Subgroups — Q₈×H(3,𝔽₄) #8091759

Order 512.

5913 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#46004	C₂	1	27	yes	maximal, complemented
2	#45494	C₂	1	18	yes	maximal, complemented
3	Q₈×#124	C₂	1	9	yes	maximal, complemented
4	Q₈×#120	C₂	1	6	yes	maximal, complemented
5	C₄×H(3,𝔽₄)	C₂	1	3	yes	maximal
6	C₂⁴×Q₈	C₂²	1	2	yes	complemented
7	C₂×H(3,𝔽₄)	C₂²	1	1	yes	characteristic
8	C₂×Q₈²	C₂²	1	3	yes	complemented
9	C₂×D₄×Q₈	C₂²	1	9	yes	—
10	#2862	C₂²	1	27	yes	complemented
11	#2848	C₂²	1	18	yes	complemented
12	#2844	C₂²	1	18	yes	complemented
13	#2840	C₂²	1	12	yes	complemented
14	#2811	C₂²	1	18	yes	complemented
15	#2804	C₂²	1	54	yes	—
16	#2763	C₂²	1	108	yes	complemented
17	#2568	C₂²	1	27	yes	complemented
18	#2563	C₂²	1	27	yes	—
19	#2492	C₂²	1	108	yes	complemented
20	#2434	C₂²	1	54	yes	complemented
21	#2409	C₂²	1	54	yes	—
22	#2379	C₂²	1	27	yes	—
23	#2375	C₂²	1	18	yes	—
24	Q₈×C₂²⋊C₄	C₂²	1	18	yes	complemented
25	C₄×#124	C₂²	1	27	yes	—
26	C₄×#120	C₂²	1	18	yes	—
27	C₄²×Q₈	C₂²	1	3	yes	complemented
28	C₂³×Q₈	—	4	24	no	—
29	C₂³×Q₈	C₂³	1	6	yes	—
30	C₂³×Q₈	C₂³	1	18	yes	—
31	C₂⁴×C₄	C₂³	1	6	yes	—
32	H(3,𝔽₄)	Q₈	1	16	yes	direct product factor
33	Q₈²	—	4	12	no	—
34	D₄×Q₈	—	4	36	no	—
35	C₂×C₄⋊Q₈	C₂³	1	27	yes	—
36	C₂×C₄⋊Q₈	C₂³	1	27	yes	—
37	C₂×#124	C₂³	1	9	yes	—
38	C₂×C₂²⋊Q₈	C₂³	1	54	yes	—
39	C₂×#120	C₂³	1	6	yes	—
40	C₂×C₄×Q₈	C₂³	1	9	yes	—
41	C₂×C₄×Q₈	C₂³	1	9	yes	—
42	C₂×C₄×D₄	C₂³	1	27	yes	—
43	#400	C₂³	1	108	yes	—
44	#395	C₂³	1	108	yes	—
45	#393	C₂³	1	108	yes	—
46	#393	C₂³	1	36	yes	—
47	#391	C₂³	1	27	yes	—
48	#391	C₂³	1	108	yes	—
49	#391	C₂³	1	108	yes	—
50	#390	C₂³	1	27	yes	—
51	#388	C₂³	1	108	yes	—
52	#386	C₂³	1	54	yes	—
53	#385	C₂³	1	108	yes	—
54	#385	C₂³	1	54	yes	—
55	#382	C₂³	1	54	yes	—
56	#382	C₂³	1	27	yes	—
57	#380	C₂³	1	54	yes	—
58	#379	C₂³	1	18	yes	—
59	C₄×C₄⋊C₄	C₂³	1	27	yes	—
60	C₄×C₂²⋊C₄	C₂³	1	54	yes	—
61	C₄³	C₂³	1	9	yes	—
62	C₂⁵	C₂⁴	1	2	yes	—
63	C₂²×Q₈	—	2	72	no	—
64	C₂²×Q₈	—	4	108	no	—
65	C₂²×Q₈	—	4	128	no	—
66	C₂²×Q₈	C₂⁴	1	18	yes	—
67	C₂²×Q₈	C₂⁴	1	18	yes	—
68	C₂²×Q₈	C₂⁴	1	12	yes	—
69	C₂²×Q₈	C₂⁴	1	3	yes	—
70	C₂²×Q₈	C₂⁴	1	1	yes	characteristic
71	C₂²×D₄	C₂⁴	1	9	yes	—
72	C₂³×C₄	—	4	36	no	—
73	C₂³×C₄	C₂⁴	1	18	yes	—
74	C₂³×C₄	C₂⁴	1	18	yes	—
75	C₄⋊Q₈	—	4	54	no	—
76	C₄⋊Q₈	—	4	54	no	—
77	#124	C₂×Q₈	1	72	yes	complemented
78	C₂²⋊Q₈	—	4	108	no	—
79	#120	C₂×Q₈	1	48	yes	complemented
80	C₄×Q₈	—	2	36	no	—
81	C₄×Q₈	—	4	18	no	—
82	C₄×D₄	—	4	54	no	—
83	C₂×C₄⋊C₄	C₂⁴	1	27	yes	—
84	C₂×C₄⋊C₄	C₂⁴	1	27	yes	—
85	C₂×C₄⋊C₄	C₂⁴	1	27	yes	—
86	C₂×C₄⋊C₄	C₂⁴	1	54	yes	—
87	C₂×C₂²⋊C₄	C₂⁴	1	54	yes	—
88	C₂×C₂²⋊C₄	C₂⁴	1	54	yes	—
89	C₂×C₂²⋊C₄	C₂⁴	1	18	yes	—
90	C₂×C₄²	C₂⁴	1	27	yes	—
91	C₂×C₄²	C₂⁴	1	27	yes	—
92	C₂×C₄²	C₂⁴	1	3	yes	—
93	#95	C₂⁴	1	108	yes	—
94	#95	C₂⁴	1	18	yes	—
95	#95	C₂⁴	1	108	yes	—
96	C₂⁴	—	4	6	no	—
97	C₂⁴	—	4	6	no	—
98	C₂⁴	C₂²×Q₈	1	8	yes	complemented
99	C₂⁴	C₂⁵	1	6	yes	—
100	C₂×Q₈	—	2	108	no	—
101	C₂×Q₈	—	4	3	no	—
102	C₂×Q₈	—	4	36	no	—
103	C₂×Q₈	—	4	3	no	—
104	C₂×Q₈	—	4	18	no	—
105	C₂×Q₈	—	4	36	no	—
106	C₂×Q₈	—	4	288	no	—
107	C₂×Q₈	—	4	96	no	—
108	C₂×Q₈	C₂²×Q₈	1	12	yes	complemented
109	C₂×Q₈	H(5,𝔽₂)	1	12	yes	—
110	C₂×D₄	—	4	9	no	—
111	C₂×D₄	—	4	9	no	—
112	C₂×D₄	C₂²×Q₈	1	36	yes	—
113	C₂²×C₄	—	2	108	no	—
114	C₂²×C₄	—	4	54	no	—
115	C₂²×C₄	—	4	96	no	—
116	C₂²×C₄	C₂⁵	1	27	yes	—
117	C₂²×C₄	C₂⁵	1	18	yes	—
118	C₂²×C₄	C₂⁵	1	9	yes	—
119	C₂²×C₄	C₂⁵	1	3	yes	—
120	C₄⋊C₄	—	2	54	no	—
121	C₄⋊C₄	—	4	27	no	—
122	C₄⋊C₄	—	4	27	no	—
123	C₄⋊C₄	—	4	54	no	—
124	C₂²⋊C₄	—	4	54	no	—
125	C₂²⋊C₄	—	4	54	no	—
126	C₂²⋊C₄	C₂²×Q₈	1	72	yes	complemented
127	C₄²	—	2	54	no	—
128	C₄²	—	4	27	no	—
129	C₄²	C₂²×Q₈	1	12	yes	complemented
130	C₂³	—	2	18	no	—
131	C₂³	—	2	18	no	—
132	C₂³	—	4	24	no	—
133	C₂³	—	4	8	no	—
134	C₂³	—	4	24	no	—
135	C₂³	C₂³×Q₈	1	12	yes	—
136	C₂³	C₂⁶	1	1	yes	characteristic
137	Q₈	—	4	12	no	—
138	Q₈	—	4	48	no	—
139	Q₈	—	4	72	no	—
140	Q₈	—	16	18	no	—
141	Q₈	H(3,𝔽₄)	1	16	yes	direct product factor
142	D₄	—	4	36	no	—
143	C₂×C₄	—	2	9	no	—
144	C₂×C₄	—	2	9	no	—
145	C₂×C₄	—	2	27	no	—
146	C₂×C₄	—	2	54	no	—
147	C₂×C₄	—	2	54	no	—
148	C₂×C₄	—	4	72	no	—
149	C₂×C₄	—	4	72	no	—
150	C₂×C₄	C₂³×Q₈	1	18	yes	—
151	C₂×C₄	C₂×H(5,𝔽₂)	1	18	yes	—
152	C₂²	—	2	36	no	—
153	C₂²	—	4	32	no	—
154	C₂²	—	4	18	no	—
155	C₂²	—	4	6	no	—
156	C₂²	C₂⁴×Q₈	1	1	yes	characteristic
157	C₂²	C₂²×H(5,𝔽₂)	1	3	yes	—
158	C₂²	2₋¹⁺⁶	1	3	yes	—
159	C₄	—	2	18	no	—
160	C₄	—	4	27	no	—
161	C₄	—	4	18	no	—
162	C₄	C₂×H(3,𝔽₄)	1	12	yes	—
163	C₂	—	4	12	no	—
164	C₂	C₂²×H(3,𝔽₄)	1	1	yes	characteristic
165	C₂	Q₈×H(5,𝔽₂)	1	3	yes	—
166	C₂	#63048	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #46004	C₂	semidirect
2. #45494	C₂	semidirect
3. Q₈×#124	C₂	semidirect
4. Q₈×#120	C₂	semidirect
6. C₂⁴×Q₈	C₂²	semidirect
8. C₂×Q₈²	C₂²	semidirect
10. #2862	C₂²	semidirect
11. #2848	C₂²	semidirect
12. #2844	C₂²	semidirect
13. #2840	C₂²	semidirect
14. #2811	C₂²	semidirect
16. #2763	C₂²	semidirect
17. #2568	C₂²	semidirect
19. #2492	C₂²	semidirect
20. #2434	C₂²	semidirect
24. Q₈×C₂²⋊C₄	C₂²	semidirect
27. C₄²×Q₈	C₂²	semidirect
32. H(3,𝔽₄)	Q₈	direct (N × H)
77. #124	C₂×Q₈	semidirect
79. #120	C₂×Q₈	semidirect
98. C₂⁴	C₂²×Q₈	semidirect
108. C₂×Q₈	C₂²×Q₈	semidirect
126. C₂²⋊C₄	C₂²×Q₈	semidirect
129. C₄²	C₂²×Q₈	semidirect
141. Q₈	H(3,𝔽₄)	direct (N × H)

Maximal subgroups

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

subgroup	index
#46004	2
#45494	2
Q₈×#124	2
Q₈×#120	2
C₄×H(3,𝔽₄)	2

Subgroups — Q8×H(3,𝔽4) #8091759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×H(3,𝔽₄) #8091759