Subgroups — C₂×#3504 #62807

Order 256.

3479 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3504	C₂	1	32	yes	maximal, direct product factor
2	C₂×#557	C₂	1	4	yes	maximal, complemented
3	C₂×#554	C₂	1	12	yes	maximal, complemented
4	C₂×#544	C₂	1	6	yes	maximal, complemented
5	C₂×#533	C₂	1	3	yes	maximal, complemented
6	C₂²×#125	C₂	1	3	yes	maximal, complemented
7	C₂×#520	C₂	1	2	yes	maximal
8	C₂×C₄×#41	C₂	1	1	yes	maximal, characteristic
9	C₂²×#41	C₂²	1	1	yes	characteristic
10	#557	C₂²	1	64	yes	complemented
11	#554	C₂²	1	192	yes	complemented
12	#544	C₂²	1	96	yes	complemented
13	#533	C₂²	1	48	yes	complemented
14	C₂×C₄⋊Q₈	C₂²	1	6	yes	—
15	C₂×C₄⋊Q₈	C₂²	1	12	yes	—
16	C₂×#125	C₂²	1	3	yes	—
17	C₂×#125	C₂²	1	6	yes	—
18	C₂×#125	C₂²	1	12	yes	—
19	C₂×#125	C₂²	1	1	yes	characteristic
20	C₂×#125	C₂²	1	12	yes	—
21	C₂×#125	C₂²	1	48	yes	complemented
22	C₂×C₂²⋊Q₈	C₂²	1	12	yes	—
23	C₂×C₂²⋊Q₈	C₂²	1	24	yes	—
24	#520	C₂²	1	32	yes	—
25	C₄×#41	C₂²	1	16	yes	—
26	C₂×C₄×Q₈	C₂²	1	2	yes	—
27	C₂×C₄×Q₈	C₂²	1	4	yes	—
28	C₂×C₄×Q₈	C₂²	1	12	yes	—
29	C₂×C₄×D₄	C₂²	1	6	yes	—
30	C₂×C₄×D₄	C₂²	1	12	yes	—
31	C₂×#117	C₂²	1	3	yes	—
32	C₂×#117	C₂²	1	6	yes	—
33	C₂²×C₄⋊C₄	C₂²	1	6	yes	—
34	C₂²×C₄⋊C₄	C₂²	1	12	yes	—
35	C₂²×C₄²	C₂²	1	3	yes	—
36	C₂×#41	Q₈	1	16	yes	complemented
37	C₂×#41	C₂³	1	8	yes	—
38	C₂²×Q₈	C₂³	1	12	yes	—
39	C₂²×Q₈	C₂³	1	1	yes	characteristic
40	C₂²×D₄	C₂³	1	3	yes	—
41	C₂³×C₄	C₂³	1	12	yes	—
42	C₂³×C₄	C₂³	1	6	yes	—
43	C₂³×C₄	C₂³	1	3	yes	—
44	C₄⋊Q₈	C₂³	1	48	yes	—
45	C₄⋊Q₈	C₂³	1	96	yes	—
46	#125	C₂³	1	24	yes	—
47	#125	C₂³	1	48	yes	—
48	#125	C₂³	1	96	yes	—
49	#125	C₂³	1	8	yes	—
50	#125	C₂³	1	96	yes	—
51	C₂²⋊Q₈	C₂³	1	96	yes	—
52	C₂²⋊Q₈	C₂³	1	192	yes	—
53	C₄×Q₈	C₂³	1	16	yes	—
54	C₄×Q₈	C₂³	1	32	yes	—
55	C₄×Q₈	C₂³	1	96	yes	—
56	C₄×D₄	C₂³	1	48	yes	—
57	C₄×D₄	C₂³	1	96	yes	—
58	#117	C₂³	1	24	yes	—
59	#117	C₂³	1	48	yes	—
60	C₂×C₄⋊C₄	C₂³	1	6	yes	—
61	C₂×C₄⋊C₄	C₂³	1	12	yes	—
62	C₂×C₄⋊C₄	C₂³	1	12	yes	—
63	C₂×C₄⋊C₄	C₂³	1	48	yes	—
64	C₂×C₄⋊C₄	C₂³	1	24	yes	—
65	C₂×C₄⋊C₄	C₂³	1	96	yes	—
66	C₂×C₄⋊C₄	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄	C₂³	1	6	yes	—
68	C₂×C₄⋊C₄	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄	C₂³	1	12	yes	—
70	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
71	C₂×C₂²⋊C₄	C₂³	1	6	yes	—
72	C₂×C₄²	C₂³	1	3	yes	—
73	C₂×C₄²	C₂³	1	12	yes	—
74	C₂×C₄²	C₂³	1	6	yes	—
75	C₂×C₄²	C₂³	1	24	yes	—
76	C₂×C₄²	C₂³	1	1	yes	characteristic
77	C₂⁴	C₂⁴	1	3	yes	—
78	#41	C₂×Q₈	1	64	yes	complemented
79	C₂×Q₈	—	4	24	no	—
80	C₂×Q₈	C₂×Q₈	1	8	yes	—
81	C₂×Q₈	C₂⁴	1	48	yes	—
82	C₂×Q₈	C₂⁴	1	4	yes	—
83	C₂×D₄	C₂×Q₈	1	24	yes	—
84	C₂×D₄	C₂⁴	1	12	yes	—
85	C₂²×C₄	—	4	12	no	—
86	C₂²×C₄	—	4	24	no	—
87	C₂²×C₄	C₂×Q₈	1	24	yes	—
88	C₂²×C₄	C₂⁴	1	2	yes	—
89	C₂²×C₄	C₂⁴	1	4	yes	—
90	C₂²×C₄	C₂⁴	1	6	yes	—
91	C₂²×C₄	C₂⁴	1	12	yes	—
92	C₂²×C₄	C₂⁴	1	24	yes	—
93	C₂²×C₄	C₂⁴	1	48	yes	—
94	C₂²×C₄	C₂⁴	1	12	yes	—
95	C₂²×C₄	C₂⁴	1	3	yes	—
96	C₂²×C₄	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄	C₂⁴	1	24	yes	—
98	C₄⋊C₄	C₂⁴	1	48	yes	—
99	C₄⋊C₄	C₂⁴	1	48	yes	—
100	C₄⋊C₄	C₂⁴	1	96	yes	—
101	C₄⋊C₄	C₂⁴	1	8	yes	—
102	C₄⋊C₄	C₂⁴	1	24	yes	—
103	C₄⋊C₄	C₂⁴	1	16	yes	—
104	C₄⋊C₄	C₂⁴	1	48	yes	—
105	C₂²⋊C₄	C₂⁴	1	48	yes	—
106	C₂²⋊C₄	C₂⁴	1	24	yes	—
107	C₄²	C₂⁴	1	4	yes	—
108	C₄²	C₂⁴	1	12	yes	—
109	C₄²	C₂⁴	1	24	yes	—
110	C₄²	C₂⁴	1	48	yes	—
111	C₂³	—	2	12	no	—
112	C₂³	C₂²×Q₈	1	12	yes	—
113	C₂³	C₂⁵	1	6	yes	—
114	C₂³	C₂⁵	1	1	yes	characteristic
115	Q₈	—	4	48	no	—
116	Q₈	C₂²×Q₈	1	16	yes	—
117	D₄	C₂²×Q₈	1	48	yes	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	24	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	4	24	no	—
123	C₂×C₄	—	4	48	no	—
124	C₂×C₄	C₂²×Q₈	1	4	yes	—
125	C₂×C₄	C₂²×Q₈	1	12	yes	—
126	C₂×C₄	C₂²×Q₈	1	48	yes	—
127	C₂×C₄	C₂⁵	1	4	yes	—
128	C₂×C₄	C₂⁵	1	8	yes	—
129	C₂×C₄	C₂⁵	1	12	yes	—
130	C₂×C₄	C₂⁵	1	24	yes	—
131	C₂×C₄	C₂⁵	1	6	yes	—
132	C₂×C₄	C₂⁵	1	2	yes	—
133	C₂²	—	2	12	no	—
134	C₂²	—	2	24	no	—
135	C₂²	C₂³×Q₈	1	12	yes	—
136	C₂²	C₂³×Q₈	1	2	yes	—
137	C₂²	#585	1	4	yes	—
138	C₂²	C₂⁶	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	2	24	no	—
141	C₄	—	2	12	no	—
142	C₄	—	2	8	no	—
143	C₄	C₂³×Q₈	1	4	yes	—
144	C₄	C₂³×Q₈	1	12	yes	—
145	C₂	—	2	12	no	—
146	C₂	#3504	1	4	yes	direct product factor
147	C₂	C₂⁴×Q₈	1	1	yes	characteristic
148	C₂	C₂×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂²×#41	C₂²
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. #3504	C₂	direct (N × H)
2. C₂×#557	C₂	semidirect
3. C₂×#554	C₂	semidirect
4. C₂×#544	C₂	semidirect
5. C₂×#533	C₂	semidirect
6. C₂²×#125	C₂	semidirect
10. #557	C₂²	semidirect
11. #554	C₂²	semidirect
12. #544	C₂²	semidirect
13. #533	C₂²	semidirect
21. C₂×#125	C₂²	semidirect
36. C₂×#41	Q₈	semidirect
78. #41	C₂×Q₈	semidirect
146. C₂	#3504	direct (N × H)

Maximal subgroups

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

subgroup	index
#3504	2
C₂×#557	2
C₂×#554	2
C₂×#544	2
C₂×#533	2
C₂²×#125	2
C₂×#520	2
C₂×C₄×#41	2

Subgroups — C2×#3504 #62807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3504 #62807