Subgroups — Q₈×#364 #497929

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 181 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₃₂	C₂	1	2	yes	maximal
2	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	#13163	C₂	1	6	yes	maximal
4	#12900	C₂	1	3	yes	maximal
5	C₄×#364	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
8	C₄²×Q₈	C₄	1	1	yes	characteristic
9	#2183	—	2	6	no	—
10	Q₈×C₁₆	—	2	2	no	—
11	C₂×#364	C₂²	1	3	yes	—
12	C₂×#364	C₂²	1	6	yes	—
13	C₂×#364	C₂²	1	1	yes	characteristic
14	#2150	C₂²	1	6	yes	—
15	C₄×M₃₂	C₂²	1	6	yes	—
16	C₈×C₄⋊C₄	C₄	1	3	yes	—
17	C₈×C₄⋊C₄	C₂²	1	3	yes	—
18	C₄×C₄⋊C₈	C₄	1	3	yes	—
19	C₄²×C₈	C₂²	1	3	yes	—
20	#1376	C₂²	1	6	yes	—
21	C₂×C₄×Q₈	—	2	1	no	—
22	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂×M₃₂	C₂³	1	6	yes	—
24	C₂×M₃₂	C₂³	1	2	yes	—
25	C₈×Q₈	C₈	1	6	yes	—
26	C₈×Q₈	C₈	1	2	yes	—
27	C₈×Q₈	D₄	1	1	yes	characteristic
28	C₈×Q₈	D₄	1	3	yes	—
29	C₈×Q₈	Q₈	1	3	yes	—
30	C₈×Q₈	Q₈	1	1	yes	characteristic
31	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
34	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
36	C₂×C₄×C₈	C₂³	1	3	yes	—
37	C₂×C₄×C₈	C₂³	1	3	yes	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	C₄×C₄⋊C₄	—	2	3	no	—
40	C₄×C₄⋊C₄	C₂×C₄	1	3	yes	—
41	C₄³	C₂×C₄	1	3	yes	—
42	#364	—	2	6	no	—
43	#364	—	4	6	no	—
44	#364	Q₈	1	4	yes	direct product factor
45	C₄⋊C₁₆	—	2	6	no	—
46	C₄⋊C₁₆	—	2	6	no	—
47	#346	—	2	6	no	—
48	C₄×C₁₆	—	2	6	no	—
49	#336	C₂×C₄	1	6	yes	—
50	C₂²×Q₈	M₁₆	1	1	yes	characteristic
51	C₂²×C₈	C₂²×C₄	1	3	yes	—
52	C₂²×C₈	C₂²×C₄	1	3	yes	—
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂⁴	1	1	yes	characteristic
55	C₄×Q₈	—	2	8	no	—
56	C₄×Q₈	C₄⋊C₄	1	3	yes	—
57	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
58	C₄×Q₈	C₂×C₈	1	3	yes	—
59	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	2	3	no	—
61	C₂×C₄⋊C₄	M₁₆	1	3	yes	—
62	C₂×C₄²	—	2	3	no	—
63	C₂×C₄²	—	2	3	no	—
64	C₂×C₄²	C₂²×C₄	1	3	yes	—
65	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄²	C₂²×C₄	1	3	yes	—
67	M₃₂	—	2	2	no	—
68	M₃₂	—	2	6	no	—
69	M₃₂	—	2	6	no	—
70	M₃₂	C₂×Q₈	1	4	yes	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	2	6	no	—
73	C₄⋊C₈	C₄⋊C₄	1	3	yes	—
74	C₄⋊C₈	C₄⋊C₄	1	6	yes	—
75	C₄⋊C₈	C₄⋊C₄	1	3	yes	—
76	C₄⋊C₈	C₂×C₈	1	6	yes	—
77	C₄⋊C₈	C₂×C₈	1	6	yes	—
78	C₄×C₈	—	2	6	no	—
79	C₄×C₈	—	2	3	no	—
80	C₄×C₈	—	2	3	no	—
81	C₄×C₈	—	4	3	no	—
82	C₄×C₈	C₂×C₈	1	6	yes	—
83	C₄×C₈	C₂×C₈	1	6	yes	—
84	C₄×C₈	C₂×D₄	1	3	yes	—
85	C₄×C₈	C₂×D₄	1	3	yes	—
86	C₄×C₈	C₂×Q₈	1	3	yes	—
87	C₄×C₈	C₂×Q₈	1	3	yes	—
88	C₄×C₈	C₂×Q₈	1	2	yes	—
89	C₄×C₈	#41	1	2	yes	—
90	#95	—	2	3	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	C₄⋊C₈	1	3	yes	—
93	C₂×Q₈	C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	3	no	—
96	C₂²×C₄	C₂×M₁₆	1	3	yes	—
97	C₂²×C₄	C₂×M₁₆	1	3	yes	—
98	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₈	—	2	3	no	—
100	C₂×C₈	—	2	3	no	—
101	C₂×C₈	—	2	3	no	—
102	C₂×C₈	—	2	3	no	—
103	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
104	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
105	C₂×C₈	C₄×Q₈	1	2	yes	—
106	C₂×C₈	C₂²×C₈	1	6	yes	—
107	C₂×C₈	C₂²×C₈	1	2	yes	—
108	C₂×C₈	#131	1	2	yes	—
109	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₈	C₂×#41	1	1	yes	characteristic
113	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄⋊C₄	—	2	12	no	—
116	C₄⋊C₄	C₄⋊C₈	1	6	yes	—
117	C₄⋊C₄	C₄⋊C₈	1	3	yes	—
118	C₄⋊C₄	C₄⋊C₈	1	3	yes	—
119	C₄²	—	2	12	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	3	no	—
122	C₄²	—	2	3	no	—
123	C₄²	—	4	6	no	—
124	C₄²	C₂×C₄⋊C₄	1	3	yes	—
125	C₄²	C₂×C₄⋊C₄	1	3	yes	—
126	C₄²	C₄×Q₈	1	2	yes	—
127	C₄²	C₂²×C₈	1	3	yes	—
128	C₄²	C₂²×C₈	1	3	yes	—
129	C₁₆	—	2	4	no	—
130	C₁₆	—	4	6	no	—
131	C₂³	C₂²×M₁₆	1	1	yes	characteristic
132	Q₈	—	2	6	no	—
133	Q₈	#364	1	4	yes	direct product factor
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	6	no	—
138	C₂×C₄	—	2	6	no	—
139	C₂×C₄	—	4	3	no	—
140	C₂×C₄	—	4	3	no	—
141	C₂×C₄	—	4	3	no	—
142	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
143	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
144	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
145	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
146	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₂×C₄	#520	1	1	yes	characteristic
149	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
150	C₂×C₄	C₂×#131	1	1	yes	characteristic
151	C₂×C₄	#568	1	1	yes	characteristic
152	C₈	—	2	3	no	—
153	C₈	—	2	3	no	—
154	C₈	—	2	6	no	—
155	C₈	C₈×Q₈	1	4	yes	—
156	C₈	D₄×Q₈	1	1	yes	characteristic
157	C₈	#550	1	1	yes	characteristic
158	C₈	#557	1	1	yes	characteristic
159	C₈	Q₈²	1	1	yes	characteristic
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
163	C₂²	Q₈×M₁₆	1	1	yes	characteristic
164	C₂²	#2991	1	1	yes	characteristic
165	C₄	—	2	4	no	—
166	C₄	—	2	3	no	—
167	C₄	—	2	3	no	—
168	C₄	—	2	3	no	—
169	C₄	—	4	6	no	—
170	C₄	C₂×#364	1	6	yes	—
171	C₄	Q₈×C₄⋊C₄	1	1	yes	characteristic
172	C₄	#2906	1	1	yes	characteristic
173	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
174	C₄	#2988	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	Q₈×C₄⋊C₈	1	1	yes	characteristic
178	C₂	C₂²×#364	1	1	yes	characteristic
179	C₂	#33401	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂³	C₂²×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
44. #364	Q₈	direct (N × H)
133. Q₈	#364	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×M₃₂	2
C₄×C₈×Q₈	2
#13163	2
#12900	2
C₄×#364	2

Subgroups — Q8×#364 #497929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#364 #497929