Subgroups — C₂×#8822 #191458

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1848	C₂	1	2	yes	maximal
2	C₂×#1566	C₂	1	4	yes	maximal
3	C₂²×C₈²	C₂	1	1	yes	maximal, characteristic
4	#8822	C₂	1	8	yes	maximal, direct product factor
5	C₂²×#108	—	2	4	no	—
6	C₂×#423	C₂²	1	4	yes	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₈	C₂²	1	2	yes	—
9	#1848	—	2	4	no	—
10	#1848	C₂²	1	8	yes	—
11	#1566	—	2	8	no	—
12	#1566	C₂²	1	16	yes	—
13	C₂×#334	—	2	8	no	—
14	C₂×C₈²	C₄	1	1	yes	characteristic
15	C₂×C₈²	C₄	1	4	yes	—
16	C₂×C₈²	C₄	1	2	yes	—
17	C₂×C₈²	C₄	1	1	yes	characteristic
18	C₂×C₈²	C₂²	1	4	yes	—
19	C₂²×M₁₆	—	2	4	no	—
20	C₂³×C₈	D₄	1	2	yes	—
21	C₂³×C₈	Q₈	1	4	yes	—
22	C₂²×C₄²	C₂³	1	1	yes	characteristic
23	C₂×#108	—	2	16	no	—
24	C₂×#108	—	4	16	no	—
25	#423	—	2	8	no	—
26	#423	—	2	4	no	—
27	#423	C₂³	1	8	yes	—
28	C₂×C₄⋊C₈	—	2	8	no	—
29	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₈	D₄	1	2	yes	—
36	C₂×C₄×C₈	D₄	1	4	yes	—
37	C₂×C₄×C₈	D₄	1	4	yes	—
38	C₂×C₄×C₈	Q₈	1	4	yes	—
39	C₂×C₄×C₈	Q₈	1	2	yes	—
40	C₂×C₄×C₈	Q₈	1	2	yes	—
41	C₂×C₄×C₈	C₂³	1	4	yes	—
42	C₂×C₄×C₈	C₂³	1	2	yes	—
43	#334	—	2	32	no	—
44	C₈²	C₂×C₄	1	4	yes	—
45	C₈²	C₂×C₄	1	8	yes	—
46	C₈²	C₂×C₄	1	4	yes	—
47	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂³×C₄	C₂×Q₈	1	2	yes	—
49	C₂×M₁₆	—	2	8	no	—
50	C₂×M₁₆	—	4	4	no	—
51	C₂×M₁₆	—	4	16	no	—
52	C₂²×C₈	—	2	16	no	—
53	C₂²×C₈	—	4	4	no	—
54	C₂²×C₈	C₄⋊C₄	1	4	yes	—
55	C₂²×C₈	C₄⋊C₄	1	8	yes	—
56	C₂²×C₈	C₄⋊C₄	1	2	yes	—
57	C₂²×C₈	C₄⋊C₄	1	2	yes	—
58	C₂²×C₈	C₄⋊C₄	1	4	yes	—
59	C₂²×C₈	C₄⋊C₄	1	4	yes	—
60	C₂²×C₈	SD₁₆	1	16	yes	—
61	C₂²×C₈	C₂×D₄	1	4	yes	—
62	C₂²×C₈	C₂×Q₈	1	8	yes	—
63	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₄	1	2	yes	—
67	C₂×C₄²	C₂×D₄	1	2	yes	—
68	C₂×C₄²	C₂×Q₈	1	2	yes	—
69	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄²	C₂⁴	1	1	yes	characteristic
72	#108	—	4	32	no	—
73	C₄⋊C₈	—	2	16	no	—
74	C₄⋊C₈	—	4	8	no	—
75	C₄×C₈	C₄⋊C₄	1	4	yes	—
76	C₄×C₈	C₄⋊C₄	1	8	yes	—
77	C₄×C₈	C₄⋊C₄	1	4	yes	—
78	C₄×C₈	C₄⋊C₄	1	2	yes	—
79	C₄×C₈	C₄⋊C₄	1	2	yes	—
80	C₄×C₈	C₄⋊C₄	1	4	yes	—
81	C₄×C₈	C₂²×C₄	1	4	yes	—
82	C₄×C₈	C₂²×C₄	1	2	yes	—
83	C₄×C₈	C₂²×C₄	1	2	yes	—
84	C₄×C₈	C₂²×C₄	1	4	yes	—
85	C₄×C₈	C₂×D₄	1	8	yes	—
86	C₄×C₈	C₂×Q₈	1	4	yes	—
87	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
90	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
93	C₂²×C₄	#127	1	1	yes	characteristic
94	C₂²×C₄	#127	1	1	yes	characteristic
95	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
96	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
97	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
98	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
99	C₂²×C₄	C₂×SD₁₆	1	8	yes	—
100	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₄	C₂²×Q₈	1	2	yes	—
102	M₁₆	—	4	16	no	—
103	M₁₆	—	8	8	no	—
104	C₂×C₈	—	2	8	no	—
105	C₂×C₈	—	2	16	no	—
106	C₂×C₈	—	2	16	no	—
107	C₂×C₈	—	4	8	no	—
108	C₂×C₈	—	8	8	no	—
109	C₂×C₈	#106	1	16	yes	—
110	C₂×C₈	#106	1	8	yes	—
111	C₂×C₈	#106	1	8	yes	—
112	C₂×C₈	#108	1	16	yes	complemented
113	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
114	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
117	C₂×C₈	C₂×SD₁₆	1	16	yes	—
118	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₄²	C₂×C₄⋊C₄	1	2	yes	—
121	C₄²	C₂×C₄⋊C₄	1	2	yes	—
122	C₄²	#127	1	2	yes	—
123	C₄²	C₄⋊Q₈	1	2	yes	—
124	C₄²	C₄⋊Q₈	1	2	yes	—
125	C₄²	C₂³×C₄	1	1	yes	characteristic
126	C₄²	C₂²×D₄	1	2	yes	—
127	C₄²	C₂²×Q₈	1	1	yes	characteristic
128	C₂³	—	2	4	no	—
129	C₂³	#384	1	1	yes	characteristic
130	C₂³	#384	1	1	yes	characteristic
131	C₂³	#498	1	4	yes	—
132	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	#384	1	2	yes	—
139	C₂×C₄	#384	1	1	yes	characteristic
140	C₂×C₄	#384	1	1	yes	characteristic
141	C₂×C₄	#384	1	2	yes	—
142	C₂×C₄	C₂×#106	1	4	yes	—
143	C₂×C₄	C₂×#106	1	4	yes	—
144	C₂×C₄	C₂×#108	1	8	yes	—
145	C₂×C₄	#492	1	4	yes	—
146	C₂×C₄	#492	1	4	yes	—
147	C₂×C₄	#498	1	4	yes	—
148	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₂²×C₄⋊C₄	1	2	yes	—
150	C₂×C₄	C₂×#127	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄⋊Q₈	1	2	yes	—
152	C₂×C₄	C₂²×SD₁₆	1	4	yes	—
153	C₈	—	2	8	no	—
154	C₈	—	8	8	no	—
155	C₈	C₂×#106	1	8	yes	—
156	C₈	C₂×#106	1	8	yes	—
157	C₈	C₂×#108	1	16	yes	complemented
158	C₂²	—	2	2	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	8	no	—
161	C₂²	#1844	1	2	yes	—
162	C₂²	#1844	1	2	yes	—
163	C₂²	#1848	1	4	yes	—
164	C₂²	C₂×#384	1	1	yes	characteristic
165	C₂²	C₂×#498	1	2	yes	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	#1844	1	2	yes	—
169	C₄	#1844	1	2	yes	—
170	C₄	#1848	1	4	yes	—
171	C₄	C₂²×#106	1	2	yes	—
172	C₄	C₂²×#108	1	4	yes	—
173	C₄	C₂×#492	1	2	yes	—
174	C₂	—	2	2	no	—
175	C₂	—	2	2	no	—
176	C₂	#8822	1	4	yes	direct product factor
177	C₂	C₂×#1844	1	1	yes	characteristic
178	C₂	C₂×#1848	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	#384
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #8822	C₂	direct (N × H)
112. C₂×C₈	#108	semidirect
157. C₈	C₂×#108	semidirect
176. C₂	#8822	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×#1848	2
C₂×#1566	2
C₂²×C₈²	2
#8822	2

Subgroups — C2×#8822 #191458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#8822 #191458