Subgroups — Q₈×#347 #493262

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 169 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	Q₈×#97	C₂	1	1	yes	maximal, characteristic
3	#12666	C₂	1	6	yes	maximal
4	#12658	C₂	1	3	yes	maximal
5	C₄×#347	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	C₄	1	12	yes	complemented
10	Q₈×C₁₆	C₄	1	4	yes	complemented
11	#2150	C₂²	1	6	yes	—
12	C₂×#347	C₂²	1	3	yes	—
13	C₂×#347	C₂²	1	6	yes	—
14	C₂×#347	C₂²	1	1	yes	characteristic
15	C₄×M₃₂	C₂²	1	6	yes	—
16	#1770	C₄	1	3	yes	—
17	#1770	C₂²	1	3	yes	—
18	#1767	C₄	1	3	yes	—
19	C₄×#97	C₂²	1	3	yes	—
20	#1375	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	—	2	3	no	—
26	C₈×Q₈	—	2	1	no	—
27	C₈×Q₈	C₂×C₄	1	2	yes	—
28	C₈×Q₈	C₂×C₄	1	6	yes	—
29	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
30	C₂×C₄⋊C₈	C₂×C₄	1	3	yes	—
31	C₂×#97	C₂×C₄	1	3	yes	—
32	C₂×#97	C₂×C₄	1	3	yes	—
33	C₂×#97	C₂³	1	3	yes	—
34	C₂×#97	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
36	C₂×C₄×C₈	C₂³	1	3	yes	—
37	C₄×C₄⋊C₄	—	2	3	no	—
38	C₄×C₄⋊C₄	C₂×C₄	1	3	yes	—
39	C₄³	C₂×C₄	1	3	yes	—
40	C₄⋊C₁₆	C₂×C₄	1	12	yes	—
41	C₄⋊C₁₆	C₂×C₄	1	12	yes	—
42	#347	—	2	12	no	—
43	#347	—	2	6	no	—
44	#347	Q₈	1	4	yes	direct product factor
45	#346	C₂×C₄	1	12	yes	—
46	C₄×C₁₆	C₂×C₄	1	12	yes	—
47	#336	C₂×C₄	1	6	yes	—
48	C₂²×Q₈	M₁₆	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₄	1	3	yes	—
50	C₂²×C₈	C₂²×C₄	1	3	yes	—
51	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂⁴	1	1	yes	characteristic
53	C₄×Q₈	—	4	4	no	—
54	C₄×Q₈	C₄²	1	3	yes	—
55	C₄×Q₈	C₄²	1	1	yes	characteristic
56	C₄×Q₈	M₁₆	1	3	yes	—
57	C₄×Q₈	M₁₆	1	1	yes	characteristic
58	C₂×C₄⋊C₄	—	2	3	no	—
59	C₂×C₄⋊C₄	M₁₆	1	3	yes	—
60	C₂×C₄²	—	2	3	no	—
61	C₂×C₄²	—	2	3	no	—
62	C₂×C₄²	C₂²×C₄	1	3	yes	—
63	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄²	C₂²×C₄	1	3	yes	—
65	M₃₂	—	2	6	no	—
66	M₃₂	—	2	6	no	—
67	M₃₂	C₂×Q₈	1	4	yes	—
68	M₃₂	#41	1	4	yes	—
69	C₂×C₁₆	C₂²×C₄	1	12	yes	—
70	C₂×C₁₆	C₂²×C₄	1	4	yes	—
71	C₄⋊C₈	—	2	3	no	—
72	C₄⋊C₈	—	2	3	no	—
73	C₄⋊C₈	C₄²	1	6	yes	—
74	C₄⋊C₈	C₄²	1	6	yes	—
75	#97	—	2	6	no	—
76	#97	—	4	3	no	—
77	#97	—	4	3	no	—
78	#97	C₂×Q₈	1	2	yes	—
79	#97	#41	1	2	yes	—
80	C₄×C₈	—	2	3	no	—
81	C₄×C₈	—	2	3	no	—
82	C₄×C₈	C₂²×C₄	1	6	yes	—
83	C₄×C₈	C₂²×C₄	1	6	yes	—
84	#95	—	2	3	no	—
85	C₂×Q₈	—	2	4	no	—
86	C₂×Q₈	#97	1	3	yes	—
87	C₂×Q₈	#97	1	1	yes	characteristic
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	3	no	—
90	C₂²×C₄	C₂×M₁₆	1	3	yes	—
91	C₂²×C₄	C₂×M₁₆	1	3	yes	—
92	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	3	no	—
96	C₂×C₈	—	2	3	no	—
97	C₂×C₈	—	2	3	no	—
98	C₂×C₈	—	2	3	no	—
99	C₂×C₈	—	2	3	no	—
100	C₂×C₈	C₂×C₄²	1	6	yes	—
101	C₂×C₈	C₄×Q₈	1	2	yes	—
102	C₂×C₈	C₂³×C₄	1	2	yes	—
103	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₈	C₂×#41	1	2	yes	—
105	C₂×C₈	C₂×#41	1	1	yes	characteristic
106	C₄⋊C₄	—	4	6	no	—
107	C₄⋊C₄	#97	1	6	yes	—
108	C₄⋊C₄	#97	1	3	yes	—
109	C₄⋊C₄	#97	1	3	yes	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	3	no	—
112	C₄²	—	2	3	no	—
113	C₄²	—	4	6	no	—
114	C₄²	—	8	3	no	—
115	C₄²	C₂×C₄²	1	3	yes	—
116	C₄²	C₂×C₄²	1	3	yes	—
117	C₄²	C₄×Q₈	1	2	yes	—
118	C₄²	C₂×M₁₆	1	3	yes	—
119	C₄²	C₂×M₁₆	1	3	yes	—
120	C₁₆	—	2	12	no	—
121	C₁₆	C₄×Q₈	1	8	yes	complemented
122	C₂³	C₂²×M₁₆	1	1	yes	characteristic
123	Q₈	—	2	6	no	—
124	Q₈	#347	1	4	yes	direct product factor
125	C₂×C₄	—	2	6	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	3	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	3	no	—
131	C₂×C₄	—	4	3	no	—
132	C₂×C₄	—	4	3	no	—
133	C₂×C₄	C₂×#97	1	3	yes	—
134	C₂×C₄	C₂×#97	1	3	yes	—
135	C₂×C₄	C₂×#97	1	3	yes	—
136	C₂×C₄	C₂×#97	1	3	yes	—
137	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₂×C₄	C₄×#41	1	1	yes	characteristic
140	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
141	C₂×C₄	#568	1	1	yes	characteristic
142	C₂×C₄	#568	1	1	yes	characteristic
143	C₈	—	2	6	no	—
144	C₈	—	2	2	no	—
145	C₈	—	4	3	no	—
146	C₈	C₂×C₄×Q₈	1	2	yes	—
147	C₈	C₄×#41	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×#97	1	1	yes	characteristic
151	C₂²	Q₈×M₁₆	1	1	yes	characteristic
152	C₂²	#2971	1	1	yes	characteristic
153	C₄	—	2	3	no	—
154	C₄	—	2	3	no	—
155	C₄	—	2	3	no	—
156	C₄	—	4	2	no	—
157	C₄	—	8	3	no	—
158	C₄	C₂×#347	1	6	yes	—
159	C₄	C₄²×Q₈	1	1	yes	characteristic
160	C₄	#2875	1	1	yes	characteristic
161	C₄	Q₈×M₁₆	1	1	yes	characteristic
162	C₄	#2971	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	Q₈×#97	1	1	yes	characteristic
166	C₂	C₂²×#347	1	1	yes	characteristic
167	C₂	#33329	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂³	C₂²×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
9. #2183	C₄	semidirect
10. Q₈×C₁₆	C₄	semidirect
44. #347	Q₈	direct (N × H)
121. C₁₆	C₄×Q₈	semidirect
124. Q₈	#347	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
Q₈×#97	2
#12666	2
#12658	2
C₄×#347	2

Subgroups — Q8×#347 #493262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#347 #493262