Subgroups — Q₈×#334 #226362

Order 512.

676 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	#9602	C₂	1	6	yes	maximal
4	#8805	C₂	1	3	yes	maximal
5	C₄×#334	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈	—	2	2	no	—
7	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₄²×Q₈	C₂²	1	1	yes	characteristic
10	#1939	—	2	6	no	—
11	#1842	C₂²	1	6	yes	—
12	C₈×C₄⋊C₄	C₄	1	3	yes	—
13	C₈×C₄⋊C₄	C₄	1	3	yes	—
14	C₄×C₄⋊C₈	C₂²	1	6	yes	—
15	C₄×C₄⋊C₈	C₂²	1	3	yes	—
16	C₄²×C₈	C₂²	1	3	yes	—
17	C₂×#334	C₂²	1	3	yes	—
18	C₂×#334	C₂²	1	6	yes	—
19	C₂×#334	C₂²	1	1	yes	characteristic
20	#1277	C₂²	1	6	yes	—
21	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
23	C₂×C₄×Q₈	Q₈	1	1	yes	characteristic
24	C₈×Q₈	—	2	4	no	—
25	C₈×Q₈	—	4	8	no	—
26	C₈×Q₈	C₈	1	8	yes	complemented
27	C₂×C₄⋊C₈	—	2	6	no	—
28	C₂×C₄⋊C₈	—	2	6	no	—
29	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
30	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊C₈	C₂³	1	6	yes	—
32	C₂×C₄⋊C₈	C₂³	1	2	yes	—
33	C₂×C₄×C₈	—	2	6	no	—
34	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
36	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
37	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
38	C₂×C₄×C₈	C₂³	1	3	yes	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₄⋊C₄	C₂×C₄	1	3	yes	—
41	C₄×C₄⋊C₄	D₄	1	3	yes	—
42	C₄×C₄⋊C₄	Q₈	1	3	yes	—
43	C₄³	C₂³	1	3	yes	—
44	#336	—	2	6	no	—
45	#336	C₂×C₄	1	6	yes	—
46	#334	—	2	6	no	—
47	#334	—	4	6	no	—
48	#334	Q₈	1	4	yes	direct product factor
49	C₂²×Q₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	—	2	2	no	—
51	C₂²×C₈	—	2	6	no	—
52	C₂²×C₈	C₂²×C₄	1	3	yes	—
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂²×C₄	1	3	yes	—
55	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₄×Q₈	—	2	4	no	—
57	C₄×Q₈	C₂×C₈	1	4	yes	—
58	C₄×Q₈	M₁₆	1	4	yes	—
59	C₄×Q₈	SD₁₆	1	8	yes	—
60	C₂×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
61	C₂×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
62	C₂×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
63	C₂×C₄²	C₂²×C₄	1	3	yes	—
64	C₂×C₄²	C₂²×C₄	1	3	yes	—
65	C₂×C₄²	C₂×D₄	1	3	yes	—
66	C₂×C₄²	C₂×D₄	1	3	yes	—
67	C₂×C₄²	C₂×Q₈	1	3	yes	—
68	C₂×C₄²	C₂×Q₈	1	3	yes	—
69	C₂×C₄²	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₈	—	2	6	no	—
71	C₄⋊C₈	—	2	6	no	—
72	C₄⋊C₈	—	2	6	no	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	4	12	no	—
75	C₄⋊C₈	—	8	6	no	—
76	C₄⋊C₈	C₂×C₈	1	12	yes	—
77	C₄⋊C₈	C₂×Q₈	1	4	yes	—
78	C₄×C₈	—	2	6	no	—
79	C₄×C₈	—	2	6	no	—
80	C₄×C₈	—	2	3	no	—
81	C₄×C₈	—	2	3	no	—
82	C₄×C₈	—	4	3	no	—
83	C₄×C₈	—	4	12	no	—
84	C₄×C₈	C₂×C₈	1	12	yes	—
85	C₄×C₈	C₂×Q₈	1	2	yes	—
86	C₄×C₈	#41	1	2	yes	—
87	#95	C₄⋊C₄	1	6	yes	—
88	C₂×Q₈	C₄⋊C₈	1	4	yes	—
89	C₂×Q₈	#106	1	4	yes	—
90	C₂×Q₈	#108	1	4	yes	—
91	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
92	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
93	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
94	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
95	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₈	—	2	3	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	3	no	—
102	C₂×C₈	—	2	3	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	3	no	—
105	C₂×C₈	—	2	3	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	4	6	no	—
109	C₂×C₈	—	4	6	no	—
110	C₂×C₈	—	4	6	no	—
111	C₂×C₈	C₄×Q₈	1	2	yes	—
112	C₂×C₈	C₄×Q₈	1	2	yes	—
113	C₂×C₈	C₂²×C₈	1	6	yes	—
114	C₂×C₈	C₂²×C₈	1	2	yes	—
115	C₂×C₈	#131	1	2	yes	—
116	C₄⋊C₄	—	2	6	no	—
117	C₄⋊C₄	C₄⋊C₈	1	12	yes	—
118	C₄⋊C₄	#106	1	12	yes	—
119	C₄²	—	2	3	no	—
120	C₄²	—	2	3	no	—
121	C₄²	—	2	6	no	—
122	C₄²	—	2	6	no	—
123	C₄²	—	4	3	no	—
124	C₄²	C₂²×C₈	1	6	yes	—
125	C₄²	C₂×M₁₆	1	6	yes	—
126	C₄²	C₂×SD₁₆	1	12	yes	—
127	C₄²	C₂²×Q₈	1	1	yes	characteristic
128	C₄²	C₂×#41	1	1	yes	characteristic
129	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
130	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂³	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	Q₈	#334	1	16	yes	direct product factor
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	3	no	—
140	C₂×C₄	—	2	3	no	—
141	C₂×C₄	—	2	3	no	—
142	C₂×C₄	C₂×C₄⋊C₈	1	6	yes	—
143	C₂×C₄	C₂×C₄⋊C₈	1	6	yes	—
144	C₂×C₄	C₂×#106	1	6	yes	—
145	C₂×C₄	C₂×#106	1	6	yes	—
146	C₂×C₄	C₂×#108	1	6	yes	—
147	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₂×C₄	#520	1	1	yes	characteristic
149	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
150	C₂×C₄	#550	1	1	yes	characteristic
151	C₂×C₄	#557	1	1	yes	characteristic
152	C₂×C₄	Q₈²	1	1	yes	characteristic
153	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
154	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
155	C₂×C₄	C₂×#131	1	1	yes	characteristic
156	C₂×C₄	#568	1	1	yes	characteristic
157	C₂×C₄	C₂²×SD₁₆	1	2	yes	—
158	C₂×C₄	#577	1	2	yes	—
159	C₈	—	2	2	no	—
160	C₈	—	2	6	no	—
161	C₈	—	4	4	no	—
162	C₈	—	4	3	no	—
163	C₈	—	8	6	no	—
164	C₈	C₈×Q₈	1	4	yes	complemented
165	C₂²	Q₈×C₄⋊C₄	1	1	yes	characteristic
166	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
167	C₂²	#2906	1	1	yes	characteristic
168	C₂²	C₂²×#106	1	1	yes	characteristic
169	C₂²	#2912	1	1	yes	characteristic
170	C₂²	C₂²×#108	1	1	yes	characteristic
171	C₂²	#2916	1	1	yes	characteristic
172	C₄	—	2	6	no	—
173	C₄	—	2	6	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	4	3	no	—
177	C₄	C₂×#334	1	12	yes	—
178	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
179	C₄	Q₈×M₁₆	1	1	yes	characteristic
180	C₄	#2988	1	1	yes	characteristic
181	C₄	#2991	1	1	yes	characteristic
182	C₄	Q₈×SD₁₆	1	2	yes	—
183	C₄	#3392	1	2	yes	—
184	C₂	C₂²×#334	1	1	yes	characteristic
185	C₂	#17788	1	1	yes	characteristic
186	C₂	#17826	1	1	yes	characteristic
187	C₂	#17835	1	1	yes	characteristic
188	C₂	Q₈×C₄⋊C₈	1	1	yes	characteristic
189	C₂	Q₈×#106	1	1	yes	characteristic
190	C₂	Q₈×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂²×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
26. C₈×Q₈	C₈	semidirect
48. #334	Q₈	direct (N × H)
132. Q₈	#334	direct (N × H)
164. C₈	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊C₈	2
C₄×C₈×Q₈	2
#9602	2
#8805	2
C₄×#334	2

Subgroups — Q8×#334 #226362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#334 #226362