Subgroups — C₂×#9633 #203992

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1574	C₂	1	2	yes	maximal
2	C₂×#1566	C₂	1	1	yes	maximal, characteristic
3	C₂×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
5	#9633	C₂	1	8	yes	maximal, direct product factor
6	C₂×#1299	C₂	1	2	yes	maximal
7	C₂×#431	C₂²	1	2	yes	—
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#423	C₂²	1	2	yes	—
10	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
12	#1574	—	2	4	no	—
13	#1574	C₂²	1	8	yes	—
14	#1566	—	2	2	no	—
15	#1566	C₂²	1	4	yes	—
16	#1564	C₄	1	4	yes	—
17	#1564	C₂²	1	4	yes	—
18	C₂×#335	—	2	2	no	—
19	C₂×#335	C₄	1	2	yes	—
20	C₂×#334	—	2	2	no	—
21	C₂×#334	—	2	1	no	—
22	C₂×#334	—	2	1	no	—
23	C₂×#334	C₄	1	2	yes	—
24	C₈×M₁₆	—	2	2	no	—
25	C₈×M₁₆	C₂²	1	4	yes	—
26	C₂×#322	—	2	1	no	—
27	C₂×C₈²	—	2	1	no	—
28	#1299	C₂²	1	8	yes	—
29	C₂²×M₁₆	—	2	1	no	—
30	C₂²×M₁₆	—	2	2	no	—
31	C₂³×C₈	Q₈	1	1	yes	characteristic
32	C₂³×C₈	Q₈	1	1	yes	characteristic
33	C₂²×C₄²	C₂³	1	1	yes	characteristic
34	#431	—	2	2	no	—
35	#431	—	2	1	no	—
36	#431	—	2	2	no	—
37	#431	—	2	1	no	—
38	#431	C₂×C₄	1	4	yes	—
39	#431	D₄	1	2	yes	—
40	#431	C₂³	1	4	yes	—
41	#431	C₂³	1	2	yes	—
42	#423	—	2	2	no	—
43	#423	—	2	2	no	—
44	#423	—	2	2	no	—
45	#423	C₂³	1	4	yes	—
46	C₂×C₄⋊C₈	—	2	2	no	—
47	C₂×C₄⋊C₈	—	2	2	no	—
48	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₈	D₄	1	2	yes	—
50	C₂×C₂²⋊C₈	—	2	2	no	—
51	C₂×C₂²⋊C₈	—	2	1	no	—
52	C₄×M₁₆	—	2	1	no	—
53	C₄×M₁₆	—	2	1	no	—
54	C₄×M₁₆	Q₈	1	2	yes	—
55	C₄×M₁₆	C₂³	1	2	yes	—
56	C₂×#97	Q₈	1	2	yes	—
57	C₂×C₄×C₈	—	2	2	no	—
58	C₂×C₄×C₈	—	2	1	no	—
59	C₂×C₄×C₈	—	2	1	no	—
60	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
63	C₂×C₄×C₈	D₄	1	1	yes	characteristic
64	C₂×C₄×C₈	D₄	1	1	yes	characteristic
65	C₂×C₄×C₈	D₄	1	2	yes	—
66	C₂×C₄×C₈	Q₈	1	2	yes	—
67	C₂×C₄×C₈	C₂³	1	2	yes	—
68	#335	—	2	8	no	—
69	#335	C₂×C₄	1	8	yes	—
70	#334	—	2	8	no	—
71	#334	—	2	4	no	—
72	#334	—	2	4	no	—
73	#334	C₂×C₄	1	8	yes	—
74	#322	—	2	4	no	—
75	C₈²	—	2	4	no	—
76	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
77	C₂³×C₄	#41	1	1	yes	characteristic
78	C₂³×C₄	#41	1	1	yes	characteristic
79	C₂×M₁₆	—	2	8	no	—
80	C₂×M₁₆	—	2	2	no	—
81	C₂×M₁₆	—	2	2	no	—
82	C₂×M₁₆	—	2	4	no	—
83	C₂×M₁₆	—	4	2	no	—
84	C₂×M₁₆	—	8	4	no	—
85	C₂²×C₈	—	2	2	no	—
86	C₂²×C₈	—	2	4	no	—
87	C₂²×C₈	—	2	4	no	—
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	—	4	2	no	—
90	C₂²×C₈	—	4	2	no	—
91	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₈	C₄⋊C₄	1	2	yes	—
95	C₂²×C₈	C₄⋊C₄	1	2	yes	—
96	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂×Q₈	1	2	yes	—
98	C₂²×C₈	C₂×Q₈	1	2	yes	—
99	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
106	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₄²	#41	1	1	yes	characteristic
109	C₂×C₄²	#41	1	1	yes	characteristic
110	C₂×C₄²	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	2	4	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄⋊C₈	—	4	2	no	—
117	C₄⋊C₈	C₂²×C₄	1	8	yes	—
118	C₄⋊C₈	C₂×D₄	1	4	yes	—
119	C₂²⋊C₈	—	2	4	no	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₂²⋊C₈	—	2	4	no	—
122	C₂²⋊C₈	—	2	2	no	—
123	#97	—	2	2	no	—
124	#97	C₂×Q₈	1	4	yes	—
125	C₄×C₈	—	2	4	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	C₄⋊C₄	1	2	yes	—
132	C₄×C₈	C₄⋊C₄	1	2	yes	—
133	C₄×C₈	C₄⋊C₄	1	2	yes	—
134	C₄×C₈	C₄⋊C₄	1	2	yes	—
135	C₄×C₈	C₂²×C₄	1	2	yes	—
136	C₄×C₈	C₂²×C₄	1	2	yes	—
137	C₄×C₈	C₂×D₄	1	2	yes	—
138	C₄×C₈	C₂×D₄	1	2	yes	—
139	C₂⁴	#125	1	1	yes	characteristic
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
149	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
150	C₂²×C₄	#121	1	1	yes	characteristic
151	C₂²×C₄	#121	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂²×C₄	#125	1	1	yes	characteristic
155	C₂²×C₄	#125	1	1	yes	characteristic
156	C₂²×C₄	#125	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
159	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
160	C₂²×C₄	C₂×#41	1	1	yes	characteristic
161	C₂²×C₄	C₂×#41	1	1	yes	characteristic
162	M₁₆	—	2	8	no	—
163	M₁₆	—	2	8	no	—
164	M₁₆	—	8	4	no	—
165	M₁₆	—	8	4	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	8	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	8	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	8	4	no	—
181	C₂×C₈	#108	1	4	yes	—
182	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
183	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
184	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
185	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
186	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₄²	C₄×D₄	1	1	yes	characteristic
189	C₄²	C₄×D₄	1	1	yes	characteristic
190	C₄²	C₄×Q₈	1	1	yes	characteristic
191	C₄²	C₄×Q₈	1	1	yes	characteristic
192	C₄²	#121	1	1	yes	characteristic
193	C₄²	#121	1	1	yes	characteristic
194	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
195	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
196	C₄²	C₄⋊Q₈	1	1	yes	characteristic
197	C₄²	C₄⋊Q₈	1	1	yes	characteristic
198	C₄²	C₂³×C₄	1	1	yes	characteristic
199	C₄²	C₂²×D₄	1	1	yes	characteristic
200	C₄²	C₂²×D₄	1	1	yes	characteristic
201	C₄²	C₂²×Q₈	1	1	yes	characteristic
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	#389	1	1	yes	characteristic
206	C₂³	#389	1	1	yes	characteristic
207	C₂³	C₂×#125	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	#389	1	1	yes	characteristic
226	C₂×C₄	#389	1	1	yes	characteristic
227	C₂×C₄	#389	1	1	yes	characteristic
228	C₂×C₄	#389	1	1	yes	characteristic
229	C₂×C₄	#389	1	1	yes	characteristic
230	C₂×C₄	#389	1	1	yes	characteristic
231	C₂×C₄	C₂×#108	1	2	yes	—
232	C₂×C₄	#430	1	2	yes	—
233	C₂×C₄	#443	1	4	yes	—
234	C₂×C₄	#471	1	4	yes	—
235	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
238	C₂×C₄	C₂×#121	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
241	C₈	—	2	8	no	—
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	—	4	8	no	—
246	C₈	—	4	4	no	—
247	C₈	—	8	4	no	—
248	C₈	C₂×#108	1	4	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	4	4	no	—
256	C₂²	#1945	1	2	yes	—
257	C₂²	#1950	1	2	yes	—
258	C₂²	#1951	1	2	yes	—
259	C₂²	C₂×#389	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	#1945	1	2	yes	—
267	C₄	#1950	1	2	yes	—
268	C₄	#1951	1	2	yes	—
269	C₄	C₂²×#108	1	1	yes	characteristic
270	C₄	C₂×#430	1	1	yes	characteristic
271	C₄	C₂×#443	1	2	yes	—
272	C₄	C₂×#471	1	2	yes	—
273	C₂	—	4	1	no	—
274	C₂	—	4	1	no	—
275	C₂	#9633	1	4	yes	direct product factor
276	C₂	C₂×#1945	1	1	yes	characteristic
277	C₂	C₂×#1950	1	1	yes	characteristic
278	C₂	C₂×#1951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	#389
prime radical	C₂⁴	#125
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. #9633	C₂	direct (N × H)
275. C₂	#9633	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×#1574	2
C₂×#1566	2
C₂×#1564	2
C₂×C₈×M₁₆	2
#9633	2
C₂×#1299	2

Subgroups — C2×#9633 #203992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#9633 #203992