Subgroups — C₂×#8827 #191463

Order 512.

726 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1837	C₂	1	2	yes	maximal
2	C₂×#1567	C₂	1	2	yes	maximal
3	C₂×#1566	C₂	1	2	yes	maximal
4	C₂²×#322	C₂	1	1	yes	maximal, characteristic
5	#8827	C₂	1	8	yes	maximal, direct product factor
6	C₂×#423	C₂²	1	4	yes	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	#1837	—	2	4	no	—
11	#1837	C₂²	1	8	yes	—
12	C₂×#341	—	2	4	no	—
13	#1567	—	2	4	no	—
14	#1567	C₂²	1	8	yes	—
15	#1566	—	2	4	no	—
16	#1566	C₂²	1	8	yes	—
17	C₂×#335	—	2	4	no	—
18	C₂×#334	—	2	4	no	—
19	C₂×#322	C₄	1	4	yes	—
20	C₂×#322	C₄	1	4	yes	—
21	C₂×#322	C₂²	1	4	yes	—
22	C₂²×M₁₆	—	2	4	no	—
23	C₂³×C₈	D₄	1	2	yes	—
24	C₂³×C₈	Q₈	1	2	yes	—
25	C₂³×C₈	Q₈	1	2	yes	—
26	C₂²×C₄²	C₂³	1	1	yes	characteristic
27	#423	—	2	4	no	—
28	#423	—	2	4	no	—
29	#423	—	2	4	no	—
30	#423	C₂³	1	8	yes	—
31	C₂×C₄⋊C₈	—	2	4	no	—
32	C₂×C₄⋊C₈	—	2	4	no	—
33	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄×C₈	D₄	1	2	yes	—
40	C₂×C₄×C₈	D₄	1	2	yes	—
41	C₂×C₄×C₈	D₄	1	2	yes	—
42	C₂×C₄×C₈	D₄	1	2	yes	—
43	C₂×C₄×C₈	D₄	1	2	yes	—
44	C₂×C₄×C₈	Q₈	1	2	yes	—
45	C₂×C₄×C₈	Q₈	1	2	yes	—
46	C₂×C₄×C₈	Q₈	1	2	yes	—
47	C₂×C₄×C₈	Q₈	1	2	yes	—
48	C₂×C₄×C₈	C₂³	1	2	yes	—
49	C₂×C₄×C₈	C₂³	1	2	yes	—
50	C₂×C₄×C₈	C₂³	1	2	yes	—
51	#341	—	2	16	no	—
52	#335	—	2	16	no	—
53	#334	—	2	16	no	—
54	#322	C₂×C₄	1	16	yes	—
55	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
57	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
58	C₂×M₁₆	—	2	8	no	—
59	C₂×M₁₆	—	4	4	no	—
60	C₂×M₁₆	—	4	16	no	—
61	C₂²×C₈	—	2	8	no	—
62	C₂²×C₈	—	4	4	no	—
63	C₂²×C₈	—	4	4	no	—
64	C₂²×C₈	—	4	4	no	—
65	C₂²×C₈	C₄⋊C₄	1	4	yes	—
66	C₂²×C₈	C₄⋊C₄	1	4	yes	—
67	C₂²×C₈	C₄⋊C₄	1	4	yes	—
68	C₂²×C₈	C₄⋊C₄	1	4	yes	—
69	C₂²×C₈	C₄⋊C₄	1	4	yes	—
70	C₂²×C₈	C₄⋊C₄	1	4	yes	—
71	C₂²×C₈	C₂×D₄	1	4	yes	—
72	C₂²×C₈	C₂×Q₈	1	4	yes	—
73	C₂²×C₈	C₂×Q₈	1	4	yes	—
74	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄²	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₈	—	2	8	no	—
87	C₄⋊C₈	—	2	8	no	—
88	C₄⋊C₈	—	4	4	no	—
89	C₄⋊C₈	—	4	4	no	—
90	C₄×C₈	C₄⋊C₄	1	4	yes	—
91	C₄×C₈	C₄⋊C₄	1	4	yes	—
92	C₄×C₈	C₄⋊C₄	1	4	yes	—
93	C₄×C₈	C₄⋊C₄	1	4	yes	—
94	C₄×C₈	C₄⋊C₄	1	4	yes	—
95	C₄×C₈	C₄⋊C₄	1	4	yes	—
96	C₄×C₈	C₂²×C₄	1	4	yes	—
97	C₄×C₈	C₂²×C₄	1	4	yes	—
98	C₄×C₈	C₂²×C₄	1	4	yes	—
99	C₄×C₈	C₂×D₄	1	4	yes	—
100	C₄×C₈	C₂×D₄	1	4	yes	—
101	C₄×C₈	C₂×Q₈	1	4	yes	—
102	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₄	#127	1	1	yes	characteristic
112	C₂²×C₄	#127	1	1	yes	characteristic
113	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
115	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
118	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
119	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
120	C₂²×C₄	#135	1	8	yes	—
121	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
124	M₁₆	—	4	16	no	—
125	M₁₆	—	8	8	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	—	2	8	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	—	2	8	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	4	8	no	—
136	C₂×C₈	—	8	8	no	—
137	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
138	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
139	C₂×C₈	C₂×C₄⋊C₄	1	8	yes	—
140	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₄²	#127	1	1	yes	characteristic
147	C₄²	#127	1	1	yes	characteristic
148	C₄²	C₄⋊Q₈	1	1	yes	characteristic
149	C₄²	C₄⋊Q₈	1	1	yes	characteristic
150	C₄²	C₄⋊Q₈	1	1	yes	characteristic
151	C₄²	C₄⋊Q₈	1	1	yes	characteristic
152	C₄²	C₂³×C₄	1	1	yes	characteristic
153	C₄²	C₂²×D₄	1	1	yes	characteristic
154	C₄²	C₂²×D₄	1	1	yes	characteristic
155	C₄²	C₂²×Q₈	1	1	yes	characteristic
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	#384	1	1	yes	characteristic
159	C₂³	#384	1	1	yes	characteristic
160	C₂³	#499	1	4	yes	—
161	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	#384	1	1	yes	characteristic
173	C₂×C₄	#384	1	1	yes	characteristic
174	C₂×C₄	#384	1	1	yes	characteristic
175	C₂×C₄	#384	1	1	yes	characteristic
176	C₂×C₄	#384	1	1	yes	characteristic
177	C₂×C₄	#384	1	1	yes	characteristic
178	C₂×C₄	#427	1	4	yes	—
179	C₂×C₄	#427	1	4	yes	—
180	C₂×C₄	#430	1	4	yes	—
181	C₂×C₄	#430	1	4	yes	—
182	C₂×C₄	#495	1	4	yes	—
183	C₂×C₄	#495	1	4	yes	—
184	C₂×C₄	#499	1	4	yes	—
185	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₄	C₂×#127	1	1	yes	characteristic
189	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
190	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
191	C₂×C₄	C₂×#135	1	4	yes	—
192	C₈	—	2	8	no	—
193	C₈	—	2	8	no	—
194	C₈	—	2	8	no	—
195	C₈	—	8	8	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	8	no	—
201	C₂²	#1846	1	2	yes	—
202	C₂²	#1846	1	2	yes	—
203	C₂²	#1849	1	2	yes	—
204	C₂²	#1849	1	2	yes	—
205	C₂²	C₂×#384	1	1	yes	characteristic
206	C₂²	C₂×#499	1	2	yes	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	#1846	1	2	yes	—
212	C₄	#1846	1	2	yes	—
213	C₄	#1849	1	2	yes	—
214	C₄	#1849	1	2	yes	—
215	C₄	C₂×#427	1	2	yes	—
216	C₄	C₂×#430	1	2	yes	—
217	C₄	C₂×#430	1	2	yes	—
218	C₄	C₂×#495	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	—	2	2	no	—
221	C₂	#8827	1	4	yes	direct product factor
222	C₂	C₂×#1846	1	1	yes	characteristic
223	C₂	C₂×#1849	1	1	yes	characteristic
224	C₂	C₂×#1849	1	1	yes	characteristic

	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
5. #8827	C₂	direct (N × H)
221. C₂	#8827	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×#1837	2
C₂×#1567	2
C₂×#1566	2
C₂²×#322	2
#8827	2

Subgroups — C2×#8827 #191463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#8827 #191463