Subgroups — C₂×#60191 #10574216

Order 512.

6602 conjugacy classes of subgroups.

The subgroup lattice has 183 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3458	C₂	1	1	yes	maximal, characteristic
2	#60191	C₂	1	32	yes	maximal, direct product factor
3	C₂×#2961	C₂	1	12	yes	maximal, complemented
4	C₂×#2960	C₂	1	12	yes	maximal, complemented
5	C₂×#2905	C₂	1	3	yes	maximal
6	C₂²×#405	C₂	1	3	yes	maximal
7	C₂³×M₁₆	C₂²	1	3	yes	—
8	C₂⁴×C₈	C₂²	1	1	yes	characteristic
9	#3458	C₄	1	16	yes	—
10	#3458	C₂²	1	16	yes	—
11	C₂×#544	C₄	1	12	yes	—
12	C₂×#543	C₄	1	4	yes	—
13	C₂×#533	C₂²	1	12	yes	—
14	C₂²×#117	C₂²	1	3	yes	—
15	#2961	C₂²	1	192	yes	complemented
16	#2960	C₂²	1	192	yes	complemented
17	C₂×#446	C₂²	1	48	yes	complemented
18	C₂×C₈×Q₈	C₂²	1	16	yes	complemented
19	#2905	C₂²	1	48	yes	—
20	C₂×#424	C₂²	1	24	yes	—
21	C₂×#423	C₂²	1	12	yes	—
22	C₂×#405	C₂²	1	12	yes	—
23	C₂×#405	C₂²	1	12	yes	—
24	C₂×#405	C₂²	1	48	yes	—
25	C₂²×#97	C₂²	1	6	yes	—
26	C₂²×C₄×C₈	C₂²	1	6	yes	—
27	C₂⁴×C₄	C₂³	1	1	yes	characteristic
28	C₂²×M₁₆	—	2	24	no	—
29	C₂²×M₁₆	C₂³	1	12	yes	—
30	C₂²×M₁₆	C₂³	1	24	yes	—
31	C₂³×C₈	Q₈	1	16	yes	complemented
32	C₂³×C₈	C₂³	1	6	yes	—
33	C₂³×C₈	C₂³	1	3	yes	—
34	C₂³×C₈	C₂³	1	3	yes	—
35	C₂³×C₈	C₂³	1	8	yes	—
36	#544	C₂×C₄	1	192	yes	—
37	#543	C₂×C₄	1	64	yes	—
38	#533	C₂×C₄	1	96	yes	—
39	#533	C₂³	1	96	yes	—
40	C₂×C₄⋊Q₈	C₂×C₄	1	24	yes	—
41	C₂×#125	C₂×C₄	1	24	yes	—
42	C₂×C₂²⋊Q₈	C₂×C₄	1	48	yes	—
43	C₂×C₄×Q₈	C₂³	1	16	yes	—
44	C₂×#117	C₂×C₄	1	12	yes	—
45	C₂×#117	C₂×C₄	1	24	yes	—
46	C₂×#117	C₂³	1	12	yes	—
47	C₂×#117	C₂³	1	24	yes	—
48	C₂²×C₄⋊C₄	C₂×C₄	1	6	yes	—
49	C₂²×C₂²⋊C₄	C₂×C₄	1	6	yes	—
50	C₂²×C₄²	C₂³	1	6	yes	—
51	#446	C₂³	1	384	yes	complemented
52	C₈×Q₈	C₂³	1	128	yes	complemented
53	#424	C₂³	1	192	yes	—
54	#423	C₂³	1	96	yes	—
55	C₂×C₄⋊C₈	C₂³	1	48	yes	—
56	#405	C₂³	1	96	yes	—
57	#405	C₂³	1	96	yes	—
58	C₂×#97	C₂³	1	24	yes	—
59	C₂×#97	C₂³	1	48	yes	—
60	C₂×C₄×C₈	C₂³	1	12	yes	—
61	C₂×C₄×C₈	C₂³	1	48	yes	—
62	C₂×C₄×C₈	C₂³	1	12	yes	—
63	C₂⁵	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Q₈	C₂²×C₄	1	16	yes	—
65	C₂³×C₄	C₂²×C₄	1	12	yes	—
66	C₂³×C₄	C₂²×C₄	1	3	yes	—
67	C₂³×C₄	C₂²×C₄	1	4	yes	—
68	C₂³×C₄	C₂×Q₈	1	8	yes	—
69	C₂³×C₄	#41	1	8	yes	—
70	C₂³×C₄	C₂⁴	1	4	yes	—
71	C₂³×C₄	C₂⁴	1	3	yes	—
72	C₂×M₁₆	—	2	48	no	—
73	C₂×M₁₆	—	2	96	no	—
74	C₂×M₁₆	C₂⁴	1	48	yes	—
75	C₂²×C₈	—	2	24	no	—
76	C₂²×C₈	C₂×Q₈	1	24	yes	—
77	C₂²×C₈	C₂×Q₈	1	24	yes	—
78	C₂²×C₈	C₂×Q₈	1	64	yes	complemented
79	C₂²×C₈	C₂⁴	1	12	yes	—
80	C₂²×C₈	C₂⁴	1	24	yes	—
81	C₂²×C₈	C₂⁴	1	3	yes	—
82	C₂²×C₈	C₂⁴	1	12	yes	—
83	C₂²×C₈	C₂⁴	1	12	yes	—
84	C₂²×C₈	C₂⁴	1	1	yes	characteristic
85	C₄⋊Q₈	C₂²×C₄	1	192	yes	—
86	#125	C₂²×C₄	1	192	yes	—
87	C₂²⋊Q₈	C₂²×C₄	1	384	yes	—
88	C₄×Q₈	C₂²×C₄	1	64	yes	—
89	C₄×Q₈	C₂⁴	1	64	yes	—
90	#117	C₂²×C₄	1	48	yes	—
91	#117	C₂²×C₄	1	96	yes	—
92	#117	C₂⁴	1	48	yes	—
93	C₂×C₄⋊C₄	C₂²×C₄	1	24	yes	—
94	C₂×C₄⋊C₄	C₂²×C₄	1	48	yes	—
95	C₂×C₄⋊C₄	C₂²×C₄	1	48	yes	—
96	C₂×C₂²⋊C₄	C₂²×C₄	1	24	yes	—
97	C₂×C₂²⋊C₄	C₂²×C₄	1	48	yes	—
98	C₂×C₄²	C₂²×C₄	1	12	yes	—
99	C₂×C₄²	C₂²×C₄	1	24	yes	—
100	C₂×C₄²	C₂⁴	1	12	yes	—
101	C₂×C₄²	C₂⁴	1	24	yes	—
102	C₄⋊C₈	C₂⁴	1	192	yes	—
103	#97	C₂⁴	1	96	yes	—
104	C₄×C₈	C₂⁴	1	48	yes	—
105	C₄×C₈	C₂⁴	1	48	yes	—
106	C₂⁴	—	4	4	no	—
107	C₂⁴	C₄×Q₈	1	8	yes	—
108	C₂⁴	C₂³×C₄	1	3	yes	—
109	C₂⁴	C₂³×C₄	1	4	yes	—
110	C₂×Q₈	—	4	32	no	—
111	C₂×Q₈	C₂³×C₄	1	64	yes	—
112	C₂²×C₄	—	4	24	no	—
113	C₂²×C₄	C₄×Q₈	1	24	yes	—
114	C₂²×C₄	C₄×Q₈	1	32	yes	—
115	C₂²×C₄	C₂³×C₄	1	24	yes	—
116	C₂²×C₄	C₂³×C₄	1	48	yes	—
117	C₂²×C₄	C₂³×C₄	1	3	yes	—
118	C₂²×C₄	C₂³×C₄	1	6	yes	—
119	C₂²×C₄	C₂³×C₄	1	12	yes	—
120	C₂²×C₄	C₂²×Q₈	1	12	yes	—
121	C₂²×C₄	C₂²×Q₈	1	16	yes	—
122	C₂²×C₄	C₂×#41	1	12	yes	—
123	C₂²×C₄	C₂×#41	1	16	yes	—
124	C₂²×C₄	C₂⁵	1	1	yes	characteristic
125	C₂²×C₄	C₂⁵	1	6	yes	—
126	M₁₆	—	2	96	no	—
127	C₂×C₈	—	2	24	no	—
128	C₂×C₈	—	2	48	no	—
129	C₂×C₈	C₂²×Q₈	1	12	yes	—
130	C₂×C₈	C₂²×Q₈	1	4	yes	—
131	C₂×C₈	C₂²×Q₈	1	48	yes	—
132	C₂×C₈	C₂²×Q₈	1	48	yes	—
133	C₂×C₈	C₂⁵	1	24	yes	—
134	C₂×C₈	C₂⁵	1	2	yes	—
135	C₂×C₈	C₂⁵	1	6	yes	—
136	C₄⋊C₄	C₂³×C₄	1	96	yes	—
137	C₄⋊C₄	C₂³×C₄	1	192	yes	—
138	C₂²⋊C₄	C₂³×C₄	1	96	yes	—
139	C₄²	C₂³×C₄	1	24	yes	—
140	C₄²	C₂³×C₄	1	48	yes	—
141	C₄²	C₂⁵	1	24	yes	—
142	C₂³	—	2	12	no	—
143	C₂³	—	4	16	no	—
144	C₂³	—	4	8	no	—
145	C₂³	C₂×C₄×Q₈	1	12	yes	—
146	C₂³	C₂×C₄×Q₈	1	16	yes	—
147	C₂³	C₂⁴×C₄	1	6	yes	—
148	C₂³	C₂⁴×C₄	1	1	yes	characteristic
149	Q₈	—	4	64	no	—
150	C₂×C₄	—	2	48	no	—
151	C₂×C₄	—	4	48	no	—
152	C₂×C₄	C₂×C₄×Q₈	1	24	yes	—
153	C₂×C₄	C₂×C₄×Q₈	1	48	yes	—
154	C₂×C₄	C₂×C₄×Q₈	1	12	yes	—
155	C₂×C₄	C₂⁴×C₄	1	48	yes	—
156	C₂×C₄	C₂⁴×C₄	1	1	yes	characteristic
157	C₂×C₄	C₂⁴×C₄	1	6	yes	—
158	C₂×C₄	C₂³×Q₈	1	12	yes	—
159	C₂×C₄	C₂³×Q₈	1	2	yes	—
160	C₂×C₄	C₂²×#41	1	2	yes	—
161	C₂×C₄	C₂²×#41	1	12	yes	—
162	C₂×C₄	C₂⁶	1	1	yes	characteristic
163	C₈	—	2	24	no	—
164	C₈	C₂³×Q₈	1	12	yes	—
165	C₈	C₂³×Q₈	1	4	yes	—
166	C₂²	—	2	24	no	—
167	C₂²	—	2	12	no	—
168	C₂²	—	4	16	no	—
169	C₂²	C₂²×C₄×Q₈	1	12	yes	—
170	C₂²	C₂²×C₄×Q₈	1	2	yes	—
171	C₂²	#3573	1	4	yes	—
172	C₂²	C₂⁵×C₄	1	1	yes	characteristic
173	C₄	—	2	48	no	—
174	C₄	C₂²×C₄×Q₈	1	2	yes	—
175	C₄	C₂²×C₄×Q₈	1	12	yes	—
176	C₄	C₂⁴×Q₈	1	1	yes	characteristic
177	C₄	C₂³×#41	1	1	yes	characteristic
178	C₂	—	2	12	no	—
179	C₂	#60191	1	4	yes	direct product factor
180	C₂	C₂³×C₄×Q₈	1	1	yes	characteristic
181	C₂	C₂×#3573	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂²	C₂⁵×C₄
Frattini, cosocle	C₂×C₄	C₂⁶
socle	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #60191	C₂	direct (N × H)
3. C₂×#2961	C₂	semidirect
4. C₂×#2960	C₂	semidirect
15. #2961	C₂²	semidirect
16. #2960	C₂²	semidirect
17. C₂×#446	C₂²	semidirect
18. C₂×C₈×Q₈	C₂²	semidirect
31. C₂³×C₈	Q₈	semidirect
51. #446	C₂³	semidirect
52. C₈×Q₈	C₂³	semidirect
78. C₂²×C₈	C₂×Q₈	semidirect
179. C₂	#60191	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#3458	2
#60191	2
C₂×#2961	2
C₂×#2960	2
C₂×#2905	2
C₂²×#405	2

Subgroups — C2×#60191 #10574216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#60191 #10574216