Subgroups — C₂×#10589118 #11796624

Order 1152.

529 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#63645	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#10589118	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4564	—	3	1	no	maximal
6	C₂×#4503	—	3	1	no	maximal
7	C₂²×C₃×Dic₆	—	2	1	no	—
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
10	C₂×C₃×Dic₁₂	—	2	1	no	—
11	C₂×C₃×Dic₁₂	—	2	2	no	—
12	C₂×C₃×Dic₁₂	C₄	1	4	yes	complemented
13	C₂×C₃×Dic₁₂	C₂²	1	2	yes	—
14	#63645	—	2	1	no	—
15	#63645	C₂²	1	2	yes	—
16	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
18	C₂²×C₃×Q₁₆	—	3	1	no	—
19	C₂²×Dic₁₂	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₁₆	—	3	1	no	—
21	C₂×#818	—	3	1	no	—
22	C₂×#818	—	3	1	no	—
23	C₂×#818	—	6	1	no	—
24	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic
25	#4564	—	3	4	no	—
26	#4503	—	3	4	no	—
27	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₃⋊Dic₃	—	4	1	no	—
29	C₂²×C₃×Dic₃	—	4	1	no	—
30	C₂×C₃×Dic₆	—	2	4	no	—
31	C₂×C₃×Dic₆	—	2	2	no	—
32	C₂×C₃×Dic₆	—	4	1	no	—
33	C₂×C₃×Dic₆	—	4	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	D₄	1	2	yes	—
37	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
38	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
39	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
40	C₃×Dic₁₂	—	2	4	no	—
41	C₃×Dic₁₂	—	2	2	no	—
42	C₃×Dic₁₂	C₂×C₄	1	4	yes	complemented
43	C₃×C₃⋊C₁₆	—	2	2	no	—
44	C₂×#358	—	9	1	no	Sylow
45	C₂²×C₃×Q₈	—	6	1	no	—
46	C₂²×Dic₆	—	2	1	no	—
47	C₂×C₃×Q₁₆	—	3	4	no	—
48	C₂×C₃×Q₁₆	—	3	2	no	—
49	C₂×C₃×Q₁₆	—	6	1	no	—
50	C₂×C₃×Q₁₆	—	6	2	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₄⋊Dic₃	—	6	2	no	—
55	C₂×C₄⋊Dic₃	—	6	1	no	—
56	C₂×C₄⋊Dic₃	—	6	1	no	—
57	C₂×Dic₁₂	—	2	2	no	—
58	C₂×Dic₁₂	—	2	1	no	—
59	C₂×Dic₁₂	Dic₃	1	4	yes	complemented
60	C₂×Dic₁₂	C₂×D₃	1	2	yes	—
61	C₂×C₃×C₁₆	—	3	2	no	—
62	C₂×C₃×C₁₆	—	6	2	no	—
63	#818	—	3	2	no	—
64	#818	—	3	2	no	—
65	#818	—	6	2	no	—
66	#818	—	6	2	no	—
67	#818	—	6	1	no	—
68	#818	—	6	1	no	—
69	C₂×C₃⋊C₁₆	—	2	2	no	—
70	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
71	C₂³×C₃²	D₈	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
74	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
75	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃⋊Dic₃	—	4	2	no	—
77	C₂×C₃⋊Dic₃	—	8	2	no	—
78	C₂×C₃×Dic₃	—	4	4	no	—
79	C₂×C₃×Dic₃	—	4	2	no	—
80	C₃×Dic₆	—	2	4	no	—
81	C₃×Dic₆	—	4	4	no	—
82	C₃×Dic₆	—	4	2	no	—
83	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
84	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂²×Q₁₆	—	3	1	no	—
87	C₂²×C₁₆	—	3	1	no	—
88	C₂×#107	—	9	1	no	—
89	#358	—	9	4	no	—
90	C₂×C₃×Q₈	—	6	4	no	—
91	C₂×C₃×Q₈	—	6	2	no	—
92	C₂×C₃×Q₈	—	12	2	no	—
93	C₂×C₃×Q₈	—	12	1	no	—
94	C₂²×C₃×C₄	—	2	1	no	—
95	C₂²×C₃×C₄	—	12	1	no	—
96	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
97	C₂²×C₃×C₄	#67	1	1	yes	characteristic
98	C₂²×Dic₃	—	4	1	no	—
99	C₂²×Dic₃	—	12	2	no	—
100	C₂²×Dic₃	—	12	1	no	—
101	C₂²×Dic₃	—	12	1	no	—
102	C₂×Dic₆	—	2	4	no	—
103	C₂×Dic₆	—	2	2	no	—
104	C₂×Dic₆	—	4	1	no	—
105	C₂×Dic₆	—	4	2	no	—
106	C₃×Q₁₆	—	3	4	no	—
107	C₃×Q₁₆	—	6	4	no	—
108	C₃×Q₁₆	—	6	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	1	no	—
112	C₂×C₃×C₈	—	2	1	no	—
113	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
114	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
116	C₂×C₃×C₈	#67	1	2	yes	—
117	C₂×C₃×C₈	#67	1	2	yes	—
118	C₂×C₃×C₈	#67	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
121	C₄⋊Dic₃	—	6	4	no	—
122	C₄⋊Dic₃	—	6	2	no	—
123	C₄⋊Dic₃	—	6	2	no	—
124	C₄⋊Dic₃	—	12	2	no	—
125	C₄⋊Dic₃	—	12	1	no	—
126	C₄⋊Dic₃	—	12	1	no	—
127	Dic₁₂	—	2	4	no	—
128	Dic₁₂	—	2	2	no	—
129	Dic₁₂	C₂×Dic₃	1	4	yes	complemented
130	C₃×C₁₆	—	6	2	no	—
131	C₃⋊C₁₆	—	2	2	no	—
132	C₂²×C₃²	#102	1	2	yes	—
133	C₂²×C₃²	SD₃₂	1	2	yes	—
134	C₂²×C₃²	Q₃₂	1	2	yes	—
135	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
136	C₃²×C₄	#102	1	2	yes	—
137	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
139	C₃⋊Dic₃	—	8	2	no	—
140	C₃×Dic₃	—	4	4	no	—
141	C₂²×Q₈	—	6	1	no	—
142	C₂×Q₁₆	—	3	4	no	—
143	C₂×Q₁₆	—	3	2	no	—
144	C₂×Q₁₆	—	6	2	no	—
145	C₂×Q₁₆	—	6	1	no	—
146	C₂²×C₈	D₃²	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	18	1	no	—
148	C₂×C₁₆	—	3	2	no	—
149	C₂×C₁₆	—	6	2	no	—
150	#107	—	9	2	no	—
151	#107	—	18	1	no	—
152	C₂³×C₃	—	2	1	no	—
153	C₂³×C₃	D₂₄	1	1	yes	characteristic
154	C₂³×C₃	#213	1	1	yes	characteristic
155	C₃×Q₈	—	6	4	no	—
156	C₃×Q₈	—	12	4	no	—
157	C₃×Q₈	—	12	2	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	12	4	no	—
163	C₂×C₃×C₄	—	12	2	no	—
164	C₂×C₃×C₄	#204	1	2	yes	—
165	C₂×C₃×C₄	D₂₄	1	1	yes	characteristic
166	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
167	C₂×C₃×C₄	#213	1	1	yes	characteristic
168	C₂×C₃×C₄	#214	1	2	yes	—
169	C₂×C₃×C₄	#217	1	2	yes	—
170	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
172	C₂×Dic₃	—	4	4	no	—
173	C₂×Dic₃	—	4	2	no	—
174	C₂×Dic₃	—	12	4	no	—
175	C₂×Dic₃	—	12	2	no	—
176	C₂×Dic₃	—	12	2	no	—
177	C₂×Dic₃	—	24	4	no	—
178	C₂×Dic₃	—	24	2	no	—
179	C₂×Dic₃	—	24	2	no	—
180	Dic₆	—	2	4	no	—
181	Dic₆	—	4	4	no	—
182	Dic₆	—	4	2	no	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	—	2	1	no	—
185	C₃×C₈	—	2	1	no	—
186	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
187	C₃×C₈	#217	1	2	yes	—
188	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
189	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
190	C₃×C₈	C₂×#67	1	1	yes	characteristic
191	C₃×C₈	C₂×#67	1	1	yes	characteristic
192	C₂×C₃²	#358	1	4	yes	complemented
193	C₂×C₃²	C₂×#102	1	1	yes	characteristic
194	C₂×C₃²	C₂×SD₃₂	1	1	yes	characteristic
195	C₂×C₃²	C₂×Q₃₂	1	1	yes	characteristic
196	C₂×Q₈	—	6	4	no	—
197	C₂×Q₈	—	6	2	no	—
198	C₂×Q₈	—	12	1	no	—
199	C₂×Q₈	—	12	2	no	—
200	C₂²×C₄	—	12	1	no	—
201	C₂²×C₄	—	36	1	no	—
202	C₂²×C₄	#626	1	1	yes	characteristic
203	Q₁₆	—	3	4	no	—
204	Q₁₆	—	6	4	no	—
205	Q₁₆	—	6	2	no	—
206	C₂×C₈	D₃×Dic₃	1	2	yes	—
207	C₂×C₈	#625	1	2	yes	—
208	C₂×C₈	#626	1	1	yes	characteristic
209	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
210	C₄⋊C₄	—	18	2	no	—
211	C₄⋊C₄	—	36	1	no	—
212	C₁₆	—	6	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	#800	1	2	yes	—
218	C₂²×C₃	Dic₂₄	1	2	yes	—
219	C₂²×C₃	#821	1	2	yes	—
220	C₂²×C₃	#828	1	2	yes	—
221	C₂²×C₃	#829	1	2	yes	—
222	C₂²×C₃	#832	1	2	yes	—
223	C₂²×C₃	C₂×D₂₄	1	1	yes	characteristic
224	C₂²×C₃	C₂×#213	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	12	4	no	—
229	C₃×C₄	#821	1	2	yes	—
230	C₃×C₄	#832	1	2	yes	—
231	C₃×C₄	C₂×#204	1	1	yes	characteristic
232	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
233	C₃×C₄	C₂×#214	1	1	yes	characteristic
234	C₃×C₄	C₂×#217	1	1	yes	characteristic
235	Dic₃	—	4	4	no	—
236	Dic₃	—	24	4	no	—
237	Dic₃	—	24	2	no	—
238	Dic₃	—	24	2	no	—
239	C₃²	C₂×#358	1	1	yes	characteristic, complemented, Sylow
240	C₂³	#3714	1	1	yes	characteristic
241	Q₈	—	6	4	no	—
242	Q₈	—	12	4	no	—
243	Q₈	—	12	2	no	—
244	C₂×C₄	—	12	4	no	—
245	C₂×C₄	—	12	2	no	—
246	C₂×C₄	—	36	2	no	—
247	C₂×C₄	—	72	2	no	—
248	C₂×C₄	#3714	1	1	yes	characteristic
249	C₂×C₄	#3716	1	2	yes	—
250	C₂×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
251	C₂×C₄	C₂×#626	1	1	yes	characteristic
252	C₈	(C₂×D₃)⋊Dic₃	1	2	yes	—
253	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
254	C₈	C₂×#625	1	1	yes	characteristic
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	4	no	—
259	C₂×C₃	#4503	1	4	yes	complemented
260	C₂×C₃	#4564	1	4	yes	complemented
261	C₂×C₃	C₂×#800	1	1	yes	characteristic
262	C₂×C₃	C₂×Dic₂₄	1	1	yes	characteristic
263	C₂×C₃	C₂×#821	1	1	yes	characteristic
264	C₂×C₃	C₂×#828	1	1	yes	characteristic
265	C₂×C₃	C₂×#829	1	1	yes	characteristic
266	C₂×C₃	C₂×#832	1	1	yes	characteristic
267	C₂²	#63549	1	2	yes	—
268	C₂²	#63551	1	2	yes	—
269	C₂²	#63563	1	2	yes	—
270	C₂²	C₂×#3714	1	1	yes	characteristic
271	C₄	—	12	4	no	—
272	C₄	—	72	2	no	—
273	C₄	#63563	1	2	yes	—
274	C₄	C₂×#3716	1	1	yes	characteristic
275	C₄	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
276	C₃	—	2	1	no	—
277	C₃	C₂×#4503	1	1	yes	characteristic, complemented
278	C₃	C₂×#4564	1	1	yes	characteristic, complemented
279	C₂	#10589118	1	4	yes	direct product factor
280	C₂	C₂×#63549	1	1	yes	characteristic
281	C₂	C₂×#63551	1	1	yes	characteristic
282	C₂	C₂×#63563	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3714
commutator	C₃²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	D₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	C₂×#358
LCS	G ▷ C₃²×C₈ ▷ 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
4. #10589118	C₂	direct (N × H)
12. C₂×C₃×Dic₁₂	C₄	semidirect
42. C₃×Dic₁₂	C₂×C₄	semidirect
59. C₂×Dic₁₂	Dic₃	semidirect
129. Dic₁₂	C₂×Dic₃	semidirect
192. C₂×C₃²	#358	semidirect
239. C₃²	C₂×#358	semidirect
259. C₂×C₃	#4503	semidirect
260. C₂×C₃	#4564	semidirect
277. C₃	C₂×#4503	semidirect
278. C₃	C₂×#4564	semidirect
279. C₂	#10589118	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×Dic₁₂	2
C₂×#63645	2
C₂²×C₃×C₃⋊C₁₆	2
#10589118	2
C₂×#4564	3
C₂×#4503	3

Subgroups — C2×#10589118 #11796624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10589118 #11796624