Subgroups — C₂×#8824 #191459

Order 512.

766 conjugacy classes of subgroups.

The subgroup lattice has 253 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×C₈²	C₂	1	1	yes	maximal, characteristic
5	#8824	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×C₈²	C₄	1	1	yes	characteristic
20	C₂×C₈²	C₄	1	4	yes	—
21	C₂×C₈²	C₄	1	1	yes	characteristic
22	C₂×C₈²	C₄	1	1	yes	characteristic
23	C₂×C₈²	C₄	1	1	yes	characteristic
24	C₂×C₈²	C₂²	1	4	yes	—
25	C₂²×M₁₆	—	2	4	no	—
26	C₂³×C₈	D₄	1	2	yes	—
27	C₂³×C₈	Q₈	1	2	yes	—
28	C₂³×C₈	Q₈	1	2	yes	—
29	C₂²×C₄²	C₂³	1	1	yes	characteristic
30	#423	—	2	4	no	—
31	#423	—	2	4	no	—
32	#423	—	2	4	no	—
33	#423	C₂³	1	8	yes	—
34	C₂×C₄⋊C₈	—	2	4	no	—
35	C₂×C₄⋊C₈	—	2	4	no	—
36	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	2	yes	—
46	C₂×C₄×C₈	D₄	1	2	yes	—
47	C₂×C₄×C₈	D₄	1	2	yes	—
48	C₂×C₄×C₈	D₄	1	2	yes	—
49	C₂×C₄×C₈	D₄	1	2	yes	—
50	C₂×C₄×C₈	Q₈	1	2	yes	—
51	C₂×C₄×C₈	Q₈	1	2	yes	—
52	C₂×C₄×C₈	Q₈	1	2	yes	—
53	C₂×C₄×C₈	Q₈	1	2	yes	—
54	C₂×C₄×C₈	C₂³	1	2	yes	—
55	C₂×C₄×C₈	C₂³	1	2	yes	—
56	C₂×C₄×C₈	C₂³	1	2	yes	—
57	#341	—	2	16	no	—
58	#335	—	2	16	no	—
59	#334	—	2	16	no	—
60	C₈²	C₂×C₄	1	4	yes	—
61	C₈²	C₂×C₄	1	4	yes	—
62	C₈²	C₂×C₄	1	4	yes	—
63	C₈²	C₂×C₄	1	4	yes	—
64	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
67	C₂×M₁₆	—	2	8	no	—
68	C₂×M₁₆	—	4	4	no	—
69	C₂×M₁₆	—	4	16	no	—
70	C₂²×C₈	—	2	8	no	—
71	C₂²×C₈	—	2	8	no	—
72	C₂²×C₈	—	2	8	no	—
73	C₂²×C₈	—	4	4	no	—
74	C₂²×C₈	C₄⋊C₄	1	2	yes	—
75	C₂²×C₈	C₄⋊C₄	1	2	yes	—
76	C₂²×C₈	C₄⋊C₄	1	4	yes	—
77	C₂²×C₈	C₄⋊C₄	1	2	yes	—
78	C₂²×C₈	C₄⋊C₄	1	2	yes	—
79	C₂²×C₈	C₄⋊C₄	1	4	yes	—
80	C₂²×C₈	C₄⋊C₄	1	4	yes	—
81	C₂²×C₈	C₄⋊C₄	1	2	yes	—
82	C₂²×C₈	C₄⋊C₄	1	2	yes	—
83	C₂²×C₈	C₂×D₄	1	4	yes	—
84	C₂²×C₈	C₂×Q₈	1	4	yes	—
85	C₂²×C₈	C₂×Q₈	1	4	yes	—
86	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
97	C₂×C₄²	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₈	—	2	8	no	—
99	C₄⋊C₈	—	2	8	no	—
100	C₄⋊C₈	—	4	4	no	—
101	C₄⋊C₈	—	4	4	no	—
102	C₄×C₈	C₄⋊C₄	1	2	yes	—
103	C₄×C₈	C₄⋊C₄	1	2	yes	—
104	C₄×C₈	C₄⋊C₄	1	4	yes	—
105	C₄×C₈	C₄⋊C₄	1	4	yes	—
106	C₄×C₈	C₄⋊C₄	1	4	yes	—
107	C₄×C₈	C₄⋊C₄	1	2	yes	—
108	C₄×C₈	C₄⋊C₄	1	2	yes	—
109	C₄×C₈	C₄⋊C₄	1	2	yes	—
110	C₄×C₈	C₄⋊C₄	1	2	yes	—
111	C₄×C₈	C₂²×C₄	1	2	yes	—
112	C₄×C₈	C₂²×C₄	1	2	yes	—
113	C₄×C₈	C₂²×C₄	1	2	yes	—
114	C₄×C₈	C₂²×C₄	1	2	yes	—
115	C₄×C₈	C₂²×C₄	1	2	yes	—
116	C₄×C₈	C₂²×C₄	1	2	yes	—
117	C₄×C₈	C₂×D₄	1	4	yes	—
118	C₄×C₈	C₂×D₄	1	4	yes	—
119	C₄×C₈	C₂×Q₈	1	4	yes	—
120	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
121	C₂²×C₄	—	2	4	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₄	#127	1	1	yes	characteristic
130	C₂²×C₄	#127	1	1	yes	characteristic
131	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
133	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
134	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
136	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
137	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄	#135	1	8	yes	—
139	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
141	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
142	M₁₆	—	4	16	no	—
143	M₁₆	—	8	8	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	—	4	8	no	—
155	C₂×C₈	—	8	8	no	—
156	C₂×C₈	#108	1	8	yes	—
157	C₂×C₈	#108	1	8	yes	—
158	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
159	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
160	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
161	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
162	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
163	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
164	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₄²	#127	1	1	yes	characteristic
171	C₄²	#127	1	1	yes	characteristic
172	C₄²	C₄⋊Q₈	1	1	yes	characteristic
173	C₄²	C₄⋊Q₈	1	1	yes	characteristic
174	C₄²	C₄⋊Q₈	1	1	yes	characteristic
175	C₄²	C₄⋊Q₈	1	1	yes	characteristic
176	C₄²	C₂³×C₄	1	1	yes	characteristic
177	C₄²	C₂²×D₄	1	1	yes	characteristic
178	C₄²	C₂²×D₄	1	1	yes	characteristic
179	C₄²	C₂²×Q₈	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	#384	1	1	yes	characteristic
183	C₂³	#384	1	1	yes	characteristic
184	C₂³	#499	1	4	yes	—
185	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	#384	1	1	yes	characteristic
197	C₂×C₄	#384	1	1	yes	characteristic
198	C₂×C₄	#384	1	1	yes	characteristic
199	C₂×C₄	#384	1	1	yes	characteristic
200	C₂×C₄	#384	1	1	yes	characteristic
201	C₂×C₄	#384	1	1	yes	characteristic
202	C₂×C₄	#427	1	4	yes	—
203	C₂×C₄	#427	1	4	yes	—
204	C₂×C₄	C₂×#108	1	4	yes	—
205	C₂×C₄	C₂×#108	1	4	yes	—
206	C₂×C₄	#495	1	4	yes	—
207	C₂×C₄	#495	1	4	yes	—
208	C₂×C₄	#499	1	4	yes	—
209	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₄	C₂×#127	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
215	C₂×C₄	C₂×#135	1	4	yes	—
216	C₈	—	2	4	no	—
217	C₈	—	2	4	no	—
218	C₈	—	2	4	no	—
219	C₈	—	2	4	no	—
220	C₈	—	8	8	no	—
221	C₈	C₂×#108	1	8	yes	—
222	C₈	C₂×#108	1	8	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	8	no	—
228	C₂²	#1846	1	2	yes	—
229	C₂²	#1846	1	2	yes	—
230	C₂²	#1848	1	2	yes	—
231	C₂²	#1848	1	2	yes	—
232	C₂²	C₂×#384	1	1	yes	characteristic
233	C₂²	C₂×#499	1	2	yes	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	#1846	1	2	yes	—
239	C₄	#1846	1	2	yes	—
240	C₄	#1848	1	2	yes	—
241	C₄	#1848	1	2	yes	—
242	C₄	C₂×#427	1	2	yes	—
243	C₄	C₂²×#108	1	2	yes	—
244	C₄	C₂²×#108	1	2	yes	—
245	C₄	C₂×#495	1	2	yes	—
246	C₂	—	2	2	no	—
247	C₂	—	2	2	no	—
248	C₂	#8824	1	4	yes	direct product factor
249	C₂	C₂×#1846	1	1	yes	characteristic
250	C₂	C₂×#1848	1	1	yes	characteristic
251	C₂	C₂×#1848	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. #8824	C₂	direct (N × H)
248. C₂	#8824	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×C₈²	2
#8824	2

Subgroups — C2×#8824 #191459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#8824 #191459