Subgroups — C₂×#9623 #203900

Order 512.

706 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1950	C₂	1	2	yes	maximal
2	C₂×#1566	C₂	1	1	yes	maximal, characteristic
3	C₂²×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
5	#9623	C₂	1	8	yes	maximal, direct product factor
6	C₂×#1301	C₂	1	2	yes	maximal
7	C₂×#431	C₂²	1	1	yes	characteristic
8	C₂²×#108	—	2	2	no	—
9	C₂×#423	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#1950	—	2	4	no	—
14	#1950	C₂²	1	8	yes	—
15	C₂×#341	—	2	2	no	—
16	#1566	—	2	2	no	—
17	#1566	C₂²	1	4	yes	—
18	C₂×#334	—	2	1	no	—
19	C₂×#334	—	2	1	no	—
20	C₂×#334	C₄	1	2	yes	—
21	C₂×#334	C₄	1	4	yes	—
22	C₂×#334	C₄	1	2	yes	—
23	C₂×#334	C₂²	1	4	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	#1301	C₂²	1	8	yes	—
29	C₂²×M₁₆	—	2	2	no	—
30	C₂²×M₁₆	D₄	1	2	yes	—
31	C₂³×C₈	—	2	2	no	—
32	C₂³×C₈	Q₈	1	1	yes	characteristic
33	C₂³×C₈	Q₈	1	1	yes	characteristic
34	C₂²×C₄²	C₂³	1	1	yes	characteristic
35	#431	—	2	1	no	—
36	#431	—	2	1	no	—
37	#431	Q₈	1	2	yes	—
38	#431	C₂³	1	2	yes	—
39	C₂×#108	—	2	8	no	—
40	C₂×#108	—	4	8	no	—
41	#423	—	2	2	no	—
42	#423	—	2	2	no	—
43	#423	—	2	2	no	—
44	#423	C₂³	1	4	yes	—
45	C₂×C₄⋊C₈	—	2	2	no	—
46	C₂×C₄⋊C₈	—	2	2	no	—
47	C₂×C₄⋊C₈	—	2	2	no	—
48	C₂×C₄⋊C₈	—	2	1	no	—
49	C₂×C₄⋊C₈	—	2	2	no	—
50	C₂×C₄⋊C₈	—	2	2	no	—
51	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₂×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₂×C₄⋊C₈	C₂³	1	4	yes	—
54	C₂×C₂²⋊C₈	Q₈	1	2	yes	—
55	C₄×M₁₆	—	2	1	no	—
56	C₄×M₁₆	—	2	1	no	—
57	C₄×M₁₆	D₄	1	2	yes	—
58	C₄×M₁₆	C₂³	1	2	yes	—
59	C₂×#97	—	2	1	no	—
60	C₂×C₄×C₈	—	2	1	no	—
61	C₂×C₄×C₈	—	2	1	no	—
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	D₄	1	1	yes	characteristic
66	C₂×C₄×C₈	D₄	1	1	yes	characteristic
67	C₂×C₄×C₈	D₄	1	2	yes	—
68	C₂×C₄×C₈	Q₈	1	2	yes	—
69	C₂×C₄×C₈	C₂³	1	2	yes	—
70	#341	—	2	8	no	—
71	#334	—	2	4	no	—
72	#334	—	2	4	no	—
73	#334	C₂×C₄	1	8	yes	—
74	#334	C₂×C₄	1	8	yes	—
75	#322	—	2	4	no	—
76	C₈²	—	2	4	no	—
77	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂×M₁₆	—	2	2	no	—
81	C₂×M₁₆	—	2	4	no	—
82	C₂×M₁₆	—	2	4	no	—
83	C₂×M₁₆	—	4	2	no	—
84	C₂×M₁₆	—	4	4	no	—
85	C₂×M₁₆	—	4	4	no	—
86	C₂×M₁₆	SD₁₆	1	8	yes	—
87	C₂×M₁₆	C₂×D₄	1	4	yes	—
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	—	2	4	no	—
90	C₂²×C₈	—	2	4	no	—
91	C₂²×C₈	—	2	4	no	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	4	no	—
94	C₂²×C₈	—	4	2	no	—
95	C₂²×C₈	—	4	4	no	—
96	C₂²×C₈	—	4	4	no	—
97	C₂²×C₈	—	4	2	no	—
98	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₄⋊C₄	1	2	yes	—
102	C₂²×C₈	C₄⋊C₄	1	2	yes	—
103	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₈	C₂×Q₈	1	2	yes	—
105	C₂²×C₈	C₂×Q₈	1	2	yes	—
106	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
113	C₂×C₄²	#41	1	1	yes	characteristic
114	C₂×C₄²	#41	1	1	yes	characteristic
115	C₂×C₄²	#41	1	1	yes	characteristic
116	C₂×C₄²	#41	1	1	yes	characteristic
117	C₂×C₄²	C₂⁴	1	1	yes	characteristic
118	#108	—	4	16	no	—
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	2	4	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	4	no	—
123	C₄⋊C₈	—	2	4	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	2	4	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	C₄⋊C₈	C₂²×C₄	1	8	yes	—
129	C₂²⋊C₈	—	2	2	no	—
130	C₂²⋊C₈	C₂×Q₈	1	4	yes	—
131	#97	—	2	2	no	—
132	#97	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	C₄⋊C₄	1	2	yes	—
138	C₄×C₈	C₄⋊C₄	1	2	yes	—
139	C₄×C₈	C₄⋊C₄	1	2	yes	—
140	C₄×C₈	C₄⋊C₄	1	2	yes	—
141	C₄×C₈	C₂²×C₄	1	2	yes	—
142	C₄×C₈	C₂²×C₄	1	2	yes	—
143	C₄×C₈	C₂×D₄	1	2	yes	—
144	C₄×C₈	C₂×D₄	1	2	yes	—
145	C₂⁴	C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
153	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
154	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
155	C₂²×C₄	#121	1	1	yes	characteristic
156	C₂²×C₄	#121	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂²×C₄	#125	1	1	yes	characteristic
160	C₂²×C₄	#125	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
164	C₂²×C₄	C₂×SD₁₆	1	4	yes	—
165	C₂²×C₄	#135	1	4	yes	—
166	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
168	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
169	M₁₆	—	2	4	no	—
170	M₁₆	—	2	8	no	—
171	M₁₆	—	4	4	no	—
172	M₁₆	—	4	4	no	—
173	M₁₆	—	8	4	no	—
174	M₁₆	C₂×SD₁₆	1	8	yes	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	4	8	no	—
190	C₂×C₈	—	4	8	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	8	4	no	—
193	C₂×C₈	#108	1	4	yes	complemented
194	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
195	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
196	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
197	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
198	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
200	C₄²	C₄×D₄	1	1	yes	characteristic
201	C₄²	C₄×D₄	1	1	yes	characteristic
202	C₄²	C₄×Q₈	1	1	yes	characteristic
203	C₄²	C₄×Q₈	1	1	yes	characteristic
204	C₄²	#121	1	1	yes	characteristic
205	C₄²	#121	1	1	yes	characteristic
206	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
207	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
208	C₄²	#125	1	1	yes	characteristic
209	C₄²	#125	1	1	yes	characteristic
210	C₄²	C₂³×C₄	1	1	yes	characteristic
211	C₄²	C₂²×D₄	1	1	yes	characteristic
212	C₄²	C₂×#41	1	1	yes	characteristic
213	C₄²	C₂×#41	1	1	yes	characteristic
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#389	1	1	yes	characteristic
217	C₂³	#389	1	1	yes	characteristic
218	C₂³	#498	1	2	yes	—
219	C₂³	#499	1	2	yes	—
220	C₂³	C₂×C₄⋊Q₈	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	#389	1	1	yes	characteristic
235	C₂×C₄	#389	1	1	yes	characteristic
236	C₂×C₄	#389	1	1	yes	characteristic
237	C₂×C₄	#389	1	1	yes	characteristic
238	C₂×C₄	#389	1	1	yes	characteristic
239	C₂×C₄	#389	1	1	yes	characteristic
240	C₂×C₄	C₂×#108	1	2	yes	—
241	C₂×C₄	#430	1	2	yes	—
242	C₂×C₄	C₄×SD₁₆	1	4	yes	—
243	C₂×C₄	C₄×SD₁₆	1	4	yes	—
244	C₂×C₄	#465	1	4	yes	—
245	C₂×C₄	#465	1	4	yes	—
246	C₂×C₄	#498	1	2	yes	—
247	C₂×C₄	#499	1	2	yes	—
248	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
251	C₂×C₄	C₂×#121	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₄	C₂×#125	1	1	yes	characteristic
254	C₂×C₄	C₂²×SD₁₆	1	2	yes	—
255	C₂×C₄	C₂×#135	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	2	2	no	—
259	C₈	—	2	8	no	—
260	C₈	—	4	4	no	—
261	C₈	—	4	8	no	—
262	C₈	—	8	4	no	—
263	C₈	C₂×#108	1	4	yes	complemented
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	#1941	1	2	yes	—
271	C₂²	#1941	1	2	yes	—
272	C₂²	#1950	1	2	yes	—
273	C₂²	#1953	1	2	yes	—
274	C₂²	C₂×#389	1	1	yes	characteristic
275	C₂²	C₂×#498	1	1	yes	characteristic
276	C₂²	C₂×#499	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	#1941	1	2	yes	—
284	C₄	#1941	1	2	yes	—
285	C₄	#1950	1	2	yes	—
286	C₄	C₂²×#108	1	1	yes	characteristic
287	C₄	C₂×#430	1	1	yes	characteristic
288	C₄	C₂×C₄×SD₁₆	1	2	yes	—
289	C₄	C₂×#465	1	2	yes	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	#9623	1	4	yes	direct product factor
294	C₂	C₂×#1941	1	1	yes	characteristic
295	C₂	C₂×#1950	1	1	yes	characteristic
296	C₂	C₂×#1953	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	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. #9623	C₂	direct (N × H)
193. C₂×C₈	#108	semidirect
263. C₈	C₂×#108	semidirect
293. C₂	#9623	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×#1950	2
C₂×#1566	2
C₂²×#334	2
C₂×C₈×M₁₆	2
#9623	2
C₂×#1301	2

Subgroups — C2×#9623 #203900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#9623 #203900