Subgroups — C₂×Q₈×C₄⋊C₄ #34112

Order 256.

1623 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈	C₂	1	2	yes	maximal
2	C₂²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄	C₂	1	16	yes	maximal, direct product factor
4	C₂×#389	C₂	1	6	yes	maximal
5	C₂×#384	C₂	1	3	yes	maximal
6	C₂×C₄×C₄⋊C₄	C₂	1	3	yes	maximal
7	C₂³×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈	—	2	16	no	—
9	C₂×C₄×Q₈	C₄	1	16	yes	complemented
10	C₂×C₄×Q₈	C₂²	1	16	yes	—
11	C₂×C₄×Q₈	C₂²	1	8	yes	—
12	C₂²×C₄⋊C₄	C₂²	1	6	yes	—
13	C₂²×C₄⋊C₄	C₂²	1	3	yes	—
14	C₂²×C₄⋊C₄	C₂²	1	6	yes	—
15	C₂²×C₄⋊C₄	C₂²	1	3	yes	—
16	C₂²×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₄²	C₂²	1	6	yes	—
18	C₂²×C₄²	C₂²	1	3	yes	—
19	#389	C₂²	1	48	yes	—
20	#384	C₂²	1	24	yes	—
21	C₄×C₄⋊C₄	C₂²	1	24	yes	—
22	C₂×#95	C₂²	1	6	yes	—
23	C₂²×Q₈	C₂×C₄	1	8	yes	—
24	C₂²×Q₈	D₄	1	8	yes	—
25	C₂²×Q₈	Q₈	1	8	yes	—
26	C₂²×Q₈	C₂³	1	4	yes	—
27	C₂³×C₄	C₂³	1	6	yes	—
28	C₂³×C₄	C₂³	1	3	yes	—
29	C₂³×C₄	C₂³	1	2	yes	—
30	C₂³×C₄	C₂³	1	3	yes	—
31	C₂³×C₄	C₂³	1	1	yes	characteristic
32	C₄×Q₈	—	2	64	no	—
33	C₄×Q₈	C₂×C₄	1	64	yes	complemented
34	C₂×C₄⋊C₄	—	2	24	no	—
35	C₂×C₄⋊C₄	—	2	12	no	—
36	C₂×C₄⋊C₄	—	4	12	no	—
37	C₂×C₄⋊C₄	C₂×C₄	1	24	yes	—
38	C₂×C₄⋊C₄	Q₈	1	8	yes	direct product factor
39	C₂×C₄⋊C₄	C₂³	1	24	yes	—
40	C₂×C₄⋊C₄	C₂³	1	12	yes	—
41	C₂×C₄⋊C₄	C₂³	1	24	yes	—
42	C₂×C₄⋊C₄	C₂³	1	12	yes	—
43	C₂×C₄⋊C₄	C₂³	1	4	yes	—
44	C₂×C₄²	—	2	24	no	—
45	C₂×C₄²	C₂×C₄	1	24	yes	—
46	C₂×C₄²	C₂³	1	24	yes	—
47	C₂×C₄²	C₂³	1	12	yes	—
48	#95	C₂³	1	24	yes	—
49	C₂⁴	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈	C₄⋊C₄	1	64	yes	direct product factor
51	C₂×Q₈	C₂²×C₄	1	16	yes	—
52	C₂×Q₈	C₂×D₄	1	16	yes	—
53	C₂×Q₈	C₂×Q₈	1	16	yes	—
54	C₂²×C₄	—	2	4	no	—
55	C₂²×C₄	—	2	6	no	—
56	C₂²×C₄	—	2	6	no	—
57	C₂²×C₄	—	2	4	no	—
58	C₂²×C₄	—	2	12	no	—
59	C₂²×C₄	—	2	12	no	—
60	C₂²×C₄	—	2	12	no	—
61	C₂²×C₄	C₂²×C₄	1	12	yes	—
62	C₂²×C₄	C₂²×C₄	1	12	yes	—
63	C₂²×C₄	C₂²×C₄	1	4	yes	—
64	C₂²×C₄	C₂×D₄	1	12	yes	—
65	C₂²×C₄	C₂×Q₈	1	8	yes	—
66	C₂²×C₄	C₂×Q₈	1	12	yes	—
67	C₂²×C₄	C₂×Q₈	1	4	yes	—
68	C₂²×C₄	#41	1	4	yes	—
69	C₂²×C₄	C₂⁴	1	12	yes	—
70	C₂²×C₄	C₂⁴	1	6	yes	—
71	C₂²×C₄	C₂⁴	1	4	yes	—
72	C₂²×C₄	C₂⁴	1	6	yes	—
73	C₂²×C₄	C₂⁴	1	2	yes	—
74	C₄⋊C₄	—	2	48	no	—
75	C₄⋊C₄	—	2	24	no	—
76	C₄⋊C₄	—	4	24	no	—
77	C₄⋊C₄	C₂²×C₄	1	48	yes	—
78	C₄⋊C₄	C₂×Q₈	1	16	yes	direct product factor
79	C₄²	—	2	48	no	—
80	C₄²	C₂²×C₄	1	48	yes	—
81	C₂³	C₂³×C₄	1	2	yes	—
82	C₂³	C₂²×D₄	1	2	yes	—
83	C₂³	C₂²×Q₈	1	2	yes	—
84	C₂³	C₂²×Q₈	1	2	yes	—
85	C₂³	C₂×#41	1	2	yes	—
86	C₂³	H(5,𝔽₂)	1	2	yes	—
87	C₂³	2₋¹⁺⁴	1	2	yes	—
88	C₂³	C₂⁵	1	1	yes	characteristic
89	Q₈	C₂×C₄⋊C₄	1	64	yes	direct product factor
90	C₂×C₄	—	2	24	no	—
91	C₂×C₄	—	2	16	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	6	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	12	no	—
97	C₂×C₄	—	2	12	no	—
98	C₂×C₄	—	2	12	no	—
99	C₂×C₄	—	4	24	no	—
100	C₂×C₄	C₂×C₄⋊C₄	1	48	yes	—
101	C₂×C₄	C₄×Q₈	1	16	yes	complemented
102	C₂×C₄	C₂³×C₄	1	12	yes	—
103	C₂×C₄	C₂³×C₄	1	12	yes	—
104	C₂×C₄	C₂³×C₄	1	4	yes	—
105	C₂×C₄	C₂²×D₄	1	12	yes	—
106	C₂×C₄	C₂²×Q₈	1	8	yes	—
107	C₂×C₄	C₂²×Q₈	1	12	yes	—
108	C₂×C₄	C₂²×Q₈	1	4	yes	—
109	C₂×C₄	C₂×#41	1	4	yes	—
110	C₂²	C₂²×C₄⋊C₄	1	4	yes	—
111	C₂²	C₂×C₄×Q₈	1	4	yes	—
112	C₂²	#520	1	4	yes	—
113	C₂²	D₄×Q₈	1	4	yes	—
114	C₂²	#550	1	4	yes	—
115	C₂²	#557	1	4	yes	—
116	C₂²	Q₈²	1	4	yes	—
117	C₂²	C₂⁴×C₄	1	1	yes	characteristic
118	C₂²	C₂³×D₄	1	1	yes	characteristic
119	C₂²	C₂³×Q₈	1	1	yes	characteristic
120	C₂²	C₂³×Q₈	1	1	yes	characteristic
121	C₂²	C₂²×#41	1	1	yes	characteristic
122	C₂²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²	C₂×2₋¹⁺⁴	1	1	yes	characteristic
124	C₄	—	2	12	no	—
125	C₄	—	2	8	no	—
126	C₄	—	4	12	no	—
127	C₄	C₂²×C₄⋊C₄	1	24	yes	—
128	C₄	C₂×C₄×Q₈	1	8	yes	complemented
129	C₂	Q₈×C₄⋊C₄	1	8	yes	direct product factor
130	C₂	C₂³×C₄⋊C₄	1	1	yes	characteristic
131	C₂	C₂²×C₄×Q₈	1	1	yes	characteristic
132	C₂	C₂×#520	1	1	yes	characteristic
133	C₂	C₂×D₄×Q₈	1	1	yes	characteristic
134	C₂	C₂×#550	1	1	yes	characteristic
135	C₂	C₂×#557	1	1	yes	characteristic
136	C₂	C₂×Q₈²	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. Q₈×C₄⋊C₄	C₂	direct (N × H)
9. C₂×C₄×Q₈	C₄	semidirect
33. C₄×Q₈	C₂×C₄	semidirect
38. C₂×C₄⋊C₄	Q₈	direct (N × H)
50. C₂×Q₈	C₄⋊C₄	direct (N × H)
78. C₄⋊C₄	C₂×Q₈	direct (N × H)
89. Q₈	C₂×C₄⋊C₄	direct (N × H)
101. C₂×C₄	C₄×Q₈	semidirect
128. C₄	C₂×C₄×Q₈	semidirect
129. C₂	Q₈×C₄⋊C₄	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×C₄×Q₈	2
Q₈×C₄⋊C₄	2
C₂×#389	2
C₂×#384	2
C₂×C₄×C₄⋊C₄	2

Subgroups — C2×Q8×C4⋊C4 #34112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₄⋊C₄ #34112