Subgroups — C₂×#10590873 #11874928

Order 1152.

785 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63869	C₂	1	2	yes	maximal
2	C₂×#63645	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#818	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	#10590873	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4878	—	3	1	no	maximal
8	C₂×#4683	—	3	1	no	maximal
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#3724	—	2	2	no	—
15	C₂×Dic₃⋊Dic₃	—	2	2	no	—
16	#63869	—	2	4	no	—
17	#63869	C₂²	1	8	yes	—
18	#63645	C₂²	1	4	yes	—
19	C₃×#818	C₂²	1	4	yes	—
20	C₃×C₈×Dic₃	C₂²	1	4	yes	—
21	#63573	C₂²	1	8	yes	—
22	C₂×#890	—	3	2	no	—
23	C₂×#870	—	3	2	no	—
24	C₂×C₃×#107	—	3	1	no	—
25	C₂×C₃×C₄×C₈	—	3	1	no	—
26	C₂×#818	—	3	1	no	—
27	C₂×#818	—	3	1	no	—
28	C₂×#818	—	6	1	no	—
29	C₂×#818	D₃	1	1	yes	characteristic
30	C₂×#817	—	3	2	no	—
31	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
32	C₂×#808	—	3	2	no	—
33	#4878	—	3	8	no	—
34	#4683	—	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	1	yes	characteristic
39	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
40	C₂×C₃²×C₈	Q₈	1	4	yes	—
41	C₂×C₃²×C₈	C₂³	1	2	yes	—
42	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
43	C₄⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
44	C₃×C₄⋊Dic₃	—	2	2	no	—
45	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
46	C₃×C₄×Dic₃	D₄	1	2	yes	—
47	C₃×C₄×Dic₃	C₂³	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	Q₈	1	4	yes	—
49	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
50	#3724	—	2	8	no	—
51	Dic₃⋊Dic₃	—	2	8	no	—
52	C₂×#499	—	9	1	no	Sylow
53	C₂²×C₃×C₈	—	2	1	no	—
54	C₂²×C₃×C₈	—	3	1	no	—
55	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×C₄⋊C₄	—	3	2	no	—
58	C₂×C₃×C₄²	—	3	1	no	—
59	C₂×C₄⋊Dic₃	—	3	2	no	—
60	C₂×C₄⋊Dic₃	—	3	2	no	—
61	C₂×C₄⋊Dic₃	—	6	2	no	—
62	C₂×C₄⋊Dic₃	—	6	2	no	—
63	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
64	C₂×Dic₃⋊C₄	—	6	2	no	—
65	C₂×Dic₃⋊C₄	—	6	2	no	—
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	#890	—	3	8	no	—
70	#890	—	6	4	no	—
71	#870	—	3	8	no	—
72	#870	—	6	4	no	—
73	C₃×#107	—	3	4	no	—
74	C₃×C₄×C₈	—	3	4	no	—
75	#818	—	3	4	no	—
76	#818	—	3	4	no	—
77	#818	—	6	4	no	—
78	#818	C₂×D₃	1	4	yes	—
79	#817	—	3	8	no	—
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	2	no	—
90	C₂×C₃×Dic₃	—	2	2	no	—
91	C₂×C₃×Dic₃	—	2	4	no	—
92	C₂×C₃×Dic₃	—	4	4	no	—
93	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
94	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
95	C₃²×C₈	C₂×Q₈	1	4	yes	—
96	C₃×C₃⋊C₈	C₂×Q₈	1	4	yes	—
97	C₂×#125	—	9	2	no	—
98	#499	—	9	8	no	—
99	C₂×#107	—	3	1	no	—
100	C₂×#107	—	9	1	no	—
101	C₂×#106	—	9	2	no	—
102	C₂×C₄×C₈	—	3	1	no	—
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	—	3	1	no	—
105	C₂²×C₃×C₄	—	3	1	no	—
106	C₂²×C₃×C₄	—	6	2	no	—
107	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂²×Dic₃	—	2	2	no	—
110	C₂²×Dic₃	—	6	2	no	—
111	C₂²×Dic₃	—	6	2	no	—
112	C₂²×Dic₃	—	12	2	no	—
113	C₂²×Dic₃	D₁₂	1	1	yes	characteristic
114	C₂²×Dic₃	D₁₂	1	1	yes	characteristic
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	3	4	no	—
118	C₂×C₃×C₈	—	3	2	no	—
119	C₂×C₃×C₈	Dic₆	1	4	yes	—
120	C₂×C₃×C₈	Dic₆	1	4	yes	—
121	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
122	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
123	C₃×C₄⋊C₄	—	3	4	no	—
124	C₃×C₄⋊C₄	—	6	2	no	—
125	C₃×C₄²	—	3	2	no	—
126	C₃×C₄²	—	3	2	no	—
127	C₄⋊Dic₃	—	2	2	no	—
128	C₄⋊Dic₃	—	3	4	no	—
129	C₄⋊Dic₃	—	3	4	no	—
130	C₄⋊Dic₃	—	6	4	no	—
131	C₄⋊Dic₃	—	6	2	no	—
132	C₄⋊Dic₃	—	6	2	no	—
133	C₄⋊Dic₃	—	6	8	no	—
134	C₄⋊Dic₃	—	12	2	no	—
135	C₄⋊Dic₃	C₂²×D₃	1	4	yes	—
136	Dic₃⋊C₄	—	6	8	no	—
137	Dic₃⋊C₄	—	6	8	no	—
138	Dic₃⋊C₄	—	6	8	no	—
139	C₄×Dic₃	D₁₂	1	2	yes	—
140	C₄×Dic₃	C₂²×D₃	1	2	yes	—
141	C₂×C₃⋊C₈	Dic₆	1	4	yes	—
142	C₂×C₃⋊C₈	C₂²×D₃	1	2	yes	—
143	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
144	C₂²×C₃²	#135	1	2	yes	—
145	C₂²×C₃²	#135	1	2	yes	—
146	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
147	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
148	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
149	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
150	C₃⋊Dic₃	—	4	4	no	—
151	C₃×Dic₃	—	2	2	no	—
152	C₃×Dic₃	—	2	2	no	—
153	C₃×Dic₃	—	4	4	no	—
154	C₂²×C₈	—	3	1	no	—
155	C₂²×C₈	D₃²	1	1	yes	characteristic
156	#125	—	9	8	no	—
157	#125	—	18	4	no	—
158	C₂×C₄⋊C₄	—	3	2	no	—
159	C₂×C₄⋊C₄	—	9	2	no	—
160	C₂×C₄⋊C₄	—	18	2	no	—
161	C₂×C₄⋊C₄	—	18	2	no	—
162	C₂×C₄²	—	3	1	no	—
163	#107	—	3	4	no	—
164	#107	—	9	4	no	—
165	#106	—	9	8	no	—
166	C₄×C₈	—	3	4	no	—
167	C₂³×C₃	—	2	1	no	—
168	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
169	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	3	2	no	—
175	C₂×C₃×C₄	—	3	2	no	—
176	C₂×C₃×C₄	—	6	2	no	—
177	C₂×C₃×C₄	—	6	2	no	—
178	C₂×C₃×C₄	—	6	4	no	—
179	C₂×C₃×C₄	—	12	4	no	—
180	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
181	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
182	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
183	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
184	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
186	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
188	C₂×Dic₃	—	2	2	no	—
189	C₂×Dic₃	—	2	2	no	—
190	C₂×Dic₃	—	2	4	no	—
191	C₂×Dic₃	—	4	4	no	—
192	C₂×Dic₃	—	6	4	no	—
193	C₂×Dic₃	—	6	4	no	—
194	C₂×Dic₃	—	12	4	no	—
195	C₂×Dic₃	—	12	4	no	—
196	C₂×Dic₃	—	12	4	no	—
197	C₂×Dic₃	—	24	4	no	—
198	C₂×Dic₃	C₂×D₁₂	1	2	yes	—
199	C₂×Dic₃	C₂×D₁₂	1	2	yes	—
200	C₃×C₈	—	2	4	no	—
201	C₃×C₈	—	3	4	no	—
202	C₃×C₈	C₂×Dic₆	1	4	yes	—
203	C₃×C₈	C₂×Dic₆	1	4	yes	—
204	C₃⋊C₈	C₂×Dic₆	1	4	yes	—
205	C₂×C₃²	#499	1	4	yes	complemented
206	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
207	C₂×C₃²	C₂×#135	1	1	yes	characteristic
208	C₂×C₃²	C₂×#135	1	1	yes	characteristic
209	C₂²×C₄	—	3	1	no	—
210	C₂²×C₄	—	3	1	no	—
211	C₂²×C₄	—	6	2	no	—
212	C₂²×C₄	—	18	2	no	—
213	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
214	C₂×C₈	—	3	2	no	—
215	C₂×C₈	—	3	4	no	—
216	C₂×C₈	#627	1	4	yes	—
217	C₂×C₈	C₂×D₃²	1	2	yes	—
218	C₄⋊C₄	—	3	4	no	—
219	C₄⋊C₄	—	6	2	no	—
220	C₄⋊C₄	—	9	4	no	—
221	C₄⋊C₄	—	18	8	no	—
222	C₄⋊C₄	—	18	8	no	—
223	C₄⋊C₄	—	18	2	no	—
224	C₄²	—	3	2	no	—
225	C₄²	—	3	2	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	#869	1	2	yes	—
231	C₂²×C₃	#888	1	2	yes	—
232	C₂²×C₃	#904	1	2	yes	—
233	C₂²×C₃	#904	1	2	yes	—
234	C₂²×C₃	#912	1	2	yes	—
235	C₂²×C₃	#919	1	2	yes	—
236	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
237	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	2	no	—
241	C₃×C₄	—	6	2	no	—
242	C₃×C₄	—	6	2	no	—
243	C₃×C₄	—	12	4	no	—
244	C₃×C₄	#869	1	2	yes	—
245	C₃×C₄	#888	1	2	yes	—
246	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
247	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
248	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
249	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
250	Dic₃	—	2	2	no	—
251	Dic₃	—	2	2	no	—
252	Dic₃	—	4	4	no	—
253	Dic₃	—	12	4	no	—
254	Dic₃	—	12	4	no	—
255	Dic₃	—	24	4	no	—
256	C₃²	C₂×#499	1	1	yes	characteristic, complemented, Sylow
257	C₂³	D₃×D₁₂	1	1	yes	characteristic
258	C₂×C₄	—	3	2	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	—	6	2	no	—
261	C₂×C₄	—	6	2	no	—
262	C₂×C₄	—	6	4	no	—
263	C₂×C₄	—	12	4	no	—
264	C₂×C₄	—	18	4	no	—
265	C₂×C₄	—	36	4	no	—
266	C₂×C₄	D₃×Dic₆	1	2	yes	—
267	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
268	C₂×C₄	C₂×#627	1	2	yes	—
269	C₂×C₄	C₂²×D₃²	1	1	yes	characteristic
270	C₈	—	3	4	no	—
271	C₈	C₂×#627	1	4	yes	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	#4683	1	4	yes	complemented
277	C₂×C₃	#4878	1	4	yes	complemented
278	C₂×C₃	C₂×#869	1	1	yes	characteristic
279	C₂×C₃	C₂×#888	1	1	yes	characteristic
280	C₂×C₃	C₂×#904	1	1	yes	characteristic
281	C₂×C₃	C₂×#904	1	1	yes	characteristic
282	C₂×C₃	C₂×#912	1	1	yes	characteristic
283	C₂×C₃	C₂×#919	1	1	yes	characteristic
284	C₂²	#63807	1	2	yes	—
285	C₂²	#63808	1	2	yes	—
286	C₂²	#63918	1	2	yes	—
287	C₂²	C₂×D₃×D₁₂	1	1	yes	characteristic
288	C₄	—	6	2	no	—
289	C₄	—	6	2	no	—
290	C₄	—	12	4	no	—
291	C₄	—	36	4	no	—
292	C₄	#63918	1	2	yes	—
293	C₄	C₂×D₃×Dic₆	1	1	yes	characteristic
294	C₄	C₂²×#627	1	1	yes	characteristic
295	C₃	—	2	1	no	—
296	C₃	C₂×#4683	1	1	yes	characteristic, complemented
297	C₃	C₂×#4878	1	1	yes	characteristic, complemented
298	C₂	#10590873	1	4	yes	direct product factor
299	C₂	C₂×#63807	1	1	yes	characteristic
300	C₂	C₂×#63808	1	1	yes	characteristic
301	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×#499
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. #10590873	C₂	direct (N × H)
205. C₂×C₃²	#499	semidirect
256. C₃²	C₂×#499	semidirect
276. C₂×C₃	#4683	semidirect
277. C₂×C₃	#4878	semidirect
296. C₃	C₂×#4683	semidirect
297. C₃	C₂×#4878	semidirect
298. C₂	#10590873	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×#63869	2
C₂×#63645	2
C₂×C₃×#818	2
C₂×C₃×C₈×Dic₃	2
C₂×#63573	2
#10590873	2
C₂×#4878	3
C₂×#4683	3

Subgroups — C2×#10590873 #11874928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10590873 #11874928