Subgroups — C₂×#10590889 #11874894

Order 1152.

913 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63918	C₂	1	2	yes	maximal
2	C₂×#63644	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#63573	C₂	1	2	yes	maximal
6	#10590889	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4855	—	3	1	no	maximal
8	C₂×#4680	—	3	1	no	maximal
9	C₂²×#627	—	2	2	no	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×Dic₃⋊Dic₃	—	2	2	no	—
16	#63918	—	2	4	no	—
17	#63918	C₂²	1	8	yes	—
18	#63644	C₂²	1	4	yes	—
19	C₃×#817	C₂²	1	4	yes	—
20	C₃×C₈×Dic₃	C₂²	1	4	yes	—
21	#63573	C₂²	1	8	yes	—
22	C₂×#888	—	3	2	no	—
23	C₂×#869	—	3	2	no	—
24	C₂×C₃×#106	—	3	1	no	—
25	C₂×C₃×C₄×C₈	—	3	1	no	—
26	C₂×#817	—	3	1	no	—
27	C₂×#817	—	3	1	no	—
28	C₂×#817	—	3	2	no	—
29	C₂×#817	—	6	1	no	—
30	C₂×#817	D₃	1	1	yes	characteristic
31	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
32	C₂×#808	—	3	2	no	—
33	#4855	—	3	8	no	—
34	#4680	—	3	8	no	—
35	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₃⋊Dic₃	—	2	2	no	—
37	C₂²×C₃×Dic₃	—	2	2	no	—
38	C₂²×C₃×Dic₃	D₄	1	2	yes	—
39	C₂×#627	—	2	8	no	—
40	C₂×#627	—	4	8	no	—
41	C₂×C₃²×C₈	Q₈	1	4	yes	complemented
42	C₂×C₃²×C₈	C₂³	1	2	yes	—
43	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
44	C₄⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
45	C₃×C₄⋊Dic₃	—	2	2	no	—
46	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
47	C₃×C₄×Dic₃	D₄	1	2	yes	—
48	C₃×C₄×Dic₃	C₂³	1	2	yes	—
49	C₂×C₃×C₃⋊C₈	Q₈	1	4	yes	complemented
50	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
51	Dic₃⋊Dic₃	—	2	8	no	—
52	C₂×#498	—	9	1	no	Sylow
53	C₂²×Dic₆	—	6	2	no	—
54	C₂²×Dic₆	—	6	2	no	—
55	C₂²×C₃×C₈	—	2	1	no	—
56	C₂²×C₃×C₈	—	3	1	no	—
57	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄⋊C₄	—	3	2	no	—
60	C₂×C₃×C₄²	—	3	1	no	—
61	C₂×C₄⋊Dic₃	—	3	2	no	—
62	C₂×C₄⋊Dic₃	—	3	2	no	—
63	C₂×C₄⋊Dic₃	—	6	2	no	—
64	C₂×C₄⋊Dic₃	—	6	2	no	—
65	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
66	C₂×Dic₃⋊C₄	—	6	2	no	—
67	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
68	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
69	#888	—	3	8	no	—
70	#888	—	6	4	no	—
71	#869	—	3	8	no	—
72	#869	—	6	4	no	—
73	C₃×#106	—	3	4	no	—
74	C₃×C₄×C₈	—	3	4	no	—
75	#817	—	3	4	no	—
76	#817	—	3	4	no	—
77	#817	—	3	8	no	—
78	#817	—	6	4	no	—
79	#817	C₂×D₃	1	4	yes	—
80	C₈×Dic₃	C₂×D₃	1	4	yes	—
81	#808	—	3	8	no	—
82	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
85	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
86	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	—	2	4	no	—
88	C₂×C₃⋊Dic₃	—	4	4	no	—
89	C₂×C₃×Dic₃	—	2	4	no	—
90	C₂×C₃×Dic₃	—	4	4	no	—
91	C₂×C₃×Dic₃	SD₁₆	1	8	yes	—
92	C₂×C₃×Dic₃	C₂×D₄	1	4	yes	—
93	#627	—	4	16	no	—
94	C₃²×C₈	C₂×Q₈	1	4	yes	complemented
95	C₃×C₃⋊C₈	C₂×Q₈	1	4	yes	complemented
96	C₂×C₄⋊Q₈	—	9	2	no	—
97	#498	—	9	8	no	—
98	C₂×#106	—	3	1	no	—
99	C₂×#106	—	9	2	no	—
100	C₂×#106	—	9	1	no	—
101	C₂×C₄×C₈	—	3	1	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	3	2	no	—
104	C₂²×C₃×C₄	—	6	2	no	—
105	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
107	C₂²×Dic₃	—	2	2	no	—
108	C₂²×Dic₃	—	6	2	no	—
109	C₂²×Dic₃	—	6	2	no	—
110	C₂²×Dic₃	—	12	2	no	—
111	C₂²×Dic₃	D₁₂	1	2	yes	—
112	C₂×Dic₆	—	6	8	no	—
113	C₂×Dic₆	—	6	8	no	—
114	C₂×Dic₆	—	12	8	no	—
115	C₂×Dic₆	—	12	8	no	—
116	C₂×C₃×C₈	—	2	4	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	—	3	4	no	—
119	C₂×C₃×C₈	—	3	2	no	—
120	C₂×C₃×C₈	Dic₆	1	4	yes	complemented
121	C₂×C₃×C₈	Dic₆	1	4	yes	complemented
122	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
123	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
124	C₃×C₄⋊C₄	—	3	4	no	—
125	C₃×C₄⋊C₄	—	6	2	no	—
126	C₃×C₄²	—	3	2	no	—
127	C₃×C₄²	—	3	2	no	—
128	C₄⋊Dic₃	—	2	2	no	—
129	C₄⋊Dic₃	—	3	4	no	—
130	C₄⋊Dic₃	—	3	4	no	—
131	C₄⋊Dic₃	—	6	4	no	—
132	C₄⋊Dic₃	—	6	8	no	—
133	C₄⋊Dic₃	—	6	2	no	—
134	C₄⋊Dic₃	—	6	2	no	—
135	C₄⋊Dic₃	—	12	2	no	—
136	C₄⋊Dic₃	C₂²×D₃	1	4	yes	—
137	Dic₃⋊C₄	—	6	8	no	—
138	C₄×Dic₃	D₁₂	1	2	yes	—
139	C₄×Dic₃	C₂²×D₃	1	2	yes	—
140	C₂×C₃⋊C₈	Dic₆	1	4	yes	complemented
141	C₂×C₃⋊C₈	C₂²×D₃	1	2	yes	—
142	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
143	C₂²×C₃²	C₂×SD₁₆	1	4	yes	—
144	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
145	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
146	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
147	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
148	C₃⋊Dic₃	—	4	4	no	—
149	C₃×Dic₃	—	4	4	no	—
150	C₃×Dic₃	C₂×SD₁₆	1	8	yes	—
151	C₂²×Q₈	—	18	2	no	—
152	C₂²×C₈	—	3	1	no	—
153	C₂²×C₈	D₃²	1	1	yes	characteristic
154	C₄⋊Q₈	—	9	8	no	—
155	C₄⋊Q₈	—	18	4	no	—
156	C₂×C₄⋊C₄	—	3	2	no	—
157	C₂×C₄⋊C₄	—	9	2	no	—
158	C₂×C₄⋊C₄	—	18	2	no	—
159	C₂×C₄²	—	3	1	no	—
160	#106	—	3	4	no	—
161	#106	—	9	8	no	—
162	#106	—	9	4	no	—
163	C₄×C₈	—	3	4	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
166	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	3	8	no	—
172	C₂×C₃×C₄	—	3	4	no	—
173	C₂×C₃×C₄	—	6	4	no	—
174	C₂×C₃×C₄	—	12	4	no	—
175	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
176	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
177	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
178	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
179	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
181	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
183	C₂×Dic₃	—	2	4	no	—
184	C₂×Dic₃	—	4	4	no	—
185	C₂×Dic₃	—	6	4	no	—
186	C₂×Dic₃	—	6	4	no	—
187	C₂×Dic₃	—	12	4	no	—
188	C₂×Dic₃	—	12	4	no	—
189	C₂×Dic₃	—	12	4	no	—
190	C₂×Dic₃	—	24	4	no	—
191	C₂×Dic₃	#204	1	8	yes	—
192	C₂×Dic₃	C₂×D₁₂	1	4	yes	—
193	Dic₆	—	12	16	no	—
194	Dic₆	—	12	16	no	—
195	C₃×C₈	—	2	4	no	—
196	C₃×C₈	—	3	4	no	—
197	C₃×C₈	C₂×Dic₆	1	4	yes	complemented
198	C₃×C₈	C₂×Dic₆	1	4	yes	complemented
199	C₃⋊C₈	C₂×Dic₆	1	4	yes	complemented
200	C₂×C₃²	#498	1	4	yes	complemented
201	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
202	C₂×C₃²	C₂²×SD₁₆	1	2	yes	—
203	C₂×Q₈	—	18	8	no	—
204	C₂×Q₈	—	36	8	no	—
205	C₂²×C₄	—	3	2	no	—
206	C₂²×C₄	—	6	2	no	—
207	C₂²×C₄	—	18	2	no	—
208	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
209	C₂×C₈	—	3	2	no	—
210	C₂×C₈	—	3	4	no	—
211	C₂×C₈	#627	1	4	yes	complemented
212	C₂×C₈	C₂×D₃²	1	2	yes	—
213	C₄⋊C₄	—	3	4	no	—
214	C₄⋊C₄	—	6	2	no	—
215	C₄⋊C₄	—	9	4	no	—
216	C₄⋊C₄	—	18	8	no	—
217	C₄⋊C₄	—	18	2	no	—
218	C₄²	—	3	2	no	—
219	C₄²	—	3	2	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	4	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	#869	1	2	yes	—
224	C₂²×C₃	#888	1	2	yes	—
225	C₂²×C₃	C₂×#204	1	4	yes	—
226	C₂²×C₃	D₃×SD₁₆	1	4	yes	—
227	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
228	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	3	8	no	—
233	C₃×C₄	—	12	4	no	—
234	C₃×C₄	#869	1	2	yes	—
235	C₃×C₄	#888	1	2	yes	—
236	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
237	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
238	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
239	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
240	Dic₃	—	4	4	no	—
241	Dic₃	—	12	4	no	—
242	Dic₃	—	12	4	no	—
243	Dic₃	—	24	4	no	—
244	Dic₃	C₂×#204	1	8	yes	—
245	C₃²	C₂×#498	1	1	yes	characteristic, complemented, Sylow
246	C₂³	D₃×D₁₂	1	1	yes	characteristic
247	Q₈	—	36	16	no	—
248	C₂×C₄	—	3	4	no	—
249	C₂×C₄	—	3	8	no	—
250	C₂×C₄	—	6	4	no	—
251	C₂×C₄	—	12	4	no	—
252	C₂×C₄	—	18	4	no	—
253	C₂×C₄	—	36	4	no	—
254	C₂×C₄	D₃×Dic₆	1	2	yes	—
255	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
256	C₂×C₄	C₂×#627	1	2	yes	—
257	C₂×C₄	C₂²×D₃²	1	1	yes	characteristic
258	C₈	—	3	4	no	—
259	C₈	C₂×#627	1	4	yes	complemented
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	—	2	4	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	#4680	1	4	yes	complemented
264	C₂×C₃	#4855	1	4	yes	complemented
265	C₂×C₃	C₂×#869	1	1	yes	characteristic
266	C₂×C₃	C₂×#888	1	1	yes	characteristic
267	C₂×C₃	C₂²×#204	1	2	yes	—
268	C₂×C₃	C₂×D₃×SD₁₆	1	2	yes	—
269	C₂²	D₃×#204	1	4	yes	—
270	C₂²	#63918	1	2	yes	—
271	C₂²	C₂×D₃×D₁₂	1	1	yes	characteristic
272	C₄	—	3	8	no	—
273	C₄	—	12	4	no	—
274	C₄	—	36	4	no	—
275	C₄	#63918	1	2	yes	—
276	C₄	C₂×D₃×Dic₆	1	1	yes	characteristic
277	C₄	C₂²×#627	1	1	yes	characteristic
278	C₃	—	2	1	no	—
279	C₃	C₂×#4680	1	1	yes	characteristic, complemented
280	C₃	C₂×#4855	1	1	yes	characteristic, complemented
281	C₂	#10590889	1	4	yes	direct product factor
282	C₂	C₂×D₃×#204	1	2	yes	—
283	C₂	C₂×#63918	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₁₂
commutator, cosocle	C₂×C₃²×C₄	C₂⁴
Frattini	C₂×C₄	C₂²×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	C₂×#498
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
6. #10590889	C₂	direct (N × H)
41. C₂×C₃²×C₈	Q₈	semidirect
49. C₂×C₃×C₃⋊C₈	Q₈	semidirect
94. C₃²×C₈	C₂×Q₈	semidirect
95. C₃×C₃⋊C₈	C₂×Q₈	semidirect
120. C₂×C₃×C₈	Dic₆	semidirect
121. C₂×C₃×C₈	Dic₆	semidirect
140. C₂×C₃⋊C₈	Dic₆	semidirect
197. C₃×C₈	C₂×Dic₆	semidirect
198. C₃×C₈	C₂×Dic₆	semidirect
199. C₃⋊C₈	C₂×Dic₆	semidirect
200. C₂×C₃²	#498	semidirect
211. C₂×C₈	#627	semidirect
245. C₃²	C₂×#498	semidirect
259. C₈	C₂×#627	semidirect
263. C₂×C₃	#4680	semidirect
264. C₂×C₃	#4855	semidirect
279. C₃	C₂×#4680	semidirect
280. C₃	C₂×#4855	semidirect
281. C₂	#10590889	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×#63918	2
C₂×#63644	2
C₂×C₃×#817	2
C₂×C₃×C₈×Dic₃	2
C₂×#63573	2
#10590889	2
C₂×#4855	3
C₂×#4680	3

Subgroups — C2×#10590889 #11874894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10590889 #11874894