1	24	yes	complemented
91	C₂²×C₄	C₂⁴	1	3	yes	—
92	C₂²×C₄	C₂⁴	1	9	yes	—
93	C₂²×C₄	C₂⁴	1	9	yes	—
94	C₂²×C₄	C₂⁴	1	36	yes	—
95	C₂²×C₄	C₂⁴	1	36	yes	—
96	C₂²×C₄	C₂⁴	1	3	yes	—
97	C₂²×C₄	C₂⁴	1	12	yes	—
98	C₂²×C₄	C₂⁴	1	3	yes	—
99	C₂²×C₄	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄	C₂⁴	1	12	yes	—
101	C₄⋊C₄	C₂⁴	1	72	yes	—
102	C₄⋊C₄	C₂⁴	1	36	yes	—
103	C₄⋊C₄	C₂⁴	1	36	yes	—
104	C₄⋊C₄	C₂⁴	1	36	yes	—
105	C₄⋊C₄	C₂⁴	1	72	yes	—
106	C₂²⋊C₄	C₂⁴	1	72	yes	—
107	C₄²	C₂⁴	1	12	yes	—
108	C₄²	C₂⁴	1	36	yes	—
109	C₄²	C₂⁴	1	36	yes	—
110	C₄²	C₂⁴	1	36	yes	—
111	C₂³	—	2	6	no	—
112	C₂³	—	2	6	no	—
113	C₂³	C₂²×Q₈	1	12	yes	—
114	C₂³	C₂⁵	1	6	yes	—
115	C₂³	C₂⁵	1	1	yes	characteristic
116	Q₈	—	2	72	no	—
117	Q₈	—	4	24	no	—
118	Q₈	C₂²×Q₈	1	16	yes	complemented
119	Q₈	C₂×#41	1	16	yes	direct product factor
120	D₄	C₂²×Q₈	1	48	yes	complemented
121	C₂×C₄	—	2	18	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	2	18	no	—
124	C₂×C₄	—	2	72	no	—
125	C₂×C₄	—	4	36	no	—
126	C₂×C₄	C₂²×Q₈	1	4	yes	—
127	C₂×C₄	C₂²×Q₈	1	12	yes	—
128	C₂×C₄	C₂²×Q₈	1	48	yes	complemented
129	C₂×C₄	C₂×#41	1	12	yes	—
130	C₂×C₄	C₂⁵	1	6	yes	—
131	C₂×C₄	C₂⁵	1	18	yes	—
132	C₂×C₄	C₂⁵	1	18	yes	—
133	C₂×C₄	C₂⁵	1	2	yes	—
134	C₂×C₄	C₂⁵	1	6	yes	—
135	C₂×C₄	C₂⁵	1	6	yes	—
136	C₂²	—	2	6	no	—
137	C₂²	—	2	6	no	—
138	C₂²	—	2	24	no	—
139	C₂²	C₂³×Q₈	1	12	yes	—
140	C₂²	C₂³×Q₈	1	2	yes	—
141	C₂²	C₂²×#41	1	2	yes	—
142	C₂²	#585	1	2	yes	—
143	C₂²	C₂⁶	1	1	yes	characteristic
144	C₄	—	2	18	no	—
145	C₄	—	2	6	no	—
146	C₄	—	2	18	no	—
147	C₄	C₂³×Q₈	1	4	yes	—
148	C₄	C₂³×Q₈	1	12	yes	—
149	C₄	C₂²×#41	1	12	yes	—
150	C₂	—	2	12	no	—
151	C₂	Q₈×#41	1	4	yes	direct product factor
152	C₂	C₂⁴×Q₈	1	1	yes	characteristic
153	C₂	C₂³×#41	1	1	yes	characteristic
154	C₂	C₂×#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
socle	C₂³	C₂⁵
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. Q₈×#41	C₂	direct (N × H)
2. C₂×Q₈²	C₂	semidirect
3. C₂×#557	C₂	semidirect
4. C₂×#554	C₂	semidirect
5. C₂×D₄×Q₈	C₂	semidirect
6. C₂×#533	C₂	semidirect
8. C₂²×C₄×Q₈	C₂	semidirect
11. Q₈²	C₂²	semidirect
12. #557	C₂²	semidirect
13. #554	C₂²	semidirect
14. D₄×Q₈	C₂²	semidirect
15. #533	C₂²	semidirect
29. C₂×C₄×Q₈	C₂²	semidirect
35. C₂×#41	Q₈	direct (N × H)
76. #41	C₂×Q₈	direct (N × H)
80. C₂×Q₈	C₂×Q₈	semidirect
81. C₂×Q₈	#41	direct (N × H)
86. C₂×D₄	C₂×Q₈	semidirect
90. C₂²×C₄	C₂×Q₈	semidirect
118. Q₈	C₂²×Q₈	semidirect
119. Q₈	C₂×#41	direct (N × H)
120. D₄	C₂²×Q₈	semidirect
128. C₂×C₄	C₂²×Q₈	semidirect
151. C₂	Q₈×#41	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#41	2
C₂×Q₈²	2
C₂×#557	2
C₂×#554	2
C₂×D₄×Q₈	2
C₂×#533	2
C₂×C₄×#41	2
C₂²×C₄×Q₈	2

Subgroups — C2×Q8×#41 #62713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#41 #62713