Subgroups — C₂×#63804 #10594695

Order 576.

566 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3798	C₂	1	1	yes	maximal, characteristic
2	C₂×#3743	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#3711	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#3710	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#63804	C₂	1	8	yes	maximal, direct product factor
9	C₂×#907	—	3	1	no	maximal
10	C₂×#901	—	3	1	no	maximal
11	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
14	C₂×#627	C₄	1	1	yes	characteristic
15	C₂×#626	C₄	1	1	yes	characteristic
16	C₂×#626	C₄	1	1	yes	characteristic
17	C₂×#625	C₄	1	1	yes	characteristic
18	#3798	C₄	1	4	yes	—
19	#3798	C₂²	1	4	yes	—
20	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	#3743	C₂²	1	4	yes	complemented
23	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₃×ΓL(1,𝔽₂₅)	C₂²	1	4	yes	complemented
26	C₃×D₃×C₈	C₂²	1	4	yes	complemented
27	#3711	C₂²	1	4	yes	complemented
28	#3710	C₂²	1	4	yes	complemented
29	D₃×C₃⋊C₈	C₂²	1	4	yes	complemented
30	C₂×#235	—	3	1	no	—
31	C₂×#235	—	3	1	no	—
32	C₂×C₃×M₁₆	—	3	1	no	—
33	C₂²×C₃×C₈	—	3	1	no	—
34	C₂×#208	—	3	1	no	—
35	C₂²×C₃⋊C₈	—	3	1	no	—
36	#907	—	3	8	no	—
37	#901	—	3	8	no	—
38	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
39	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
40	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
41	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
42	C₂×ΓL(1,𝔽₂₅)	D₃	1	1	yes	characteristic, complemented
43	C₂×D₃×C₈	—	3	1	no	—
44	C₂×D₃×C₈	—	3	1	no	—
45	C₂×D₃×C₈	—	3	1	no	—
46	C₂×D₃×C₈	D₃	1	1	yes	characteristic, complemented
47	C₂²×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₃×D₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₃×D₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
53	C₄×C₃⋊D₃	C₂³	1	2	yes	—
54	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
56	C₃×C₄×D₃	C₂×C₄	1	2	yes	—
57	C₃×C₄×D₃	C₂×C₄	1	2	yes	—
58	C₃×C₄×D₃	C₂³	1	2	yes	—
59	C₃×C₄×D₃	C₂³	1	2	yes	—
60	#627	C₂×C₄	1	4	yes	—
61	#626	C₂×C₄	1	4	yes	—
62	#626	C₂×C₄	1	4	yes	—
63	#625	C₂×C₄	1	4	yes	—
64	C₃²×C₈	C₂³	1	2	yes	—
65	C₃⋊(C₃⋊C₈)	C₂³	1	2	yes	—
66	C₃×C₃⋊C₈	C₂³	1	2	yes	—
67	C₃×C₃⋊C₈	C₂³	1	2	yes	—
68	C₂×#131	—	9	1	no	Sylow
69	C₂²×C₃×C₄	—	3	1	no	—
70	C₂²×C₃×C₄	—	3	1	no	—
71	C₂×#67	—	3	1	no	—
72	C₂×#67	—	3	1	no	—
73	C₂×#67	—	3	1	no	—
74	C₂×#67	—	3	1	no	—
75	#235	—	3	4	no	—
76	#235	—	3	4	no	—
77	#235	—	3	4	no	—
78	#235	—	3	4	no	—
79	C₂×D₁₂	—	3	1	no	—
80	C₂×D₁₂	—	3	1	no	—
81	C₂×C₄×D₃	—	3	1	no	—
82	C₂×C₄×D₃	—	3	1	no	—
83	C₂×C₄×D₃	—	6	1	no	—
84	C₂×C₄×D₃	C₂×D₃	1	1	yes	characteristic
85	C₂×C₄×D₃	C₂×D₃	1	1	yes	characteristic
86	C₂×Dic₆	—	3	1	no	—
87	C₂×Dic₆	—	3	1	no	—
88	C₃×M₁₆	—	3	4	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	3	1	no	—
91	C₂×C₃×C₈	—	3	1	no	—
92	C₂×C₃×C₈	—	3	4	no	—
93	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
95	#208	—	3	4	no	—
96	C₂×C₃⋊C₈	—	3	1	no	—
97	C₂×C₃⋊C₈	—	3	1	no	—
98	C₂×C₃⋊C₈	—	3	4	no	—
99	C₂×C₃⋊C₈	—	6	1	no	—
100	C₂×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
101	C₂×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
102	ΓL(1,𝔽₂₅)	—	3	4	no	—
103	ΓL(1,𝔽₂₅)	—	3	4	no	—
104	ΓL(1,𝔽₂₅)	—	3	4	no	—
105	ΓL(1,𝔽₂₅)	—	6	4	no	—
106	ΓL(1,𝔽₂₅)	C₂×D₃	1	4	yes	complemented
107	D₃×C₈	—	3	4	no	—
108	D₃×C₈	—	3	4	no	—
109	D₃×C₈	—	3	4	no	—
110	D₃×C₈	C₂×D₃	1	4	yes	complemented
111	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₃⋊D₃	—	2	2	no	—
113	C₂×C₃⋊D₃	C₂²×C₄	1	2	yes	—
114	C₂×C₃×D₃	—	2	2	no	—
115	C₂×C₃×D₃	—	2	2	no	—
116	C₂×C₃×D₃	C₂²×C₄	1	2	yes	—
117	C₂×C₃×D₃	C₂²×C₄	1	2	yes	—
118	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
119	C₃²×C₄	C₂⁴	1	1	yes	characteristic
120	C₃⋊Dic₃	C₂²×C₄	1	2	yes	—
121	C₃×Dic₃	C₂²×C₄	1	2	yes	—
122	C₃×Dic₃	C₂²×C₄	1	2	yes	—
123	C₂×#41	—	9	1	no	—
124	#131	—	9	8	no	—
125	C₂×M₁₆	—	3	1	no	—
126	C₂×M₁₆	—	9	1	no	—
127	C₂×M₁₆	—	9	1	no	—
128	C₂²×C₈	—	3	1	no	—
129	C₂²×C₈	—	9	1	no	—
130	C₂²×C₈	—	9	1	no	—
131	C₂³×C₃	—	3	1	no	—
132	C₂³×C₃	—	3	1	no	—
133	C₂²×D₃	—	3	1	no	—
134	C₂²×D₃	—	3	1	no	—
135	C₂²×D₃	—	6	1	no	—
136	C₂²×D₃	C₄×D₃	1	1	yes	characteristic
137	C₂²×D₃	C₄×D₃	1	1	yes	characteristic
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	3	1	no	—
140	C₂×C₃×C₄	—	3	1	no	—
141	C₂×C₃×C₄	—	3	2	no	—
142	C₂×C₃×C₄	—	3	2	no	—
143	C₂×C₃×C₄	—	3	2	no	—
144	C₂×C₃×C₄	—	3	2	no	—
145	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
147	#67	—	3	4	no	—
148	#67	—	3	4	no	—
149	#67	—	3	4	no	—
150	#67	—	3	4	no	—
151	C₂×Dic₃	—	3	1	no	—
152	C₂×Dic₃	—	3	1	no	—
153	C₂×Dic₃	—	6	1	no	—
154	C₂×Dic₃	C₄×D₃	1	1	yes	characteristic
155	C₂×Dic₃	C₄×D₃	1	1	yes	characteristic
156	D₁₂	—	3	4	no	—
157	D₁₂	—	3	4	no	—
158	C₄×D₃	—	3	2	no	—
159	C₄×D₃	—	3	2	no	—
160	C₄×D₃	—	3	2	no	—
161	C₄×D₃	—	3	2	no	—
162	C₄×D₃	—	6	2	no	—
163	C₄×D₃	—	6	2	no	—
164	C₄×D₃	C₄×D₃	1	2	yes	—
165	C₄×D₃	C₄×D₃	1	2	yes	—
166	C₄×D₃	C₂²×D₃	1	2	yes	—
167	C₄×D₃	C₂²×D₃	1	2	yes	—
168	Dic₆	—	3	4	no	—
169	Dic₆	—	3	4	no	—
170	C₃×C₈	—	2	2	no	—
171	C₃×C₈	—	3	2	no	—
172	C₃×C₈	—	3	2	no	—
173	C₃×C₈	C₂²×D₃	1	2	yes	—
174	C₃×C₈	C₂²×D₃	1	2	yes	—
175	C₃⋊C₈	—	3	2	no	—
176	C₃⋊C₈	—	3	2	no	—
177	C₃⋊C₈	—	6	2	no	—
178	C₃⋊C₈	C₂²×D₃	1	2	yes	—
179	C₃⋊C₈	C₂²×D₃	1	2	yes	—
180	C₂×C₃²	#131	1	2	yes	complemented
181	C₂×C₃²	C₂³×C₄	1	1	yes	characteristic
182	C₃⋊D₃	—	2	2	no	—
183	C₃×D₃	—	2	2	no	—
184	C₃×D₃	—	2	2	no	—
185	#41	—	9	4	no	—
186	#41	—	9	4	no	—
187	C₂×Q₈	—	9	1	no	—
188	C₂×D₄	—	9	1	no	—
189	C₂×D₄	—	9	1	no	—
190	C₂×D₄	—	9	1	no	—
191	C₂²×C₄	—	3	1	no	—
192	C₂²×C₄	—	3	1	no	—
193	C₂²×C₄	—	9	1	no	—
194	M₁₆	—	3	4	no	—
195	M₁₆	—	9	4	no	—
196	M₁₆	—	9	4	no	—
197	C₂×C₈	—	3	1	no	—
198	C₂×C₈	—	3	1	no	—
199	C₂×C₈	—	3	4	no	—
200	C₂×C₈	—	9	4	no	—
201	C₂×C₈	—	9	4	no	—
202	C₂×C₈	—	9	1	no	—
203	C₂×C₈	D₃²	1	1	yes	characteristic
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	3	2	no	—
206	C₂²×C₃	—	3	2	no	—
207	C₂²×C₃	—	6	2	no	—
208	C₂²×C₃	—	6	2	no	—
209	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
210	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
211	C₂×D₃	—	2	2	no	—
212	C₂×D₃	—	2	2	no	—
213	C₂×D₃	—	3	2	no	—
214	C₂×D₃	—	3	2	no	—
215	C₂×D₃	—	6	2	no	—
216	C₂×D₃	—	6	2	no	—
217	C₂×D₃	—	6	2	no	—
218	C₂×D₃	—	12	2	no	—
219	C₂×D₃	C₂×C₄×D₃	1	2	yes	—
220	C₂×D₃	C₂×C₄×D₃	1	2	yes	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	3	2	no	—
224	C₃×C₄	—	3	2	no	—
225	C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
226	C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
227	C₃×C₄	C₂³×D₃	1	1	yes	characteristic
228	C₃×C₄	C₂³×D₃	1	1	yes	characteristic
229	Dic₃	—	3	2	no	—
230	Dic₃	—	3	2	no	—
231	Dic₃	—	6	2	no	—
232	Dic₃	C₂×C₄×D₃	1	2	yes	—
233	Dic₃	C₂×C₄×D₃	1	2	yes	—
234	C₃²	C₂×#131	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	3	1	no	—
236	C₂³	—	3	1	no	—
237	C₂³	—	9	1	no	—
238	Q₈	—	9	4	no	—
239	D₄	—	9	4	no	—
240	D₄	—	9	4	no	—
241	D₄	—	9	4	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	3	1	no	—
244	C₂×C₄	—	3	2	no	—
245	C₂×C₄	—	3	2	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	9	1	no	—
249	C₂×C₄	—	9	2	no	—
250	C₂×C₄	—	9	2	no	—
251	C₂×C₄	C₂×D₃²	1	1	yes	characteristic
252	C₈	—	3	2	no	—
253	C₈	—	3	2	no	—
254	C₈	—	9	2	no	—
255	C₈	C₂×D₃²	1	2	yes	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	6	2	no	—
259	C₂×C₃	—	6	2	no	—
260	C₂×C₃	#901	1	2	yes	complemented
261	C₂×C₃	#907	1	2	yes	complemented
262	C₂×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
263	C₂×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
264	D₃	—	2	2	no	—
265	D₃	—	2	2	no	—
266	D₃	—	6	2	no	—
267	D₃	—	6	2	no	—
268	D₃	—	12	2	no	—
269	C₂²	—	3	2	no	—
270	C₂²	—	3	2	no	—
271	C₂²	—	6	2	no	—
272	C₂²	—	6	2	no	—
273	C₂²	—	9	2	no	—
274	C₂²	—	18	2	no	—
275	C₂²	C₄×D₃²	1	1	yes	characteristic
276	C₄	—	3	2	no	—
277	C₄	—	3	2	no	—
278	C₄	—	9	2	no	—
279	C₄	C₄×D₃²	1	1	yes	characteristic
280	C₄	C₂²×D₃²	1	1	yes	characteristic
281	C₃	—	2	1	no	—
282	C₃	C₂×#901	1	1	yes	characteristic, complemented
283	C₃	C₂×#907	1	1	yes	characteristic, complemented
284	C₂	—	6	2	no	—
285	C₂	—	6	2	no	—
286	C₂	—	18	2	no	—
287	C₂	#63804	1	2	yes	direct product factor
288	C₂	C₂×C₄×D₃²	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃²
commutator	C₂×C₃²	C₂³×C₄
Frattini	C₄	C₂²×D₃²
Fitting	C₂×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₂²×C₄
prime radical	C₂×#3798	C₂
cosocle	C₃²×C₄	C₂⁴
hypercenter	C₂×C₈	D₃²
hypocenter	C₃²	C₂×#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#3743	C₂	semidirect
3. C₂×C₃×ΓL(1,𝔽₂₅)	C₂	semidirect
4. C₂×C₃×D₃×C₈	C₂	semidirect
5. C₂×#3711	C₂	semidirect
6. C₂×#3710	C₂	semidirect
7. C₂×D₃×C₃⋊C₈	C₂	semidirect
8. #63804	C₂	direct (N × H)
22. #3743	C₂²	semidirect
25. C₃×ΓL(1,𝔽₂₅)	C₂²	semidirect
26. C₃×D₃×C₈	C₂²	semidirect
27. #3711	C₂²	semidirect
28. #3710	C₂²	semidirect
29. D₃×C₃⋊C₈	C₂²	semidirect
42. C₂×ΓL(1,𝔽₂₅)	D₃	semidirect
46. C₂×D₃×C₈	D₃	semidirect
106. ΓL(1,𝔽₂₅)	C₂×D₃	semidirect
110. D₃×C₈	C₂×D₃	semidirect
180. C₂×C₃²	#131	semidirect
234. C₃²	C₂×#131	semidirect
260. C₂×C₃	#901	semidirect
261. C₂×C₃	#907	semidirect
282. C₃	C₂×#901	semidirect
283. C₃	C₂×#907	semidirect
287. C₂	#63804	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×#3798	2
C₂×#3743	2
C₂×C₃×ΓL(1,𝔽₂₅)	2
C₂×C₃×D₃×C₈	2
C₂×#3711	2
C₂×#3710	2
C₂×D₃×C₃⋊C₈	2
#63804	2
C₂×#907	3
C₂×#901	3

Subgroups — C2×#63804 #10594695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#63804 #10594695